Damage and Failure of Composite Materials

Damage and Failure of Composite MaterialsUnderstanding damage and failure of composite materials is critical for reliable and cost-effective engineering design. Bringing together materials mechanics and modeling, this book provides a complete guide to damage, fatigue, and failure of composite materials. Early chapters focus on the underlying principles governing composite damage, reviewing basic equations and mechanics theory, before describing mechanisms of damage such as cracking, breakage, and buckling. In subsequent chapters, the physical mechanisms underlying the formation and progression of damage under mechanical loads are described with ample experi- mental data, and micro- and macro-level damage models are combined. Finally, fatigue of composite materials is discussed using fatigue-life diagrams. While there is a special emphasis on polymer matrix composites, metal and ceramic matrix composites are also described. Outlining methods for more reliable design of composite structures, this is a valuable resource for engineers and materials scientists in industry and academia. Ramesh Talreja is a Professor of Aerospace Engineering at Texas A&M University. He earned his Ph.D. and Doctor of Technical Sciences degrees from the Technical University of Denmark. He has contributed extensively to the fields of damage, fatigue, and failure of composite materials by authoring numerous books and book chapters as well as by editing several encyclopedic works. Chandra Veer Singh is an Assistant Professor of Materials Science and Engineering at the University of Toronto. He earned his Ph.D. in aerospace engineering from Texas A&M University, and worked as a post-doctoral Fellow at Cornell University. His research expertise is in damage mechanics of composite materials, atomistic modeling, and computational materials science. His industry experience includes R&D at GE Aircraft Engines. Damage and Failure of Composite Materials RAMESH TALREJA Texas A&M University CHANDRA VEER SI NGH University of Toronto cambri dge uni versi ty press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sa˜ o Paulo, Delhi, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521819428 #R. Talreja and C. V. Singh 2012 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2012 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication Data Talreja, R. Damage and failure of composite materials / Ramesh Talreja, Chandra Veer Singh. p. cm. Includes bibliographical references. ISBN 978-0-521-81942-8 (Hardback) 1. Composite materials–Fatigue. 2. Composite materials–Fracture. I. Singh, Chandra Veer. II. Title. TA418.9.C6T338 2012 620.1 0 126–dc23 2011035578 ISBN 978-0-521-81942-8 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface page ix 1 Durability assessment of composite structures 1 1.1 Introduction 1 1.2 Historical development of damage mechanics of composites 3 1.3 Fatigue of composite materials 5 References 7 2 Review of mechanics of composite materials 9 2.1 Equations of elasticity 9 2.1.1 Strain–displacement relations 9 2.1.2 Conservation of linear and angular momenta 10 2.1.3 Constitutive relations 11 2.1.4 Equations of motion 15 2.1.5 Energy principles 15 2.2 Micromechanics 17 2.2.1 Stiffness properties of a unidirectional lamina 18 2.2.2 Thermal properties of a unidirectional lamina 19 2.2.3 Constitutive equations for a lamina 20 2.2.4 Strength of a unidirectional lamina 21 2.3 Analysis of laminates 24 2.3.1 Strain–displacement relations 25 2.3.2 Constitutive relationships for the laminate 26 2.3.3 Stresses and strains in a lamina within a laminate 28 2.3.4 Effect of layup configuration 28 2.4 Linear elastic fracture mechanics 29 2.4.1 Fracture criteria 30 2.4.2 Crack separation modes 31 2.4.3 Crack surface displacements 32 2.4.4 Relevance of fracture mechanics for damage analysis 33 References 34 3 Damage in composite materials 36 3.1 Mechanisms of damage 37 3.1.1 Interfacial debonding 37 3.1.2 Matrix microcracking/intralaminar (ply) cracking 39 3.1.3 Interfacial sliding 39 3.1.4 Delamination/interlaminar cracking 41 3.1.5 Fiber breakage 42 3.1.6 Fiber microbuckling 42 3.1.7 Particle cleavage 44 3.1.8 Void growth 44 3.1.9 Damage modes 45 3.2 Development of damage in composite laminates 46 3.3 Intralaminar ply cracking in laminates 49 3.4 Damage mechanics 50 References 52 4 Micro-damage mechanics 57 4.1 Introduction 57 4.2 Phenomena of single and multiple fracture: ACK theory 58 4.2.1 Multiple matrix cracking 61 4.2.2 Perfectly bonded fiber/matrix interface: a modified shear lag analysis 65 4.2.3 Frictional fiber/matrix interface 67 4.3 Stress analysis (boundary value problem) for cracked laminates 68 4.3.1 Complexity and issues 68 4.3.2 Assumptions 71 4.4 One-dimensional models: shear lag analysis 73 4.4.1 Initial shear lag analysis 74 4.4.2 Interlaminar shear lag analysis 77 4.4.3 Extended shear lag analysis 79 4.4.4 2-D shear lag models 80 4.4.5 Summary of shear lag models 80 4.5 Self-consistent scheme 84 4.6 2-D stress analysis: variational methods 87 4.6.1 Hashin’s variational analysis 87 4.6.2 Effect of residual stresses 96 4.6.3 [0 m /90 n ] s vs. [90 n /0 m ] s laminates 97 4.6.4 Improved variational analysis 97 4.6.5 Related works 101 4.6.6 Comparison between 1-D and 2-D stress-based models 101 4.7 Generalized plain strain analysis – McCartney’s model 104 vi Contents 4.8 COD-based methods 110 4.8.1 3-D laminate theory: Gudmundson’s model 111 4.8.2 Lundmark–Varna model 117 4.9 Computational methods 119 4.9.1 Finite element method (FEM) 120 4.9.2 Finite strip method 121 4.9.3 Layerwise theory 123 4.10 Other methods 124 4.11 Changes in thermal expansion coefficients 125 4.12 Summary 126 References 126 5 Macro-damage mechanics 134 5.1 Introduction 134 5.2 Continuum damage mechanics (CDM) of composite materials 138 5.2.1 RVE for damage characterization 139 5.2.2 Characterization of damage 141 5.2.3 A thermodynamics framework for materials response 144 5.2.4 Stiffness–damage relationships 148 Case 1: Cracking in one off-axis orientation 152 Case 2: Cross-ply laminates 152 Evaluation of material constants 153 5.3 Synergistic damage mechanics (SDM) 155 5.3.1 Two damage modes 156 5.3.2 Three damage modes 165 5.4 Viscoelastic composites with ply cracking 170 5.5 Summary 176 References 177 6 Damage progression 179 6.1 Introduction 179 6.2 Experimental techniques 180 6.3 Experimental observations 185 6.3.1 Initiation of ply cracking 185 6.3.2 Crack growth and multiplication 187 6.3.3 Crack shapes 189 6.3.4 Effect of cracking 189 6.3.5 Loading and environmental effects 191 6.3.6 Cracking in multidirectional laminates 193 6.4 Modeling approaches 194 6.4.1 Strength-based approaches 194 6.4.2 Energy-based approaches 198 6.4.3 Strength vs. energy criteria for multiple cracking 210 vii Contents 6.5 Randomness in ply cracking 211 6.6 Damage evolution in multidirectional laminates 217 6.7 Damage evolution under cyclic loading 223 6.8 Summary 229 References 230 7 Damage mechanisms and fatigue-life diagrams 237 7.1 Introduction 237 7.2 Fatigue-life diagrams 237 7.3 On-axis fatigue of unidirectional composites 238 7.4 Effects of constituent properties 241 7.5 Unidirectional composites loaded parallel to the fibers 242 7.5.1 Polymer matrix composites (PMCs) 242 7.5.1.1 Experimental studies of mechanisms 247 7.5.2 Metal matrix composites (MMCs) 250 7.5.3 Ceramic matrix composites (CMCs) 252 7.6 Unidirectional composites loaded inclined to the fibers 257 7.7 Fatigue of laminates 259 7.7.1 Angle-ply laminates 260 7.7.2 Cross-ply laminates 261 7.7.3 General multidirectional laminates 263 7.8 Fatigue-life prediction 265 7.8.1 Cross-ply laminates 266 7.8.2 General laminates 273 7.9 Summary 273 References 274 8 Future directions 276 8.1 Computational structural analysis 276 8.2 Multiscale modeling of damage 278 8.2.1 Length scales of damage 280 8.2.2 Hierarchical multiscale modeling 282 8.2.3 Implication on multiscale modeling: Synergistic damage mechanics 286 8.3 Cost-effective manufacturing and defect damage mechanics 287 8.3.1 Cost-effective manufacturing 288 8.3.2 Defect damage mechanics 291 8.4 Final remarks 296 References 298 Author index 301 Subject index 303 viii Contents Preface The field of composite materials has advanced steadily from the early develop- ments during the 1970s when laminate plate theory and anisotropic failure criteria were in focus to today’s diversification of composite materials to multifunctional and nanostructured composite morphologies. Throughout the 1970s and 1980s several books appeared along with courses that were developed and taught at advanced levels dealing with mechanics of composite materials and structures. The failure analysis was mostly limited to descriptions of strength that extended previous continuum descriptions of metal yielding and failure. Beginning around the mid-1980s, micromechanics and continuum damage mechanics were applied to multiple cracking observed in composite materials. Under the overall descrip- tion of “damage mechanics” a flurry of activities took place as evidenced by conferences and symposia. Other than several conference proceedings that recorded such activities, a collection of seminal contributions to the field appeared in a volume (Damage Mechanics of Composite Materials, R. Talreja, ed., Composite Materials Series, R.B. Pipes, series ed., Vol. 9, Amsterdam: Elsevier Science Publishers, 1994). The two main avenues of approach to damage in composite materials and its effect on materials response, now referred to as micro-damage mechanics (MIDM) and macro-damage mechanics (MADM), were presented in a balanced form in that volume. In the years since then, many developments have taken place that have brought this field to such level of maturity that a book coherently presenting the material was felt to be timely. It is hoped that this book will help provide impetus for teaching advanced courses in composite damage at universities as well as support short courses for professional development of engineers in industry. The wealth of material covered can also help new research- ers in advancing the field further. To this end, the last chapter provides some guidance in identifying gaps and needs for further work. The structure of the book is as follows. Chapter 1 lays down the overall strategy for durability assessment of composite structures, emphasizing the needs and motivating the content of the book to follow. Chapter 2 provides an easy reference to the basic continuum mechanics topics that are felt to be relevant to the subsequent treatment. Chapter 3 describes the mechanisms of damage that under- lie the phenomena aimed for modeling. Many of the physical observations described there are viewed to be vital to developing proper understanding of the complex field of damage in composite materials. Chapters 4 and 5 deal with the two main approaches stated above, i.e., the MIDM and the MADM. Selected works from the literature, including the authors’ own, are given as much treatment as was found justified to generate coherency without overly including details. While these two chapters focus on descriptions of damage and the constitutive property changes caused by it, Chapter 6 is devoted to the progression of damage. The crack multiplication is a distinctive feature of damage in composite materials that distinguishes it from single crack growth in monolithic materials, and there- fore justifies treatment in a chapter by itself. Chapter 7 is on fatigue of composite materials. This field suffers from the historical treatments of metal fatigue and is unfortunately the least understood part of damage in composite materials. Multi- axial fatigue illustrates the situation well where the literature displays little under- standing of the mechanisms underlying failure. While a separate book on fatigue of composite materials is needed to do full justice to the field, a single chapter here is added to draw attention to the mechanisms-based concepts for proper interpret- ation and modeling. Finally, Chapter 8 presents a summary of the book and points to the directions in which further advances are seen to be necessary. Particular emphasis is given to the computational incorporation of damage mod- eling in durability assessment as well as taking account of the manufacturing- induced defects in an integrated manner. Although the authors have written this book, the credit goes to many research- ers who have worked on various aspects of damage in composite materials. Their collective contributions have made it possible for us to present what we have seen as a coherent story at this time. The field is evolving, and future versions of the story will hopefully spur further development and, most importantly, transfer of this knowledge to industry will take place. x Preface 1 Durability assessment of composite structures 1.1 Introduction Composite structures for mechanical and aerospace applications are designed to retain structural integrity and remain durable for the intended service life. Since the early 1970s important advances have been made in characterizing and model- ing the underlying mechanical behavior and developing tools and methodologies for predicting the fracture and fatigue of composite materials. This book provides an exposition of the concepts and analyses related to this area and presents recent results. The next chapters treat damage in composite materials as observed by a variety of techniques, followed by modeling at the micro and macro levels. Fatigue is treated separately because of its particular complexities that require systematic interpretation schemes developed for the purpose. A chapter is added in the beginning to provide convenient access to the mechanics concepts needed for the modeling analyses in later chapters. Here we present an overview of the durability assessment process for composite structures. Figure 1.1 depicts the connectivity and flow of the elements of this process. To begin, one usually conducts stress analysis of the component using the “initial” constitutive behavior of the composite along with the service loading on the component as input. In contrast to monolithic materials, such as metals, the constitutive behavior of a composite can change due to damage incurred in service. The stress analysis combined with prior experience allows identifying critical sites (“hot spots”) in the component that are prone to be the sites of failure. Further examination of these sites in terms of the local stress/strain/ temperature excursions combined with the composite material composition at those sites helps to identify the possible mechanisms of damage that can result. Examples of such mechanisms are microcracking of the matrix, delamination (separation of layers at interfaces), aging (of the polymer matrix), etc. Chapter 3 describes these mechanisms in some detail. The next step is to analyze the conse- quences of the mechanisms on the material response and in turn on the structural performance. Chapters 4 and 5 deal with different models to predict the damage- induced material response changes. Since the scales at which damage occurs are small in comparison to the characteristic geometrical size of the “hot spots,” models must account for the multiple length (or size) scales. The differentiation of scales is conventionally described as “micro” (the scale of damage) and “macro” (the scale at which structural response is characterized). Since connectiv- ity between these scales must be established, an intermediate scale called “meso” is defined as needed by the particular model used. In micromechanics the concept of “representative volume element” (RVE) has been proposed. The size of this element is commonly taken to be the meso scale. Chapters 4 and 5 describe the three scales in the context of different models. Chapter 6 is focused on the initiation and progression of damage. Together the three chapters provide the content of the subject known as “damage mechanics,” which as indicated in Figure 1.1 is central to durability assessment. The common output of the damage mechanics models is a description of the material response, often described as “stiffness degradation,” caused by damage. This description necessarily involves averaging over the so-called RVE. Thus the materials response, or averaged constitutive behavior description, forms the new input to the stress analysis that was conducted initially using pristine (undamaged) material properties. The resulting iterative process of stress analysis should be an inherent feature of composite structural analysis, although the industry practice currently does not fully implement this procedure. Another output of the damage mechanics analysis is “strength degradation,” i.e., reduction in the load-bearing capability of the structure due to damage. Depending on the functional requirements of a given structure, degradation of stiffness or strength would be the path to loss of structural integrity. A typical example of a stiffness-critical structure is an aircraft wing that must deform appropriately to perform its aerodynamic function, while a fuselage is strength-critical as its design requirement is to contain the pressure within it. While monolithic materials such as metals fail due to unstable growth of a crack, the heterogeneous internal structure of a composite leads to formation of multiple cracks. A generic heterogeneous solid is illustrated in Figure 1.2 in three states: pristine (undamaged) to the left in the figure shows a representative region Figure 1.1. A durability analysis scheme for composite structural components. 2 Durability assessment of composite structures of the solid within which heterogeneities (reinforcements) are indicated symbolic- ally as filled circles, and two states that have multiple cracks resulting from debonding of reinforcements (middle figure) and from local failure of the matrix induced by defects and/or stress concentrations. Consider the external loading on a composite structure resulting in tractions t on the surface bounding the repre- sentative region of the composite shown. If the response to these tractions in terms of the bounding surface displacements is given by u in the pristine state, then the surface displacements of the multiple cracks (commonly expressed as crack opening displacements, COD, and crack sliding displacement, CSD) within the volume will change this to u 1 or u 2 depending on the type of damage (see Figure 1.2). The local environment around the cracks influences the COD and CSD of distributed cracks within the volume. This local environment is typically described as a “constraint” (i.e., moderation) to the crack surface displacements and is expressed in terms of the variables of heterogeneities. If the heterogeneous solid with multiple cracks is homogenized over the representative region, then the stress–strain response averaged over the RVE is given by the averaged stiffness properties that change (degrade) with increasing number of cracks and the con- straint to the crack surface displacements. This stiffness degradation is the subject of damage mechanics, as discussed above in describing the durability assessment procedure depicted in Figure 1.1. 1.2 Historical development of damage mechanics of composites Although the field of solid mechanics applied to heterogeneous solids was developed in the late 1950s and early 1960s, and became known as micromechanics, the specific situations encountered in composite materials such as those with continuous fiber reinforcements were not addressed until much later. The concepts developed in micromechanics turned out to be useful for multiple cracking in composite materials and are recommended as essential background (see the text by Nemat-Nasser and Hori [1]). However, the first pioneering work that clarified the phenomenon of multiple cracking in the presence of fiber/matrix interfaces in reinforced composites was by Aveston et al. [2] published as a conference proceedings paper in 1971. u t t t u 1 u 2 Figure 1.2. A heterogeneous solid in pristine (undamaged) state (left) and in two possible multiple cracking states (middle and right). 3 1.2 Historical development of damage mechanics of composites This work, which became known as the ACK theory, treated multiple parallel cracks normal to fibers in a matrix with all fibers in one direction loaded in tension along fibers. The model produced an expression for the overall strain at which multiple cracking occurs based on a simplified stress analysis and energy balance concepts. The expression provided a basis for assessing the roles of fiber and matrix properties, their volume fractions, and the fiber diameter in resisting multiple cracking. The ACK model was motivated by the observation of multiple cracking in brittle matrix composites such as cement reinforced with steel wires. For polymer matrix composites, the application of the ACK theory was at first not clear since a ply with unidirectional fibers of glass or carbon does not have the right conditions for multiple cracking when loaded in tension along fibers. Garrett and Bailey in 1977 [3] found that the multiple cracking observed in cross-ply laminates of glass fiber-reinforced polyester under axial tension could in fact be described well by the case of fully bonded interfaces treated by Aveston and Kelly [4] in a follow on paper to the ACK model. This required replacing the matrix and fibers in that model by the transverse and longitudinal plies, respectively. Garrett and Bailey then repeated with appropri- ate modification the one-dimensional stress analysis and energy balance consider- ations used in [4]. Thus began a long series of works that applied the one-dimensional stress analysis, known as shear lag analysis, which assumes axial load transfer from cracked to uncracked plies by the shear stress at the interfaces. The inadequacy of the shear lag analysis to properly provide stresses in the cracked cross-ply laminate was a severe limitation until a variational analysis- based two-dimensional approximation appeared in the English literature [5]. This spurred further work of more accuracy [6] and extension to partially debonded frictional interfaces [7], while extension to cracked plies of other than transverse orientation required other approaches [8]. The analyses that use local ply stress solutions to evaluate overall stiffness degradation are grouped together in “micro- damage mechanics” (MIDM) and are treated in Chapter 4. In some ways parallel to the MIDM emerged another approach that became known as continuum damage mechanics (CDM). Its beginnings are not attrib- uted to composite materials but to metals undergoing creep. Kachanov in 1958 [9] put forth a concept of a field of internal material discontinuity responsible for distributed local stress enhancement leading to overall creep strain. Later, the internal state was called damage and a (hidden) scalar variable D was associated with it. The continuum now had an internal damage state and because of its irreversible nature its treatment required thermodynamics, in particular the Second Law, which places conditions on the entropy changes. Kachanov’s work stayed relatively unknown until Lemaitre and Chaboche [10] applied it to analysis of various structural materials with distributed cavities and cracks. Krajcinovic [11] further enhanced the field by connecting it to concepts known from fracture mechanics and plasticity and by elaborating the thermodynamics implications. For composite materials of technological interests that are constructed with specific symmetries such as orthotropic, the first work to apply CDM was by Talreja [12] and its companion paper that 4 Durability assessment of composite structures validated the stiffness degradation relationships by experimental data [13]. The CDM concept for composites is illustrated in Figure 1.3. The reinforcements in a composite are regarded as a stationary microstructure and are homogenized as an anisotropic medium in which the damage entities such as cracks are embedded. Further homogenization is done by smearing out the damage entities into an internal field, which is represented by a pair of vectors, whose dyadic product averaged over all damage entities in a RVE provides the characterization of damage. Since the early papers [12, 13] the CDM field for composite materials has developed steadily, more recently in a version named as synergistic damage mechanics (SDM) where the micromechanics is judi- ciously applied to enhance the applicability of CDM. All this is the subject of Chapter 5 on macro-damage mechanics (MADM). As depicted in Figure 1.1, the stress analysis of critical structural sites requires stress–strain relationships that reflect the presence of damage. These relationships are developed by combining stiffness degradation and damage evo- lution. The subject of damage evolution is complex with its own challenges. Therefore Chapter 6 is devoted exclusively to its treatment. 1.3 Fatigue of composite materials It is natural to assume that the complexities of damage in composite materials observed under quasi-static loading would be enhanced when the loading is applied in a cyclic manner. The experience with metal fatigue indicates that the fracture surface of a sample failed in fatigue shows distinctly different features than if failed in the application of a monotonically increasing load. The fracture surface of a unidirectional fiber-reinforced composite loaded along fibers mono- tonically or cyclically does not give clear indication of mechanisms preceding Stationary microstructure Homogenization of stationary microstructure Homogenized continuum with damage Damage entity Evolving microstructure RVE a n P Figure 1.3. Illustration of the CDM concept for composites. 5 1.3 Fatigue of composite materials failure in either case. In more general fiber architectures, such as laminates and woven fabric composites, following the events from the first (initiation) to the last (separation by breakage) is generally difficult. However, as advances in nondes- tructive observation techniques are made, increasing clarity in mechanisms is emerging. In the early years of composite fatigue studies in the 1970s, little was understood of mechanisms and consequently the assumptions made in predictive models were speculative at best. One study of a unidirectional glass/epoxy composite made assumptions of fatigue mechanisms that led to reasonable explanation of the trends in fatigue life [14]. Following that work, a systematic conceptual framework for interpret- ation of fatigue damage and failure was proposed by Talreja [15] for more general cases of loading as well as for more general fiber orientations. The framework took the form of a two-dimensional plot called a “fatigue life dia- gram” in which regions of dominant mechanisms were separated. The diagram is not meant to be a data-fitted S-N curve (historically known as a Wo¨ hler dia- gram) but as a means of interpreting the roles of fibers, matrix, and interfaces as well as of laminate configuration parameters such as ply orientation, sequence, and thickness. Since the unidirectional composite (or ply) is a basic unit in laminates, the fatigue life diagram for this composite under tension–tension loading forms the baseline diagram from which more general cases evolve. This diagram is illustrated in Figure 1.4 and discussed in detail in Chapter 7. As shown, the vertical axis of the diagram is the maximum strain attained at the first application of maximum stress in a load controlled fatigue test. This quantity forms a proper reference to the loading condition and provides upper and lower limits to the fatigue behavior. Thus the strain to failure (of fiber) forms the upper limit while the strain corres- ponding to the fatigue limit (primarily a matrix property) forms the lower limit. These strain values can always be converted to applied stress, but plotting these in the diagram allows a systematic and proper interpretation of the roles of the constituents. The regions indicated in the fatigue life diagram provide clarity of the governing mechanisms dictated by the constituent properties. The construc- tion of the diagram for unidirectional composites was initially based on systematic arguments and logical deduction. Physical evidence to support the diagram was later presented by an elaborate and tedious experimental study [16]. The fatigue life diagram can also serve the purpose of facilitating mechanisms- based life prediction modeling. For cross-ply laminate this was demonstrated in [17]. Generally the path to predictive modeling with account of the underlying damage mechanisms is long and hard. Consequently, the literature has a prepon- derance of studies that resort to “failure criteria” that are mostly extensions of those for static failure with assumed procedures without fundamental validation. The models are therefore not reliable enough to extend beyond the cases that formed the impetus for the proposed schemes. Chapter 7 treats the subject of composite damage with emphasis on mechan- isms. It is not exhaustive in the sense of including the literature on models for life 6 Durability assessment of composite structures prediction. A recent paper [18] has a fairly thorough examination of the main models for multiaxial fatigue. It reveals the frustrating situation of lack of reliabil- ity of the models. In Chapter 7 the main findings of this review are discussed and a mechanisms-based methodology is proposed. References 1. S. Nemat-Nasser and M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials. (Amsterdam: Elsevier, 1993). 2. J. Aveston, G. A. Cooper, and A. Kelly, Single and multiple fracture. In The Properties of Fiber Composites. (Surrey, UK: IPC Science and Technology Press, National Phys- ical Laboratory, 1971), pp. 15–26. 3. K. W. Garrett and J. E. Bailey, Multiple transverse fracture in 90 degrees cross-ply laminates of a glass fiber-reinforced polyester. J Mater Sci, 12:1 (1977), 157–68. 4. J. Aveston and A. Kelly, Theory of multiple fracture of fibrous composites. J Mater Sci, 8:3 (1973), 352–62. 5. Z. Hashin, Analysis of cracked laminates: a variational approach. Mech Mater, 4:2 (1985), 121–36. 6. J. Varna and L. A. Berglund, Multiple transverse cracking and stiffness reduction in cross-ply laminates. J Compos Tech Res, 13:2 (1991), 97–106. 7. N. V. Akshantala and R. Talreja, A mechanistic model for fatigue damage evolution in composite laminates. Mech Mater, 29 (1998), 123–40. 8. L. N. McCartney, Model to predict effects of triaxial loading on ply cracking in general symmetric laminates. Compos Sci Technol, 60 (2000), 2255–79 (see Errata in Compos Sci Technol, 62:9 (2002), 1273–4). 9. L. M. Kachanov, On the creep rupture time. Izv Akad Nauk SSR, Otd Tekhn Nauk, 8 (1958), 26–31. 10. J. Lemaitre and J. L. Chaboche, Mechanique des Materiaux Solide. (Paris: Dunod, 1985). 11. D. Krajcinovic, Continuous damage mechanics. Appl Mech Rev, 37 (1984), 1–5. 12. R. Talreja, A continuum-mechanics characterization of damage in composite- materials. Proc R Soc London A, 399:1817 (1985), 195–216. Figure 1.4. Fatigue life diagram of a unidirectional fiber-reinforced composite subjected to cyclic tension in the fiber direction. 7 References 13. R. Talreja, Transverse cracking and stiffness reduction in composite laminates. J Compos Mater, 19 (1985), 355–75. 14. C. K. H. Dharan, Fatigue failure mechanisms in a unidirectionally reinforced compos- ite material. In Fatigue in Composite Materials, ASTM STP 569. (Philadelphia, PA: ASTM, 1975), pp. 171–88. 15. R. Talreja, Fatigue of composite materials: damage mechanisms and fatigue-life dia- grams. Proc R Soc London A, 378 (1981), 461–75. 16. E. K. Gamstedt and R. Talreja, Fatigue damage mechanisms in unidirectional carbon fibre-reinforced plastics. J Mater Sci, 34 (1999), 2535–46. 17. N. V. Akshantala and R. Talreja, A micromechanics based model for predicting fatigue life of composite laminates. Mater Sci Eng A, 285 (2000), 303–13. 18. M. Quaresimin, L. Susmel, and R. Talreja, Fatigue behaviour and life assessment of composite laminates under multiaxial loadings. Int J Fatigue, 32 (2010), 2–16. 8 Durability assessment of composite structures 2 Review of mechanics of composite materials In this chapter the fundamental aspects of elasticity, strength, and fracture of composite solids are reviewed. Although this information is available in numerous texts, more comprehensively and in greater detail than here, a brief exposition is provided for convenient reference. For further in-depth treatment, the reader may consult, e.g., [1–5] for theory of elasticity and continuum mechanics, [6–12] for mechanics of composite materials, and [13–17] for fracture mechanics. 2.1 Equations of elasticity 2.1.1 Strain–displacement relations Figure 2.1 illustrates the initial and deformed configurations of a body whose representative material point P is described with respect to a fixed rectangular Cartesian frame by coordinates X j and x i respectively, j, i = 1, 2, 3. The compon- ents of displacement of the point are given by u i = x i ÷X j d ij ; (2:1) where X j are the coordinates of the material point in the initial undeformed configuration, x i are the coordinates of the material point in the final deformed configuration, and d ij is the Kronecker delta. The Lagrangian description of displacement at time t is expressed in terms of the X j coordinates as u i = x i X 1 ; X 2 ; X 3 ; t ( ) ÷X j d ij : (2:2) The components of the Green–Lagrange strain tensor are given by E ij = 1 2 u i;j ÷u j;i ÷u i;k u j;k _ _ ; (2:3) where u i;j = @u i @X j ; etc., and repeated indices imply summation. When u i;j ¸ ¸ ¸ ¸ ¸ 1, E ij reduces to the infinitesimal strain tensor e ij given by e ij = 1 2 @u i @x j ÷ @u j @x i _ _ = 1 2 u i;j ÷u j;i _ _ : (2:4) From Eq. (2.4) it is seen that the strain tensor is symmetric. Thus, there are six independent strain components, which in the infinitesimal version are three normal strains (e 11 , e 22 , and e 33 ), and three shear strains (e 12 = e 21 , e 23 = e 32 , and e 13 = e 31 ). To ensure single-valued displacements u i , the strain components e ij cannot be assigned arbitrarily but must satisfy certain integrability or compatibility condi- tions, given by e ij;kl ÷e kl;ij ÷e ik;jl ÷e jl;ik = 0: (2:5) Of the 81 equations included in Eq. (2.5), only six are independent. The remainder are either identities or repetitions due to symmetry of e ij . For the special case of plane stress conditions, the only surviving compatibility equation is e 11;22 ÷e 22;11 ÷2e 12;12 = 0: (2:6) 2.1.2 Conservation of linear and angular momenta Ingeneral, the forces exertedona continuumbody are body forces andsurface forces. Body forces, such as gravitational and magnetic forces, act on all particles within the volume of the body and are described in terms of force intensity per unit mass or per unit volume, while surface forces are contact forces that act across an internal surface or an external (bounding) surface. The continuum description of surface forces is given by the traction vector t acting on a surface element dS with a unit normal n (see Figure 2.2(a)). Let dP be the total force exerted on dS by the material points on the side of dS toward which n is pointing. The traction vector t is then defined as t = lim dS÷0 dP dS : (2:7) At an internal point P there are infinitely many surface elements, each with a different unit normal vector. According tothe Cauchy theorema tractionvector onany of these Initial configuration Deformed configuration X x u X 2 X 1 O P PЈ X 3 Figure 2.1. Initial and deformed geometry of a continuum body. 10 Review of mechanics of composite materials planes can be expressed in terms of the traction vectors on three orthogonal planes passingthroughthe point P. InaCartesianreference frame, the three planes are chosen parallel to the coordinate planes and the resultant traction vectors on these planes are decomposed along the three coordinate axes. These 3 × 3 = 9 components taken together form the components of the second rank stress tensor associated with the considered point P. They are indicated in Figure 2.2(b) where their positive directions are shown. Inindex notation, they are denoted by s ij , where the first index refers to the directionof the unit normal onthe surface (the face of the cube inFigure 2.2(b)) andthe second index stands for the direction of the resolved traction component. The stress components withtwoequal indices, e.g., s 11 , are callednormal stresses while those with unequal indices, e.g., s 23 , are termed shear stresses. The traction vector components are related to the stress tensor components by the following equation t i = s ij n j ; (2:8) where n j are components of the normal vector associated with the traction vector. The conservation of linear momentum at a material point inside the continuum body gives the following relation s ji; j ÷ f i = r€ u i ; (2:9) where f i are components of the body force vector, and r is the mass density. For quasi-static problems the right-hand side of Eq. (2.9) vanishes, and if the body forces are neglected, the equations of equilibrium reduce to s ji; j = 0: (2:10) When there are no body moments, the conservation of angular momentum results in the symmetry of the stress tensor, i.e., s ij = s ji : (2:11) 2.1.3 Constitutive relations For an elastic material there exists a positive-definite, single-valued, potential function of strains e kl , defined as s 22 s 23 s 21 s 13 s 11 s 32 s 31 P n t dS (a) (b) s 33 s 12 Figure 2.2. (a) Traction vector; (b) a volume element with components of stress tensor. 11 2.1 Equations of elasticity U = _ e kl 0 s ij de ij : (2:12) This function is termed as the “strain energy density.” U is independent of the loading path and thus a function of final strains only. Differentiating Eq. (2.12) with respect to the strains, the stress tensor can be written as s ij = @U @e ij : (2:13) If we consider a linear elastic material, then U can be written as a quadratic function in e kl U e kl ( ) = 1 2 C ijkl e ij e kl ; (2:14) where C ijkl is a fourth-order tensor of material stiffness coefficients known as the stiffness tensor. Using Eqs. (2.13) and (2.14), one obtains the generalized Hooke’s law s ij = C ijkl e kl : (2:15) A potential function of stresses known as the complementary energy density is defined as U + s ij _ _ = s ij e ij ÷U: (2:16) Differentiation of Eq. (2.16) with respect to stress tensor yields the relation e ij = @U + @s ij : (2:17) Analogous to Eq. (2.14), U * can also be represented as a quadratic function as U + s ij _ _ = 1 2 S ijkl s ij s kl ; (2:18) where S ijkl are components of the compliance tensor. Using Eqs. (2.17) and (2.18), one obtains the inverse constitutive law e ij = S ijkl s kl : (2:19) In all, the stiffness matrix C ijkl has 81 coefficients. However, not all of these coefficients are independent. Note first that the symmetry of the strain compon- ents (e kl = e lk ) leads to C ijkl = C ijlk , which reduces the number of coefficients from 81 to 54. Similarly, the symmetry of the stress tensor further reduces the number of these coefficients to 36. Finally, differentiating Eq. (2.14) twice with respect to strains, one obtains C ijkl = @ 2 U @e ij @e kl : (2:20) 12 Review of mechanics of composite materials Since the order of differentiation in the above equation is arbitrary, one infers that C ijkl = C klij ; (2:21) which reduces the number of independent material coefficients to 21. The coefficient matrix C ijkl is expressed in compact form by using the Voigt notation, in which stress and strain tensor components are denoted using a single subscript, whereas two subscripts are used to denote the stiffness tensor. With this, the constitutive relation, Eq. (2.15), can be written as s p = C pq e q ; p, q = 1,2, . . ., 6, or in expanded matrix form as s 1 s 2 s 3 s 4 s 5 s 6 _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ = C 11 C 12 C 13 C 14 C 15 C 16 C 21 C 22 C 23 C 24 C 25 C 26 C 31 C 32 C 33 C 34 C 35 C 36 C 41 C 42 C 43 C 44 C 45 C 46 C 51 C 52 C 53 C 54 C 55 C 56 C 61 C 62 C 63 C 64 C 65 C 66 _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ e 1 e 2 e 3 e 4 e 5 e 6 _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ ; (2:22) where C pq = C qp , and s 1 = s 11 ; s 2 = s 22 ; s 3 = s 33 ; s 4 = s 23 ; s 5 = s 31 ; s 6 = s 12 e 1 = e 11 ; e 2 = e 22 ; e 3 = e 33 ; e 4 = 2e 23 ; e 5 = 2e 31 ; e 6 = 2e 12 C 11 = C 1111 ; C 22 = C 2222 ; : : : ; etc: (2:23) These constitutive relationships are for an anisotropic material. If material sym- metry exists, then further reduction occurs in the number of independent coeffi- cients of the stiffness matrix. It should be noted that the stiffness matrix in the Voigt notation does not follow the transformation rule for tensors. The fourth- order stiffness tensor C ijkl transforms as C / ijkl = ‘ ip ‘ jq ‘ kr ‘ ls C pqrs ; (2:24) where ‘ ij is the matrix of direction cosines associated with coordinate trans- formation from one coordinate system (x 1 , x 2 , x 3 ) to another (x / 1 ; x / 2 ; x / 3 ). A material with one plane of symmetry is called monoclinic, and if this plane is parallel to the x 1 –x 2 plane then it can be shown that the constitutive relation is given by s 1 s 2 s 3 s 4 s 5 s 6 _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ = C 11 C 12 C 13 0 0 C 16 C 21 C 22 C 23 0 0 C 26 C 31 C 32 C 33 0 0 C 36 0 0 0 C 44 C 45 0 0 0 0 C 54 C 55 0 C 61 C 62 C 63 0 0 C 66 _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ e 1 e 2 e 3 e 4 e 5 e 6 _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ : (2:25) Here, the stiffness matrix has 13 independent material coefficients. If a mater- ial has two mutually orthogonal planes of symmetry, then the plane 13 2.1 Equations of elasticity orthogonal to these planes is also a plane of symmetry. In this case, the material symmetry is described as orthotropic, and the number of independent constants in the stiffness matrix reduces to nine. The stress–strain relations when the symmetry planes are parallel to the three coordinate planes take the following form s 1 s 2 s 3 s 4 s 5 s 6 _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ = C 11 C 12 C 13 0 0 0 C 21 C 22 C 23 0 0 0 C 31 C 32 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 55 0 0 0 0 0 0 C 66 _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ e 1 e 2 e 3 e 4 e 5 e 6 _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ : (2:26) In terms of the engineering elastic constants the inverse strain–stress relations for the orthotropic case become as follows e 1 e 2 e 3 e 4 e 5 e 6 _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ = 1 E 1 ÷n 21 E 2 ÷n 31 E 3 0 0 0 ÷n 12 E 1 1 E 2 ÷n 32 E 3 0 0 0 ÷n 13 E 1 ÷n 23 E 2 1 E 3 0 0 0 0 0 0 1 G 23 0 0 0 0 0 0 1 G 31 0 0 0 0 0 0 1 G 12 _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ s 1 s 2 s 3 s 4 s 5 s 6 _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ ; (2:27) where E 1 , E 2 , E 3 are Young’s moduli in the three material symmetry directions (x 1 , x 2 , x 3 ) respectively, n ij ; i ,= j; are the six Poisson’s ratios defined in the conventional way, e.g., n 12 =÷ e 2 / e 1 with s 1 applied, and G 23 , G 31 , and G 12 are shear moduli in the x 2 –x 3 , x 1 –x 3 , and x 1 –x 2 planes, respectively. The compliance matrix in Eq. (2.27), being the inverse of a symmetric matrix, is also symmetric. From this symmetry follows the “reciprocal” relationship, n ij E i = n ji E j (no sumon i; j); (2:28) which can be used to eliminate three of the six Poisson’s ratios. If the material is isotropic in a plane, i.e., with same elastic properties in all directions in the plane, it is called transversely isotropic. Let the x 2 –x 3 plane be the plane of isotropy, i.e., E 3 = E 2 ; n 31 = n 12 ; G 31 = G 12 ; G 23 = E 2 2 1 ÷n 23 ( ) : The com- pliance tensor is then given by 14 Review of mechanics of composite materials S [ [ = S 11 S 12 S 12 0 0 0 S 12 S 22 S 23 0 0 0 S 12 S 23 S 22 0 0 0 0 0 0 2 S 22 ÷S 23 ( ) 0 0 0 0 0 0 S 66 0 0 0 0 0 0 S 66 _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ = 1 E 1 ÷n 12 E 1 ÷n 12 E 1 0 0 0 ÷n 12 E 1 1 E 2 ÷n 23 E 2 0 0 0 ÷n 12 E 1 ÷n 23 E 2 1 E 2 0 0 0 0 0 0 E 2 2 1 ÷n 23 ( ) 0 0 0 0 0 0 1 G 12 0 0 0 0 0 0 1 G 12 _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ : (2:29) As seen above, a transversely isotropic material has five independent stiffness coefficients, viz. E 1 , E 2 , n 23 , n 12 , and G 12 . For a completely isotropic material there are only two independent material coefficients, namely the Young’s modulus (E) and Poisson’s ratio (n) or, alternatively, the Lame constants (l and m). The constitutive relations can now be written as s ij = le kk d ij ÷2me ij ; (2:30) where d ij is the Kronecker delta. Alternatively, e ij = 1 E 1 ÷n ( )s ij ÷ns kk d ij _ ¸ : (2:31) 2.1.4 Equations of motion The equations governing the motion of a deformable body can be obtained by combining kinematic relations, Eq. (2.4), equilibrium equations, Eq. (2.10), and the constitutive relations, Eq. (2.15). For the particular case of linear elastic isotropic materials, they can be written as l ÷m ( )u j; ji ÷mu i; jj ÷f i = r€ u i : (2:32) These equations are known as Navier’s equations. The displacement field obtained from these equations is unique and results into the determination of strains and stresses by use of kinematic and constitutive relations. 2.1.5 Energy principles Energy principles for a continuum body allow formulating the relationships between stresses, strains or deformations, displacements, material properties, and external effects in the form of energy or work done by internal and external forces. They are also useful for obtaining approximate solutions of complex boundary value problems, e.g., finite element methods. Detailed treatment of these concepts can be found in [18–20]. 15 2.1 Equations of elasticity Principle of virtual work In the context of an elastic boundary value problem, consider a solid continuum body (Figure 2.3), occupying a volume V and bounded by surface S = S t ÷ S u , to be in static equilibrium under prescribed body forces f i over volume V, surface tractions t i on S t , and displacements u i over remaining portion of the boundary S u . For a statically admissible stress field ~ s ij (such that ~ s ij; j = 0 in V, and ~ t i = ~ s ij n j on S t ) and a kinematically admissible displacement field ^ u i (such that ^e ij = 1 2 ^ u i; j ÷ ^ u j;i _ _ ), the principle of virtual work states _ S ~ t i ^ u i dS ÷ _ V f i ^ u i dV = _ V ~ s ij ^e ij dV: (2:33) It should be noted that the displacement field ^ u i and the stress field ~ s ij are completely independent of each other. Principle of minimum potential energy For a kinematically admissible displacement field ^ u i , the potential energy of a linear elastic continuum body under the action of conservative forces f i and prescribed surface tractions t i on S t is defined as Å ^ u i ( ) = 1 2 _ V ^ s ij ^e ij dV ÷ _ S t i ^ u i dS ÷ _ V f i ^ u i dV: (2:34) The principle of minimum potential energy states that among all the kinematically admissible displacement fields the actual displacement field minimizes the poten- tial energy. Thus, if u i represents the actual displacement field, then Å ^ u i ( ) _ Å u i ( ) : (2:35) V t j n i S t S u u i f i Figure 2.3. A continuum body loaded with body forces inside its volume, and traction and displacement on the boundary. 16 Review of mechanics of composite materials Principle of minimum complementary energy For a statically admissible stress field ^ s ij , the complementary potential energy of a linear elastic body is defined as Å + ^ s ij _ _ = 1 2 _ V ^ s ij ^e ij dV ÷ _ S ^ t i u i dS; (2:36) where ^ t i = ^ s ij n j is the reaction on S u . The principle of minimum complementary energy states that among all the statically admissible stress fields the actual stress field minimizes the complementary potential energy. Thus, if u i represents the actual displacement field, then Å + ^ s ij _ _ _ Å + s ij _ _ : (2:37) For actual stress, strain, anddisplacement fields, additionof Eqs. (2.34) and(2.36) yields Å u i ( ) ÷Å + s ij _ _ = _ V s ij e ij dV ÷ _ S t i u i dS ÷ _ V f i u i dV: (2:38) The right-hand side of Eq. (2.38) vanishes by virtue of the principle of virtual work. Hence, Å u i ( ) = ÷Å + s ij _ _ : (2:39) Using Eqs. (2.35), (2.37), and (2.39), we obtain the lower and upper bounds to the potential energy of a continuum body ÷Å ^ u i ( ) _ ÷Å u i ( ) = Å + s ij _ _ _ Å + ^ s ij _ _ : (2:40) For the purpose of illustration, the potential and complementary energies for a typical load–displacement response are shown in Figure 2.4. 2.2 Micromechanics Micromechanics is a well-developed advanced field that treats the response of a heterogeneous solid based on the behavior of its constituents and their geometrical configurations. For a detailed exposition the reader may refer to, e.g., [8]. Here a brief summary of simple micromechanics estimates of the linear elastic properties P Õ * P, D Õ D Figure 2.4. A typical load–displacement diagram. 17 2.2 Micromechanics of a unidirectional fiber-reinforced composite is provided. These estimates are useful in selecting fibers and matrix materials and their volume fractions. In many structural applications a unidirectional composite, fabricated as a thin layer, called lamina or ply, is used as a basic unit and a laminate is constructed by stacking these layers as illustrated in Figure 2.5. 2.2.1 Stiffness properties of a unidirectional lamina Linear elastic properties of a lamina can be referred to a coordinate system (x 1 , x 2 , x 3 ) where the x 1 -axis is aligned with fibers, x 2 -axis is transverse to fibers in the plane of the lamina, and the x 3 -axis is normal to the plane of lamina (see Figure 2.6). Noting that the lamina has orthotropic symmetry, the nine independent elastic constants, as described in Section 2.1.3 above, in this reference system are the three Young’s moduli (E 1 , E 2 , E 3 ), the three Poisson’s ratios (n 12 , n 13 , n 23 ), and the three shear moduli (G 12 , G 13 , G 23 ). For a subset of these constants that represents in-plane properties, i.e., E 1 , E 2 , n 12 , n 21 , and G 12 , in the x 1 –x 2 plane, the following expressions hold E 1 = E f V f ÷E m V m ; (2:41) n 12 = n f V f ÷n m V m ; (2:42) 1 E 2 = V f E f ÷ V m E m ; (2:43) 1 G 12 = V f G f ÷ V m G m ; (2:44) where E, n, G, and V stand for the Young’s modulus, Poisson’s ratio, shear modulus, and volume fraction, respectively, with the subscripts f and m indicating fibers and matrix, respectively. The minor Poisson’s ratio n 21 can be estimated using the reciprocal relationship n 21 = n 12 (E 2 /E 1 ). Equations (2.41) and (2.42) have the form of the familiar rule of mixtures and Eqs. (2.43) and (2.44) follow that rule for the inverse of the respective properties. A unidirectional lamina Lamination Laminate (a) (b) (c) Figure 2.5. Stacking of a number of laminae makes up a laminate. 18 Review of mechanics of composite materials The first two expressions predict the experimental properties usually well while the third and fourth expressions are found to be less accurate. Halpin and Kardos [21] and Halpin and Tsai [22] proposed semi-empirical relationships based on numer- ical computations. These relations can be used in place of Eqs. (2.43) and (2.44), and are together expressed as p p m = 1 ÷xV f 1 ÷V f ; (2:45) where = p f p m ÷1 p f p m ÷x : (2:46) Here p represents E 2 or G 12 , and p f and p m are the corresponding moduli for fiber and matrix, respectively. The fitting parameter x needs to be determined by comparing predictions with experimental data. More advanced micromechanics approaches, such as Hashin–Shtrikman vari- ational bounds [23–29], Mori–Tanaka model [30], composite sphere and cylinder assemblage model [31, 32], self-consistent method [33], method of cells [34–36], etc. have also been developed in the past four decades. Interested readers are referred to texts on micromechanics, e.g., [8, 37, 38], for detailed treatment of these approaches. 2.2.2 Thermal properties of a unidirectional lamina Simple micromechanics estimates for the linear coefficient of thermal expansion of a lamina can be obtained in the same way as the linear elastic properties. The expressions obtained are as follows a 1 = 1 E 1 a f E f V f ÷a m E m V m ( ); a 2 = 1 ÷n f ( )a f V f ÷ 1 ÷n m ( )a m V m ÷a 1 n 12 ; (2:47) x 2 x 1 x y x 3 , z Figure 2.6. Coordinate systems for a unidirectional ply. The material system is denoted by x 1 , x 2 , x 3 ; while the laminate system is denoted by x, y, z. 19 2.2 Micromechanics where a 1 and a 2 are the thermal expansion coefficients in the fiber and transverse directions, respectively, and E 1 and n 12 are given by Eqs. (2.41) and (2.42). 2.2.3 Constitutive equations for a lamina A lamina is thin compared to other dimensions of the entire laminate. Therefore, the lamina can be assumed to be in a state of generalized plane stress. Conse- quently, all the through-thickness stress components are zero, i.e., s 4 = s 5 = s 6 = 0. In such a case, the constitutive relation for an individual lamina referred to the three axes of symmetry can be written in Voigt notation as s 1 s 2 s 6 _ _ _ _ _ _ = Q 11 Q 12 0 Q 12 Q 22 0 0 0 Q 66 _ _ _ _ e 1 e 2 e 6 _ _ _ _ _ _ ; (2:48) with Q 11 = E 1 1÷n 12 n 21 ; Q 22 = E 2 1÷n 12 n 21 ; Q 12 = n 12 E 2 1÷n 12 n 21 = n 21 E 1 1÷n 12 n 21 ; Q 66 =G 12 : (2:49) The inverse constitutive relation for the lamina is given by e ij = S ijkl s kl = e 1 e 2 e 6 _ _ _ _ _ _ = 1 E 1 ÷ n 21 E 2 0 ÷ n 12 E 1 1 E 2 0 0 0 1 G 12 _ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ _ s 1 s 2 s 6 _ _ _ _ _ _ : (2:50) The above constitutive relations are written in the lamina coordinate system (i.e., with x 1 along the fiber direction, x 2 normal to the fiber direction, and x 3 along the lamina thickness). The constitutive relation for the lamina in another coordinate system (x–y–z), which, for instance, could be aligned with the coordinate system chosen for the laminate, is s xx s yy s xy _ _ _ _ _ _ = Q 11 Q 12 Q 16 Q 12 Q 22 Q 26 Q 16 Q 26 Q 66 _ _ _ _ e xx e yy 2e xy _ _ _ _ _ _ ; (2:51) where Q ij are known as reduced stiffness coefficients. These are related to Q ij , defined by Eq. (2.49), by the transformation rules for stresses and strains. Thus, s 1 s 2 s 6 _ _ _ _ _ _ = m 2 n 2 2mn n 2 m 2 ÷2mn ÷mn mn m 2 ÷n 2 _ _ _ _ s xx s yy s xy _ _ _ _ _ _ = T [ [ s xx s yy s xy _ _ _ _ _ _ (2:52) e 1 e 2 e 6 _ _ _ _ _ _ = m 2 n 2 mn n 2 m 2 ÷mn ÷2mn 2mn m 2 ÷n 2 _ _ _ _ e xx e yy e xy _ _ _ _ _ _ = T [ [ ÷1 _ _ T e xx e yy e xy _ _ _ _ _ _ (2:53) 20 Review of mechanics of composite materials where m = cos y, n = sin y, where y is the angle between the x- and x 1 -axes (Figure 2.6). Then by inverting Eqs. (2.52) and (2.53), substituting these in Eq. (2.51), and on using Eq. (2.48) one obtains Q _ ¸ = T [ [ ÷1 Q [ [ T [ [ T _ _ ÷1 : (2:54) [T] ÷1 is simply given by changing y to –y, i.e., [T(y)] ÷1 = [T(÷y)]. Expanding the above relation, we have Q 11 = Q 11 m 4 ÷2 Q 12 ÷2Q 66 ( )m 2 n 2 ÷Q 22 n 4 ; Q 22 = Q 11 n 4 ÷2 Q 12 ÷2Q 66 ( )m 2 n 2 ÷Q 22 m 4 ; Q 12 = Q 11 ÷Q 22 ÷4Q 66 ( )m 2 n 2 ÷Q 12 m 4 ÷n 4 _ _ ; Q 16 = Q 11 ÷Q 12 ÷2Q 66 ( )m 3 n ÷ Q 12 ÷Q 22 ÷2Q 66 ( )mn 3 ; Q 26 = Q 11 ÷Q 12 ÷2Q 66 ( )mn 3 ÷ Q 12 ÷Q 22 ÷2Q 66 ( )m 3 n; Q 66 = Q 11 ÷Q 22 ÷2Q 12 ÷2Q 66 ( )m 2 n 2 ÷Q 66 m 4 ÷n 4 _ _ : (2:55) The transformation rules described above enable us to express engineering moduli for the lamina referred to arbitrary in-plane axes (x–y) in terms of moduli in the principal (x 1 –x 2 ) directions as 1 E x = m 4 E 1 ÷ n 4 E 2 ÷ 1 G 12 ÷ 2n 12 E 1 _ _ m 2 n 2 ; 1 E y = n 4 E 1 ÷ m 4 E 2 ÷ 1 G 12 ÷ 2n 12 E 1 _ _ m 2 n 2 ; n xy E x = n 12 E 1 ÷ 1 ÷2n 12 E 1 ÷ 1 E 2 ÷ 1 G 12 _ _ m 2 n 2 ; 1 G xy = 1 G 12 ÷4m 2 n 2 1 ÷n 12 E 1 ÷ 1 ÷n 21 E 2 ÷ 1 G 12 _ _ : (2:56) To account for thermal stresses, we need to modify strains in Eq. (2.51) to include thermal strains, as e xx e yy 2e xy _ _ _ _ _ _ = e 0 xx e 0 yy 2e 0 xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ ÷ e th xx e th yy 2e th xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ ; (2:57) where the superscripts 0 and th denote mechanical and thermal strains, respectively, with e th xx e th yy 2e th xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ = T [ [ e e th 1 e th 2 2e th 12 _ _ _ _ _ _ = T [ [ e a 1 a 2 0 _ _ _ _ _ _ ÁT : (2:58) 2.2.4 Strength of a unidirectional lamina Phenomenological failure (strength) criteria that use experimental data to deter- mine material constants have been proposed for composite materials along the 21 2.2 Micromechanics lines of those used for metals such as the von Mises yield criterion. Failure mechanisms in composite materials are, however, significantly more complex, resulting in a large number of criteria. Here, some common criteria will be stated for reference; the interested reader is encouraged to consult [39] for more in-depth treatment. For a unidirectional fiber-reinforced lamina, the five basic strength parameters under in-plane loading are as follows: X = ultimate tensile strength in the fiber direction X / = ultimate compressive strength in the fiber direction Y = ultimate tensile strength transverse to fibers Y / = ultimate compressive strength transverse to fibers S = ultimate shear strength in the lamina plane. These parameters are obtained by experimental testing. See, e.g., [39, 40] for further details. Maximum stress theory According to this theory, a lamina fails if s 1 = X s 1 > 0; ÷X / s 1 < 0; _ s 2 = Y s 2 > 0; ÷Y / s 2 < 0; _ s 6 [ [ = S: (2:59) For combined loading, theoretical predictions of the theory are inaccurate because the maximum stress criterion does not account for stress interactions. For an off- axis normal loading, i.e., loading axis inclined to fibers, this theory can be applied by transforming the stresses to the principal material directions and then using the criteria in Eq. (2.59). Maximum strain theory This theory states that failure occurs when e 1 = X e e 1 > 0; ÷X / e e 1 < 0; _ e 2 = Y e e 2 > 0; ÷Y / e e 2 < 0; _ e 6 [ [ = S e ; (2:60) where X e = X=E 1 ; X / e = X / =E 1 ; Y e = Y=E 2 ; Y / e = Y / =E 2 ; and S e = S=G 12 are the ultimate failure strains analogous to the stress-based parameters mentioned above. 22 Review of mechanics of composite materials Distortional energy (Tsai–Hill) criterion This criterion is based on the distortional energy failure (yield) theory of von Mises. Hill [41] further developed this yield criterion for anisotropic materials and Azzi and Tsai [42] modified it to describe failure of a composite lamina as follows s 2 1 X 2 ÷ s 1 s 2 X 2 ÷ s 2 2 Y 2 ÷ s 2 6 S 2 = 1; (2:61) where s 1 and s 2 are the tensile normal stresses along fibers and normal to fibers, respectively, and s 6 is the in-plane shear stress. When the normal stresses are compressive, the compressive strength values in Eq. (2.61) are to be used. Tsai–Wu criterion A polynomial function of stress components can be formulated with the multiply- ing terms of the polynomial expressing strength properties. Restricted to quadratic terms of in-plane stress components, such a function is known as the Tsai–Wu criterion [43] and can be expressed as F 1 s 1 ÷F 2 s 2 ÷F 11 s 2 1 ÷F 22 s 2 2 ÷F 66 s 2 6 ÷2F 12 s 1 s 2 = 1: (2:62) The product terms s 1 s 6 and s 2 s 6 are not present in Eq. (2.62) because the multiplying coefficients to these terms can be shown to vanish. Also, the linear term in s 6 is absent because of the shear strength being independent of the sign of shear stress, which renders the coefficient of this term to be zero. The six material constants in the Tsai–Wu criterion require two tests (tension and compression) in the fiber direction, two similar tests normal to fibers, an in- plane shear test, and a biaxial normal load test. Hashin’s criterion Hashin [44] formulated three-dimensional failure criteria for unidirectional fiber composites in terms of quadratic stress polynomials. The terms used in the polynomials were functions of the stress invariants for transversely isotropic symmetry. Thus the cross-sectional plane of a unidirectional fiber composite was assumed as an isotropic plane. For relatively thick layers this may be a good assumption. A unidirectional fiber composite was assumed to fail in one of four possible separate modes: tensile fiber mode (s 1 > 0), compressive fiber mode (s 1 < 0), tensile matrix mode (s 2 + s 3 > 0), and compressive matrix mode (s 2 + s 3 < 0) For a thin unidirectional fiber composite layer (lamina), the four failure criteria are given by 23 2.2 Micromechanics s 2 X _ _ 2 ÷ s 6 S _ _ 2 = 1; s 1 > 0; s 1 = ÷X / ; s 1 < 0; s 2 Y _ _ 2 ÷ s 6 S _ _ 2 = 1; s 2 > 0; s 2 2S / _ _ 2 ÷ Y / 2S / _ _ 2 ÷1 _ _ s 2 Y / ÷ s 6 S _ _ 2 = 1; s 2 < 0; (2:63) where S / is the strength in transverse shear, while S here is the same in axial shear. The difference between the two shear strengths is not fully unambiguous. Over the years, a wide variety of failure criteria have been proposed. There is no single failure theory that seems to capture all the complexities of composite failure. A world-wide failure exercise was conducted to evaluate applicability of most theories by comparing their predictions with test data [45]. 2.3 Analysis of laminates Laminates used in most engineering applications are fabricated by stacking plies in different orientations. An example of a laminate with layup [0/90/45] s is shown in Figure 2.7. A commonly used method of determining stresses and strains for such laminates is based on the classical laminate plate theory (CLPT). More advanced theories are treated in [9, 46]. The geometrical conditions needed for the application of the CLPT are: (a) the individual plies are of uniform thickness, (b) they are perfectly bonded to their neighboring plies, and (c) the total thickness of the laminate follows the so-called thin plate assumption, which states that the thickness dimension is much smaller than other structural dimensions (width and length). The kinematic assumptions of the CLPT derive from the Kirchhoff assumptions, which state that (a) a line element normal to the mid-plane in the undeformed state 0˚ 90˚ 45˚ 45˚ 90˚ 0˚ y z x z 0 z 1 z 2 z n q Figure 2.7. Stacking of unidirectional plies in different orientations to make a multidirectional [0/90/45] s laminate. The subscript s denotes that the laminate is symmetric about the mid-plane. 24 Review of mechanics of composite materials of the plate remains straight and perpendicular to the mid-plane after deformation, and (b) such a line element does not change its length when the plate deforms. 2.3.1 Strain–displacement relations The Kirchoff assumptions stated above lead to the x-, y-, and z-displacements u, v, and w, respectively, in the coordinate system shown in Figure 2.7 as follows u(x; y; z) = u 0 (x; y) ÷z @w 0 (x; y) @x ; v(x; y; z) = v 0 (x; y) ÷z @w 0 (x; y) @y ; w(x; y; z) = w 0 (x; y); (2:64) where (u 0 , v 0 , w 0 ) are the displacements of the laminate mid-plane. The corres- ponding strain–displacement relations are given by e xx = @u @x = @u 0 @x ÷z @ 2 w 0 @x 2 ; e yy = @v @y = @v 0 @y ÷z @ 2 w 0 @y 2 ; e zz = @w @z = 0; e xy = 1 2 @u @y ÷ @v @x _ _ = 1 2 @u 0 @y ÷ @v 0 @x _ _ ÷z @ 2 w 0 @x@y ; e xz = 1 2 @u @z ÷ @w @x _ _ = 0; e yz = 1 2 @v @z ÷ @w @y _ _ = 0: (2:65) The nonzero equations can be written in the following form e xx e yy g xy _ _ _ _ _ _ = e 0 xx e 0 yy g 0 xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ ÷z k xx k yy k xy _ _ _ _ _ _ ; (2:66) where e 0 xx ; e 0 yy ; g 0 xy _ _ are the mid-plane strains and k xx ; k yy ; k xy _ _ are the laminate curvatures, given by e 0 xx e 0 yy g 0 xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ = @u 0 @x @v 0 @y @u 0 @y ÷ @v 0 @x _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ _ and k xx k yy k xy _ _ _ _ _ _ = ÷ @ 2 w 0 @x 2 ÷ @ 2 w 0 @y 2 ÷2 @ 2 w 0 @x@y _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ : (2:67) 25 2.3 Analysis of laminates 2.3.2 Constitutive relationships for the laminate Using the lamina constitutive relations described earlier, the constitutive equation for the kth (k = 1, 2, . . .) layer of the laminate can be written as s ¦ ¦ (k) = Q _ ¸ (k) e ¦ ¦ (k) : (2:68) In the above equation, the square bracket represents a 3×3 matrix and the curly bracket is for a 3×1 vector. The strains in the kth ply are given by e ¦ ¦ (k) = e 0 _ _ ÷z k ¦ ¦: (2:69) The thermal strains can be added to these strains, such that e ¦ ¦ (k) = e 0 _ _ ÷z k ¦ ¦ ÷ a ¦ ¦ k ÁT: (2:70) The kth ply stresses on using Eq. (2.68) can now be written as s ¦ ¦ (k) = Q _ ¸ (k) e 0 _ _ ÷ a ¦ ¦ k ÁT _ _ ÷z Q _ ¸ (k) k ¦ ¦: (2:71) At the laminate level the force and moment resultants are defined as N [ [ = N xx N yy N xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ = _ h=2 ÷h=2 s xx s yy s xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ dz; M [ [ = M xx M yy M xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ = _ h=2 ÷h=2 s xx s yy s xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ z dz: (2:72) In terms of ply stresses that generally vary from ply to ply, we have N xx N yy N xy _ _ _ _ _ _ = N k=1 _ z k÷1 z k s xx s yy s xy _ _ _ _ _ _ dz; (2:73) which gives us N xx N yy N xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ ÷ N th xx N th yy N th xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ = N k=1 _ z k÷1 z k Q 11 Q 12 Q 16 Q 12 Q 22 Q 26 Q 16 Q 26 Q 66 _ ¸ _ _ ¸ _ e 0 xx e 0 yy g 0 xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ dz ÷ N k=1 _ z k÷1 z k Q 11 Q 12 Q 16 Q 12 Q 22 Q 26 Q 16 Q 26 Q 66 _ ¸ _ _ ¸ _ k xx k yy k xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ z dz; (2:74) where the force resultants due to thermal stresses are given by 26 Review of mechanics of composite materials N th xx N th yy N th xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ = N k=1 _ z k÷1 z k Q 11 Q 12 Q 16 Q 12 Q 22 Q 26 Q 16 Q 26 Q 66 _ _ _ _ a x ÁT a y ÁT 0 _ _ _ _ _ _ dz: (2:75) The relations in Eq. (2.74) can be rewritten in more compact form by using matrices [A] and [B] as follows N xx N yy N xy _ _ _ _ _ _ ÷ N th xx N th yy N th xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ = A 11 A 12 A 16 A 12 A 22 A 26 A 16 A 26 A 66 _ _ _ _ e 0 xx e 0 yy g 0 xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ ÷ B 11 B 12 B 16 B 12 B 22 B 26 B 16 B 26 B 66 _ _ _ _ k xx k yy k xy _ _ _ _ _ _ : (2:76) Similarly, M xx M yy M xy _ _ _ _ _ _ = N k=1 _ z k÷1 z k s xx s yy s xy _ _ _ _ _ _ zdz; (2:77) i.e., M xx M yy M xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ ÷ M th xx M th yy M th xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ = N k=1 _ z k÷1 z k Q 11 Q 12 Q 16 Q 12 Q 22 Q 26 Q 16 Q 26 Q 66 _ ¸ _ _ ¸ _ e 0 xx e 0 yy g 0 xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ zdz ÷ N k=1 _ z k÷1 z k Q 11 Q 12 Q 16 Q 12 Q 22 Q 26 Q 16 Q 26 Q 66 _ ¸ _ _ ¸ _ k xx k yy k xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ z 2 dz; (2:78) where M th xx M th yy M th xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ = N k=1 _ z k÷1 z k Q 11 Q 12 Q 16 Q 12 Q 22 Q 26 Q 16 Q 26 Q 66 _ _ _ _ a x ÁT a y ÁT 0 _ _ _ _ _ _ zdz: (2:79) Introducing a new matrix [D], Eq. (2.78) can be rewritten as M xx M yy M xy _ _ _ _ _ _ ÷ M th xx M th yy M th xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ = B 11 B 12 B 16 B 12 B 22 B 26 B 16 B 26 B 66 _ _ _ _ e 0 xx e 0 yy g 0 xy _ ¸ _ ¸ _ _ ¸ _ ¸ _ ÷ D 11 D 12 D 16 D 12 D 22 D 26 D 16 D 26 D 66 _ _ _ _ k xx k yy k xy _ _ _ _ _ _ : (2:80) The material coefficients (A ij , B ij , D ij ) are known as the extensional stiffness, the extension–bending coupling stiffness, and the bending stiffness coefficients, respect- ively. These are given by A ij ; B ij ; D ij _ _ = _ h 2 ÷ h 2 Q ij 1; z; z 2 _ _ dz; (2:81) 27 2.3 Analysis of laminates or A ij = N k=1 Q ij z k÷1 ÷z k ( ) ; B ij = 1 2 N k=1 Q ij z 2 k÷1 ÷z 2 k _ _ ; D ij = 1 3 N k=1 Q ij z 3 k÷1 ÷z 3 k _ _ : (2:82) The laminate constitutive relations can now be written in compact form as N ¦ ¦ M ¦ ¦ _ _ = A [ [ B [ [ B [ [ D [ [ _ _ e 0 _ _ k ¦ ¦ _ _ (2:83) where {N}, {M} include thermal resultants. 2.3.3 Stresses and strains in a lamina within a laminate The laminate constitutive relations in Eq. (2.83) can be reverted to yield the mid- plane strains and curvatures in terms of the stress and moment resultants. A partial inversion is first done by inverting Eq. (2.76) and substituting into Eq. (2.80) to obtain e 0 M _ _ = A + B + C + D + _ _ N k _ _ ; (2:84) where A + = A ÷1 ; B + = ÷A ÷1 B; C + = BA ÷1 = ÷ B + ( ) T ; D + = D ÷BA ÷1 B; (2:85) and the brackets for matrix/vector representation have been dropped for conveni- ence. Solving Eq. (2.84) for k and its substitution back gives e 0 k _ _ = A / B / B / D / _ _ N M _ _ ; (2:86) where A / = A + ÷B + D + ( ) ÷1 B + ( ) T ; B / = B + D + ( ) ÷1 ; D / = D + ( ) ÷1 : (2:87) Once mid-plane strains and curvatures are known, the strains and stresses in each lamina can be determined using Eqs. (2.70) and (2.68), respectively. 2.3.4 Effect of layup configuration The sequence of ply layup has a significant impact on the stiffness properties of the designed laminate. Some interesting ply configurations are described below. 28 Review of mechanics of composite materials v Balanced laminate: If for each +y-ply, we have another identical ply of the same thickness, but –y orientation, we have A 16 = A 26 = 0. Such laminates are known as balanced laminates. If additionally these plies are at the same distance about the mid-plane (one above and another below the mid-plane), then D 16 = D 26 = 0. An example of a balanced laminate is [0/+45/÷45/90 2 /0] T , where the subscript T denotes “total” laminate sequence. v Symmetric laminate: If a laminate has plies stacked in such a way that through its thickness the plies are symmetrical about the mid-plane, then B ij = 0. Thus, such laminates will not exhibit any extension–bending coupling, e.g., [0/±30/ 45 2 /90 2 /45 2 /±30/0] T = [0/±30/45 2 /90] s , where the subscript s represents sym- metry about mid-plane. v Cross-ply laminate: If the plies are stacked in two orthogonal directions, e.g., in longitudinal (0 · ) and transverse (90 · ) directions, the laminate thus built is known as a cross-ply laminate, e.g., [0 2 /90 4 /0] T . v Quasi-isotropic laminate: If plies of identical properties and thickness are oriented in such a way that the angle between any two adjacent plies is equal to p/n, where n is the number of plies equal to or greater than three, then [A] becomes directionally independent, thereby showing isotropy in the in-plane material properties. This construction does not imply that the matrices [B] and [D] are also isotropic, e.g., [0/±45/90] s . For the special case of a symmetric balanced laminate, the in-plane engineering moduli can be obtained from the [A], [B], and [D] matrices using E x = 1 h A 11 ÷ A 2 12 A 22 _ _ ; E y = 1 h A 22 ÷ A 2 12 A 11 _ _ ; n xy = A 12 A 22 ; G xy = A 66 h ; (2:88) where h is the total laminate thickness. The laminate analysis summarized here is inadequate at and near free edges in laminates. The stress state in a laminate near a free boundary is three-dimensional and cannot be assumed to be well described by plane stress or plane strain assumptions. The through-thickness normal and shear stresses can, in some cases, be significant and could cause laminate failure. 2.4 Linear elastic fracture mechanics The basic concepts of fracture mechanics are useful in analyzing damage and failure in composite materials. Here we will briefly review those concepts and list some of the commonly used results from the linear elastic fracture mechanics. For more detailed treatment of fracture mechanics comprehensive texts, e.g., [13, 15, 17, 47] are suggested. 29 2.4 Linear elastic fracture mechanics 2.4.1 Fracture criteria The traditional strength of the materials approach to structural design and mater- ial selection is based on the notion of yield or failure stress (strength) of a given material. The fracture mechanics approach instead recognizes the presence of material flaws whose unstable growth could cause catastrophic failure. To deter- mine conditions for this type of failure the local stress field in the vicinity of flaws (modeled as cracks) is analyzed. The singularity of this stress field is characterized by the so-called stress intensity factor and its critical value is associated with unstable crack growth. Alternatively, energy balance considerations are used to find the so-called energy release rate, per unit increment in the crack surface, and its critical value is associated with the condition of unstable crack growth. For linear elastic materials undergoing brittle failure the two approaches produce the same failure criteria. The stress intensity criterion Consider an infinite plate with a through thickness edge crack of size a subjected to a remote tensile stress as shown in Figure 2.8. For a linear elastic plate material the stress field in close vicinity of the crack tip is given by s xx = K I ffiffiffiffiffiffiffiffi 2pr _ cos y 2 _ _ 1 ÷sin y 2 _ _ sin 3y 2 _ _ _ _ ; s yy = K I ffiffiffiffiffiffiffiffi 2pr _ cos y 2 _ _ 1 ÷sin y 2 _ _ sin 3y 2 _ _ _ _ ; t xy = K I ffiffiffiffiffiffiffiffi 2pr _ cos y 2 _ _ sin y 2 _ _ cos 3y 2 _ _ ; (2:89) where r and y are as shown in the figure and K I, known as the stress intensity factor, is given by K I = s ffiffiffiffiffiffi pa _ : (2:90) where the subscript I denotes the opening mode (mode I). It can be noted that the stress field is singular at the crack tip with a r 1/2 singularity. The condition of failure, i.e., unstable crack growth, is assumed when K I _ K IC : (2:91) K IC , known as the critical stress intensity factor, or fracture toughness, is a parameter representing the material resistance to fracture, and can be obtained experimentally. The energy criterion In the energy-based approach, one considers a cracked body and examines the changes brought about by an incremental crack growth in the potential energy of 30 Review of mechanics of composite materials applied forces – the stored elastic strain energy and the crack surface energy. The condition for unstable crack growth is then expressed as G _ G C ; (2:92) where G is the energy available for crack growth per unit of crack surface area, called the energy release rate, and G C is its critical value, which depends on the material in which the crack is advancing. G C is viewed as the resistance to crack growth induced by the material. For a linear elastic material undergoing small- scale yielding at the crack front, the energy release rate is found to be related to the stress intensity factor, described above, as G = K 2 I E / ; (2:93) where E / = E for plane stress condition, and E / = E 1 ÷n 2 for plane strain condition. 2.4.2 Crack separation modes A crack is activated, i.e., it produces stresses at its front, when the two crack surfaces separate. The separation can take place in combination of three a x y r s yy t xy q s xx s s Figure 2.8. Edge crack in a plate in tension. 31 2.4 Linear elastic fracture mechanics independent modes, denoted as modes I, II, and III, illustrated in Figure 2.9. In mode I, also called the crack opening mode, the two crack surfaces separate symmetrically about the crack plane. Mode II is a sliding mode, in which the two crack surfaces remain in contact and slide past each other in the crack plane. Finally, mode III, described as the tearing mode, is driven by out-of-plane shear, resulting in displacement of the two crack surfaces in the x 3 -direction. Any displacement of the crack surfaces for a general loading can be viewed as a superposition of these three modes. Denoting the stress intensity factors in individ- ual modes as K I , K II , and K III , the energy release rate for mixed-mode is given by G = K 2 I E / ÷ K 2 II E / ÷ 1 ÷n E K 2 III ; (2:94) where K I = s 11 ffiffiffiffiffiffi pa _ ; K II = s 12 ffiffiffiffiffiffi pa _ ; K III = s 13 ffiffiffiffiffiffi pa _ ; (2:95) where the stresses s 11 , etc. refer to the axes shown in Figure 2.9. 2.4.3 Crack surface displacements The displacement jump across the two crack surfaces, expressed as Áu i = u ÷ i ÷u ÷ i ; (2:96) where u ÷ i and u ÷ i represent the displacements of the upper and lower crack surfaces, respectively, is a quantity of interest in fracture analysis. For the opening mode of crack separation, mode I, illustrated in Figure 2.10, i = 2, this quantity is described as crack opening displacement (COD). For an infinite isotropic homo- geneous medium the COD value is given by Áu 2 = k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ÷ x 1 a _ _ 2 _ ; (2:97) which describes an elliptical crack opening profile. (a) (b) (c) x 3 x 2 x 1 Figure 2.9. Crack separation modes: (a) opening; (b) sliding; and (c) tearing. 32 Review of mechanics of composite materials 2.4.4 Relevance of fracture mechanics for damage analysis Fracture mechanics developed and matured well before damage mechanics emerged. For both fields, the impetus came from the need to analyze failure of metals. Fracture mechanics initially addressed brittle failure from sharp defects based on idealized stress analysis of cracks. In contrast, damage mechanics was concerned with the effect of distributed voids and cracks on the average response of a solid. For composite materials, the complexity of failure processes involving a multitude of cracks gave rise to further development of damage mechanics. Today, damage mechanics of composite materials stands on its own as a mature field solidly founded in thermodynamics and having a variety of analytical and computational methodologies associated with it. Fracture mechanics has aided the development of damage mechanics of composite materials in providing energy-based concepts for addressing evolution of failure states. However, the stress analysis of cracks, char- acterized by stress intensity factors, is less relevant to composite damage analysis. Other than a few cases where single crack growth is a dominant failure mechanism, such as delamination emanating from free edges in laminates, crack front singular- ities are of little interest. Indeed, individual cracks constituting damage modes are usually arrested at interfaces. Therefore, their growth is of little interest. Instead, energy dissipation occurs due to crack multiplication. Therefore, an appropriate energy-based analysis, needed to treat this type of situation, does not resort to stress intensity factors, as is the case in brittle fracture of metals. x 1 x 2 u 2 + u 2 – – s s + Figure 2.10. Crack opening displacement for a crack of size “2a.” 33 2.4 Linear elastic fracture mechanics References 1. Y. C. Fung, A First Course in Continuum Mechanics. (Englewood Cliffs, NJ: Prentice- Hall, Inc., 1977). 2. L. E. Malvern, Introduction to the Mechanics of a Continuous Medium. (Englewood Cliffs, NJ: Prentice-Hall, Inc., 1969). 3. I. S. Sokolnikoff, Mathematical Theory of Elasticity, 2ndedn. (NewYork: McGraw-Hill, 1956). 4. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity. (New York: Dover Publications, 1944). 5. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity. (NewYork: McGraw-Hill, 1951). 6. R. M. Jones, Mechanics of Composite Materials, 2nd edn. 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There are, however, significant differences in design procedures depending on whether the material used is a so-called monolithic material, e.g., a metal or a ceramic, or whether it is a composite material with distinctly different constituents. The hetero- geneity of microstructure as well as the anisotropy of properties provide signifi- cantly different characteristics to composite materials in how they deform and fail when compared to metals or ceramics. This chapter will review those characteris- tics. However, before proceeding we need to introduce certain definitions. Fracture: Conventionally, fracture is understood to be “breakage” of material, or at a more fundamental level, breakage of atomic bonds, manifesting itself in formation of internal surfaces. Examples of fracture in composites are fiber breakage, cracks in matrix, fiber/matrix debonds, and separation of bonded plies (delamination). The field known as fracture mechanics deals with condi- tions for formation and enlargement of the surfaces of material separation. Damage: Damage, on the other hand, refers to a collection of all the irreversible changes brought about in a material by a set of energy dissipating physical or chemical processes, resulting from the application of thermomechanical load- ings. Damage may inherently be manifested by atomic bond breakage. Unless specified differently, damage is understood to refer to distributed changes. Examples of damage in composites are multiple fiber-bridged matrix cracking in a unidirectional composite, multiple intralaminar cracking in a laminate, local delamination distributed in an interlaminar plane, and fiber/matrix inter- facial slip associated with multiple matrix cracking. These damage mechanisms will be explained in some detail later in this chapter. The field of damage mechanics deals with conditions for initiation and progression of distributed changes as well as with consequences of those changes on the response of a material (and by implication, a structure) to external loading. Failure: The inability of a given material system (and consequently, a structure made from it) to perform its design function. Fracture is one example of a possible failure; but, generally, a material could fracture (locally) and still perform its design function. Upon suffering damage, e.g., in the form of multiple cracking, a composite material may still continue to carry loads and, thereby, meet its load-bearing requirement but fail to deform in a manner needed for its other design requirements, such as vibration characteristics and deflection limits. It is a common practice for engineers to predict composite failure based on any of the multitude of lamina failure criteria described in the previous chapter. These criteria only predict the final event of failure, and generally cannot characterize the damage mechanisms leading to the final failure. In reality, the failure event in a composite structure is preceded and influenced by the progressive occurrence and interaction of various damage mechanisms. Structural integrity: The ability of a load-bearing structure to remain intact and functional upon the application of loads. In contrast to metals, remaining intact (not breaking up in pieces) for composites is not necessarily the same as remaining functional. For instance, composites can lose their functionality by suffering degradation in their stiffness properties while still carrying significant loads. Durability: A term very close in meaning to structural integrity. Specifically, durability is defined as the ability of a structure to retain adequate properties (strength, stiffness, and environmental resistance) throughout its life to the extent that any deterioration can be controlled and repaired [1]. The long-term durability of a composite structure is an important design requirement in civil, infrastructure, and aircraft industries. 3.1 Mechanisms of damage The heterogeneous microstructure of composites, the large differences between constituent properties, the presence of interfaces as well as directionality of reinforcement that induces anisotropy in overall properties, are reasons for the complexities observed in geometrical features of micro-level failure (microcracks) in composites. Additionally, when interfaces are present, such as between fibers and matrix and between plies in a laminate, the stress transfer via interfaces provides conditions for multiple cracking (to be discussed later). The wealth of observations reported in the literature on various cracking processes, collectively referred to as “damage mechanisms,” are summarized below for the purpose of treatments in later chapters related to deformation and failure of composite materials at a “macro” level. 3.1.1 Interfacial debonding The performance of a fiber-reinforced composite is markedly influenced by the properties of the interface between the fiber and the matrix resin. The 37 3.1 Mechanisms of damage adhesion bond at the interfacial surface affects the macroscopic mechanical properties of the composite. The interface plays a significant role in stress transfer between fiber and matrix. For instance, if the fibers are weakly held by the matrix, the composite starts to form a matrix crack at a relatively low stress. On the other hand, if the fibers are strongly bonded to the matrix, the matrix cracking is delayed and the composite fails catastrophically because of fiber fracture as the matrix cracks. The constraint between the fiber and the matrix also influences other damage mechanisms such as interfacial slipping, and fiber pull-out. Controlling interfacial properties can thus provide a way to control the performance of a composite structure. In unidirectional compos- ites, debonding occurs at the interface between fiber and matrix when the interface is weak. Figure 3.1 shows debond surfaces observed in a fiber- reinforced composite [2]. The longitudinal interfacial debonding behavior of single-fiber composites has been studied in detail by the use of the pull-out [3–7] and fragmentation [8–11] tests. The mechanics of fiber/matrix interfacial debonding in a unidirectional fiber-reinforced composite is depicted in Figure 3.2. When fracture strain of the fiber is greater than that of the matrix, a crack originating at a point of stress concentration, e.g., voids, air bubbles, or inclusions, in the matrix is either halted by the fiber, if the stress is not high enough (Figure 3.2(b)), or it may pass around the fiber without destroying the interfacial bond. As the applied load increases, the fiber and matrix deform differentially, resulting in a buildup of large local stresses in the fiber. This causes local Poisson contraction and even- tually when the shear stress developed at the interface exceeds the interfacial shear strength, debonding extending over a distance along the fiber results (Figure 3.2(c)). Shear lag and cohesive zone models are commonly used approaches to predict initiation of debonding and stress transfer at the interface [3, 5, 12–17]. 20µm Figure 3.1. Debonds in a fiber-reinforced composite. Reprinted, with kind permission, from Compos Sci Technol, Vol. 59, E. K. Gamstedt and B. A. Sjo¨ gren, Micromechanisms in tension-compression fatigue of composite laminates containing transverse plies, pp. 167–78, copyright Elsevier (1999). 38 Damage in composite materials 3.1.2 Matrix microcracking/intralaminar (ply) cracking Fiber-reinforced composites offer high strength and stiffness properties in the longitudinal direction. Their properties, however, in the transverse directions are generally low. As a result, they readily develop cracks along fibers. These cracks are usually the first observed form of damage in fiber-reinforced composites [19]. In laminates with plies in different fiber orientations, these cracks can form from defects in a given ply and grow traversing the thickness of the ply and running parallel to the fibers in that ply. The terms matrix microcracks, transverse cracks, intralaminar cracks, and ply cracks are invariably used to refer to these very same cracks. Such cracks are found to be caused by tensile loading, fatigue loading, as well as by changes in temperature or by thermal cycling. They can originate from fiber/matrix debonds or manufacturing-induced defects such as voids and inclu- sions [20] (see Figure 3.3). Matrix cracks can also form in ceramic matrix compos- ites (CMC), and in short fiber composites (SFC). The field of damage mechanics deals with prediction of formation, growth, and effects of matrix cracks on overall material behavior. Analysis, design, and behavior of composites subjected to intralaminar cracking will be dealt with in detail in the subsequent chapters. Figure 3.4 illustrates matrix cracks observed on the free edges of continuous fiber and woven fabric polymer composite laminates induced due to fatigue loading [21, 22]. Although matrix cracking does not cause structural failure by itself, it can result in significant degradation in material stiffness and can also induce more severe forms of damage, such as delamination and fiber breakage, and give pathways for entry of fluids. 3.1.3 Interfacial sliding Interfacial sliding between constituents in a composite can take place by differential displacement of the constituents. One example of this is when fibers and matrix in a composite are not bonded together adhesively but by a “shrink-fit” mechanism due to difference in thermal expansion properties of the constituents. On (a) (b) (c) Figure 3.2. Mechanics of interfacial debonding in a simple composite: (a) perfect laminate; (b) differential deformation of fiber and matrix crack causes high stresses at fiber/matrix interface; (c) shear stress exceeds the interfacial shear strength nucleating a debond. Reprinted from [18], with kind permission from Maney Publishing. 39 3.1 Mechanisms of damage thermomechanical loading, the shrink-fit (residual) stresses can be removed, leading to a relative displacement (sliding) at the interface. The relief of interfacial normal stress can also occur when a matrix crack tip approaches or hits the interface. Debond Matrix Void Fiber 5 µm 10 µm (a) (b) Figure 3.3. Matrix crack initiation from: (a) fiber debonds; (b) void results. Reprinted, with kind permission, from Compos Sci Technol, Vol. 57, C. A. Wood and W. L. Bradley, Determination of the effect of seawater on the interfacial strength of an interlayer E-glass/ graphite/epoxy composite by in situ observation of transverse cracking in an environmental SEM, pp. 1033–43, copyright Elsevier (1997). (a) (b) Figure 3.4. Examples of matrix cracks observed in (a) continuous fiber and (b) woven fabric polymer composite laminates [22]. Part (a) reprinted, with kind permission, from Compos Sci Technol, Vol. 68, D. T. G. Katerelos, J. Varna, and C. Galiotis, Energy criterion for modeling damage evolution in cross-ply composite laminates, pp. 2318–24, copyright Elsevier (2008). 40 Damage in composite materials When the two constituents are bonded together adhesively, interfacial sliding can occur subsequent to debonding if a compressive normal stress on the interface is present. The debonding can be induced by a matrix crack, or it can result from growth of interfacial defects. Thus, interfacial sliding that follows debonding can be a separate damage mode or it can be a damage mode coupled with matrix damage. Interfacial sliding in ceramic matrix composites (CMCs) can be significant if the temperature change imposed is high and the thermal expansion mismatch between the fibers and matrix is also large. When the matrix in a CMC cracks, the resulting interfacial debonding affects interfacial sliding, causing interactive effect on the composite deformation [23]. 3.1.4 Delamination/interlaminar cracking Interlaminar cracking, i.e., cracking in the interfacial plane between two adjoining plies in a laminate, causes separation of the plies (laminae) and is referred to as delamination. In composite laminates, delamination can occur at cut (free) edges, such as at holes, or at an exposed surface through the thickness. When loaded in the plane, the laminate develops through-thickness normal and shear stresses at the traction-free surface extending a short distance into the laminate plane. These stresses can result in local cracking in the interlaminar planes. Delaminations can also form as a result of low-velocity impact [24–26]. In contrast to metals, in polymer composite laminates delamination can occur below the surface of a structure under a relatively light impact, such as that from a dropped tool, while the surface appears undamaged to visual inspection [25, 27, 28]. The growth of delamination cracks under the subsequent application of external loads leads to a rapid deterioration of the mechanical properties and may cause catastrophic failure of the composite structure [29, 30]. Another source of delamination is the local interlaminar cracking induced by ply cracks. This delamination can grow and separate the region between two adjacent ply matrix cracks as illustrated in Figure 3.5. Delamination can be a substantial problem in designing composite structures as it can diminish the role of strong fibers and make the weaker matrix properties govern the structural strength [31]. In initiating delamination the critical material property is the interlaminar strength, which is determined by the matrix [26, 31]. Once the interlaminar cracks are formed, their growth is determined by the interlaminar fracture toughness, which is also governed by the matrix. If delamination is viewed as decohesion of the cohesive zone between the separating plies, then both the matrix strength and the fracture toughness act as material parameters [32]. As a design approach, delamination can be reduced either by improving the interlaminar strength and fracture toughness or by modifying the fiber architecture to reduce the driving forces for delamination [33, 34]. 41 3.1 Mechanisms of damage 3.1.5 Fiber breakage The failure (separation) of a fiber-reinforced composite ultimately comes from breakage of fibers. In a unidirectional composite loaded in tension along fibers the individual fibers fail at their weak points and stress redistribution between fibers and matrix occurs, affecting other fibers in the local vicinity of the broken fibers and possibly breaking some. The fiber/matrix interface transfers the stress from the broken fiber back to the fiber at a certain distance, making another fiber break possible if the strength is exceeded by the stress. The fiber breakage process is of a statistical nature because of the nonuniformity of fiber strength along the fiber length and the stress redistribution. When plies of unidirectional fibers are stacked in a laminate, the stress on fibers is enhanced in the vicinity of ply cracks in the adjacent plies, causing a narrow distribution of fiber failure sites [35]. A greater number of fiber breaks per unit volume is found closer to the interface where the ply crack terminates than away from the interface where the local stress concen- tration falls off [35]. The ultimate tensile strength of a ply within a general laminate is difficult to predict from the tensile strength of fibers due to the statistical nature of fiber failure and the progression of fiber failures [36, 37]. Fracture (crack growth) properties such as the fracture toughness of a composite depend not only on the failure properties of the constituents but significantly also on the efficiency of bonding across the interface [38]. 3.1.6 Fiber microbuckling When a unidirectional composite is loaded in compression, the failure is governed by a mechanism known as microbuckling of fibers. There are two idealized basic modes of microbuckling deformation, denoted “extensional” and “shear” modes [39], illustrated in Figure 3.6, depending upon whether the fibers deform “out of phase” or “in phase” with one another. The corresponding compressive strength for the onset of instability is given as Figure 3.5. Interlaminar delamination crack formed due to joining of two adjacent matrix cracks in a fiber-reinforced composite laminate. 42 Damage in composite materials s c ¼ 2V f ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V f E f E m 3ð1 À V f Þ s ; ð3:1Þ for the extension mode, and s c ¼ G m 1ÀV f ; ð3:2Þ for the shear mode, where E and G denote Young’s modulus and shear modulus, respectively, and the subscripts f and m designate fiber and matrix, respectively. These expressions for idealized deformation modes do not generally agree with experimental data for compression strength. It has been argued that in practical composites the manufacturing process tends to cause misalignment of fibers, which can induce localized kinking of fibers. The kinking process is driven by local shear, which depends on the initial misalignment angle f 0 [40]. The critical compressive stress corresponding to instability then is given by s c ¼ t y f 0 ; ð3:3Þ where t y represents the in-plane shear strength (yielding). Budiansky [41] con- sidered the kink band geometry (see Figure 3.7) and derived the following estimate for the kink band angle b in terms of the transverse modulus E T and shear modulus G of a composite layer: ð ffiffiffi 2 p À 1Þ 2 G À s c E T <tan 2 b< G À s c E T : ð3:4Þ To account for shear deformation effects, Niu and Talreja [42] modeled the fiber as a generalized Timoshenko beam with the matrix as an elastic foundation. It was observed that not only an initial fiber misalignment but also any misalignment in the loading system can affect the critical stress for kinking. Extensional Mode Shear Mode Figure 3.6. Extensional and shear modes of fiber microbuckling. 43 3.1 Mechanisms of damage 3.1.7 Particle cleavage If brittle particles (e.g., ceramics) are placed in a ductile but strong and tough matrix, particle cleavage is the main mode of damage in initial stages of deform- ation. This mode of damage is found in particulate metal matrix composites. Cleavage refers to the breakage of the reinforcing particle. The cleavage crack typically forms perpendicular to the global maximum principal stress. The damage analysis has been performed assuming viscoplastic material behavior [43]. Failure of many practically relevant particulate two-phase composites can typically be attributed to cleavage fracture of the brittle particles followed by ductile crack growth in the matrix [44]. To account for particle geometry and distribution, statistical methods are employed to predict inclusion fracture. To fully character- ize brittle fracture of a particle embedded in a ductile metallic matrix, careful computational modeling (FEA) sometimes becomes necessary (see, e.g., [45–47]). 3.1.8 Void growth A composite structure may contain an appreciable amount of manufacturing- induced defects. For polymer matrix composites, the defects induced during manufacturing can be in the fiber architecture, e.g., fiber misalignment, irregular fiber distribution in the cross section, and broken fibers; in the matrix, e.g., voids; or at the fiber/matrix interface, e.g., disbonds and delaminations. Voids are one of the primary defects found virtually in all types of composite materials. The formation of voids is controlled by manufacturing parameters, such as vacuum pressure, cure temperature, cure pressure, and resin viscosity. The presence of voids, even at low volume fractions, is found to have a significant detrimental effect on the overall material behavior. The flexural, trans- verse, and shear properties are affected the most. Their shape, size, and distribu- tion also play role in material degradation. Micromechanics homogenization methods, such as Mori–Tanaka [48], are commonly used to estimate the average composite property assuming voids as inclusions with zero properties. More sophisticated methods have also been developed to analyze the effect of voids on overall composite elastic and failure properties [49]. Voids can also lead to appreciable inelastic deformations in the material locally, which can act as precursors to initiation of damage processes, such as crazing, b W j 0 s s Figure 3.7. Kink band geometry assumed in Budiansky [41]. Reprinted, with kind permission, from Computers & Structures, Vol. 16, B. Budiansky, Micromechanics, pp. 3–12, copyright Elsevier (1983). 44 Damage in composite materials shear yielding, fibrillation, and local fracture. These damage processes in the final stage may have significant influence on the deformation response and failure properties of the composite material. In composites with metallic and polymer matrices, the matrix phase undergoes ductile fracture due to nucleation, growth and coalescence of voids and cavities. These voids grow and expand due to high local inelastic strains and high stress triaxiality in the matrix. Ductile fracture models, such as Rice–Tracy [50] can be used to model the initiation and growth of voids in ductile matrices [51]. These voids can sometimes coalesce to form matrix cracks, and may also cause fiber matrix debonds. 3.1.9 Damage modes The damage mechanisms described above have different characteristics depending on a variety of geometric and material parameters. Each mechanism has different governing length scales and evolves differently when the applied load is increased. Interactions between individual mechanisms further complicate the damage pic- ture. As the loading increases, stress transfer takes place from a region of high damage to that of low damage, and the composite failure results from the critical- ity of the last load-bearing element or region. For clarity of treatment, the full range of damage can be separated into damage modes, treating them individually followed by examining their interactions. Which damage mechanisms become active in a given life period of a composite structure depends mainly on the properties of the base material (e.g., matrix), architecture, orientation, distribution, and volume fraction of the reinforcing agent (fiber), the properties of the interface, and loading and environmental conditions. Intralaminar and interlaminar cracking, fiber fracture, and microbuckling are the dominant damage mechanisms in long fiber composites. Short fiber composites show three basic mechanisms of interfacial failure [52], as depicted in Figure 3.8: Mode a: Localized matrix yielding at the interface due to the stress concen- tration at the fiber end (see Figure 3.8(a)). Typically, this occurs in combination with debonding of the fiber end and the formation of a penny-shaped crack. Mode b: If the interface is relatively weak, an interface crack propagates from the debonded fiber end (Figure 3.8(b)). This is different than the fiber end penny-shaped crack and remains closed upon increase in tensile loading on the composite, and the load transfer occurs by frictional stress transfer. Mode g: If the interface is relatively strong, a conical matrix crack propagates from the debonded fiber end at an angle y c to the fiber axis (Figure 3.8(c)). This matrix crack opens with increasing applied load and suppresses load transfer across the crack faces. For particulate composites, the major damage mechanisms are dewetting (debonding) of the particle and cavity nucleation [53] (see Figure 3.9). At a critical tensile load, the particles separate from the matrix causing dewetting. Dewetting 45 3.1 Mechanisms of damage of the particle eventually leads to cavity formation which grows on subsequent loading. Dewetting introduces volume dilatation and results in nonlinearity in the stress–strain behavior. For well-bonded particles, cavities and cracks may form entirely within the matrix [54]. Damage modes in continuous fiber laminates are thus rich in complexity. These will be described below in the context of their evolution with loading. 3.2 Development of damage in composite laminates A schematic description of damage development in composite laminates in tension is depicted in Figure 3.10, where the five identifiable damage mechanisms are indicated in the order of their occurrence. Although the figure is developed on the basis of fatigue experiments [55–60], it provides the basic details for quasi- static loading as well. In the early stage of damage accumulation, multiple matrix cracking dominates in the layers which have fibers aligned transverse to the applied load direction. (a) (b) (c) q c q c Figure 3.8. Failure mechanisms of interfacial failure in short fiber/epoxy composites: (a) mode a; (b) mode b; (c) mode g. Reprinted, with kind permission, from Compos Sci Technol, Vol. 60, S. Sirivedin, D. N. Fenner, R. B. Nath, and C. Galiotis, Matrix crack propagation criteria for model short-carbon fibre/epoxy composites, pp. 2835–47, copyright Elsevier (2000). 46 Damage in composite materials Static tensile tests on cross-ply laminates have shown that the transverse matrix cracks can initiate as early as at 0.4% applied strain depending upon the laminate configuration. They initiate from the locations of defects such as voids, or areas of high fiber volume fraction or resin rich areas. Ply cracks grow unstably through the width direction and quickly span the specimen width. As the applied load is increased (or the specimen is loaded cyclically), more and more cracks appear. The accumulation of ply cracks in a cracked ply is depicted in Figure 3.11. Initially these cracks are irregularly spaced and isolated from each other, i.e., have no interaction among themselves. However, as cracks become closer they start inter- acting, i.e., the in-between tensile stresses diminish and can no longer build up to earlier levels. Thus further increase in load is required to produce new cracks. This is well illustrated in Figure 3.12 by plots of diminishing crack spacing versus load or number of cycles. The configuration to which crack density saturates, often reached only under fatigue loading, has been termed the “characteristic damage state” (CDS) [57–59]. This state seems to mark the termination of the intralaminar cracking. The uniqueness of the CDS for a given laminate irrespective of the loading path has, however, not been found to hold in all cases [61]. Subsequent loading causes initiation of cracks transverse to the primary (intra- laminar) cracks lying in plies adjacent to the ones with those primary cracks (see Figure 3.10). These cracks, known as secondary cracks, are small in size and they can cause interfacial debonding, thereby initiating interlaminar cracks. The inter- laminar cracks are also initially small, isolated and distributed in the interlaminar planes. Subsequently, some interlaminar cracks merge into strip-like zones leading to large scale delaminations. This results into loss of the integrity of the laminate s s s s s s Matrix Particle Debond Cavity Figure 3.9. Damage mechanisms in particulate composites. Reprinted, with kind permission, from Int J Solids Struct, Vol. 32, G. Ravichandran and C. T. Liu, Modeling constitutive behavior of particulate composites undergoing damage, pp. 979–90, copyright Elsevier (1995). 47 3.2 Development of damage in composite laminates in those regions. Further development of damage is highly localized, increasing unstably, and involving extensive fiber breakage. The final failure event is mani- fested by the formation of a failure path through the locally failed regions and is therefore highly stochastic. The damage prior to localization is sometimes referred to as sub-critical damage. The intralaminar (ply) cracking in this stage causes loss of stiffness properties in the laminate and can by itself lead to loss of functionality (failure) of the composite structure. The field of “damage mechanics” addresses the initi- ation and progression of the sub-critical damage. Later chapters will be devoted to this subject. The next section discusses the phenomenon of multiple cracking and its effects on overall (average) laminate response. 100 PERCENT OF LIFE 5. Fracture D A M A G E CDS 0º 0º 1. Matrix Cracking 0º 0º 0º 0º 0º 0º 3. Delamination 2. Crack coupling- Interfacial debonding 4. Fiber Breakage Figure 3.10. Development of damage in composite laminates [62]. Figure 3.11. Accumulation of intralaminar cracks in an off-axis ply of a composite laminate. Based on X-ray radiographs reported in [64]. 48 Damage in composite materials 3.3 Intralaminar ply cracking in laminates One of the earliest observations of ply cracking in laminates was reported by Broutman and Sahu [65]. However, the first major explanation of multiple matrix cracking was proposed by Aveston, Cooper, and Kelly [66, 67]. They argued that multiple fracture occurs in a fibrous composite when one of the constituents (fiber or matrix) fractures at a much lower elongation than the other and when the unbroken constituent is able to take the additional load; otherwise single fracture results. Later, a group of experimentalists, Garret, Bailey, Parvizi, and colleagues [68–74] carried out significant tests to analyze the ply cracking behavior in cross-ply laminates. These experiments showed the initiation of microcracking in glass-fiber reinforced polyster and epoxy cross-ply laminates. It was observed that for thick 90 -plies, transverse cracks initiated at the edge of the specimen and propagated instantly through the width of entire cross section. As the 90 -plies were made thinner, the strain to initiate transverse cracks increased (see Figure 3.13). For very thin 90 -plies (< 0.1 mm), cracks were suppressed and the laminates failed before crack initiation. One of these experiments [74] involved a microscopy study into the origins of matrix cracks and revealed that they nucleate from the processing flaws, voids and the regions of high fiber volume fraction, and progress through fiber-matrix debonding. The thickness effect on crack initiation can be explained in terms of the “constraint” posed by uncracked plies over displacement of crack surfaces in cracked plies (Figure 3.13). On one hand, as the thickness of (cracked) 90 -plies 0 0 0 0.2 4.0 6.0 8.0 10.0 C r a c k s p a c i n g ( m m ) 2.0 0.4 0.6 0.8 1.0 100 200 300 400 500 600 700 Applied stress (MPa) Cycles (× 10 6 ) Fatigue data Quasistatic data Figure 3.12. Spacing of cracks in À45 -plies of [0/90/Æ 45] s graphite/epoxy laminates as a function of quasi-static and fatigue loading [57]. Reprinted, with kind permission, from Damage in Composite Materials, ASTM STP 775, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428. 49 3.3 Intralaminar ply cracking in laminates increases, the relative constraint from (uncracked) 0 -plies decreases leading to ply cracking at lower applied strains. On the other hand, thicker 0 -plies exert a larger constraint on opening of cracks in 90 -plies, thereby delaying the crack initiation in those plies. For qualitative understanding, Talreja [75] classified this constraint in four categories: A – no constraint; B – low constraint; C – high constraint; and D – full constraint. The stress–strain behavior for each of these four cases varies greatly and is illustrated in Figure 3.14. On one extreme, it resembles an elastic- rigid plastic like deformation behavior for constraint type A, and on the other end a linear elastic behavior for constraint type D. Over the past four decades, numerous approaches to analyze ply cracking in composite laminates have been developed. They can be categorized into two broad categories: micromechanics-based models (Chapter 4), and continuum damage models (Chapter 5). Based on the dimensionality of the boundary value problem, micromechanics-based models can be sub-divided into one-dimensional [59, 76–81], two-dimensional [82–85], and three-dimensional [86–88]. The next chapter is devoted to these approaches. 3.4 Damage mechanics Damage mechanics can be broadly defined as the “subject dealing with mechanics- based analyses of microstructural events in solids responsible for changes in their 0 2.0 4.0 6.0 8.0 2n Number of 90° plies 0 0.25 0.50 0.75 1.00 D C B Gr./Ep. [0 4 / 90 n ] s A 0° Ply failure 90° Ply failure e FPF % Figure 3.13. The strain at first ply failure (e FPF ) as a function of the number of transverse plies in [0 4 /90 n ] s laminates. Source: [75]. 50 Damage in composite materials response to external loading.” The general objectives of damage mechanics analy- sis are as follows: 1. Understand the conditions for initiation of the first damage event. 2. Predict the evolution of progressive damage. 3. Characterize and quantify damage in the structure. 4. Analyze the effect of damage on thermomechanical response, e.g., by express- ing stiffness properties as a function of damage. 5. Assess failure (criticality of damage) and durability of the structure. 6. Provide input into overall structural analysis and design. This chapter provided an overview of damage development in composites. The next three chapters will describe analysis methods for quasi-static loading. Chapter 4 will describe “micro-damage mechanics” (MIDM), whereas Chapter 5 will describe the “macro-damage mechanics” (MADM). Evolution of damage will be covered in Chapter 6. Damage in fatigue and models for lifetime prediction will be treated in Chapter 7. e FPF e FPF e FPF e c e c e c e c e e e s s s s (a) (b) (c) (d) Figure 3.14. Schematic stress–strain response of cross-ply laminates at different constraints to transverse cracking: (a) single fracture, no constraint; (b) multiple fracture, low constraint; (c) multiple fracture, high constraint; (d) multiple fracture, full constraint (with crack suppression). Source: [75]. 51 3.4 Damage mechanics References 1. 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The other approach, on the contrary, looks at the overall response at the macro or structural scale by using some internal variables to characterize damage, and thus can be termed as “macro-damage mechanics” (MADM). These terms were originally coined by Hashin [1]. MADM is the same as “continuum damage mechanics” (CDM), which is still the commonly used terminology. MIDM for composite materials is derived from an older and more mature field called micromechanics that deals with overall properties of heterogeneous mater- ials (see, e.g., [2]). In micromechanics one views heterogeneities such as inclusions and voids as “microstructure” and estimates overall properties by various methods, e.g., averaging schemes such as self-consistent and differential schemes, or variational methods to obtain bounds to average properties. Microcracks are treated as limiting geometry of microvoids, such as ellipsoidal voids with one dimension much smaller than the other two. As illustrated in the previous chapter, “damage” in composite materials has significant complexities concerning the geometry as well as evolution characteristics such as multiplication of cracks within a fixed volume. For these reasons a simple extension of micromechanics to damage in composites is generally not possible. A separate field identified as MIDM has therefore emerged. This chapter will treat the features of MIDM that have been developed to specifically treat certain cases of damage in composite materials. Since determining local (micro-level) stress or displacement fields is a necessary feature of micromechanics, it is expected that not all cases within the wide range of damage in composites can be handled by MIDM. However, this limitation can be alleviated by incorporating computational solutions of the local stress or displacement fields, thereby broadening classical micromechanics to include so-called computational micromechanics. In the most recent versions of MIDM this strategy has been used. More on this will be discussed toward the end of this chapter. In the following we will first treat the aspect of damage in composite mater- ials that is due to the presence of continuous interfaces between dissimilar materials, such as fibers and matrix or plies oriented differently in a laminate. In fact this aspect is fundamental to understanding damage in composite materials. Historically, it was first analyzed in a classical work by Aveston, Cooper, and Kelly [3] who explained conditions that lead to failure of a composite from a single crack versus when multiple cracks precede the final failure. Their work has become known as the ACK theory. Although the case treated by them is of simple geometry and loading, namely a unidirectional fiber-reinforced brittle matrix composite loaded in tension along fibers, it explains the basic mechanism underlying multiple cracking in a wide range of cases. The simplified stress analysis and the associated energy balance consider- ations in the ACK paper have later been extended to include more accurate solutions, but little further insight into the multiple cracking mechanism has resulted by these efforts. 4.2 Phenomena of single and multiple fracture: ACK theory The mode of failure (separation in two or more pieces) in homogeneous mater- ials such as metals and ceramics may be described as “single fracture” in the sense that the failure is attributable to a single source – a crack. Heterogeneous materials, on the other hand, can fail in the mode of single fracture or sustain multiple fractures of one of the phases before ultimately separating in two or more pieces. The latter phenomenon is known as “multiple fracture” and can commonly occur in fibrous composites with brittle matrices, such as cement plaster, glass, etc. Damage in composite materials usually initiates with matrix cracking. Figure 4.1 depicts the isotropic stress–strain response of an unreinforced glass and of a fiber-reinforced glass in tension loading along fibers. Other than the enhancement of the stress–strain response in the fiber direction, a striking aspect is the nonlinearity shown by the reinforced specimen. This nonlinearity is a type of ductility which occurs due to multiple cracking of the matrix (glass) [4]. Although the phenomenon of multiple fracture was observed earlier, e.g., by Cooper and Sillwood [5] it was systematically investigated in a landmark paper by Aveston, Cooper, and Kelly [3]. Their work, the ACK theory, is the basis of the treatment presented below. Consider a unidirectional fibrous composite loaded in tension along fibers as shown in Figure 4.2 and assume the following: 1. Fibers are of the same diameter and uniformly distributed in the matrix. 2. All the fibers are aligned parallel to one another. 3. There are no pre-existing flaws in the matrix such as voids and cracks. 4. Both the matrix and the fibers are linearly elastic. 58 Micro-damage mechanics Assuming that fibers and matrix have different failure strain in tension, when one of the constituents fails, the other will either fail simultaneously or continue deforming by carrying the additional load. In the latter case, the constituent that failed first will fail again at a different site. Thus, there are two necessary condi- tions for multiple fracture to occur in a composite: 1. One of the constituents has a lower failure strain than the other. 2. When the weaker constituent fails, i.e., when it no longer carries any load, the stronger constituent must be able to carry the additional load thrown upon it. If P c is the total tensile load on the composite and P f and P m represent the load taken up by the fibers and matrix, respectively, then by force balance we have P c = P f ÷ P m : (4:1) s m s f s c Figure 4.2. A unidirectional fibrous composite loaded in tension. 0 200 S t r e s s / M P a 400 600 800 1000 0.4 0.8 1.2 Strain (%) Unreinforced glass Fibre- reinforced glass Figure 4.1. The stress–strain curves for borosilicate glass alone (dotted line) and reinforced with aligned carbon fibers. Reprinted from [4] with kind permission from Royal Society Publishing, London. 59 4.2 Phenomena of single and multiple fracture: ACK theory Dividing by the composite’s cross-sectional area A, we get P c A = P f A ÷ P m A ; (4:2) or P c A = P f A f A f A ÷ P m A m A m A : (4:3) Assuming unit composite length, we obtain s c = s f V f ÷ s m V m ; (4:4) where V f and V m are the volume fractions of fibers and matrix, respectively. Depending upon which phase fails first, two cases as shown in Figure 4.3 arise. Case 1 is whenfibers have a lower breaking strainthan the matrix (e fu <e mu ), while Case 2 is for the opposite (e mu < e fu ). In the first case, fibers will undergo multiple fracture if the matrix is able to carry the additional load thrown upon it due to fiber failure, i.e., if P mu > P c [ e fu ; i:e:; if s mu V m > s fu V f ÷ s / m V m (4:5) where P mu is the maximum load that can be carried by the matrix, s fu and s mu are the tensile strength values for fibers and matrix, respectively, and s / m = E m e fu = (s mu =e mu )e fu is the stress inthe matrix requiredtoproduce a strainequal to the breaking strain of the fibers. In this case, the fibers will be successively fractured intoshorter lengths until the matrix attains its failure strainandat that instant the whole compositefails. Ontheother hand, for Case2, thematrixwill undergomultiplefractureif P fu > P c [ e mu ; i:e:; if s fu V f > s mu V m ÷ s / f V f ; (4:6) A Multiple fracture of fibers s s s f Single fracture V f s m σ f σ ′ f s ′ m (a) A Multiple fracture of matrix Single fracture V f (b) s m Figure 4.3. Single and multiple fractures in a unidirectional composite. Fracture stress is plotted against the fiber volume fraction: (a) case 1: e mu > e fu ; (b) case 2: e mu < e fu . 60 Micro-damage mechanics where P mu is the maximum load that can be carried by the matrix, s fu and s mu are the tensile strength values for fibers and matrix, respectively, and s / f = E f e mu = (s fu =e fu )e mu is the stress in the fibers required to produce a strain equal to the breaking strain of the matrix. 4.2.1 Multiple matrix cracking From this point on we will focus on the case of multiple matrix cracking, assuming e mu < e fu . In addition to the assumptions described in the previous section, the following analysis will assume: 1. Fibers remain intact throughout entire loading history. 2. Matrix cracks extend in the entire cross section. 3. Fibers debond completely between adjacent matrix cracks. If we concentrate on the matrix region between two fibers, the force P m shed by the matrix, subsequent to its failure, is carried by fibers in the cracked cross section, and is transferred back to the matrix over a distance x / . This load transfer takes place through shear at the fiber/matrix interface with constant shear stress t. The mechanism of interfacial load transfer is illustrated in Figure 4.4. The load balance between the total matrix load P m and the total shear force at the fiber-matrix interfaces yields P m = s mu A m = t 2pr x / N; (4:7) where r is the fiber radius and N is the number of fibers in the composite cross section of area A. P m xЈ τ Figure 4.4. Mechanism of load transfer at the fiber/matrix interface. 61 4.2 Phenomena of single and multiple fracture: ACK theory Thus, x / = s mu t _ _ A m 2prN : (4:8) Now, A m 2prN = A m =A 2prN=A = V m :r 2(N:pr 2 =A) = V m :r 2V f : Hence, x / = s mu t _ _ V m V f r 2 : (4:9) The assumptionof constant shear stress at the interface simplifies the analysis. However, its inaccuracycanbe notedbyrealizingthat the shear stress at the point where the matrix crack meets the interface must vanish if the crack surface is to remain traction free. Stress distribution in fibers and matrix The balance of forces for a piece of fiber of length Dx / , over which the change in the fiber stress is Ds f , yields Ás f :pr 2 = t 2pr Áx / : (4:10) Thus the rate of load transfer is a constant given by Ás f Áx / = t r ; (4:11) and consequently the fiber stress s f varies linearly, and correspondingly s m also varies linearly, along the fiber axis. The maximum stress in fiber occurs at the matrix crack and can be determined as s f;max = s f ÷ P mu A f = s f ÷ s mu A m A f = s f ÷ s mu V m V f : (4:12) Additional strain due to cracking The strain in fibers increases from e mu at cracking to the maximum value given by e f; max = s f;max E f = s f E f ÷ s mu E f V m V f = e f ÷ s mu E m E m E f V m V f = e mu 1 ÷ a ( ); (4:13) with a = E m E f V m V f : (4:14) The mean strain over crack spacing 2x / is equal to e mean;2x / = 1 2 e mu ÷ e mu 1 ÷ a ( ) [ [ = e mu 1 ÷ a 2 _ _ : (4:15) When the crack spacing reduces to x / , the mean strain increases to e mean;x / = e mu 1 ÷ 3a 4 _ _ : (4:16) 62 Micro-damage mechanics Energy considerations in multiple cracking Consider the unidirectional composite of Figure 4.2 at fixed applied load P c = s c A. Let its initial configuration be denoted as state 1 and let state 2 refer to its configuration with multiple matrix cracking. The energy changes in going from state 1 to state 2 are described below. “Supply” of energy 1. DW: the work done per unit cross-sectional area A by an external (fixed) load through specimen extension caused by cracking is given by ÁW = 1 A P c 2Áx / = s c 2 e mu a 2 x / _ _ = E c e mu e mu ax / = E c e 2 mu ax / : (4:17) 2. DU m : the reduction in elastic strain energy of matrix over distance 2x / is given by ÁU m = 2 _ x / 0 1 2 E m V m e 2 mu ÷ 1 2 E m V m e mu x x / _ _ 2 _ _ dx = 2 3 E m V m e 2 mu x / = E m V m 3t e 3 mu ar: (4:18) “Consumption” of energy 1. Energy spent in formation of matrix cracks. If g m is the surface energy per unit area of crack surface, then the energy spent in formation of a matrix crack per unit cross-sectional area A is 2g m A m A = 2g m V m : (4:19) 2. Energy spent in fiber/matrix interfacial debonding. If we take G II to be the energy released per unit area of the debond surface, then the debond energy g db per unit cross-sectional area A can be expressed as g db A = G II 2pr 2x / N; (4:20) i.e., g db = G II 2prN A 2x / = 2G II A f Ar V m V f s mu t r = 2G II V m s mu t : (4:21) 3. U s : the energy spent in sliding of the matrix onto the fiber surface over a distance 2x / , per unit cross-sectional area A is given by U s = 1 A N 2 _ x / 0 Áv t 2pr dx (4:22) 63 4.2 Phenomena of single and multiple fracture: ACK theory where Dn is the sliding displacement at x. This sliding displacement is equal to the difference in displacements of the fiber and the matrix. It can be found by integrating the strain in matrix and fiber: _ x / 0 tÁv dx = t _ x / 0 e mu 1 ÷ a ( )x ÷ a 2 x 2 x / ÷ x / 1 ÷ a 2 _ _ ÷x 2 x / ÷ x / 2 _ _ dx = te mu x / 2 6 1 ÷ a ( ): (4:23) Thus, U s = E f E m V m 6t e 3 mu ra 1 ÷ a ( ): (4:24) 4. DU f : the increase in the elastic energy of fibers due to additional extension caused by additional fiber stress, per cross-sectional area A, is given by ÁU f = U (2) f ÷ U (1) f = 2 _ x / 0 1 2 E f V f e mu a 1 ÷ x x / _ _ e mu _ _ 2 ÷ 1 2 E f V f e 2 mu _ _ dx = E f V f e 2 mu x / a 1 ÷ a 3 _ _ = E f E m V m 2t e 3 mu ra 1 ÷ a 3 _ _ : (4:25) Conditions for multiple matrix cracking 1. Stress in the matrix is greater than or equal to the matrix failure stress, i.e., s m _ s mu or e m _ e mu : (4:26) 2. The “supply” of energy in going from state 1 to state 2 is greater than or equal to the “consumption” of energy, i.e., 2g m V m ÷ g db ÷ U s ÷ ÁU f _ ÁW ÷ ÁU m : (4:27) Substituting Eqs. (4.17)–(4.25) derived above into Eq. (4.27), one obtains 2V m g m ÷ G II s mu t _ _ _ E c E f e 3 mu a 2 r 6t : (4:28) It can be argued that the energy term G II is much smaller than the other energy contributions. Assuming G II = 0 then gives 2V m g m _ E c E f e 3 mu a 2 r 6t : (4:29) Thus, the strain required to cause multiple matrix cracking is given by the following expression e muc = 12tg m E f V 2 f E c E m V m r _ _ 1=3 : (4:30) 64 Micro-damage mechanics Stress–strain response When the composite is loaded to an applied strain level equal to the failure strain of the matrix, multiple cracking in the matrix starts occurring. If the matrix has a well- defined single-valued breaking strain, the cracking will continue at a constant applied stress E c e mu until the matrix is broken down into a set of blocks of length between x / and 2x / . The composite stress–strain behavior subsequent to multiple fracture of the matrix is shown in Figure 4.5. During multiple matrix cracking (Point A to Point B), the mean strain varies between e mu 1 ÷ a 2 _ _ and e mu 1 ÷ 3a 4 _ _ while going from a crack spacing of 2x / to x / . The total strain at the limit of multiple cracking e mc is therefore e mu 1 ÷ a 2 _ _ <e mc <e mu 1 ÷ 3a 4 _ _ : (4:31) When the applied load is increased, fibers are stretched further and start slipping through the blocks of matrix (point Bto Point C). Since the matrix can no longer take any load the Young’s modulus of the specimen is reduced to E f V f . The composite will eventually fail at a stress s fu V f . The failure strain of the composite e cu is given by e fu ÷ ae mu 2 _ _ <e cu < e fu ÷ ae mu 4 _ _ : (4:32) 4.2.2 Perfectly bonded fiber/matrix interface: a modified shear lag analysis The previous analysis was based on the assumption that the fibers debond com- pletely from the matrix during the fracture process. In reality the displacements of matrix and fibers are interrelated. The complete debonding scenario can be viewed as one extreme, the other extreme being the fully bonded case. For the latter, a modified shear lag model [6] is applicable. 1+ A C fu f V s s D O B 1+ 4 3a 4 2 2 E c E f V f ae mu ae mu mu e mu e mu e fu e fu e e fu e ∼ ∼ a Figure 4.5. The stress–strain response subsequent to multiple fracture for a unidirectional composite according to the ACK theory. 65 4.2 Phenomena of single and multiple fracture: ACK theory For both cases the fundamental equation governing load transfer between fibers and matrix is obtained from a simple force balance and is, for discrete fibers of radius r in a continuous matrix, given by dF dy = 2V f t i r ; (4:33) where dF is the load per cross-sectional area A transferred over the distance dy and t i is the shear stress acting at the interface. For the unbonded case, the load transfer from fiber to matrix is given by Eq. (4.11), whereas for the bonded case the fibers in the plane of the first crack will be subjected to an additional stress, Ds 0 , given by Ás 0 = s a V f ÷ E f e mu ; (4:34) where s a is the applied stress. This additional stress has its maximum value at the plane of the matrix crack and decays with distance from the crack surface. Using a modified shear lag analysis, Aveston and Kelly [6] showed that the variation of this additional stress along the fiber is given by Ás y ( ) = Ás 0 e ÷ ffiffi w _ y ; (4:35) with w = 2G m E c E f E m V m _ _ 1 r 2 ln R r _ _ ; (4:36) where G m is the shear modulus of the matrix. Carrying out a force balance over an element of fiber of length dy, dÁs pr 2 ÷ t i 2pr dy = 0; (4:37) from which the rate of change of stress in the fiber is given by dÁs dy = ÷ 2 r t i : (4:38) Differentiating Eq. (4.35) w.r.t. y and substituting into Eq. (4.38), the shear stress at the interface between the fibers and matrix is given by t i = r 2 Ás 0 we ÷ ffiffi w _ y : (4:39) Substituting Eq. (4.39) into Eq. (4.33), and integrating, the load F per cross- sectional area A transferred to the matrix over any distance l from the crack surface can be found as F = V f Ás 0 1 ÷ e ÷ ffiffi w _ l _ _ : (4:40) 66 Micro-damage mechanics If Ds 0 _ s mu (V m /V f ), the matrix will undergo further cracking into blocks of length between l and 2l, where l is obtained by setting F = s mu V m into Eq. (4.40), l = ÷ 1 ffiffiffi w _ ln 1 ÷ s mu Ás 0 V m V f _ _ : (4:41) The energetic considerations in this case result into the crack initiation strain for the matrix, as given by e mu = 2g m V m ffiffiffi w _ aE c _ _ 1=2 : (4:42) The effective Young’s modulus for the matrix can be determined by averaging the stress distribution in the matrix, i.e., E m = 1 e mu 1 s _ s 0 Ás m y ( )dy; (4:43) where “2s” is the mutual crack spacing, and the stress distribution in the matrix is given by s m y ( ) = E m e mu ÷ Ás V f V m : (4:44) Using Eqs. (4.34), (4.43), and (4.44), E m can be derived as [7] E m = E m ÷ E m s ffiffiffi w _ e ÷s ffiffi w _ ÷ 1 _ ¸ : (4:45) The effective modulus for the unidirectional composite can then be determined using the rule of mixture E c = E f V f ÷ E m V m : (4:46) 4.2.3 Frictional fiber/matrix interface As suggested by Wang and Parvizi-Majidi [7], another important case is to assume that the fiber/matrix interface is frictional. For this case, the load transfer between fiber and matrix is assumed constant. After the development of matrix cracking, the additional stress transferred to the fiber at the crack surface should be transferred back to the matrix by a constant interfacial shear stress over the distance of the limiting matrix crack spacing, x. Accordingly, the stress in the matrix away from the crack surface will vary linearly, from a zero value at the crack to a maximum of s m at a distance x from the crack. This stress will stay at its maximum level for distances longer than x. The average stress and the stiffness values in the matrix can therefore be represented by the following equations s m = s m 2s ÷ x x ; E m = E m 2s ÷ x x : (4:47) The overall stiffness for the composite can then be calculated from Eq. (4.46). 67 4.2 Phenomena of single and multiple fracture: ACK theory The one-dimensional analyses of stress transfer used in the ACK theory and its later version are inadequate to give accurate prediction, e.g., of the strain to onset of multiple cracking. The main value of these works was in explaining the phe- nomenon of multiple cracking. In subsequent sections we will address the problem of multiple cracking more generally and describe a wide range of approaches taken since the appearance of the ACK theory. 4.3 Stress analysis (boundary value problem) for cracked laminates 4.3.1 Complexity and issues Damage analysis of cracked laminates of a general layup is a highly complex task. The major issues in analyzing damage in a multidirectional laminate are discussed below: 1. Anisotropy and heterogeneity: The commonly used laminate analysis is based on the assumption that the plies are homogeneous and anisotropic, with the effective properties of a unidirectional composite. This is a valid assumption for undamaged laminates for the usual cases of membrane force and moment loading when the variation of the primary, in-plane, laminate stresses through the ply thickness is constant or linear. But the presence of intralaminar cracks may cause local conditions such as high stress gradients to invalidate assump- tions needed for homogenization of plies [1]. Furthermore, an in-plane stress analysis will be inadequate to account for interactions between cracks in different plies. 2. Stress singularity: An assumption of ideal cracks within plies will produce stress singularity. In reality, however, the crack tips will be blunted due to the presence of finitely sized fibers near the ply interfaces and by local flow of the matrix at crack tips. Hence, the actual stress field in the vicinity of the crack tip may not be accurately given by the usual stress analysis of cracks in homogeneous elastic medium. In such instances, numerical approaches such as the finite element method may become necessary. 3. Interaction between cracks: At sufficient crack densities, the stress fields around two adjacent cracks in a ply start interacting, thereby relaxing the region between those two cracks. This crack interaction affects the crack surface displacements as well as the overall stress fields. Accurate modeling of crack interaction is a complex task. 4. Three-dimensionality of the boundary value problem: Following all the previous points, and realizing that in a real scenario the ply cracks in a general laminate may be curved, irregularly spaced, or not fully grown through the laminate width, a complex 3-D boundary value problem (BVP) arises. In the simpler case of cracked cross-ply laminate, assuming that cracks are periodic, straight, and fully grown through the ply thickness and width, the resulting BVP can be reduced to a generalized plane strain problem. For the off-axis ply cracking, however, the stress analysis problem 68 Micro-damage mechanics is still a truly 3-D BVP, and any reduction to a 2-D problem will not produce accurate predictions. 5. Defining RVE size: In homogenizing a heterogeneous solid such as a composite ply the RVE size must be large enough to contain sufficiently many fibers to provide average properties. With fibers of typically 0.01 mm in diameter, and a ply thickness of typically 0.125 mm, the RVE extending across a ply thickness may or may not suffice, depending on the fiber volume fraction and fiber distribution irregularity, but it is implicitly assumed to suffice in the classical laminate theory. However, when cracks appear within a ply, the local stress gradients increase sharply, leading to a breakdown of the homogenized ply properties. Away from the cracks, nevertheless, the properties hold. In obtaining the overall (average) composite properties with multiple cracks, the RVE size must be large enough to contain a representation of the cracks. To satisfy this requirement the RVE must extend in the laminate length direction while it is limited in the thickness direction by the laminate thickness. 6. Multiscale considerations: Connected with the RVE issue is the consideration of the multiscale nature of the stress and failure analysis. The RVE scale is the so-called meso (intermediate) scale, while the scale of discrete entities within the RVE is the micro scale. The scale at which structural analysis is conducted is the macro scale. Most of the MIDM is concerned with determining changes of mesoscale properties using stress fields at the microscale. In fact the analyses described in the following mostly assume uniform distribution of cracks, giving repeating unit cells for BVP solutions. 7. Constraint effects: In a cracked laminate, stress perturbations are caused by the surface displacements of the ply cracks in response to the applied loading. These surface displacements do not occur freely, as they would if the cracks were to lie in a homogeneous ply of infinite thickness, but are affected by constraint from the neighboring plies. Understanding these constraint effects is the key in determining the effective properties of cracked laminate. They will be discussed in further detail in a subsequent chapter. 8. Complexity of off-axis ply cracking: Unlike in cross-ply laminates, intralami- nar cracking in off-axis plies of orientations other than 90 · can be complex. Microscopic observations suggest that depending on the off-axis angle, ply thickness, and orientation of neighboring plies these cracks may be partially grown and erratic in shape and size, and of nonuniform distribution [8, 9]. Raman Spectroscopy experiments on [0/45] s laminates show that a crack developing in the 45 · -ply behaves differently from a similar crack in the 90 · - ply of a cross-ply laminate [10], which seemed to suggest that the initiation and propagation strains for a 45 · crack were different. For the laminates containing a 90 · -ply, the cracks usually initiate in that ply under axial tension, while cracks in other off-axis plies initiate at higher loading. The observations on multidir- ectional laminates indicate that the angle between the two adjacent plies may have significant effects on damage initiation and progression. When this angle is small, partially formed cracks are observed before propagation in the fiber 69 4.3 Stress analysis (boundary value problem) for cracked laminates direction. However, fully developed cracks mainly form in the cases of large intersecting angles [11, 12]. The damage development in the 60 · -ply of a [0/60 2 / 90] s laminate is shown in Figure 4.6. Moreover, shear–extension coupling may introduce some additional complexity in analysis of off-axis laminates [13, 14]. 9. Randomness in cracking process: In general, damage models assume a uni- form distribution of transverse cracks, i.e., they are assumed to be periodic and self-similar. In reality the variation in crack spacing can be considerable, particularly at large crack spacing. Recently, there have been some develop- ments to account for the influence of the spatial nonuniformity of matrix cracking on the stress transfer and the effective mechanical properties of cracked cross-ply laminates [15–17]. 10mm 90Њ 60Њ (a) σ x = 315[MPa] ε x = 0.71% σ x = 347[MPa] ε x = 0.80% σ x = 388[MPa] ε x = 0.89% σ x = 428[MPa] ε x = 1.00% (b) (c) (d) Figure 4.6. Consecutive matrix cracking behavior in contiguous plies in a [0/60 2 /90] s laminate. Parts (a)–(d) show damage state at different levels of applied strain. Reprinted, with kind permission, from Int J Solids Struct, Vol. 42, T. Yokozeki, T. Aoki, and T. Ishikawa, Consecutive matrix cracking in contiguous plies of composite laminates, pp. 2785–802, copyright Elsevier (2005). 70 Micro-damage mechanics 10. Multiple damage mechanisms: In general, composite laminates can display a multitude of damage modes. Here, our focus is on ply cracking, which usually occurs much before other damage mechanisms such as delamina- tions and fiber fracture. The influence of manufacturing-induced defects such as voids and fiber clusters may further complicate the analysis. These interactions may become important for failure analysis and have been considered in some studies [18–24]. The above list of complex issues concerning cracked laminates, even for simple configurations, makes the task of stress and failure analysis daunting. The accuracy of stress analysis by itself may not always be the answer to the engineering problem at hand. Often approximations and simplifications need to be made in a judicious manner guided by the application. In the following the efforts made at analyzing damage and its effects are treated in increasing order of complexity. 4.3.2 Assumptions Consider a cross-ply laminate loaded in tension (Figure 4.7(a)). At a certain value of the load, it develops transverse cracks in the 90 · -plies. As described in Chapter 3 these cracks instantaneously traverse the thickness of the 90 · -plies and span the composite specimen width, i.e., the extent of the uniformly distributed load in the width direction. Thus, regarding the width direction to be infinite in extent, the boundary value problem can be reduced to a generalized plane strain problem, as indicated in Figure 4.7(b). The crack spacing 2l shown in the figure is assumed to be periodic. Experimental observations indicate that although the crack spacing is nonuniform in the early stage of the cracking process, it attains uniformity soon and maintains it until crack saturation. In addition to the assumptions concerning the crack geometry and spacing, the ply material is assumed to be homogeneous, linearly elastic, with symmetry prop- erties dictated by the fiber direction. Assuming all fibers to be parallel, the cross- sectional plane becomes a plane of symmetry. The two mutually orthogonal planes that are orthogonal to the cross-sectional plane, namely the mid-plane and the through-thickness plane, can both be assumed as planes of symmetry, rendering the ply an orthotropic symmetry. Alternatively, one can assume the cross-sectional plane to be the plane of isotropy, which would make the ply transversely isotropic. Boundary value problem for cracked cross-ply laminates As stated above, the cracked cross-ply laminate can be treated as a two- dimensional BVP shown in Figure 4.7(b). The laminate coordinate system “x–y–z” is shown in the figure. For the two-dimensional geometry shown in Figure 4.7(b) the “x–z” axes are placed with the origin located midway between the cracks, as indi- cated. The local material coordinate system in a given lamina is “x 1 –x 2 –x 3 ,” denoted simply as “1–2–3.” 71 4.3 Stress analysis (boundary value problem) for cracked laminates The following are specified for the BVP: v Ply material – transversely isotropic: E 1 = Longitudinal Young’s modulus E 2 = Transverse Young’s modulus n 12 = Major Poisson’s ratio G 12 = In-plane shear modulus n 21 = n 12 E 2 E 1 = Minor Poisson’s ratio E x = Longitudinal Young’s modulus for the laminate E x0 or E c = Longitudinal Young’s modulus for the virgin (undamaged) laminate E 0 x0 = Longitudinal Young’s modulus for the 0 · -ply (undamaged laminate) E 90 x0 = Longitudinal Young’s modulus for the 90 · -ply (undamaged laminate) (4:48) v Geometry: A = Cross-sectional area 2t 0 = Thickness of 0 · -ply 2t 90 = Thickness of 90 o -ply 2h = 2 t 0 ÷ t 90 ( ) = Total laminate thickness 2l = Average spacing between two adjacent cracks (4:49) t 0 t 90 h x z 2l 90º 0º 0º σ c (a) (b) N xx N xx x z y O Figure 4.7. Construction of a unit cell for stress analysis of a cracked cross-ply laminate: (a) cracked laminate in tension; (b) equivalent 2-D unit cell. 72 Micro-damage mechanics v Loading: N xx =Applied tensile load per unit width in x-direction (in-plane stress resultant) (4:50) v Stresses and strains: s c = Total applied stress for laminate along x-direction s 0 xx = Total x-direction stress in 0 · -ply s 90 xx = Total x-direction stress in 90 · -ply s 0 xx0 = Initial (virgin-laminate) x-direction stress in 0 · -ply s 90 xx0 = Initial (virgin-laminate) x-direction stress in 90 · -ply (4:51) Other notations will be defined as and when needed. The boundary value problem can be stated as: For a cracked cross-ply laminate loaded in tension determine the displacement and stress fields which satisfy equilibrium and boundary conditions, and further determine its effective stiffness properties for a fixed state of damage (given crack spacing). That is, determine s ij which satisfy the following conditions: 1. Force balance: N xx = s c A: (4:52) 2. Equilibrium conditions: s ij; j = 0: (4:53) 3. Boundary and continuity conditions: Laminate mid-plane symmetry: s 90 xz x; 0 ( ) = 0 Traction continuity across interface: s 90 xz x; t 90 ( ) = s 0 xz x; t 90 ( ) s 90 zz x; t 90 ( ) = s 0 zz x; t 90 ( ) Traction-free boundary: s 0 xz x; h ( ) = 0 s 0 zz x; h ( ) = 0 Traction-free crack surfaces: s 90 xz ±l; z ( ) = 0; ÷t 90 _ z _ t 90 s 90 xx ±l; z ( ) = 0; ÷t 90 _ z _ t 90 : (4:54) 4.4 One-dimensional models: shear lag analysis A class of one-dimensional models that has been useful in the analysis of multiple cracking is the so-called shear lag models. Although the stress analysis is inherently inaccurate in these models, their ability to capture the stress 73 4.4 One-dimensional models: shear lag analysis transfer at the interface by shear stress makes them useful. Historically, the shear lag analysis was first used by Cox [25] to describe the stress transfer between a fiber and a matrix for discontinuous fiber composites. He considered an axisymmetrical model of a single fiber embedded in the matrix. Later, Aveston and Kelly [6] modified the model for predicting strain to initiate multiple matrix cracking in a unidirectional fiber composite with brittle matrix. A recent work considering axisymmetrical model has shown that Cox’s original result can be derived by a series of approximations to the elasticity theory [26]. Garrett, Bailey, and Parvizi [27, 28] and Manders et al. [29] adopted Cox’s approach for the particular case of transverse cracking. Most of the early studies on transverse cracking used shear lag analysis for the unit cell shown in Figure 4.7(b). There have been many modifications and extensions of the same analysis. For a more detailed discussion on shear lag analysis, the reader is referred to [30–58]. All shear lag analyses are based on the following basic concept: In the plane of a transverse crack the transverse ply does not carry the axial load, while away from the crack a part of this load is transferred back to the transverse ply by axial shear at the interface between the cracked transverse ply and the adjacent uncracked ply. The shear lag analysis is essentially a 1-D analysis and it uses the following basic assumptions: 1. The axial shear stress, t xy a dn/dx, where v is the displacement in the axial, y-direction. This violates the relationship of linear elasticity, t xy = G xy dn dx ÷ du dy _ _ . Hence, this is equivalent to assuming that du/dy = 0 or du/dy ¸ dn/dx. 2. The axial normal stress remains constant over the thickness of the transverse ply after cracking. In other words, concentration of local stress near cracks is neglected. 3. Cracks remain sufficiently far apart so that their mutual interactions can be neglected. The shear lag analyses will be presented here with consistent notations and expressions that may differ in form from those given in the original articles. 4.4.1 Initial shear lag analysis Here we describe the analysis presented by Garrett, Bailey, and Parvizi [27, 43] and modified by Manders et al. [29] to account for the presence of neighboring cracks. For convenience, we follow a treatment similar to the one described in a review paper by Berthelot [41]. The objective is to determine the variation of the axial (x-direction) stress in the transverse plies on crack formation in these plies. To begin, one 74 Micro-damage mechanics assumes that the x-displacement in the 0 · -ply is constant through its thick- ness while the corresponding displacement of the transverse plies increases linearly from the 0/90 ply interface towards the laminate mid-plane. In accordance with the first shear lag assumption stated above, the interfacial shear stress is given by t = G 90 xz u 90 ÷ u 0 t 90 _ _ ; (4:55) where u 0 is the x-displacement in the 0 · -ply, u 90 is the x-displacement at mid-plane, and G 90 xz is the transverse shear modulus of the 90 · -ply. The equilibrium of axial forces in an element of the 90 · -ply (see Figure 4.8) gives t = t 90 ds 90 xx dx ; (4:56) where s 90 xz is the axial normal stress, assumed to be constant in the 90 · -ply across the thickness (z-direction). Substituting Eq. (4.56) into Eq. (4.55), one obtains ds 90 xx dx = G 90 xz u 90 ÷ u 0 t 2 90 _ _ : (4:57) The axial stresses in the 0 · - and 90 · -plies are related to the applied stress s c as ls 0 xx ÷ s 90 xx = 1 ÷ l ( )s c ; (4:58) where the ply thickness ratio l is defined as l = t 0 t 90 : (4:59) Lastly, the stress–strain relations in the 0 · - and 90 · -plies are: s 0 xx = E 0 x0 e 0 xx ; with e 0 xx = du 0 dx ; s 90 xx = E 90 x0 e 90 xx ; with e 90 xx = du 90 dx ; (4:60) 90º 0º 0º 90 xx σ x dx τ τ 90 xx σ + 90 xx σ d Figure 4.8. Stresses acting on an element of 90 · -ply. 75 4.4 One-dimensional models: shear lag analysis where E 0 x0 = E 1 and E 90 x0 = E 2 are the initial axial Young’s moduli of the 0 · - and 90 · -plies, respectively. Differentiating Eq. (4.57) with respect to x and using Eqs. (4.58)–(4.60), one obtains the differential equation for axial stress in the transverse ply as d 2 s 90 xx dx 2 ÷ b 2 t 2 90 s 90 xx = ÷ b 2 t 2 90 E 90 x0 E x0 s c ; (4:61) with b 2 = G 90 xz0 1 E 90 x0 ÷ 1 lE 0 x0 _ _ ; (4:62) where G 90 xz0 is the initial in-plane shear modulus of the 90 · -ply and E x0 = E c is the axial modulus of the undamaged laminate given by the rule of mixtures as E x0 = lE 0 x0 ÷ E 90 x0 1 ÷ l : (4:63) This value can be estimated accurately by using the classical laminates plate theory. Since the crack surfaces are traction free, the axial stress in the transverse ply vanishes on the crack planes (x = ±l). Hence, the solution of Eq. (4.61) is given by s 90 xx = s c E 90 x0 E x0 1 ÷ cosh b x t 90 cosh b l t 90 _ _ _ _ _ _ _ _ : (4:64) Thus b appears as a load transfer parameter, and is sometimes known as the shear lag parameter. A quite similar shear lag analysis was conducted by Dvorak et al. [44], except that they included residual thermal stresses and in place of Eq. (4.55) assumed that the shear stress is given by t = K u 90 ÷ u 0 ( ); (4:65) where K is a shear parameter, which is to be determined from experimental data. Dvorak et al. [44] suggested determining b from the experimentally meas- ured stress at first ply failure. The differential equation for the axial stress in this case is d 2 s 90 xx dx 2 ÷ b 2 t 2 90 s 90 xx = ÷ b 2 t 2 90 s 90 xxR ÷ E 90 x0 E x0 s c _ _ ; (4:66) where the shear lag parameter is now given by b 2 = Kt 90 t 0 E 0 x0 ÷ t 90 E 90 x0 _ _ t 0 E 0 x0 E 90 x0 = Kt 90 1 E 90 x0 ÷ 1 lE 0 x0 _ _ ; (4:67) 76 Micro-damage mechanics and the solution obtained for axial stress in the transverse ply is s 90 xx = s 90 xxR ÷ s c E 90 x0 E x0 _ _ 1 ÷ cosh b x t 90 cosh b l t 90 _ _ _ _ _ _ _ _ ; (4:68) where s 90 xxR is the axial residual thermal stress in the 90 · -ply. 4.4.2 Interlaminar shear lag analysis Based on extensive experimental observations of cracks, Highsmith and Reifsnider [30] found that shear deformations in a given ply were restricted to a thin region near interfaces between adjacent plies. This region is resin rich and thus is less stiff in its response to shear stress than the central portion of the ply. Transverse cracks were observed to extend up to the region, but not into it. Based on these observa- tions, interlaminar shear lag models were developed [30, 45], in which shear stresses were assumed to develop only within this thin region, whose thickness and shear modulus are unknown. The unit cell for such models is shown in Figure 4.9. We present here the interlaminar shear lag analysis reported by Fukunaga et al. [45], who incorporated the thermal residual stresses and Poisson’s effect into the analysis. The displacements in the x- and y-directions of each ply are still assumed to be constant across thickness, as above, but are expressed as u 0 = e c x ÷ U 0 x ( ); v 0 = ÷ A 12 A 22 e c y ÷ V 0 x ( ); u 90 = e c x ÷ U 90 x ( ); v 90 = ÷ A 12 A 22 e c y ÷ V 90 x ( ); (4:69) where the coefficients A ij are the in-plane stiffness components of half the cross-ply laminate. These are the same as the components of the [A] matrix in classical laminates plate theory, and can be expressed in terms of the reduced stiffness components Q ij of the 0 · - and 90 · -plies as A 11 = Q 11 t 0 ÷ Q 22 t 90 ; A 12 = Q 12 h; A 22 = Q 22 t 0 ÷ Q 11 t 90 : (4:70) t 0 t 90 h x z 2l 90 º 0 º 0 º σ c t s Figure 4.9. Unit cell for interlaminar shear lag analysis. 77 4.4 One-dimensional models: shear lag analysis These relations suppose that the 0 · - and 90 · -plies have the same elastic properties. The strain e c is the average strain of the laminate and is given by e c = s c E x ; (4:71) where the axial Young’s modulus of the laminate E x is given by E x = A 11 A 22 ÷ A 2 12 hA 22 : (4:72) The equilibrium of the axial forces yields Q 11 d 2 u 0 dx 2 ÷ G t 0 t s u 90 ÷ u 0 ( ) = 0; Q 22 d 2 u 90 dx 2 ÷ G t 90 t s u 90 ÷ u 0 ( ) = 0; (4:73) where G and t s represent the shear modulus and the thickness of the “resin-rich” layers, respectively. Next, the balance of the in-plane shear loads gives Q 66 d 2 v 0 dx 2 ÷ G t 0 t s v 90 ÷ v 0 ( ) = 0; Q 66 d 2 v 90 dx 2 ÷ G t 90 t s v 90 ÷ v 0 ( ) = 0: (4:74) The solution of the equilibrium equations, Eqs. (4.73) and (4.74), determines the displacement field, from which the strains are determined using the strain–dis- placement relations. Finally, the stress components are found by using the consti- tutive relations. The axial stress in the 90 · -ply is obtained as s 90 xx = s 90 xxR ÷ s c Q 22 E 0 x _ _ 1 ÷ Q 12 A 12 Q 22 A 22 _ _ 1 ÷ cosh b x t 90 cosh b l t 90 _ _ _ _ _ _ _ _ ; (4:75) where the shear lag parameter b is given by b 2 = G t 90 t s 1 Q 22 ÷ 1 lQ 11 _ _ : (4:76) The interlaminar shear lag models by Highsmith and Reifsnider [30] differ from Fukunaga’s analysis [45] in their definition of the shear-lag parameter. For the Highsmith–Reifsnider model, b 2 = G t 90 t s 1 E 90 x0 ÷ 1 lE 0 x0 _ _ : (4:77) It can be seen that the expressions for the axial stress in the 90 · -ply, Eqs. (4.64), (4.68), and (4.75), are very similar in different shear lag models. Usually, the definition of the shear lag parameter is different. The biggest limitation of the 78 Micro-damage mechanics interlaminar shear lag analysis is that the thickness of the boundary layer must be assumed in a somewhat arbitrary fashion. 4.4.3 Extended shear lag analysis Limand Hong [38] modified the interlaminar shear lag model of Fukunaga et al. [45] to include the effect of crack interaction. The governing differential equations are the the same as in Fukunaga’s analysis (Eqs. (4.73), (4.74)). However, the boundary conditions are used such that they are valid for any crack spacing. With this modification, the axial stress in the 90 · -ply is given by s 90 xx = s 90 xx0 1 ÷ a 1 e bx=t 90 ÷ a 2 e ÷bx=t 90 _ _ _ _ ; (4:78) where a 1 = 1 ÷ e ÷2bl=t 90 e 2bl=t 90 ÷ e ÷2bl=t 90 ; a 2 = e 2bl=t 90 ÷ 1 e 2bl=t 90 ÷ e ÷2bl=t 90 ; (4:79) and the shear lag parameter b is given by Eq. (4.76). The solution in Eq. (4.78) is different than in Eq. (4.75) due to differences in the assumed boundary conditions. It is noted that s 90 xx0 in Eq. (4.78) is the total stress in the transverse ply before cracking, and thus includes thermal residual stress, if any. To account for the progressive shear in the 90 · ply, Steif [46] developed a shear lag theory, where he used an x-displacement that was constant through the 0 · layer thickness but varied parabolically through the 90 · layer thickness. This analysis leads to an expression of the axial stress in the transverse ply similar to that given in Eq. (4.64), with a modification in the shear lag parameter: b 2 = 3G 90 xz0 1 E 90 x0 ÷ 1 lE 0 x0 _ _ ; (4:80) which is three times the value of b 2 as given by Eq. (4.62). Later, the same approach was applied by Ogin et al. [47, 48] to study the stiffness reduction of glass-fiber cross-ply laminates subjected to quasi-static or fatigue loading. The transverse shear effects in both 0 · - and 90 · -plies have been considered by Nuismer and Tan [49] and Lee, Daniel and Yaniv [31, 56]. These authors consider stress approaches based on the assumption of linear shear stress variation in each layer through the thickness, which extends the initial shear lag analysis. Lee and Daniel [31] relate these linear variations to parabolic variations of the x-displacements through the thicknesses of the 0 · and 90 · -plies. These approaches also lead to modified expressions for the load transfer parameter. For example, the analysis by Lee and Daniel yields the following expression for the shear lag parameter b 2 = 3G 90 xz0 1 ÷ l G 90 xz0 G 0 xz0 1 E 90 x0 ÷ 1 lE 0 x0 _ _ : (4:81) It can be seen that Eq. (4.81) reduces to Eq. (4.80) when G 0 xz0 ¸ G 90 xz0 . 79 4.4 One-dimensional models: shear lag analysis 4.4.4 2-D shear lag models Flaggs [32], and Nuismer and Tan [33, 49] developed “two-dimensional” shear lag models. Contrary to the authors’ claims, these models are essentially equivalent to 1-D models with a minor modification to correct for the contraction due to the Poisson’s effect [42]. The analysis by Flaggs [32] accounts for both applied normal and shear load- ings and results into a system of two coupled second-order differential equations. For only normal applied loading, it reduces to a single ODE identical to general 1-D analysis equations, except that the shear lag parameter is given by b 2 = 2 1 lQ 0 xx0 ÷ Q 90 yy0 ÷ Q 90 xy0 Q 0 xy0 =Q 0 xx0 _ _ Q 90 xx0 Q 90 yy0 ÷ Q 90 xy0 _ _ 2 _ _ _ _ _ _ 1 k 2 ÷ 1 2 _ _ 1 G 90 xz0 ÷ l 2G 0 xz0 ; (4:82) where k is a transverse shear correction factor and the Q’s are stiffness coefficients for the virgin laminate. The Flaggs analysis therefore uses a minor correction for the Poisson’s contraction introduced by approximate inclusion of the transverse coordinate (y). Another 2-D elasticity analysis was performed by Nuismer and Tan [33, 49]. This analysis uses the shear lag parameter defined as b 2 = 1 Q 90 xx0 ÷ 1 lQ 0 xx0 1 3G 90 xz0 ÷ l 3G 0 xz0 : (4:83) The analysis also yields a nonzero o (see Eq. (4.85) described in the next subsec- tion), given by o = 1 l t 90 t / i ÷ b 2 s 90 x0 _ ¸ ; (4:84) where t / i is the slope of the interfacial shear stress at the crack location. This analysis is also equivalent to other 1-D models with a minor correction for the Poisson’s ratio. More recently, there have been efforts to modify and extend the 1-D and 2-D shear lag models to enable predictions for laminates of other than cross-ply layups, such as the equivalent constraint models [13, 51–53] and 2-D displacement analysis [54, 55]. These models also suffer from deficiencies that are common in the shear lag models described above. 4.4.5 Summary of shear lag models It can be shown [42] that all one-dimensional stress analyses discussed above can be reduced to a generalized form of Garrett and Bailey’s equation [27], 80 Micro-damage mechanics d 2 Ás dx 2 ÷ b 2 Ás = o P ( ); (4:85) where Ds represents the total stress transferred from the 90 · -layer to the 0 · -layer, and is given by Ás = s 0 xx ÷ s 0 xx0 ; (4:86) and x = x/t 90 is a dimensionless x-coordinate, b is the shear lag parameter, and o(P) is a function which may depend on the laminate structure, crack spacing, and the applied load (P). The corresponding boundary conditions, equivalent to traction-free crack surfaces, are Ás x = ±r ( ) = s 90 xx0 l ; (4:87) where r = l/t 90 denotes the crack spacing normalized by the cracked ply thickness. Other than in the one-dimensional model by Nuismer and Tan [33, 49], the function o is chosen to be zero. It can be clearly observed that all the one- dimensional shear lag models are essentially the same; the difference being in the choice of the shear lag parameter. The different shear lag models are summarized in Table 4.1. The solution of the differential equation (4.85) provides Ás = ÷ o b 2 ÷ s 90 xx0 l ÷ o b 2 _ _ cosh bx cosh br : (4:88) With this, the stress in the 0 · -layer is given by s 0 xx = s 0 xx0 ÷ o b 2 ÷ s 90 xx0 l ÷ o b 2 _ _ cosh bx cosh br ; (4:89) and in the 90 · -layer, it is given by s 90 xx = s 90 xx0 ÷ lo b 2 _ _ 1 ÷ cosh bx cosh br _ _ : (4:90) Now, the rate of load transfer from the 90 · -layer to the 0 · -layer, analogous to Eq. (4.56), can be rewritten as dÁs dx = t l : (4:91) It is noted here that the difference in Eqs. (4.56) and (4.91) is due to the definition of Ds (see Eq. (4.86)). Differentiating Eq. (4.88) with respect to x, and comparing the result with Eq. (4.91), gives the interfacial shear stress as t i = s 0 xx0 ÷ lo b 2 _ _ cosh bx cosh br : (4:92) 81 4.4 One-dimensional models: shear lag analysis The stress predictions fromvarious shear lag models are compared in Figure 4.10 for [0/90 2 ] s carbon/epoxy (Hercules AS4/3501–6) laminate with lamina properties [31, 56]: t ply = 0.154 mm, E 1 = 130 GPa, E 2 = 9.7 GPa, G 12 = 5.0 GPa, G 13 = 3.6 GPa, n 12 =0.3, andn 13 =0.5. The axial normal stress inthe 90 · -plies increases fromzeroon the crack to a maximum value midway between two cracks, as the load is transferred fromthe 0 · -plies back into the 90 · -plies. The traction-free boundary condition at the crack surfaces is thus fulfilled accurately for the normal stress in all models. However, none of the shear lag models represents interfacial shear stress accurately. The shear stress has a maximumon the crack surface and decays towards zero as we move away Table 4.1 A summary of different shear lag models to analyze cracked cross-ply laminates Shear lag model Feature Shear lag parameter (b 2 ) o(P) Garret and Bailey [27], Manders et al. [29] Simplest shear lag model G 90 xz0 1 E 90 x0 ÷ 1 lE 0 x0 _ _ 0 Laws and Dvorak [36] Use first ply failure data to determine b Kt 90 1 E 90 x0 ÷ 1 lE 0 x0 _ _ 0 Steif [46], Ogin et al. [47, 48] Parabolic displacement profile 3G 90 xz0 1 E 90 x0 ÷ 1 lE 0 x0 _ _ 0 Highsmith and Reifsnider [30] Use an effective shear transfer layer G t 90 t s 1 E 90 x0 ÷ 1 lE 0 x0 _ _ 0 Fukunaga et al. [45] Use an effective shear transfer layer G t 90 t s 1 Q 22 ÷ 1 lQ 11 _ _ 0 Limand Hong [38] Use an effective shear transfer layer, account for crack interaction G t 90 t s 1 Q 22 ÷ 1 lQ 11 _ _ 0 Nuismer and Tan [33, 49] Account for Poisson’s effect 1 Q 90 xx0 ÷ 1 lQ 0 xx0 1 3G 90 xz0 ÷ l 3G 0 xz0 1 l t 90 t / i ÷ b 2 s 90 x0 _ ¸ Flaggs [32] 2-D shear lag analysis to account for both transverse and shear loading 2 1 lQ 0 xx0 ÷ Q 90 yy0 ÷ Q 90 xy0 Q 0 xy0 Q 0 xx0 Q 90 xx0 Q 90 yy0 ÷ Q 90 xy0 _ _ 2 _ _ _ _ _ _ _ _ _ _ 1 k 2 ÷ 1 2 _ _ 1 G 90 xz0 ÷ l 2G 0 xz0 0 Lee and Daniel [31] Parabolic displacement variation, account for Poisson’s effect 3G 90 xz0 1 ÷ l G 90 xz0 G 0 xz0 1 E 90 x0 ÷ 1 lE 0 x0 _ _ 0 82 Micro-damage mechanics from the crack. The nonzero shear stress on the crack surfaces is a clear violation of the boundary condition. This makes all 1-D analyses inherently inaccurate. Let us now turn our attention to the stiffness degradation. The treatment provided here follows the approach covered in a book chapter by Nairn and Hu [42]. Before cracking, both the 0 · - and 90 · -plies share the applied load. But after cracking, the 90 · -layers at the crack planes do not carry any load. The total applied load at these planes is carried by the 0 · -plies. Away from cracks, the 90 · -plies take up load again through the shear transfer mechanism. To determine the modulus of the cracked laminate, let us first determine the total x-displacement of the load bearing layer in the region between two cracks in the transverse ply. It can be found by integrating the axial strain as u P ( ) = _ l ÷l e 0 xx dx = t 90 _ ÷r ÷r s 0 xx E 0 x ÷ n 0 xz s 0 zz E 0 x ÷ n 0 xy s 0 yy E 0 x _ _ dx; (4:93) where P is the applied load. It is noted that the thermal strain is ignored because for linear thermoelastic materials the modulus is independent of the residual stresses. The compliance of a unit cell of the laminate with cracks is defined as C = u P ( ) ÷ u 0 ( ) P : (4:94) The effective modulus is then given by E = 2l hWC : (4:95) 60 –60 –4 –3 –2 –1 0 ξ 1 2 Manders et al. Laws & Dvorak Fukunaga et al. Lee & Daniel Steif 3 4 –40 –20 0 ( M P a ) 9 0 9 0 x x Ј x z σ σ 20 40 Figure 4.10. Variation of the axial normal and interfacial stresses between two cracks in a [0/90 2 ] s laminate from shear lag analysis assuming normalized crack spacing r = 4 and applied stress of 50 MPa. 83 4.4 One-dimensional models: shear lag analysis Assuming a plane stress state, s 0 yy = 0: (4:96) Moreover there is no load applied in the thickness (z) direction, i.e., _ ÷r ÷r s 0 zz dx = 0: (4:97) Thus, for both one-dimensional analysis and two-dimensional plane stress analy- sis the total displacement reduces to u P ( ) = t 90 _ ÷r ÷r s 0 xx E 0 x dx: (4:98) Substituting stress from Eq. (4.89) for 1-D shear lag analyses, we obtain 1 E x = 1 E 0 c 1 ÷ E 90 x0 lE 0 x0 tanh br br _ _ ÷ o P ( ) ÷ o 0 ( ) E 0 x0 b 2 s 0 1 ÷ tanh br br _ _ : (4:99) Assuming o(P)÷o(0) = 0, as is the case for most one-dimensional analyses, the effective axial modulus is then 1 E x = 1 E 0 c 1 ÷ E 90 x0 lE 0 x0 tanh br br _ _ : (4:100) Thus, the axial modulus for the cracked laminate normalized with its virgin value is given by E x E 0 c = 1 ÷ E 90 x0 lE 0 x0 tanh br br _ _ ÷1 : (4:101) The predictions for stiffness reductionusing various shear lag models as a function of crack density for [0/90 3 ] s glass/epoxy (Scotch Ply 1003) laminate are shown in Figure 4.11. It should be noted, however, that these predictions may vary depending upon the parameters used in the models. It is important to note that the original shear lag model by Garrett, Bailey, and Parvizi [27, 43], or equivalently by Manders et al. [29], is reasonably accurate. More complicated shear lag models not only involve more complicated analysis, and adjustable parameters, but also do not yield more accurate predictions. The basic deficiency of shear lag models lies in their one-dimensionality of stress field, and no significant improvement is essentially possible. 4.5 Self-consistent scheme The self-consistent method is widely employed for estimating properties of elastic solids containing entities such as inclusions, voids, and cracks; see for example Nemat-Nasser and Hori [2]. Using this method generally requires assuming 84 Micro-damage mechanics an infinite solid. For composite laminates containing ply cracks, Laws and Dvorak [59] estimated elastic properties using this method. They first assumed that a unidirectional ply (within a laminate) with cracks along its fibers can be modeled as an infinite solid with the same fiber volume fraction and crack density. Taking then the estimated average properties of the infinite solid as those of the cracked ply, they replaced the cracked ply in a composite laminate by a homogeneous ply of the degraded properties of the infinite solid. Figure 4.12 illustrates the assumed scheme. Elastic properties of the cracked composite laminate were then calculated by the classical laminate plate theory. This procedure in principle allows calculating stiffness degradation of any composite laminate with given crack density in any ply, i.e., for any orientation of cracks. It is doubtful, however, that the assumption of an infinite solid will give accurate elastic property estimates for plies that are typically of a fraction of a millimeter in thickness. The predicted properties of various laminates with cracks in different plies and with different crack density values were reported in Laws and Dvorak [59] but these properties were not compared with experimental data or independent numerical computations. Readers interested in details of the self-consistent method for composite mater- ials with cracks are referred to Laws et al. [60]. We provide here a brief account of the concepts involved. Essentially, Hill [61] and Budiansky [62] developed the self-consistent method for composite materials, i.e., homogeneous solids with inclusions. The idea in this method is that a single inclusion is embedded in a homogeneous elastic solid which has the yet-unknown overall properties of the heterogeneous solid and the local fields thus estimated are then used to obtain the overall properties. Laws et al. [60] use a key result obtained for ellipsoidal inclusions by Eshelby [63, 64] that the elastic field within the inclusion is uniform. This result was used for two models: a three-phase system and a two-phase system. 1.0 0.9 0.8 0.7 R e l a t i v e S t i f f n e s s 0.6 0.5 0.0 0.1 0.2 0.3 Hashin Garrett & Bailey Reifsnider Nuismer & Tan Glass/Epoxy Flaggs [0/90 3 ]s Ogin et al. 0.4 0.5 Microcrack Density (1/mm) 0.6 0.7 0.8 0.9 1.0 Figure 4.11. Variation of normalized Young’s modulus as a function of crack density for a [0/90 3 ] s glass/epoxy (Scotch Ply 1003) laminate. The experimental data are from [30]. Reprinted, with kind permission, from Damage Mechanics of Composite Materials, J. A. Nairn and S. Hu, Micromechanics of damage: a case study of matrix microcracking, pp. 187–243, copyright Elsevier (1994). 85 4.5 Self-consistent scheme In the former the fibers and cracks are considered as two phases (special cases of inclusions) embedded in an elastic matrix, while in the latter the fibers and matrix are smeared into a homogeneous solid in which cracks are embedded. Cracks in both models are introduced as slits that are regarded as limiting cases of elliptical voids. When the fiber diameter is much smaller than the crack size, the two-phase model is more appropriate. Thus if Q is the estimated stiffness matrix of the cracked ply, and Q 0 is its value without the cracks present, then the two matrices are related by [60] Q = Q 0 ÷ 1 4 bQ 0 ÃQ; (4:102) where b is a crack density parameter and L is a function of the aspect ratio of the elliptical crack and the stiffness coefficient matrix Q. It is seen from this equation that the estimation procedure for stiffness of a cracked ply is iterative and it requires quantities L that are estimated by assuming the homogenized composite ply to be of infinite extent. Variations of the self-consistent method described above have also appeared in the literature. Noteworthy work is that of Hoiseth and Qu [65, 66]. They followed the differential self-consistent method and derived an incremental differential equation describing the effective axial modulus for the cracked laminate by representing the change in strain energy due to an increase in terms of the number of cracks in a lamina. If r = t 90 /l denotes the normalized crack density, then the axial modulus is given by (a) (b) Figure 4.12. Self-consistent scheme: (a) cracked laminate containing fibers of small diameter and periodic cracks (b) homogenized cracked lamina. 86 Micro-damage mechanics d E c dr = ÷ d 4t 90 2 E c r ( ) ÷ E 1 1 ÷ n 12 n 21 _ _ ; (4:103) subject to the initial condition E c 0 ( ) = E c0 ; (4:104) where d is the average crack opening displacement (COD), defined as d = 1 t 90 _ t 90 0 d z ( ) dz; (4:105) with d(z) being the COD variation across the z-direction (thickness). The authors obtained it numerically using finite element calculations on a unit cell. For cross- ply laminates, their predictions compared well with the independent FE simula- tions that they carried out themselves. For multidirectional laminates, these approaches become complex and do not yield accurate estimates. 4.6 2-D stress analysis: variational methods 4.6.1 Hashin’s variational analysis An improved 2-D stress analysis can be obtained using the principle of minimum complementary energy applied on a cracked laminate volume. Hashin [67] con- sidered cross-ply laminates and used this principle to solve the boundary value problem. He constructed an admissible stress field assuming that the normal stresses in the loading direction are constant over the ply thickness. The admissible stress field satisfies equilibrium and boundary and interface conditions. The boundary volume problem thus solved provides the stresses and the reduced stiffness coefficients for the cracked cross-ply laminate. We followthe general notations for geometry, material, and stress components as given in Section 4.3. Consider a symmetric cross-ply laminate containing a parallel array of transverse cracks and subjected to a uniform axial load (Figure 4.13) N xx per unit laminate width. Let us first investigate the stress states before and after cracking. Before cracking, all shear stresses and the transverse normal stress in the entire laminate are zero, i.e., s 0 xx0 ; s 90 xx0 ,= 0; s 0 yy0 ; s 90 yy0 ,= 0; s 0 zz0 = s 90 zz0 = 0; s 0 yz0 = s 90 yz0 = 0; s 0 xz0 = s 90 xz0 = 0; s 0 xy0 = s 90 xy0 = 0; (4:106) where superscripts denote the layer orientations. The nonzero axial stresses in the 0 · - and 90 · -layers for virgin laminate are constant throughout individual layers and can be obtained using the laminate theory as s 0 xx0 = E 0 x0 E x0 s c ; s 90 xx = E 90 x0 E x0 s c ; (4:107) where s c = N xx /2h is the normal tensile stress on the laminate. 87 4.6 2-D stress analysis: variational methods Now on sufficient tensile load, the 90 · -ply develops continuous intralaminar cracks in the fiber direction that extend through the whole laminate width. These cracks will cause perturbation in the stress fields in both layers. In a homogeneous elastic material, there is theoretically a crack-tip singularity in the stress field. However, for transverse matrix cracking in composite laminates, the stresses at a crack tip are finite because the fibers have a finite size and will blunt the crack tip. In Hashin’s analysis, he assumes the following: 1. The perturbations in axial stresses Ás 90 xx and Ás 0 xx are constant through the thickness of layers. Thus, Ás 90 xx = Ás 90 xx x ( ); Ás 0 xx = Ás 0 xx x ( ): (4:108) 2. There are no perturbations in s yy , s xy , and s yz , i.e., Ás 90 yy = Ás 90 xy = Ás 90 yz = Ás 0 yy = Ás 0 xy = Ás 0 yz = 0: (4:109) The stress components for layer m (m = 0, and 90 for 0 · -ply and 90 · -ply, respectively) of the cracked laminate can be expressed as s m ij = s m ij0 ÷ Ás m ij (4:110) where the subscript 0 indicates the uncracked laminate. The stress perturbations can be expressed as Ás 90 xx = ÷s 90 xx0 f 90 x ( ); Ás 0 xx = ÷s 0 xx0 f 0 x ( ); (4:111) where f 90 and f 0 are unknown perturbation functions. z x y O t 0 t 0 t 90 t 90 h h N xx Figure 4.13. A cross-ply laminate loaded in axial tension. 88 Micro-damage mechanics The equilibrium equations for the plies are given by @s m xx @x ÷ @s m xy @y ÷ @s m xz @z = 0; @s m yx @x ÷ @s m yy @y ÷ @s m yz @z = 0; @s m zx @x ÷ @s m zy @y ÷ @s m zz @z = 0: (4:112) Using Eqs. (4.108)–(4.110), one obtains @Ás m xx @x ÷ @Ás m xz @z = 0; @Ás m zx @x ÷ @Ás m zz @z = 0: (4:113) Substituting Eq. (4.111) into Eq. (4.113) and integrating, one gets Ás 90 xz = s 90 xx0 f / 90 x ( )z ÷ f 90 x ( ) [ [; Ás 90 zz = ÷s 90 xx0 1 2 f // 90 x ( )z 2 ÷ f / 90 x ( )z ÷ g 90 x ( ) _ _ ; (4:114) for m = 90 (90 · -ply), and Ás 0 xz = s 0 xx0 f / 0 x ( )z ÷ f 0 x ( ) [ [; Ás 0 zz = ÷s 0 xx0 1 2 f // 0 x ( )z 2 ÷ f / 0 x ( )z ÷ g 0 x ( )[; _ (4:115) for m = 0 (0 · -ply), where f 0 (x), f 90 (x), g 0 (x), and g 90 (x) are unknown functions and primes denote derivatives with respect to x. To obtain the relation between f 90 and f 0 , consider equilibrium in the x-direction of the undamaged laminate N xx = _ h ÷h s xx0 dz = 2 s 90 xx0 t 90 ÷ s 0 xx0 t 0 _ _ : (4:116) If the same membrane force is applied to the cracked laminate, the equilibrium in the x-direction will give N xx = 2 s 90 xx0 t 90 ÷ s 0 xx0 t 0 _ _ ÷ 2 s 90 xx0 t 90 f 90 x ( ) ÷ s 0 xx0 t 0 f 0 x ( ) _ ¸ : (4:117) From Eqs. (4.116) and (4.117), we get s 90 xx0 t 90 f 90 x ( ) ÷ s 0 xx0 t 0 f 0 x ( ) = 0; (4:118) i.e., f 0 x ( ) = ÷ s 90 xx0 s 0 xx0 1 l f 90 x ( ); (4:119) where l = t 0 /t 90 is the ply thickness ratio (Eq. (4.59)). 89 4.6 2-D stress analysis: variational methods The boundary value problem for solving the three stress components s xx , s xz , and s zz is defined on the unit cell shown in Figure 4.14, in which the origin of the coordinate system is placed at the mid-plane of the laminate, and midway between two cracks. Clearly z = 0 is the plane of symmetry. Hence, the shear stress s xz at all points along this plane is zero. At the interface between the cracked 90 · -ply and the uncracked 0 · -ply (z =t 90 ), the shear stress s xz and normal stress s zz must be continu- ous. Moreover, the surface at z = h is free from any external loading, i.e., the shear stress s xz and normal stress s zz are zero on that surface. Finally, the crack surfaces x = ±l are traction free, i.e., s 90 xz = Ás 0 xz = 0 and s 90 xx = s 90 xx0 ÷ Ás 90 xx = 0 at x = ±l. Thus, all the applicable boundary conditions for the total stresses are written as Symmetry : s 90 xz x; 0 ( ) = 0; (a) Interface : s 90 xz x; t 90 ( ) = s 0 xz x; t 90 ( ); (b) s 90 zz x; t 90 ( ) = s 0 zz x; t 90 ( ); (c) Free boundary : s 0 xz x; h ( ) = 0; (d) s 0 zz x; h ( ) = 0; (e) Traction free : s 90 xz ±l; z ( ) = 0; ÷t 90 _ z _ t 90 ; (f) s 90 xx ±l; z ( ) = 0; ÷t 90 _ z _ t 90 ; (g) (4:120) where 2l is the distance between any two adjacent cracks. These conditions are the same as those given earlier without derivation in Eq. (4.54) Denoting f 90 (x) by f(x), putting boundary conditions, Eq. (4.120), into Eqs. (4.114)–(4.115), and using Eq. (4.119), after some mathematical manipulations, the resulting stress field in the cracked laminate is given by x z l l h t 0 t 90 Figure 4.14. A repeating unit cell for a cracked cross-ply laminate. 90 Micro-damage mechanics s 90 xx = s 90 xx0 1 ÷ f x ( ) [ [; s 90 xz = s 90 xx0 f / x ( )z; s 90 zz = s 90 xx0 f // x ( ) 1 2 1 ÷ l ( )t 2 90 ÷ z 2 _ ¸ ; (4:121) in the 90 · -ply, and s 0 xx = s 0 xx0 ÷ s 90 xx0 1 l f x ( ); s 0 xz = s 90 xx0 f / x ( ) 1 l 1 ÷ l ( )t 90 ÷ z [ [; s 0 zz = s 90 xx0 f // x ( ) 1 2l 1 ÷ l ( )t 90 ÷ z [ [ 2 ; (4:122) in the 0 · -ply. Using the crack surface boundary conditions, Eq. (4.120f–g), we get additional conditions for the unknown perturbation function f(x) At x = ±l : s 90 xx = 0 = s 90 xx0 1 ÷ f x ( ) [ [; s 0 xz = 0 = s 90 xx0 f / x ( )z : (4:123) Thus, f ±l ( ) = l; f / ±l ( ) = 0; ÷t 90 _ z _ t 90 : (4:124) This completes the description of the resulting boundary value problem. The object- ive is to estimate f(x) in order to determine the stress field in the cracked laminate. The stress field in Eqs. (4.121)–(4.122) represents an admissible stress field as it satisfies all equilibrium and boundary conditions. Thus, the principle of minimum complementary energy (see Section 2.1.5) can be applied to obtain the unknown function f(x). The procedure for doing this, given in Hashin [67], is described next. For the problem at hand, we can define an admissible stress field ¯ s ij within the volume V of the cracked body with associated tractions ¯ T i which satisfy only equilibrium and the traction boundary conditions. Since the tractions are not altered during the cracking process, ~ T i = T 0 i on S = S T ; (4:125) where T i 0 are the tractions applied on the uncracked body. Also, the crack surfaces S c are traction free, i.e., ~ T i = 0 on S = S c : (4:126) If we write ~ s ij = s 0 ij ÷ s / ij ; ~ T i = T 0 i ÷ T / i ; (4:127) where s 0 ij are stresses in the uncracked laminate, s / ij are the perturbation stresses due to cracking, and T / is the additional traction due to cracking. Then from the traction boundary conditions in Eqs. (4.125) and (4.126), we obtain 91 4.6 2-D stress analysis: variational methods T / i = 0 on S = S T ; T / i = ÷T 0 i on S = S c : (4:128) The complementary energy functional is given by (see Eq. (2.36), Section 2.1.5) Å + 0 = 1 2 _ V S ijkl s 0 ij s 0 kl dV ÷ _ S u T 0 i ^ u i dS (4:129) for the uncracked body, where S ijkl is the compliance tensor, and ~ Å + = 1 2 _ V S ijkl ~ s ij ~ s kl dV ÷ _ S u ~ T i ^ u i dS (4:130) for the crackedbody. Since only tractionboundary conditions are applied, the second term is zero (S u = 0; S T = S t ). However, for the sake of completeness, we will follow the proof given in Appendix 1 of [67], which applies to mixed boundary conditions. Substituting the stress field given in Eq. (4.127) into Eq. (4.130) and using Eq. (4.129), the complementary energy for the cracked solid can be expressed as ~ Å + = Å + 0 ÷ _ V S ijkl s 0 ij s / kl dV ÷ 1 2 _ V S ijkl s / ij s / kl dV ÷ _ S u T / i ^ u i dS : (4:131) Consider the first integral in Eq. (4.131). The stresses s / ij satisfy equilibrium since s 0 ij and ¯ s ij do. Also, the strain field in the uncracked body is given by e 0 ij = S ijkl s 0 kl : (4:132) Therefore, by virtual work (see Eq. (2.33), Section 2.1.5) J = _ V S ijkl s 0 kl s / ij dV = _ S u T / i u 0 i dS ÷ _ S c T / i u 0 i dS / ; (4:133) where the tractions in the second surface integral are defined with respect to the inward normal to S c . Using the boundary condition in Eq. (4.128), we have J = _ S u T / i ^ u i dS ÷ _ S c T 0 i u 0 i dS : (4:134) The second integral is taken over the two adjacent surfaces of each crack. Since the normal on one surface is the negative of the normal on the other surface and since T 0 i and u 0 i are continuous across the crack surface, the integrals on the two crack surfaces cancel one another and therefore the total integral vanishes. Introducing the remainder of Eq. (4.134) into Eq. (4.131) we obtain the complementary energy for a cracked solid as ~ Å + = Å + 0 ÷ 1 2 _ V S ijkl s / ij s / kl dV = Å + 0 ÷ Å / (4:135) 92 Micro-damage mechanics where П / 0 is the complementary energy due to perturbation stresses. The effective elastic compliances of the undamaged and damaged laminates can be expressed in terms of the complementary energy functionals. To obtain the relation, consider homogeneous boundary conditions on the cracked body T i = s ij n j ; s ij = constant; (4:136) and let s ij = 1 V _ V s ij dV: (4:137) Then, from Eq. (4.130) we have Å + = 1 2 S + ijkl s ij s kl V; (4:138) where S + ijkl is the effective elastic compliance tensor for the homogenized medium[68]. The laminate under consideration is loaded by a membrane force N xx on two horizontal edges. Thus, volume and the average stress for this laminate are given by V = 2Ah s xx = N xx 2h = s c ; (4:139) where A is the area of the plane normal to the x-axis. The complementary energy for cracked and uncracked laminates takes the form Å + 0 = 1 2 s 2 c E x0 2Ah Å + = 1 2 s 2 c E x 2Ah; (4:140) where E x0 , E x are the longitudinal moduli of undamaged and damaged laminates, respectively. Now, from the principle of minimum complementary energy, it follows that if the admissible stress system, Eqs. (4.121)–(4.122), with boundary conditions, Eq. (4.124), is introduced into Eq. (4.135) then ~ Å + _ Å + : (4:141) Using Eqs. (4.135) and (4.140) into Eq. (4.141), we obtain 1 2 s 2 c E x0 2Ah ÷ Å / _ 1 2 s 2 c E x 2Ah; (4:142) where П / represents the perturbationincomplementary energy due tocracking. Hence, 1 E x _ 1 E x0 ÷ Å / s 2 c Ah : (4:143) Therefore, the variational boundary value problem results in the following calcu- lus of variations problem: Find f(x) that minimizes П / . 93 4.6 2-D stress analysis: variational methods The complementary energy change П / is the sum of energies brought about by perturbation stresses in the individual laminae, i.e., Å / = 2 _ l ÷l _ t 90 0 W 90 dz dx ÷ 2 _ l ÷l _ h t 90 W 0 dz dx; (4:144) where W 90 and W 0 are stress energy densities due to perturbation stresses in the 90 · - and 0 · -layers, respectively. The stress energy density in a transversely iso- tropic unidirectional fiber composite is given by W = 1 2 s ij e ij = 1 2 s 2 11 E 1 ÷ s 2 22 ÷ s 2 33 _ _ E 2 ÷ 2n 12 s 11 s 22 ÷ s 33 ( ) E 1 ÷ 2n 23 s 22 s 33 E 2 ÷ s 2 23 G 23 ÷ s 2 12 ÷ s 2 13 _ _ G 12 _ _ ; (4:145) where 1 is in the fiber direction and 2, 3 are transverse directions, and the elastic moduli are defined in Eq. (4.48). To determine the perturbation stress energy, we can express stress energy densities in the 90 · - and 0 · -plies in terms of the perturb- ation stresses in the corresponding plies. The stresses in each lamina are given in terms of perturbation stresses by s 22 = Ás 90 xx ; s 23 = Ás 90 xz ; s 33 = Ás 90 zz ; s 11 = s 12 = s 13 = 0 (4:146) for the 90 · -ply, and s 11 = Ás 0 xx ; s 13 = Ás 0 xz ; s 33 = Ás 0 zz ; s 12 = s 22 = s 23 = 0 (4:147) for the 0 · -ply. Thus, the energy densities in the two sets of laminae are W 90 = 1 2 Ás 90 xx _ _ 2 E 2 ÷ Ás 90 zz _ _ 2 E 2 ÷ 2n 23 Ás 90 xx Ás 90 zz E 2 ÷ Ás 90 xz _ _ 2 G 23 _ _ ; W 0 = 1 2 Ás 0 xx _ _ 2 E 1 ÷ Ás 0 zz _ _ 2 E 2 ÷ 2n 12 Ás 0 xx Ás 0 zz E 1 ÷ Ás 0 xz _ _ 2 G 12 _ _ : (4:148) Consider again the laminate region bounded by two transverse planes through adjacent cracks, Figure 4.14. Due to symmetry about z = 0, it is sufficient to take one half of the laminate ÷l _ x _ l, 0 _ z _ h with unit width in y direction. Further, the symmetry across x results into f x ( ) = f ÷x ( ); (4:149) Substituting Eqs. (4.121), (4.122), and (4.148) into Eq. (4.144), and carrying out the integration along z, П / is given by Å / = s 90 xx0 _ _ 2 _ l ÷l t 90 C 00 f 2 ÷ t 3 90 C 02 ff // ÷ t 3 90 C 11 f / 2 ÷ t 5 90 C 22 f // 2 _ _ dx; (4:150) 94 Micro-damage mechanics where C 00 = 1 E 2 ÷ 1 lE 1 ; C 02 = l ÷ 2 3 _ _ u 23 E 2 ÷ l 3 u 12 E 1 ; C 11 = 1 3 1 G 23 ÷ l G 12 _ _ ; C 22 = l ÷ 1 ( ) 3l 2 ÷ 12l ÷ 8 _ _ 1 60E 2 : (4:151) Introducing a nondimensional geometry parameter x = x/t 90 , Eq. (4.150) can be rewritten as Å / = s 90 xx0 _ _ 2 t 2 90 _ r ÷r C 00 f 2 ÷ C 02 f d 2 f dx 2 ÷ C 11 df dx _ _ 2 ÷ C 22 d 2 f dx 2 _ _2 _ _ dx; (4:152) where r = l/t 90 denotes the crack spacing normalized with the cracked ply thickness. The Euler–Lagrange equation for f(x) is d 4 f dx 4 ÷ p d 2 f dx 2 ÷ qf = 0; (4:153) where the coefficients p and q are given by p = C 02 ÷ C 11 C 22 ; q = C 00 C 22 : (4:154) This is a fourth-order ODE whose characteristic equation is r 4 ÷ pr 2 ÷ q = 0 : (4:155) The roots of the characteristic equation are r = ± a 1 ÷ ia 2 ( ); i = ffiffiffiffiffiffiffi ÷1 _ ; a 1 = q 1=4 cos y 2 ; a 2 = q 1=4 sin y 2 ; y = tan ÷1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4q p 2 ÷ 1 ¸ ; (4:156) provided that 4q/p 2 > 1. The general solution to Eq. (4.153) will involve terms with e ±a 1 x cos a 2 x and e ±a 1 x sin a 2 x. However, it is more convenient to use hyperbolic functions instead of exponentials, because of symmetry condition, Eq. (4.149). Since only even product functions are admissible solutions, the solution of the fourth-order ODE, Eq. (4.153), leads to the following expression for f(x) f = A 1 cosh a 1 x cos a 2 x ÷ A 2 sinh a 1 x sin a 2 x; (4:157) where A 1 and A 2 are constants determined from the boundary conditions as A 1 = 2 a 1 cosh a 1 r sin a 2 r ÷ a 2 sinh a 1 rcos a 2 r ( ) a 1 sin 2a 1 r ÷ a 2 sinh 2a 2 r ; A 2 = 2 a 2 cosh a 1 r sin a 2 r ÷ a 1 sinh a 1 rcos a 2 r ( ) a 1 sin 2a 1 r ÷ a 2 sinh 2a 2 r ; (4:158) 95 4.6 2-D stress analysis: variational methods when 4q/p 2 < 1, f(x) is given by [68] f = a 2 / cosh a 1 / x sinh a 1 / r a 2 / coth a 1 / r ÷ a 1 / coth a 2 / r ( ) ÷ a 1 / cosh a 2 / x sinh a 2 / r a 1 / coth a 2 / r ÷ a 2 / coth a 1 / r ( ) ; (4:159) where a / 1 ; a / 2 are defined as a 1 / ; a 2 / = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÷p 2 ± ffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 4 ÷ q _ ¸ ; (4:160) assuming that p < 0, which usually holds good for typical glass/epoxy and graphite epoxy materials [69]. Once f is known, the stress field can be calculated from Eqs. (4.121)–(4.122). The effective stiffness coefficients for the cracked laminate are then determined by calculating average stresses and strains. It is important to note here that variational analysis gives lower bounds of stiffness properties for the cracked laminates, as canbe seenfromthe inequality inEq. (4.143). 4.6.2 Effect of residual stresses As a laminate is cooled from the cure temperature, thermal residual stresses are generated due to differential thermal expansion of the 0 · and 90 · -plies. These residual stresses can be easily incorporated by laminate theory for uncracked laminate. The effect of thermal stresses on crack initiation was investigated in detail by [70]. The variational stress formulation for cracked cross-ply laminate including thermal residual stresses is covered in [69, 71]. The complementary energy analogous to Eq. (4.152) is then given by [69] Å / = Å / 0 ÷ t 2 90 _ r ÷r C 00 c 2 ÷ C 02 c d 2 c dx 2 ÷ C 11 dc dx _ _ 2 ÷ C 22 d 2 c dx 2 _ _2 ÷ 2Áa ÁTc ÷ C 2T d 2 c dx 2 _ _ dx; (4:161) where c = s 90 xx0 ÷ Áa ÁT C 00 _ _ f ÷ Áa ÁT C 00 ; (a) Áa = a 22 ÷ a 11 ; ÁT = T 0 ÷ T ref ; (b) C 2T = a 22 ÁT ÷ n 23 s c E c _ _ 2 3 ÷ l _ _ ÷ a 22 ÁT ÷ n 12 s c E c _ _ l 2 3 ; (c) (4:162) where a 11 and a 22 are the coefficients of thermal expansion in the longitudinal and transverse directions, respectively, T 0 is the service temperature, and T ref is the stress-free (reference/curing) temperature of the laminate. The corresponding Euler–Lagrange equation for this case is 96 Micro-damage mechanics d 4 c dx 4 ÷ p d 2 c dx 2 ÷ qc = Áa ÁT C 22 : (4:163) The solution for f remains the same as in Eqs. (4.158)–(4.160), from which c can be obtainedusingEq. (4.162a). The stresses canthenbe calculatedusingEqs. (4.121)–(4.122). 4.6.3 [0 m /90 n ] s vs. [90 n /0 m ] s laminates The stiffness changes due to cracking in [0 m /90 n ] s laminates are different than in [90 n / 0 m ] s laminates because in the former case we have cracks in an internal 90 · -layer, whereas the cracks in the latter are exposed to the free boundary. For the same area of crack surfaces, the tractions on the crack surfaces (see Eq. (4.128)) do more work in closing (or opening) the crack surfaces in the latter case, and, therefore, the perturbation complementary energy has a higher value. This results in a lower effective axial stiffness as compared to laminates with cracks in internal plies. Nairn [69] carried out a variation of the solution for transverse cracking in [90 n /0 m ] s laminates. The procedure is the same as for [0 m /90 n ] s laminates. The obtained solution for f is also the same, except that some of the constants appearing in Eqs. (4.150) and (4.162) are different. The newconstants are as follows C 02 = 1 ÷ 2 3 l _ _ n 12 E 1 ÷ n 23 3E 2 ; C 22 = l ÷ 1 ( ) 3 ÷ 12l ÷ 8l 2 _ _ 1 60E 2 ; C 2T = 1 3 n 23 s c E c ÷ a 22 ÁT _ _ ÷ n 12 s c E c ÷ a 22 ÁT _ _ l ÷ 2 3 l 2 _ _ : (4:164) A comparison of the stress profiles in [0/90 2 ] s and [90 2 /0] s laminates is illustrated in Figure 4.15(b)–(c). A discussion follows later. 4.6.4 Improved variational analysis Some major improvements toHashin’s variational analysis were suggestedbyVarnaand Berglund [71–73] with the most refined model described in [73]. In Hashin’s approach, the axial stress variation across the thickness is assumed constant, whereas the Varna– Berglund approach determines these by further application of the principle of minimum complementary energy. The stress functions in the 90 · - and 0 · -plies are chosen as È 90 = s 90 xx0 z 2 2 ÷ c x ( ) z 2 2 ÷ A + _ _ _ _ ÷ c 1 x ( )’ 2 z ( ) _ _ t 2 90 ; È 0 = s 0 xx0 z 2 2 ÷ s 90 xx0 ’ 1 z ( )c x ( ) _ _ ÷ s 90 xx0 c 1 x ( )’ 3 z ( ) _ _ t 2 90 ; (4:165) where s 0 xx0 ands 90 xx0 are the axial stresses in the 0 · - and 90 · -plies of uncracked laminate, respectively, x = x=t 90 and z = z=t 90 are nondimensional coordinates, 97 4.6 2-D stress analysis: variational methods and A* is a constant. In the above expressions, the arbitrary function ’ 2 (z) repre- sents nonuniformity in the x-axis stress distribution in the 90 · -layer close to the cracks along z, whereas c 1 (x) characterizes this stress variation along the x-direc- tion. For the 0 · -layer, the redistribution in the stress field is described by c 1 (x) and ’ 3 (z) along the x- and z-directions, respectively. If we neglect ’ 2 (z); ’ 3 (z); c 1 (x) and set A + = ÷h=2t 90 and ’ 1 z ( ) = ÷(h ÷ z) 2 =2t 0 t 90 we revert back to Hashin’s model. For the current model, the stress components in layer m are given by 10 20 30 40 –10 –20 –30 –40 (interfacial) (MPa) xz s 90 xx s 90 x Crack Crack –4 –3 (a) –2 –1 4 3 2 1 50 60 90 xz σ 90 xx σ (interfacial) (interfacial) 90 zz σ 60 (MPa) 40 30 20 10 –40 –30 –20 –10 –3 –4 Crack Crack –2 1 2 3 4 x 50 –1 (b) 70 (MPa) (interfacial) 60 90 xx s 90 zz s (interfacial) 90 xz s 50 Crack Crack –1 –4 4 x 40 30 20 10 –30 –20 –10 –2 –3 3 2 1 (c) Figure 4.15. Profile of stresses between two adjacent ply cracks in a carbon/epoxy cross-ply laminate: (a) using 1-D shear lag analysis for [0/90 2 ] s ; (b) using 2-D variational analysis for [0/90 2 ] s ; and (c) using 2-D variational analysis for a [90 2 /0] s laminate. The normal stress s 90 xx is same for the entire 90 · -ply group, whereas shear stress s 90 xz and transverse stress s 90 zz are plotted at the 0/90 ply interface x = 1. Reprinted, with kind permission, from Damage Mechanics of Composite Materials, J. A. Nairn and S. Hu, Micromechanics of damage: a case study of matrix microcracking, pp. 187–243, copyright Elsevier (1994). 98 Micro-damage mechanics s m xx = @ 2 È m @z 2 s m zz = @ 2 È m @x 2 s m xz = ÷ @ 2 È m @x@z : (4:166) Thus, (a) in the 90 · -layer s 90 xx = s 90 xx0 1 ÷ c x ( ) ÷ c 1 x ( )’ // 2 z ( ) [ [; s 90 xz = s 90 xx0 ÷c / x ( )z ÷ c / 1 x ( )’ / 2 z ( ) [ [; s 90 zz = s 90 xx0 c // x ( ) z 2 2 ÷ A + _ _ ÷ c // 1 x ( )’ 2 z ( ) _ _ ; (4:167) (b) in the 0 · -layer s 0 xx = s 0 xx0 ÷ s 90 xx0 c x ( )’ // 1 z ( ) ÷ c 1 x ( )’ // 3 z ( ) [ [; s 0 xz = s 90 xx0 c / x ( )’ / 1 z ( ) ÷ c / 1 x ( )’ / 3 z ( ) [ [; s 0 zz = s 90 xx0 ÷c // x ( )’ 1 z ( ) ÷ c // 1 x ( )’ 3 z ( ) [ [ : (4:168) The boundary and interface conditions remain the same as given in Eq. (4.120), which, after using Eqs. (4.167) and (4.168), results in ’ / 1 1 ( ) = 0; ’ 1 1 ( ) = A + ÷ 1 2 ; ’ / 1 h ( ) = ’ 1 h ( ) = 0; ’ / 3 h ( ) = ’ 3 h ( ) = 0; ’ / 2 1 ( ) = ’ / 3 1 ( ) = 1; ’ 2 1 ( ) = ’ 3 1 ( ); c ±r ( ) = 1; c / ±r ( ) = c / 1 ±r ( ) = c 1 ±r ( ) = 0; (4:169) where h = h=t 90 . Clearly, this model is complex and requires determining the constant A*, and the functions c x ( ); c 1 x ( ); and ’ 1 z ( ); ’ 2 z ( )’ 3 z ( ). However, the authors found that the following choice of functions ’ 1 z ( ); ’ 2 z ( ); ’ 3 z ( ) is useful ’ 1 z ( ) = 1 ÷ cosh Á 1 z ÷ h ( ) Á 1 sinh Á 1 l ; ’ 2 z ( ) = A ÷ z 2n 2n ; ’ 3 z ( ) = 1 ÷ cosh Á 3 z ÷ h ( ) Á 3 sinh Á 3 l ; (4:170) with A a constant and D 1 , D 3 , and n arbitrary shape parameters, where n is an integer. Using the boundary conditions described in Eq. (4.169), we obtain A + = ÷ 1 2 ÷ 1 ÷ cosh Á 1 l Á 1 sinh Á 1 l ; A = ÷ 1 2n ÷ 1 ÷ cosh Á 3 l Á 3 sinh Á 3 l : (4:171) Now, the total complementary energy of the cracked laminate system is given by Å / = Å / 0 ÷ _ r ÷r v c; c / ; c // ; c 1 ; c / 1 ; c // 1 ) d x; ( (4:172) 99 4.6 2-D stress analysis: variational methods where П / 0 is the complementary energy of the virgin laminate, and v is the comple- mentary energy density due to perturbation stresses. Using the minimization procedure as described above, we finally obtain the following system of ordinary differential equations with constant coefficients C 0 22 c /// ÷ C 0 02 ÷ C 0 11 _ _ c // ÷ C 0 00 c ÷ C 01 22 c /// 1 ÷ C R c // 1 ÷ C 01 00 c 1 = 0; ÷ C 01 22 c /// ÷ C R c // ÷ C 01 00 c ÷ C 1 22 c /// 1 ÷ C 1 02 ÷ C 1 11 _ _ c // 1 ÷ C 1 00 c 1 = 0; (4:173) with the boundary conditions as in Eq. (4.169) and constants given by C 0 00 = 1 ÷ E 2 E 1 I 1 ; C 0 22 = 1 20 ÷ A + 3 ÷ A + 2 ÷ I 2 ; C 0 11 = E 2 3G 23 ÷ E 2 G 12 I 3 ; C 0 02 = ÷2n 23 A + ÷ 1 6 _ _ ÷ 2n 12 E 2 E 1 I 4 ; C 1 00 = I T 1 ÷ E 2 E 1 I L 1 ; C 1 22 = I T 2 ÷ I L 2 ; C 1 11 = E 2 G 23 I T 3 ÷ E 2 G 12 I L 3 ; C 1 02 = ÷2n 23 I T 4 ÷ ÷2n 12 E 2 E 1 I L 4 ; C 01 00 = 1 ÷ E 2 E 1 I C 1 ; C 01 22 = F 3 ÷ I C 2 ; C 01 11 = E 2 G 23 F 1 ÷ E 2 G 12 I C 3 ; C 01 02 = ÷2n 23 F 2 ÷ 2n 12 E 2 E 1 I C2 4 ; C 01 20 = ÷2n 23 F 4 ÷ 2n 12 E 2 E 1 I C1 4 ; C R = 1 2 C 01 20 ÷ C 01 02 _ _ ÷ C 01 11 ; (4:174) where I 1 = _ h 1 ’ // 1 z ( ) [ [ 2 dz; I 2 = _ h 1 ’ 1 z ( ) [ [ 2 dz; I 3 = _ h 1 ’ / 1 z ( ) [ [ 2 dz; I 4 = _ h 1 ’ 1 z ( )’ // 1 z ( )dz; I T 1 = _ 1 0 ’ // 2 z ( ) [ [ 2 dz; I T 2 = _ 1 0 ’ 2 z ( ) [ [ 2 dz; I T 3 = _ 1 0 ’ / 2 z ( ) [ [ 2 dz; I T 4 = _ 1 0 ’ 2 z ( )’ // 2 z ( )dz; F 1 = _ 1 0 z’ 2 z ( )dz; F 2 = _ 1 0 ’ 2 z ( )dz; F 3 = _ 1 0 ’ 2 z ( ) z 2 2 ÷ A + _ _ dz; F 4 = 1 2 ÷ A + ÷ ’ 2 1 ( ) ÷ F 2 ; I L 1 = _ h 1 ’ // 3 z ( ) [ [ 2 dz; I L 2 = _ h 1 ’ 3 z ( ) [ [ 2 dz; I L 3 = _ h 1 ’ / 3 z ( ) [ [ 2 dz; I L 4 = _ h 1 ’ 3 z ( )’ // 3 z ( )dz; I C 1 = _ h 1 ’ 1 // z ( )’ // 3 z ( )dz; I C 2 = _ h 1 ’ 1 z ( )’ 3 z ( )dz; I C 3 = _ h 1 ’ 1 / z ( )’ / 3 z ( )dz; I C1 4 = _ h 1 ’ 1 z ( )’ // 3 z ( )dz; I C2 4 = _ h 1 ’ // 1 z ( )’ 3 z ( )dz: (4:175) 100 Micro-damage mechanics The minimum value of the complementary energy corresponding to the obtained solution is given by [73] as Å / min = Å / 0 ÷ s 90 xx0 t 90 _ _ 2 2E 2 C 01 22 É /// 1 r ( ) ÷ C 0 22 É /// r ( )[ : _ (4:176) The stresses resulting from cracking can be calculated by solving the system of coupled ODEs in Eq. (4.173) to get the perturbation functions, and then putting them into Eqs. (4.167)–(4.168). The average of the stresses will yield the average stiffness properties of the cracked laminate. For this model, the longitudinal Young’s modulus normalized with its virgin state value is given by [73] E x E x0 = 1 1 ÷ E 2 lE 1 f r ( ) r ; (4:177) where f r ( ) = 1 2 _ ÷r ÷r É x ( ) ÷ É 1 x ( ) [ [ d x : (4:178) The normalized effective Poisson’s ratio for the cracked cross-ply laminate is given by n xy n 0 xy = 1 ÷ 1 ÷ E 2 E 1 _ _ t 90 h f r ( ) r 1 ÷ E 2 lE 1 f r ( ) r : (4:179) 4.6.5 Related works There have been some further developments based on the variational analysis procedure. The analysis of transverse cracks in cross-ply laminates undergoing shear loading was covered by Hashin in his original paper [67]. Later he utilized the variational approach to predict the thermal expansion coefficients for the cracked cross-ply laminate [74]; this is covered later in Section 4.11. In another paper [75], he also analyzed the case of orthogonally cracked cross-ply lamin- ates. Another significant analysis of cross-ply laminates was carried out by Kuriakose and Talreja [76] where they analyzed cross-ply laminates subjected to bending moments. 4.6.6 Comparison between 1-D and 2-D stress-based models Comparison of stress predictions between 1-D shear lag [27] and 2-D variational analysis [67] for a [0/90 2 ] s carbon epoxy laminate are shown in 101 4.6 2-D stress analysis: variational methods Figure 4.15(a)–(b). The laminate is subjected to an applied mechanical stress of 100 MPa, and a temperature change of DT = –125 · C. The normalized crack spacing is r = 4. As seen in Figure 4.15(b), s 90 xx = 0 at x= ± r as the crack surfaces are traction free and the transverse plies carry no axial load at the crack planes. As we move away from the crack surfaces, s 90 xx increases as stress is transferred back into the 90 · -layer from the 0 · -layer. This stress reaches a maximum midway between two transverse cracks and its value depends on the crack spacing. When the cracks are sufficiently far apart (i.e., no interaction between adjacent cracks), s 90 xx will reach s 90 xx0 at this position (x = 0). The distributions of s 90 xx are qualitatively similar for shear lag and variational models. However, the maximum value of s 90 xx is different in the two cases. Also, variational predictions correctly show that this stress has zero slope at crack surfaces (x = ± r), while the shear lag models display a nonzero slope. Shear lag analysis predicts a maximum value for the interfacial shear stress t i = s 90 xz z = t 90 ( ) at the crack surfaces, in violation of the boundary conditions that require t i = 0 at crack planes (Figure 4.15(a)). In addition, it cannot provide the transverse normal stress s 90 zz . The one-dimensional nature of the shear lag models results in no distinction between [0/90] s and [90/0] s laminates, while experimental observations show differences in the cracking behavior and resultant stiffness degradation in the two cases. These differences are appropriately predicted by the variational solution shown for a [90 2 /0] s laminate in Figure 4.15(c), where the stress variations shown are different from the [0/90 2 ] s case. The interfacial trans- verse stress s 90 zz near crack surfaces is compressive in [0/90 2 ] s while it is tensile in [90 2 /0] s laminate. This suggests that cracks in a [90 2 /0] s laminate will tend to promote mode I delamination. In an alternative way, Varna and coauthors [77–79] investigated shear lag and variational methods from the viewpoint of average crack opening displacement (COD). They noted that the basic difference between different models is how they model the average COD, which is defined as u = 1 t 90 _ t 90 0 u z ( ) dz : (4:180) Consider transverse cracking in a symmetric and balanced [S/90 n ] s laminate, consisting of two balanced sublaminates as outer layers and 90 · layers in the middle. The longitudinal modulus and Poisson’s ratio for the laminate are given by E xx = s 0 e S xx ; n xy = ÷ e S yy e S xx ; (4:181) where s 0 is the applied stress, the overbars denote volume averages, and superscript S denotes sublaminate. The average COD for this laminate is given by 102 Micro-damage mechanics u = l e S xx ÷e 90 xx _ _ : (4:182) Following [79], the stiffness properties for the cracked laminate can be shown to be given by E xx E xx0 = 1 1 ÷ arR l _ _ ; u xy u xy0 = 1 ÷ crR l _ _ 1 ÷ arR l _ _ ; (4:183) where r = 1/2l is the crack density; a, c, and g are known functions of laminate material and thickness; R l ( ) is the average stress perturbation function; and l = l=t 90 is the half-crack spacing normalized with the 90 · -layer thickness. Different approaches model R l ( ) differently: 1. Shear lag models: R l ( ) = 2 b tanh b l ( ); (4:184) where b is the shear lag parameter. Different shear lag models use different definitions for this parameter [78]. 2. Variational methods: R l ( ) = 4a 1 a 2 a 2 1 ÷ a 2 2 cosh 2a 1 l ( ) ÷ cos 2a 2 l ( ) a 2 sinh 2a 1 l ( ) ÷ a 1 sin 2a 2 l ( ) ; (4:185) where the constants a 1 , a 2 are defined as per model. The comparisons of different models for [S/90 4 ] s laminates, as performed by Joffe and Varna [79] are shown in Figures 4.16–4.19 for different values of y. For the 2-D-0 model, the expression for the stress perturbation function is the same as for Hashin’s variational analysis, except that the coefficients are now given as [79] C 00 = 1 E 2 ÷ 1 E S x I 1 ; C 02 = ÷2 n 32 E 2 1 6 ÷ A + _ _ ÷ 2 n S xz E S x I 4 ; C 11 = 1 2G 23 ÷ 1 G S xz I 3 : (4:186) This comparison confirms that variational analyses consistently provide a lower bound to stiffness properties and are significantly better than shear lag models. The variational analysis predictions are also very close to FE predictions. 103 4.6 2-D stress analysis: variational methods 4.7 Generalized plain strain analysis – McCartney’s model These models may be seen as a further development of stress analysis-based damage models for cracked laminates. The cracked cross-ply laminates, symmetric about the laminate mid-plane, were considered in a generalized plane strain condition, thereby reducing the directional dependence of essentially a 3-D stress field in the cracked laminate. Assuming a regular array of fully grown parallel cracks in 90 · -ply, McCartney [80] developed a theory of stress transfer between 0 · - and 90 · -plies while retaining all relevant stress and strain components. The equilibrium equations, as in Hashin’s formulation, are given by Eq. (4.112). Assuming lamina properties to be transversely isotropic (or orthotropic), the 1.0 (a) 0.9 0.8 E x / E x 0 0.7 0.6 0.5 0.4 0.0 0.1 Sh.L.-2 [0 2 /90 4 ] s Hashin FE 2D-0 Experiment 0.2 0.3 Crack density (cr/mm) 0.4 0.5 0.6 0.7 1.0 (b) [0 2 /90 4 ] s 0.9 0.8 0.7 V x y / V x y 0 0.6 0.5 0.4 0.0 0.1 0.2 0.3 Crack density (cr/mm) 0.4 0.5 Sh.L.-2 Hashin FE 2D-0 Experiment 0.6 0.7 Figure 4.16. Variation of elastic stiffness moduli with crack density for a [0 2 /90 4 ] s glass/epoxy laminate: (a) normalized Young’s modulus; (b) normalized Poisson’s ratio versus crack density. Reprinted, with kind permission, from Compos Sci Technol, Vol. 59, R. Joffe and J. Varna, Analytical modeling of stiffness reduction in symmetric and balanced laminates due to cracks in 90 degrees layers, pp. 1641–52, copyright Elsevier (1999). 104 Micro-damage mechanics thermomechanical constitutive relations for 0 · - and 90 · -plies in material (lamina) coordinate system can be written as e m 11 = s m 11 E m 11 ÷ n m 21 s m 22 E m 22 ÷ n m 31 s m 33 E m 33 ÷ a m 11 ÁT; e m 22 = ÷ n m 12 s m 11 E m 11 ÷ s m 22 E m 22 ÷ n m 32 s m 33 E m 33 ÷ a m 22 ÁT; (4:187) e m 33 = ÷ n m 13 s m 11 E m 11 ÷ n m 23 s m 22 E m 22 ÷ s m 33 E m 33 ÷ a m 33 ÁT; e m 12 = s m 12 2G m 12 ; e m 13 = s m 13 2G m 13 ; e m 23 = s m 23 2G m 23 ; 1.0 (a) 0.9 0.8 0.7 E x / E x 0 0.6 0.5 0.4 0.0 0.1 0.2 Sh.L.-2 Hashin FE 2D-0 [ + 15/90 4 ] s Experiment 0.3 Crack density (cr/mm) 0.4 0.5 0.6 0.7 _ 1.0 (b) 0.9 0.8 0.7 V x y / V x y 0 0.6 0.5 0.4 0.0 0.1 Sh.L.-2 Hashin FE 2D-0 Experiment 0.2 0.3 [±15/90 4 ] s Crack density (cr/mm) 0.4 0.5 0.6 0.7 Figure 4.17. Variation of elastic stiffness moduli with crack density for a [±15/90 4 ] s glass/ epoxy laminate: (a) normalized Young’s modulus; (b) normalized Poisson’s ratio versus crack density. Reprinted, with kind permission, from Compos Sci Technol, Vol. 59, R. Joffe and J. Varna, Analytical modeling of stiffness reduction in symmetric and balanced laminates due to cracks in 90 degrees layers, pp. 1641–52, copyright Elsevier (1999). 105 4.7 Generalized plain strain analysis – McCartney’s model where m = 0, 90 for 0 · -ply and 90 · -ply, respectively, and E ii , G ij , n ij ; i, j = 1, 2, 3 represent the Young’s modulus, shear modulus, and Poisson’s ratio, respectively, in corresponding directions and planes. Following the plane strain assumption, the displacement field for the damaged laminate has the following form u m = u m (x; z); n m = e c T y; w m = w m x; z ( ); (4:188) where e c T is the uniform strain in the cracked laminate along the transverse (y) direction. This representation of the displacement field is valid if ply cracks form in planes normal to the x-direction, and are well away from the laminate edges. 1.0 (a) 0.9 0.8 0.7 0.6 0.5 0.4 0.0 0.1 0.2 0.3 Shear lag model: [±30/90 4 ] s Crack density (cr/mm) 0.4 0.5 1 3 2 0.6 0.7 E x / E x 0 Experiment 1.0 0.9 (b) 0.8 0.7 V x y / V x y 0 0.6 0.5 0.4 0.0 0.1 0.2 0.3 0.4 Crack density (cr/mm) 0.5 0.6 0.7 Sh.L.-2 Hashin FE 2D-0 Experiment [±30/90 4 ] s Figure 4.18. Variation of elastic stiffness moduli with crack density for a [±30/90 4 ] s glass/ epoxy laminate: (a) normalized Young’s modulus; (b) normalized Poisson’s ratio versus crack density. Reprinted, with kind permission, from Compos Sci Technol, Vol. 59, R. Joffe and J. Varna, Analytical modeling of stiffness reduction in symmetric and balanced laminates due to cracks in 90 degrees layers, pp. 1641–52, copyright Elsevier (1999). 106 Micro-damage mechanics Now, if we additionally assume that the longitudinal stress components in both the 0 · - and 90 · -plies are independent of the transverse coordinate z, the relevant stress components can be written in the following functional form s 0 xx = s 0 xx0 ÷ C x ( ); s 90 xx = s 90 xx0 ÷ lC x ( ); (4:189) where s 0 xx0 = Q 0 11 e c ÷ n 0 12 E 0 22 E 0 11 ÷ a / 0 11 ÁT _ _ ; s 90 xx0 = Q 90 11 e c ÷ n 0 12 e c T ÷ a / 90 22 ÁT _ _ ; (4:190) where e c is the average longitudinal strain applied to the laminate, and Q 0 11 and Q 90 11 are the reduced stiffnesses for the 0 · - and 90 · -plies, respectively, given by (a) 1.0 0.9 0.8 0.7 E x / E x 0 0.6 0.5 0.4 0.0 0.1 0.2 0.3 0.4 Sh.L.-2 Hashin 2D-0 Experiment FE Crack density (cr/mm) 0.5 0.6 0.7 [±40/90 4 ] s 1.0 (b) 0.9 0.8 0.7 0.6 0.5 0.4 0.0 0.1 0.2 0.3 Crack density (cr/mm) 0.4 0.5 0.6 0.7 Sh.L.-2 Hashin FE 2D-0 Experiment [±40/90 4 ] s V x y / V x y 0 Figure 4.19. Variation of elastic stiffness moduli with crack density for a [±40/90 4 ] s glass/ epoxy laminate: (a) normalized Young’s modulus; (b) normalized Poisson’s ratio versus crack density. Reprinted, with kind permission, from Compos Sci Technol, Vol. 59, R. Joffe and J. Varna, Analytical modeling of stiffness reduction in symmetric and balanced laminates due to cracks in 90 degrees layers, pp. 1641–52, copyright Elsevier (1999). 107 4.7 Generalized plain strain analysis – McCartney’s model Q m 11 = E m 11 1 ÷ n m 12 n m 21 ; m = 0; 90 (4:191) and a / 0 11 and a / 90 22 are given as a / 0 11 = a 0 11 ÷ n 0 12 E 0 22 E 0 11 a 0 22 ; a / 90 22 = a 90 11 ÷ n 0 12 a 90 11 : (4:192) To fully represent the stress state in the cracked laminate, the basic task is to determine the unknown function C(x) appearing in Eq. (4.189). The boundary conditions for the stress field are: symmetry about the mid-plane, symmetry about the plane parallel to cracks lying midway between two adjacent cracks, stress continuity at the interface between the 0 · - and 90 · -plies, and zero traction at crack surfaces and the external surfaces of the outer plies (similar to Hashin’s analysis, see Eq. (4.120)). Additionally, by force balance we have s c h = _ t 90 0 s 90 xx dz ÷ _ h t 90 s 0 xx dz : (4:193) Also, since there is no load applied on the laminate in the transverse direction, the average transverse stress must add to zero, i.e., _ t 90 0 _ l 0 s 90 yy dxdz ÷ _ h t 90 _ l 0 s 90 yy dxdz = 0 : (4:194) The stress field in the cracked laminate can be obtained by substituting Eq. (4.189) into the equilibrium equations, Eq. (4.112), and integrating and applying bound- ary conditions. Thus, the transverse normal and shear stresses in the laminate are given by s 0 xz = C / x ( ) 1 ÷ l ( )t 90 ÷ z [ [; s 90 xz = lC / x ( )z; s 0 zz = 1 2 C // x ( ) 1 ÷ l ( )t 90 ÷ z [ [ 2 ; s 90 zz = 1 2 C // x ( ) 1 ÷ l ( )t 2 90 ÷ z 2 _ ¸ : (4:195) It can be observed that these relations are quite similar in form to those obtained by Hashin (Eqs. (4.121) and (4.122)). In fact, they are identical when C(x) = (1=l)s 90 xx f(x). The complete stress field is given by combining Eqs. (4.189) and (4.195). The displacement field, given by Eq. (4.188), can be obtained by using the stress field into constitutive relations, Eq. (4.187), and integrating. However, the stress–strain relations must be satisfied in an average sense. Hence, the stress and strain fields are averaged over each ply thickness. These averaged relations lead to the following fourth-order differential equation for the unknown func- tion C(x) FC IV x ( ) ÷ GC // x ( ) ÷ HC x ( ) = 0; (4:196) where the coefficients F, G, H are given by 108 Micro-damage mechanics F = 1 20E 0 22 ÷ 2 15E 90 22 t 90 t 0 t 90 t 0 _ _ 2 ÷ 5 2 t 90 t 0 ÷ 15 8 _ _ ; G = 1 3 1 G 0 12 ÷ 1 G 0 23 t 90 t 0 ÷ n 0 12 E 0 11 ÷ n 90 23 E 90 22 2 t 90 t 0 ÷ 3 _ _ _ _ ; H = 1 E 0 11 ÷ t 90 t 0 1 E 0 22 : (4:197) When the inner and outer plies are made of the same material the differential equation, Eq. (4.196), has the same form as the Euler–Lagrange equation derived by Hashin. It is worth noting that the Reissner energy functional has a stationary value when the constitutive relations and equilibrium equations are satisfied in an average sense. Thus, McCartney’s approach is quite analogous to Hashin’s approach: the former uses a displacement formulation (equivalent to minimiza- tion of Reissner’s energy functional), while the latter uses a stress formulation (equivalent to minimization of complementary energy functional). For a cracked cross-ply laminate, McCartney’s 2-D approach yields the following relations for the effective longitudinal modulus E xx = E xx0 1 ÷ t 90 l E 0 22 E 0 11 È ; (4:198) where È = 2Ãpq p 2 ÷ q 2 ÷ cosh 2 pl t 0 ÷ cosh 2 ql t 0 _ _ ; 1 à = q sinh pl t 0 cosh pl t 0 ÷ p sin ql t 0 cos ql t 0 ; p = ffiffiffiffiffiffiffiffiffiffi r ÷ s 2 _ ; q = ffiffiffiffiffiffiffiffiffiffiffiffiffi r ÷ s [ [ 2 _ ; r = G 2F > 0; s = ffiffiffiffi H F _ > 0 : (4:199) If the plies are thick, the averaging of stresses andstrains is not advisable as it canresult in inaccurate predictions. To improve the approach, McCartney [81–86] used the so- called “ply-refinement” procedure, in which each ply is divided thickness-wise into N segments, and the resulting stress–strain relations are satisfied, in an average sense, over each ply segment. Consequently, the unknown functions C i (x) for the ith ply segment (i = 1,2, . . . , N), satisfy the following N homogeneous differential equations N i=1 F ij C IV i x ( )÷ N i=1 G ij C // i x ( ) ÷ N i=1 H ij C i x ( ) = 0; j = 1; 2; : : : ; N; (4:200) where the coefficients F ij , G ij , and H ij are calculated numerically using a suitable ODE solver. McCartney [82, 84, 86] has later generalized this approach for multi- layered cross-ply laminates, and also for triaxial loading. 109 4.7 Generalized plain strain analysis – McCartney’s model McCartney [82, 84, 86] has also shown that for cracked cross-ply laminates, the moduli are interrelated. Similar interrelationships among the Poisson’s ratio, transverse modulus, and longitudinal modulus have also been independently derived by Nairn [87]. If we define a damage parameter or normalized stiffness parameter as D = 1 E xx ÷ 1 E xx0 ; (4:201) then other moduli for the cracked cross-ply laminate are interrelated as n xy0 E xx0 ÷ n xy E xx = k 1 D; 1 E yy ÷ 1 E yy0 = k 2 1 D; n xz0 E xx0 ÷ n xz E xx = k 2 D; 1 E zz ÷ 1 E zz0 = k 2 2 D; n yz0 E yy0 ÷ n yz E yy = k 1 k 2 D; (4:202) and a xx ÷ a xx0 = k 3 D; a yy ÷ a yy0 = k 1 k 3 D; a zz ÷ a zz0 = k 2 k 3 D ; (4:203) where k 1 = E xx0 E yy0 B ÷ n xy0 E yy0 =E xx0 _ _ 1 ÷ n xy0 B ; k 2 = E xx0 A ÷ n xz0 ÷ n xz0 E xx0 =E yy0 _ _ B 1 ÷ n xy0 B ; k 3 = E xx0 s xx0 ÷ Ba yy0 ÷ C _ ¸ 1 ÷ n xy0 B ; (4:204) with A = n 13 E 22 ÷ n 23 n 12 E 11 ; B = n 12 ; C = a 22 ÷ n 12 a 11 : (4:205) These interrelationships can in principle reduce the burden of evaluating change in each modulus subsequent to matrix cracking. Instead, the evaluation of degrad- ation in longitudinal modulus is sufficient for predicting other moduli. However, this result has yet to be verified experimentally. 4.8 COD-based methods One way to view the elastic response changes caused by the presence of cracks in a medium is to consider the additional overall (global) strain of the RVE contrib- uted by the crack surface displacements of the individual cracks within the RVE. Contrarily, if none of the cracks in the RVE conducts surface displacements, the overall elastic response of the RVE will not change. This observation motivates the focus on crack surface displacements. Although generally these displacements 110 Micro-damage mechanics can be described as crack opening displacement (COD) and crack sliding displace- ment (CSD), one commonly refers only to COD. We will in the following discuss stiffness relations in terms of average COD that can be calculated either analytic- ally [88–96] or numerically [97–101]. 4.8.1 3-D laminate theory: Gudmundson’s model Consider a general three-dimensional laminate consisting of N plies (see Figure 4.20). Each ply is defined by its material properties, layup angle, and thickness. For uncracked laminate the global average stresses s ij and strains e ij are defined as [102] s ij = N k=1 V k s k ij ; e ij = N k=1 V k e k ij ; (4:206) where s k ij ; e k ij are the average stresses and strains, respectively, in the kth ply (k = 1, 2, . . ., N), and V k stands for the volume fraction of the kth ply such that N k=1 V k = 1. Note that for uncracked laminates the ply average stresses and strains are the same as the individual (constant) ply stresses. For the sake of simplicity, we will denote the tensors with a ~. Now, let us partition the stresses, strains, and thermal expansion coefficients into in-plane and out-of-plane parts s ~ = s ~ I s ~ O _ _ ; e ~ = e ~ I e ~ O _ _ ; s ~ = s ~ I s ~ O _ _ ; (4:207) where s ~ I = s xx s yy s xy _ _ _ _ ; e ~ I = e xx e yy 2e xy _ _ _ _ ; a ~ I = a xx a yy 2a xy _ _ _ _ ; (4:208) denote the in-plane stresses, strains, and thermal expansion coefficients, and s ~ O = s zz s xz s yz _ _ _ _ ; e ~ O = e zz 2e xz 2e yz _ _ _ _ ; a ~ O = a zz 2a xz 2a yz _ _ _ _ ; (4:209) k k+1 t k l k l k Figure 4.20. A general 3-D laminate with cracks in some off-axis layers. 111 4.8 COD-based methods denote the out-of-plane stresses, strains, and thermal expansion coefficients. The consti- tutive law between global average stress tensor and average strain tensor is given by e ~ = S ~ s ~ ÷ a ~ ÁT = e ~ I e ~ O _ _ = S ~ II S ~ IO S ~ IO _ _ T S ~ OO _ ¸ _ _ ¸ _ s ~ I s ~ O _ _ ÷ a ~ I a ~ O _ _ ÁT; (4:210) where S ~ is the compliance tensor, and the superscript T represents the trans- pose. Similarly, the relation between ply stress tensor and ply strain tensor can be written as e ~ k = S k ~ s k ~ ÷s k(r) ~ _ _ ÷ a ~ ÁT = e ~ I k e ~ O k _ _ _ _ = S ~ I k S ~ IO k S ~ IO k _ _ T S ~ OO k _ ¸ _ _ ¸ _ s ~ I k ÷ s ~ I k(r) s ~ O k _ _ _ _ ÷ s ~ I k s ~ O k _ _ _ _ ÁT; (4:211) where s ~ k(r) denote residual stresses present due to reasons other than thermal mismatch, such as chemical shrinkage during the manufacturing process. The global average in-plane and out-of-plane residual stresses vanish due to equilib- rium. The compatibility and equilibrium conditions in the laminate give us e ~ I k = e ~ I ; s ~ O k = s ~ O : (4:212) Substituting Eq. (4.212) into Eq. (4.210), one obtains s ~ I k = S ~ II k _ _ ÷1 e ~ I ÷ S ~ IO k s ~ O ÷ a ~ I k ÁT _ _ ÷ s ~ I k(r) : (4:213) Using Eq. (4.206), and using the fact that the volume average of residual stresses vanish, the above equation can be rewritten as e ~ I = S ~ II s ~ I ÷ S ~ IO s ~ O ÷ a ~ I ÁT; (4:214) where S ~ II = N k=1 V k S ~ II k _ _ ÷1 _ _ ÷1 ; S ~ IO = S ~ II N k=1 V k S ~ II k _ _ ÷1 S ~ IO k _ _ ; a ~ I = S ~ II N k=1 V k S ~ II k _ _ ÷1 a ~ I k _ _ : (4:215) 112 Micro-damage mechanics In a similar way, substituting Eq. (4.212) into Eq. (4.211), and applying Eq. (4.206), we get e ~ O = S ~ IO _ _ T s ~ I ÷ S ~ OO s ~ O ÷ a ~ O ÁT; (4:216) where S ~ OO = S ~ IO _ _ T S ~ II _ _ ÷1 S ~ IO ÷ N k=1 V k S ~ OO k ÷ S ~ IO k _ _ T S ~ II k _ _ ÷1 S ~ IO k _ _ ; a ~ O = S ~ IO _ _ T S ~ II _ _ ÷1 a ~ I ÷ N k=1 V k a ~ O k ÷ S ~ IO k _ _ T S ~ k II ) ÷1 a ~ I k _ _ : _ (4:217) Eqs. (4.215) and (4.217) relate the effective laminate properties to the local ply level properties. The above equations form the basis of a 3-D laminate theory. The corresponding relations for a 2-D laminate theory are obtained by considering only the in-plane tensors. To obtain the effective laminate properties for a cracked laminate, define the nondimensional crack density in the kth ply as r k = t k l k ; (4:218) where t k and l k denote the thickness and average crack spacing, respectively, in the kth ply. The process of transverse cracking reduces the elastic energy of the laminate and the change in elastic energy is associated with the release of tractions on the crack surfaces. It was shown by Gudmundson and Ostlund [92] that the coupling terms between the energy of the uncracked laminate and the change in elastic energy due to matrix cracking vanish (see also the proof by Hashin [67], which is described earlier in the chapter, Eq. (4.135)). It is noted that for a cracked laminate, the effective strains are different from the average strains, whereas there is no distinction between global effective and average stresses. The effective strains are the strains measured on a global scale, whereas average strains come from averaging strains in individual plies over the RVE. The difference between the effective strains and average strains is equal to strain increment caused by crack opening displacements. Thus, the average stresses are defined as (see Eq. (4.206)) s (a) ij = N k=1 V k s k(a) ij ; (4:219) where the superscript (a) denotes average variables. The global effective strains are defined as e (e) ij = 1 2V _ À out u i n j ÷ u j n i _ _ dÀ; (4:220) 113 4.8 COD-based methods where u i , i = 1, 2, 3 indicates the displacement vector, n i is the unit normal vector on the outer boundary surface G out of the representative volume V, and the superscript (e) denotes the effective variables. In the same way the effective ply strains can be defined as [102] e k(e) ij = 1 2V k _ À kout u k i n k j ÷ u k j n k i _ _ dÀ; (4:221) where V k is the volume of the ply k and the surface integral is performed on the outer boundary of ply k. Obviously, 1 2V _ À out u i n j ÷ u j n i _ _ dÀ = N k=1 V k 1 2V k _ À kout u k i n k j ÷ u k j n k i _ _ dÀ _ _ : (4:222) Hence, from Eqs. (4.220)–(4.222), one gets e (e) ij = N k=1 V k e k(e) ij : (4:223) Applying the divergence theorem on Eq. (4.221), we get e k(e) ij = 1 2V k _ V k u k i;j ÷ u k j;i _ _ dÀ ÷ 1 2V k _ À kc u k i n k j ÷ u k j n k i _ _ dÀ; (4:224) where the first integral is equal to the average ply strain, given by e (a) ij = 1 2V k _ V k u k i;j ÷ u k j;i _ _ dÀ; (4:225) and the second integral is the strain increment due to matrix cracks Áe k ij is given by Áe k ij = ÷ 1 2V k _ À kc u k i n k j ÷ u k j n k i _ _ dÀ = r k 2t k Á u k i n k j ÷ Á u k i n k j _ _ ; (4:226) where Á u k i = 1 t k _ t k 0 u k(÷) i ÷ u k(÷) i _ _ dt k = 1 t k _ t k 0 Á u k i dt k (4:227) is the average crack opening displacement for ply k. Thus, we can write effective ply strains as e k(e) ij = e k(a) ij ÷ Áe k ij : (4:228) Finally, after applying the proper boundary conditions, Gudmundson and Zang [89] arrive at the following expressions for effective properties of cracked laminate 114 Micro-damage mechanics S ~ II(c) = S ~ II _ _ ÷1 ÷ N k=1 n k r k A ~ k _ _ T N i=1 b ~ ki A ~ k _ _ ÷1 ; S ~ IO(c) =S ~ II(c) S ~ II _ _ ÷1 S ~ IO ÷ N k=1 n k r k A ~ k _ _ T N i=1 b ~ ki B ~ i _ _ ÷1 ; S ~ OO(c) = S ~ IO(c) _ _ T S ~ II(c) _ _ ÷1 S ~ IO(c) ÷ S ~ IO _ _ T S ~ II _ _ ÷1 S ~ IO ÷S ~ OO ÷ N k=1 n k r k B k ~ _ _ T N i=1 b ~ ki B ~ i ; a ~ I(c) =a ~ I ÷S ~ II(c) N k=1 n k r k A ~ k _ _ T N i=1 b ~ ki C ~ i ; a ~ O(c) =a ~ O ÷ S ~ IO(c) _ _ T S ~ II(c) _ _ ÷1 a ~ I(c) ÷a ~ I _ _ ÷ N k=1 n k r k B ~ k _ _ T N i=1 b ~ ki C ~ i ; (4:229) where matrices S ~ II ; S ~ IO ; S ~ OO ; S ~ II ; a ~ I ; and a ~ O are given in Eqs. (4.215) and (4.217), b ~ is the matrix containing average crack opening displacements (COD), and matrices A ~ ; B ~ ; C ~ are given by A ~ k = N ~ I k S ~ II k _ _ ÷1 ; B ~ k = N ~ O k ÷ N ~ I k S ~ k II ) ÷1 S ~ k IO ; C ~ k = A ~ k a ~ I ÷ a ~ k _ _ ; _ (4:230) where the matrices N ~ k I and N ~ k O represent the unit normal vector on the crack surfaces in ply k, i.e., N ~ I k = n k 1 0 n k 2 0 n k 2 n k 1 0 0 0 _ _ _ _ N ~ k O = 0 0 0 0 0 0 0 n k 1 n k 2 _ _ _ _ : (4:231) The expressions in Eq. (4.229) for stiffness of the cracked laminates are exact assuming the homogenization procedure. However, the main problem is the determination of average COD for the cracked laminate. The heterogeneity in the composite laminates and constraint effects on crack surfaces from the sur- rounding uncracked plies make it impossible to derive exact analytical solutions. Gudmundson and coworkers, however, made the following assumptions in order to evaluate the average crack opening displacements: 1. The surface displacements of a ply crack in a finite-thickness laminate are equal to those of a crack in an infinite, homogeneous transversely isotropic medium. The stress intensity factors for an infinite row of equidistant cracks in an infinite homogeneous isotropic medium under the action of uniform tractions on crack surfaces, given by Benthem and Koiter [104] and Tada et al. [105], are assumed to hold for the current case of transversely isotropic medium. 2. There is no effect of orientation of a cracked ply. This means that the matrices b ~ ki in Eq. (4.229) may not be accurate for off-axis cracks in laminates. 115 4.8 COD-based methods 3. The crack density is low (r ¸ 1). 4. There is no coupling between crack opening displacements of different plies. Hence the matrices b ~ ki are diagonal. With these assumptions, the matrices b ~ ki are given by b ~ ki = 0 ~ ; for all k ,= i; or b ~ kk = b k 1 0 0 0 b k 2 0 0 0 b k 3 _ _ _ _ ; (4:232) where b k 1 ; b k 2 ; andb k 3 are determined by Gudmundson using a numerical integra- tion and are given by b k 1 = 4 p g 1 ln cosh pr k 2 _ _ _ _ r k ( ) 2 ; b k 2 = p 2 g 2 10 j=1 a j 1 ÷ r k ( ) j ; b k 3 = p 2 g 3 9 j=1 b j 1 ÷ r k ( ) j÷2 ; (4:233) for cracks in an internal ply, where g 1 = 1 2G 12 ; g 2 = g 3 = 1 ÷ u 12 u 21 E 2 ; (4:234) and a j and b j are numerical parameters given in Table 4.2. For surface cracks (or cracks in external plies), we have b k(s) 1 = 8 p g 1 ln cosh pr k _ _ _ ¸ 2r k ( ) 2 ; b k(s) 2 = 2 1:12 ( ) 2 p 2 g 2 10 j=1 c j 1 ÷ r k ( ) j _ _ ; (4:235) Table 4.2 Numerical parameters used in calculation of COD matrix coefficients b k 1 ; b k 2 ; and b k 3 j a b c 1 0.63666 0.63666 0.25256 2 0.51806 –0.08945 0.27079 3 0.51695 0.15653 –0.49814 4 –1.04897 0.13964 8.62962 5 8.95572 0.16463 –51.2466 6 –33.0944 0.06661 180.9631 7 74.32002 0.54819 –374.298 8 –103.064 –1.07983 449.5947 9 73.60337 0.45704 –286.51 10 –20.3433 – 73.84223 116 Micro-damage mechanics where c j are another set of numerical parameters also given in Table 4.2. It is clear despite the assumptions used in COD calculation, the above approach is quite complex and difficult to implement in a numerical scheme. Similar relations are also available for the combined extension and bending loading scenario [94, 95]. 4.8.2 Lundmark–Varna model Lundmark and Varna [98, 99, 106, 107] have recently derived effective properties using a homogenization very similar to that in Gudmundson’s approach. How- ever, their relationships are simpler and they use the crack surface displacements numerically obtained from FEM simulations, which makes their model much more accurate. However, their model has been checked against experimental data only for cross-ply laminates. Here, we provide the main relations derived by them. Accordingly, the average strains in a ply, analogous to (4.228), are defined as e ij _ _ a k = e ij _ _ LAM ÷ b ij _ _ k ; (4:236) where the superscript a means “average” and e ij _ _ a k = e 11 e 22 2e 12 _ _ _ _ _ _ a k ; e ij _ _ LAM = e 11 e 22 2e 12 _ _ _ _ _ _ LAM ; b ij _ _ k = b 11 b 22 2 b 12 _ _ _ _ _ _ k ; (4:237) where b ij _ _ k represents the Vakulenko–Kachanov tensor, defined by b ij _ _ k = 1 2V k _ À kc u k i n k j ÷ u k j n k i _ _ dÀ; (4:238) which is same as Áe k ij in Eq. (4.226). In terms of crack surface displacements, b ij _ _ k for a cracked ply is derived as b _ _ k = ÷r kn E 2 A [ [ k U [ [ k A [ [ k Q _ ¸ k e 0 ¦ ¦ LAM ÷ a 0 ¦ ¦ k ÁT _ _ ; (4:239) where [A] k is the transformation matrix for ply k and [U] k is the displacement matrix given by U [ [ k = 2 = 0 0 0 0 u k 2an 0 0 0 E 2 G 12 u k 1an _ _ _ _ ; (4:240) where u k 1an and u k 2an are the normalized average crack face sliding and opening displacements, respectively, and are given by u k 1an = u k 1a G 12 t k s k 120 ; u k 2an = u k 2a E 2 t k s k 20 ; (4:241) 117 4.8 COD-based methods with u k 1a and u k 2a being the average crack surface displacements, defined as u k 1a = 1 2t k _ t k 2 ÷ t k 2 Áu 1 z ( ) dz; u k 2a = 1 2t k _ t k 2 ÷ t k 2 Áu 2 z ( ) dz; (4:242) where Du 1 and Du 2 are the relative separation of the two crack faces along and normal to the crack surface. Now, the average stress–strain relationships for the kth ply in the global coordinate system are s ¦ ¦ a k = Q _ ¸ k e ¦ ¦ a k ÷ a ¦ ¦ k ÁT _ ¸ : (4:243) Since, the average stresses for the laminate remain equal to the applied stresses, we have s ¦ ¦ LAM = s ¦ ¦ a = N k=1 s ¦ ¦ a k t k H ; (4:244) where H is the total thickness of the laminate. Substituting Eqs. (4.236) and (4.239) into Eq. (4.244), we obtain s ¦ ¦ LAM = Q 0 [ [ LAM e ¦ ¦ LAM ÷ e 0 ¦ ¦ LAM th _ _ ÷ 1 H N k=1 t k Q _ ¸ k b _ _ k ; (4:245) where [Q 0 ] LAM is the stiffness matrix for the undamaged laminate, and e 0 ¦ ¦ LAM th = 1=H ( ) N k=1 t k a ¦ ¦ k ÁT are the thermal strains in the undamaged laminate. Substituting Eq. (4.239) into Eq. (4.245), the average thermomechanical response of the damaged laminate is given by s ¦ ¦ LAM = Q 0 [ [ LAM e ¦ ¦ LAM ÷ e 0 ¦ ¦ LAM th _ _ ÷ 1 HE 2 N k=1 r kn t k Q _ ¸ k A [ [ k U [ [ k A [ [ k Q _ ¸ k e 0 ¦ ¦ LAM ÷ a 0 ¦ ¦ k ÁT _ _ : (4:246) For purely mechanical response, the stiffness matrix of the damaged laminate is given by Q [ [ LAM = I [ [ ÷ 1 HE 2 N k=1 r kn t k Q _ ¸ k A [ [ k U [ [ k A [ [ k Q _ ¸ k S 0 [ [ LAM _ _ ÷1 Q 0 [ [ LAM ; (4:247) where [S 0 ] LAM = ([Q 0 ] LAM ) ÷1 is the compliance tensor for the undamaged laminate. The only remaining unknown in this model is [U] k , Eq. (4.240). To determine the crack surface displacements, Lundmark and Varna [98] suggest using actual FE calculations on a unit cell of cracked laminate. For cross-ply laminates, they carried out a set of such FE calculations for varying values of ply thickness and 118 Micro-damage mechanics stiffness, and by curve fitting they provided the following power law-type expres- sions for the normalized average crack opening displacements u 2an = A ÷ B E 2 E s x _ _ p ; (4:248) where A, B, and p are constants that depend on the thickness ratio of cracked to uncracked plies, and the type of crack (in an internal or external ply). For example, for an internal crack in a GFRP [S n /90 m ] s laminate, A = 0:52; B = 0:3075 ÷ 0:1652 t 90 ÷ 2t s 2t s _ _ ; p = 0:0307 t 90 2t s _ _ 2 ÷ 0:0626 t 90 2t s _ _ ÷ 0:7037; (4:249) where t 90 and t s are the thicknesses of cracked 90 · and each uncracked sublaminate layer. Figure 4.21 shows comparison of the model with experimental data for a [±30/90 4 ] s laminate with cracks in the central 90 · layer. 4.9 Computational methods For simulating the effect of damage in composite laminates, in general, many computational methods have been utilized, such as the finite element method (FEM), the finite difference method (FDM), and the boundary element method (BEM). For the particular problem of transverse cracking, the most common numerical tool is FEM. Some simpler numerical tools have also been devised, e.g., the finite strip method by Li et al. [108], and the layerwise theory of Reddy [109]. In this subsection, we will briefly describe some of these developments. 0 Crack density (cracks/mm) [30/–30/90 4 ] s + experimental model 0.2 0.4 0.6 0.6 0.7 0.8 0.9 E x / E x 0 1.0 (a) (b) 0 Crack density (cracks/mm) [30/–30/90 4 ] s + experimental model 0.2 0.4 0.6 0.6 0.7 0.8 0.9 1.0 n x y / n x y 0 Figure 4.21. Reductions in longitudinal modulus (a) and Poisson’s ratio (b) for a graphite/ epoxy [±30/90 4 ] s laminate using the Lundmark–Varna model [98]. Reprinted, with kind permission, from P. Lundmark and J. Varna, Int J Damage Mech, Vol. 14, pp. 235–59, copyright # 2005 by Sage Publications. 119 4.9 Computational methods 4.9.1 Finite element method (FEM) FEM is the most widely used numerical approach to analyze damage in composite laminates. Its advantages are that it can model highly complex situations, and provides quite accurate results. As can be seen from the preceding discussion, the analytical models of cracking are severely limited in scope with respect to laminate layup and loading scenario. Nonetheless, all numerical methods have a unique limitation: every time the laminate geometry, loading, or material changes, simu- lations need to be carried out afresh, which might involve new mesh generation, and computation. Hence, FE modeling of cracked laminates is time consuming and does not by itself provide much insight into the damage mechanisms. Even with these limitations it can be successfully used for calibration/verification of analytical models as well as for computation of parameters or constants useful in analytical methods (for example, it can be used to evaluate constants used in Talreja’s continuum damage model, see the next chapter for details). FE modeling can also be utilized to simulate complex experimental situations, a task equivalent to carrying out “numerical experiments,” and eliminating the need for cumber- some experimentation whenever possible. The first task in FE modeling is to define a geometrical model of the cracked laminate. Mostly, a representative unit cell is developed assuming periodic array of self-similar cracks (see Figure 4.7(a)). For cross-ply laminates, a 3-D repeating unit can be reduced to a 2-D plane stress/strain model (see Figure 4.7(b)). Furthermore, the symmetry of the resulting boundary value problem can reduce the size of the modeled unit cell. For example, cracking in a [0/90] s laminate can be modeled using a quarter of a representative unit cell as illus- trated in Figure 4.22. For multidirectional laminates with cracks in more than one orientation, how- ever, a repeating unit cannot be defined uniquely because of differences in direc- tionality and mutual spacing of cracks in different plies. A repeating unit cell can be defined for up to two off-axis cracking modes with cracked surfaces represented on nonorthogonal boundaries [110]. The RVE in such cases is therefore 3-D and skewed (nonorthogonal in the x–y plane; z being the thickness direction). A reduction to two dimensions is often not possible in such cases. As is common in micromechanics, the displacement, strain, and stress fields must be periodic across repeating unit cells. Thus the following periodic boundary conditions [111] are applied on the FE model u i x a ÷ Áx a ( ) = u i x a ( ) ÷ Áx b @u i @x b _ _ ; e ij x a ÷ Áx a ( ) = e ij x a ( ); s ij x a ÷ Áx a ( ) = s ij x a ( ); (4:250) where u i , ¸@u i /@u b ), and Dx b , i, b = 1,2,3 represent the displacements, the volume averaged displacement gradients, and the vector of periodicity, respectively. For a 120 Micro-damage mechanics skewed RVE, periodic BC might be complicated to enforce. For a detailed discussion about skewed RVE, periodic BCs, and symmetries in composites, the reader is referred to [110, 112–114]. Once the FE results are available, the overall stiffness properties of the damaged laminate can be calculated by volume averaging of stresses and strains inside the RVE. For example, for a cracked laminate undergoing in-plane loading, the in- plane elasticity properties can be calculated by E x = s xx ¸ ) e xx ¸ ) ; E y = s yy ¸ _ e yy ¸ _ ; G xy = s xy ¸ _ g xy ¸ _ ; n xy = ÷ e yy ¸ _ e xx ¸ ) ; (4:251) where <g> denotes the volume average of a field g. Important numerical studies related to analysis of cracked composite laminates can be found in [110– 113, 115, 116]. 4.9.2 Finite strip method To enable predictions for laminated composites with arbitrary layups [110, 117, 118], Li et al. [108] devised an approximate numerical approach based on the generalized plain strain formulation. Although there can be more than one cracked ply, all the cracked plies have to be of the same orientation. Hence, this method cannot be used for laminates with cracks in multiple orientations. Consider first an uncracked laminate. Given the generalized strains {e}, e ¦ ¦ = e xx0 ; e yy0 ; 2e xy0 ; k xx ; k yy ; 2k xy _ ¸ T ; (4:252) the displacement field based on the classical laminate theory without rigid body displacements can be written as u 0 = xe xx0 ÷ ye xy0 ÷ xzk xx ÷ yzk xy ; v 0 = ye yy0 ÷ xe xy0 ÷ yzk yy ÷ xzk xy ; w 0 = ÷ x 2 2 k xx ÷ y 2 2 k yy ÷ xyk xy ÷ o z ( ); (4:253) t 0 t 90 h x z 2l 90˚ 0˚ 0˚ (b) (a) s c s c Figure 4.22. Building FE model for a cracked cross-ply laminate: (a) 2-D representative unit cell; (b) FE model (1/4 cell). 121 4.9 Computational methods where o is an arbitrary integration function which depends on boundary conditions. When transverse cracks appear, perturbations to this displacement field are induced. These perturbations can simply be superimposed on the displacements due to the linearity of the problem. Assuming sufficiently long cracks, the perturbations are independent of y. Hence, the displacement field for a cracked laminate reads u = u 0 ÷ U x; z ( ); v = v 0 ÷ V(x; z); w = w 0 ÷ W(x; z); (4:254) where U, V, and W denote changes due to the presence of cracks. The resulting strains are calculated using infinitesimal deformation theory. Our objective then is to solve for U, V, and W. The approach in the finite strip method is to divide the planar region of a typical segment of the cracked laminate into a finite number of strip elements parallel to the x-axis. In each element, M nodal lines are introduced along which displacements are functions of x only and the displacements in the strip elements are then interpolated by polynomials in the z-direction. For a typical element, the displacement field can be expressed as U V o ÷ W _ _ _ _ _ _ e = 3 i=1 N i z ( ) U i V i W i _ _ _ _ _ _ e ; (4:255) where the superscript e denotes the element number, and N i are shape functions, the same as those used in the one-dimensional finite element analysis [119], which are given by N 1 = 1 2 z 1 ÷ z ( ); N 2 = 1 ÷ z 2 ; N 1 = 1 2 z 1 ÷ z ( ); (4:256) with ÷1 _ z _ 1 being the nondimensional coordinate along the z-axis z = 3 i=1 N i z ( )z i ; (4:257) where z i is the z-coordinate of the ith nodal line in the element. Going through the usual FEM formulation, Li et al. arrived at the following set of differential equations for the resulting variational problem ÷ K 2 [ [ € y _ _ ÷ K 1 [ [ ÷ K 1 [ [ T _ _ _ y _ _ ÷ K 0 [ [ y ¦ ¦ = F 0 ¦ ¦; (4:258) and a set of boundary conditions, where [K 2 ], [K 1 ], and [K 0 ] are the matrices defined in Li et al. [108] and y ¦ ¦ = U V W _ _ _ _ _ _ : (4:259) 122 Micro-damage mechanics No direct solution to the differential Eq. (4.258) exists. An approximate solution can be obtained by taking the nodal displacements as a Fourier series of unknown constants, as U n = U h n x l ÷ K n k=1 U k n sin kpx l ; V n = V h n x l ÷ K n k=1 V k n sin kpx l ; W n = W h n x l _ _ 2 ÷ W 0 n ÷ K n k=1 U k n cos kpx l ; (4:260) for n = 1,2, . . ., M, and the coefficients U h n ; V h n ; W h n ; U k n ; V k n ; V k n ; and W 0 n are unknown constants to be determined, K n is the order where the Fourier series is truncated for the approximate solution, and M is the number of nodal lines in the laminate. Substituting Eq. (4.260) into Eq. (4.258), and rearranging, the following algebraic equations similar to FEM are obtained A [ [  ¦ ¦ = È ¦ ¦ : (4:261) The solution of these algebraic equations along with the appropriate boundary conditions gives the displacement field, from which the strain field can be obtained. 4.9.3 Layerwise theory The layerwise theory of Reddy [109] is motivated by a desire to develop a computational model that is more efficient than the conventional 3-D finite element models [119] and can incorporate damage effects such as transverse cracks and delaminations in a layered medium. It is based on 3-D kinematics where the displacement field within each layer is expanded using the Lagrange family of finite elements. For a comparable mesh, the layerwise theory typically takes less computational time as compared to the conventional 3-D FEM while providing the same level of accuracy. In the theory, the whole laminate is divided into a number of subdivisions across its thickness. The displacement field in the laminate is written as u i x; y; z ( ) = N J=1 U J i x; y ( )È J z ( ); i = 1; 2; 3; (4:262) where N is the number of subdivisions through the thickness of the laminate and F J are global interpolation functions defined in terms of the Lagrange interpolation functions associated with the layers connected to the Jth interface through the laminate thickness, and U J i are the nodal displacements. Independent interpolation functions for u 1 , u 2 , and u 3 can also be used whenever necessary. The strain field is determined using the von Karman nonlinear theory. Then the governing equations 123 4.9 Computational methods for the nodal displacements are derived using the principle of virtual displacements [120, 121]. The resulting system of partial differential equations can be converted to a systemof equations analogous to the FEMin a procedure similar to that followed in previous subsection. Solution of this system gives us the nodal displacement, from which strain and stress fields can be determined. Further details on the approach and its implementation can be found in [122–127]. Recent papers by Na and Reddy [128, 129] provide direct implementation of the layerwise theory for transverse cracking and delaminations, respectively, in cross-ply laminates. 4.10 Other methods There have been other developments in analyzing cracked laminates. The detailed treatment is not covered here and we will highlight the important aspects of these approaches. The readers are referred to the cited articles for further details. It should be pointed out, however, that these approaches are mostly extensions of the previously developed ideas, and do not really improve the predictions. More- over, many of these approaches are complex, semi-analytical and thus difficult to implement, and may sometimes need high computational times. Close to the development of the variational analysis by Hashin, Aboudi and coworkers [130, 131] developed a three-dimensional semi-analytical method. In this approach, an approximate analytical solution for the displacement field is sought using a series expansion in the form of Legendre polynomials. In a somewhat similar way, Lee and Hong [132] developed a refinement of the shear lag approach using a series polynomial expansion. The approach accounted for the crack opening displacement, thermal stresses, and Poisson’s effects. However, the improvement over the traditional shear lag methods was insignificant. Another similar effort by Gamby and Rebiere [133] used the transverse shear stresses in the 0 · - and 90 · -layers as double Fourier series in x and z (longitudinal and transverse coordinates). Vanishing tractions at the crack surfaces were used as boundary conditions. To obtain the stresses and strains in the cracked laminate, the use of constitutive equations and the equilibrium equations was made. Finally, the coefficients of the Fourier series were derived using the principle of minimum complementary energy. Quite recently, Zhang et al. [134, 135] have developed a “state space method” to analyze cracked off-axis laminates. In this approach they use displacement fields and the stresses to be given by the Fourier series expansions. The coefficient terms in expansions are obtained numerically using the equilibrium equations, boundary conditions, and ply refinement. Following on the lines of modeling of thin shells, Shoeppner and Pagano [136– 138] developed a method to model the thermoelastic response of flat laminated composites. This approach employs the idea that the stress field in an axisym- metric cylinder approaches that in a long flat coupon as the radius to thickness 124 Micro-damage mechanics ratio approaches infinity. Thus, the stress field in a flat laminate is nearly the same as in a large radius axisymmetric hollow layered cylinder model. To obtain the stress fields, they used Reissner’s variational principle along with the equilibrium equations. The predictions matched quite well with FE simulations. However, the governing equations are very complex, need numerical tools to solve them, and this limits the usability of the approach. The models of McCartney and Shoeppner–Pagano are compared in [139]. 4.11 Changes in thermal expansion coefficients Hashin [74] and Nairn [140] have developed variational models to predict changes in thermal expansion coefficients due to ply cracking. They can be derived in a relatively straightforward way following Hashin’s treatment (refer to [74] for details). The longitudinal expansion coefficient for a cracked cross-ply laminate is given by a 90 xx = a c ÷ a 11 ÷ a 22 1 ÷ l 1 ÷ B 0 C 0 _ _ k 90 x f; (4:263) where a c is the longitudinal thermal expansion coefficient for an uncracked composite laminate, and f = 1 2l _ l ÷l f x ( ) dx; B 0 = ÷ 1 ÷ l ( ) n 12 E 1 ; (4:264) Lundmark and Varna [98] also developed a model for thermal expansion coeffi- cients. Following the analysis covered in Section 4.8.2 and noting that only thermal loading is present, the strain field in the laminate is given by (setting mechanical loads to zero in Eq. (4.245)) e ¦ ¦ LAM = e 0 ¦ ¦ LAM th ÷ S 0 [ [ LAM 1 H N k=1 t k Q _ ¸ k b _ _ k : (4:265) Substituting b _ _ k from Eq. (4.239) and dividing by DT yields a ¦ ¦ LAM = I [ [ ÷ N k=1 t k r kn D [ [ k _ _ a ¦ ¦ LAM 0 ÷ 1 H N k=1 t k r kn D [ [ k a ¦ ¦ k ; (4:266) where D [ [ k = S 0 [ [ LAM 1 E 2 Q _ ¸ k T [ [ T k U [ [ k T [ [ k Q _ ¸ k : (4:267) The model predictions for longitudinal thermal expansion coefficient are plotted in Figure 4.23 for a [0/90] s carbon/epoxy laminate. For comparison, experimental and model data from Kim et al. [141] are also shown. 125 4.11 Changes in thermal expansion coefficients 4.12 Summary This chapter has provided an exposition of the main concepts and methods related to evaluating the effects of multiple cracking in composite materials on their deform- ational response. Beginning with the early approaches, known as shear lag methods, where one-dimensional stress analysis is used, and progressing to computational methods that accurately determine the local stress fields, the range is covered as much as possible. While the field is still evolving and more methods are appearing in the literature, it is hoped that the treatments and discussions provided here are useful for researchers in the field to gain an appreciation of this class of approaches. A key consideration in selecting an approach is the purpose at hand. For material selection purposes a quick assessment of the elastic modulus may be needed, in which case the shear lag approach may be adequate, while for structural analysis purposes an evaluation of all properties and the suitability of the approach as an integral part of a structural analysis scheme would be the factors of consideration. In any case, the micro-damage mechanics (MIDM) treated here is one part of the total damage mechanics picture. The next chapter on macro-damage mechanics (MADM) pro- vides another perspective on the complex problemof damage in composite materials. References 1. Z. Hashin, Analysis of damage in composite materials. 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Assuming the average response of the solid to be linear elastic, the stress–strain relationships can be written as s ij = C ijkl e kl ; (5:1) where s ij ; e kl ; and C ijkl denote the tensors of stress, strain, and stiffness, respectively, at a point of the continuum body in its undamaged state. At some loading state, the body may initiate and evolve damage in the form of voids, microcracks, cavities, etc. (Figure 5.1, middle). These damage entities will cause perturbations in the local stress and strain states of the continuum body. The stress–strain response averaged over a representative volume element (RVE) at a fixed damage state is still given by Eq. (5.1) but now C ijkl denotes the stiffness tensor of the homogenized continuum body containing damage. The main objective of the CDM approach is to character- ize C ijkl for the continuum body in which damage entities have been homogenized. A concept central to CDM is the “internal” state representing the damage entities that have been homogenized in the continuum (Figure 5.1, right). Historically, this concept is credited to Kachanov [1, 2] who introduced the so-called “continuity” as a varying state of a metal in an attempt to describe its degradation due to creep. Accordingly, a single scalar variable, denoted as f, was used to describe the continu- ity with values equal to 1 in the virgin state, and 0 at rupture. The loss of “continuity” signifies material deterioration, and was assumed to occur at a rate given by df dt = ÷A s f _ _ m ; (5:2) where A and m are material constants and s is the maximum principal tensile stress. The variable f can be viewed as representing surface density of discontinuities in the material, and can then be used to describe an “effective” stress. Thus in Eq. (5.2), the effective stress is equal to s/f, which represents the current level of stress causing further material degradation. Later, Robotnov [3] interpreted “discontinuity,” defined by the parameter o = 1 ÷ f, as a measure of net area reduction (see Figure 5.2). Accordingly, o is defined as o = S D S = S ÷ S + S ; (5:3) where the “damaged” cross section S D = S – S * , in which S is the cross-sectional area in the undamaged state and S * is the net area of the damaged specimen which excludes the area held by the damage entities (discontinuities). Thus the net area of the specimen effectively carries the applied load. The damage parameter o = 0 signifies an undamaged state, while o = o c represents a critical state corresponding Damage Initiation Homogenization Undamaged homogenized continuum Continuum with damage Homogenized damaged continuum S u S t t i u i S u S t t i u i S u S t t i u i Figure 5.1. The basic concept of internal variable damage mechanics. Damaged state (0<ϕ<1) Equivalent fictitious undamaged state (ϕ=0) Area S Area S* Force, F=s.S Force, F=s ∗ .S ∗ Figure 5.2. The concept of effective stress for isotropic damage in uniaxial loading. 135 5.1 Introduction to the rupture of the specimen. Following this approach, Robotnov [3] postulated the creep strain rate as de dt = B s 1 ÷ o _ _ n ; (5:4) where B and n are constants. It should be noted here that Kachanov’s effective stress, as defined, increases to infinity at failure, while Robotnov’s effect- ive stress, as interpreted, would increase only by a few percent since the volume fraction of voids or discontinuities at failure is found by microstructural studies for metals to be small. Robotvov’s concept of effective stress was also used later by Murakami and Ohno [4] for creep damage of polycrystalline metals and by Lemaitre and co- workers [5, 6] for effective elastic properties. Denoting the quantity in parenthesis in Eq. (5.4) by ~ s = s 1 ÷ o ; (5:5) Hooke’s law can be expressed in two equivalent forms as s = ~ Ee e ; ~ s = Ee e ; (5:6) where e e is the elastic strain and ~ E is the effective Young’s modulus, which is related to the damage parameter o as follows ~ s = s 1 ÷o = E ~ E s = o = 1 ÷ ~ E E : (5:7) In a somewhat similar way, for brittle creep damage, Lemaitre and Chaboche [7] defined the damage variable in terms of the strain rate. Using a power law, the strain rate during secondary creep (undamaged state), _ e s , can be described by _ e s = s l _ _ N ; (5:8) where l is a material constant and the exponent N is found from experimental tests. The damage variable follows easily from the measurement of the strain rate during tertiary creep, _ e, and the use of effective stress concept as o = 1 ÷ _ e _ e s _ _ 1=N : (5:9) Similar to the effective stress concept, the “strain equivalence principle” has also been proposed by Lemaitre and coworkers [6, 8], which states that “any strain constitutive equation for a damaged material may be derived in the same way as for a virgin material except that the usual stress is replaced by the effective stress.” In creep of metals as well as in brittle cracking of ceramics, rocks, concrete, etc. the voids and cracks usually form along some preferred orientations, e.g., on grain 136 Macro-damage mechanics boundaries and in weak planes. It is obvious that a scalar damage variable cannot account for the directional dependence of the effects of voids and cracks. To characterize this effect, Murakami and Ohno [4] considered an arbitrarily oriented plane in a solid containing arbitrarily distributed voids and defined a second-order tensor to describe the anisotropic net area reduction. This tensor was assumed to represent the directional nature of discontinuities and was called the damage tensor. The Murakami–Ohno damage tensor V relates the elemental area dA of an arbitrarily oriented plane with the unit normal n lying in the damaged configur- ation to the elemental area dA* of the same plane with the unit normal n* lying in a fictitious configuration with reduced net load carrying area. Thus, n + dA + = I ÷V ( )n dA; (5:10) where I is the identity tensor of second order. The effective stress tensor s* then is the magnified (net area) stress in the fictitious configuration which is related to the stress in the damaged configuration by s + = I ÷V ( ) ÷1 s: (5:11) Following (5.7), Chaboche [9–11] generalized the relation between elastic con- stants and the damage variable to the three-dimensional case, replacing o by a damage tensor D. This damage tensor, in general, is a nonsymmetric fourth-order tensor, and is defined by the following transformation E ~ = I ÷ D ( ) : E; (5:12) where I is the identity tensor of fourth order, E and E ~ are the elasticity tensor for the undamaged and damaged materials, respectively, and (:) stands for the second- order contraction. Thus, the damage tensor is given by D = I ÷ E ~ E ÷1 ; (5:13) and the effective stress tensor is given by s + = I ÷ D ( ) ÷1 : s: (5:14) It is worth noting that the second-order damage tensor defined by Murakami and Ohno [4], Eq. (5.10), is based on the notion of effective net area reduction, while the fourth-order damage tensor defined by Chaboche [9–11], Eq. (5.13), uses the notion of the effective stress defined by Eq. (5.6). The damage tensor D defined by (5.13) has been the focus of much attention in several works on continuum damage mechanics. Ju [12], for instance, has dis- cussed the notion of isotropic and anisotropic damage variables and has demon- strated that isotropic damage does not necessarily imply a scalar damage variable. From a mechanics study of changes in the elasticity compliance tensor due to microcracks he found that the fourth-order tensor D is isotropic for the case of isotropic damage, e.g., when the cracks are perfectly randomly distributed in all 137 5.1 Introduction directions. However, for preferred directions in crack orientations the damage tensor will, in general, be anisotropic. 5.2 Continuum damage mechanics (CDM) of composite materials The damage variables defined on the basis of the effective stress concept, or the associated notion of the reduced net area, do not contain any specific details of the damage entities. In the Lemaitre–Chaboche damage tensor, for instance, the components of the damage tensor are determined from the elastic compli- ance changes. Thus, e.g., two sets of damage entities of different characteristic sizes and concentrations leading to the same elastic compliance changes will be represented by the same variables. Such equivalency of different sets of damage entities cannot, in general, be expected to hold for all effects of damage. Furthermore, in media with, say, two levels of microstructure, e.g., in compos- ites, the damage entities forming at the lower size-scale may be affected by the higher size-scale microstructure as well as by the difference in the symmetry properties of the two microstructures. To illustrate this aspect, consider an ellipsoidal particle embedded in a compliant matrix and surrounded by a parallel array of stiff fibers (Figure 5.3). Let the particle debond from the matrix under appropriate loading such that an ellipsoidal damage entity is formed. The local stress perturbation induced by the surface displacements of this damage entity, and the resulting overall elastic compliance, will depend on the local microstructure details such as inclination of the fibers with respect to the principal planes of the damage entity and the fiber stiffness. Thus the same microstructure arranged differently can produce a different effect on the overall elastic response. This example serves to illustrate that a characterization of damage in terms of the elastic compliance changes will not necessarily be unique. Figure 5.3. Schematic illustration of anisotropic constraint effect in composites. An ellipsoid entity is shown surrounded by fibers or plies in a composite. The axes of the ellipsoid are inclined with respect to the axes of the constraining elements. Reprinted, with kind permission, from Damage Mechanics of Composite Materials, R. Talreja, Damage characterization by internal variables, pp. 53–78, copyright Elsevier (1994). 138 Macro-damage mechanics In the alternative characterization of damage, adopted here, the damage entities are represented by appropriate specific variables, and averages of these variables over an RVE then give the damage variables. The degree of charac- terization, i.e., to what extent the details of damage entities are represented, depends on the type of variables chosen. Thus, a scalar variable will only represent the size but not the shape or orientation characteristics of the damage entities. A vector variable can account for the size and orientation, but the positive and negative senses of a vector introduce ambiguities in the damage characterization that cause difficulties in the description of damage evolution. With two vectors simultaneously associated with a damage entity these difficul- ties are overcome. At the same time, the resulting second-order tensor gives mathematical convenience of formulating response functions, which are com- monly expressed in terms of the second-order stress and strain tensors, as discussed below. Talreja’s original paper on CDM used a vectorial description of damage [13]. Although the ambiguities concerning the sense of damage vector could be remed- ied by imposing an additional condition, it was found that using a second-order tensor instead avoided this step [14–17]. Here we will follow the tensorial description. 5.2.1 RVE for damage characterization Any continuum description of a solid entails homogenization since materials are inherently heterogeneous. For polycrystalline metals, for instance, the scale of heterogeneity (e.g., grain size) is often small compared to the scale at which material response characteristics (e.g., the elastic constants) are measured, allowing the stress and strain states to be defined as continuous fields. For commonly used fiber-reinforced solids, such as glass/epoxy and carbon/epoxy, the fiber diameter of approximately 10 micrometers allows treating these materials as a homogeneous continuum with good accuracy. When internal surfaces in composite materials form, their characteristic dimensions and mutual distances between them can be orders of magnitude larger than the scale of heterogeneities underlying the homogenized pristine composite. Furthermore, on application of external loads the internal surfaces are subject to evolution (enlargement and multiplication), as discussed above. This war- rants a separate (and different) homogenization of the composite with internal surfaces (collectively called damage). Figure 5.4 depicts a homogenization procedure for a composite solid contain- ing damage. The heterogeneities in the pristine (undamaged) composite are referred to as “stationary microstructure” and are homogenized first. This may also be called “classical” homogenization. Textbooks on mechanics of composite materials usually begin with this homogenization. In fact, the classical laminate theory goes one step further by developing homogeneous constitutive relations for laminates consisting of stacked layers of homogenized unidirectionally 139 5.2 Continuum damage mechanics (CDM) of composite materials reinforced composite (ply or lamina). Returning to Figure 5.4, the second hom- ogenization pertains to the internal surfaces, collectively named as damage or “evolving microstructure” to highlight their ability to permanently change by processes of energy dissipation. Homogenization of the evolving microstructure necessitates employing the notion of a representative volume element (RVE), which will be discussed next. A general and thorough exposition of the RVE notion in the context of micro- mechanics is given in [18]. Here, we shall apply this notion to the particular case of composite materials with damage. With reference to Figure 5.4 again, a generic point P in the homogenized composite with damage has associated with it a damage state (in addition to stress and strain states), which is given by an appropriate volume-averaged measure of the presence of internal surfaces that affect the constitutive behavior (stress–strain relations) at the point P. The volume over which the averaging is performed must be representative of the neighborhood of point P that can be associated with P. This neighborhood is the RVE, whose volume is not fixed but depends on the geometrical configuration (size, spacing, etc.) of the internal surfaces around P. As this configuration changes under applied loading, the RVE size changes. With the notion of RVE at hand, the damage state at P can be defined by a set of variables obtained by averaging over the RVE. The choice of the variables is guided by the type of internal surfaces formed, and in this respect P a i n j Homogenization of stationary microstructure Homogenization of evolving microstructure Characterization of a damage entity RVE for damage characterization Fully homogenized continuum P V Step 2 Step 1 Stationary microstructure Evolving microstructure Continuum after homogenizing the stationary microstructures Figure 5.4. Homogenization of a continuum body with heterogeneous stationary structure and evolving damage entities. A tensorial characterization of a damage entity is depicted on the right. 140 Macro-damage mechanics the knowledge of damage mechanisms discussed above is useful. In general, the variables can be scalars, vectors, or tensors of second order or higher. Settling on which variables to employ is a matter of finding a balance between capturing sufficient physics of the damage process and usefulness of the ensu- ing formulation of constitutive relations. In the following the second-order characterization of damage in composite materials adopted by Talreja [14, 15, 17] is described. 5.2.2 Characterization of damage As shown in Figure 5.4, a single internal surface within a RVE, called a damage entity from now on, can be characterized by two vectors: a unit outward normal n at a point on the surface, and an “influence” vector a at the same point. A dyadic product of the two vectors, integrated over the surface S, is denoted the damage entity tensor, and given by d ij = _ S a i n j dS; (5:15) where the components of the vectors are with reference to a Cartesian coordinate system. The dyadic product assures consistency of the signs of the two vectors. A characterization of this type was first proposed by Vakulenko and Kachanov [19] for flat cracks, where the influence vector a represented the displacement jump across the crack surface. The physical significance of this characterization is that it represents the oriented nature of the presence of internal surfaces. As illustrated by the examples of damage in Chapter 3, common internal surfaces are cracks (flat or curved) generated by interface debonding and matrix failure. The unit normal vector at a point on the damage entity carries the information on orientation of the surface (with respect to the frame of reference), while the other vector represents an appropriate influence induced by activation of the considered point on the surface. This influence is generally also directed in nature. For the case of mechanical response, the appropriate influence would be the displacement of the activated point on the damage entity surface. For a nonmechanical response, such as thermal or electrical conductivity, the perturbation induced by an internal surface can also be cast as a vector-valued quantity. Integrating the dyadic product in Eq. (5.15) over the damage entity surface provides the total net effect of the entity. For example, if the entity is a flat crack, then taking a as the displacement vector in the integral gives the crack surface separation times the crack surface area. This product may be viewed as an affected volume associated with the crack. For a penny-shaped crack with the two surfaces separating symmetrically about the initial crack plane, the sole surviving term of the damage entity tensor represents an ellipsoidal-shaped volume. 141 5.2 Continuum damage mechanics (CDM) of composite materials Referring once again to Figure 5.4, the RVE associated with a generic point P carries a sufficiently large number of the discrete damage entities to represent the collective effect on the homogenized constitutive response at the point. The number of damage entities needed for this representation, and the consequent RVE size, depend on the distribution of the entities. For instance, if the entities are sparsely distributed, then the RVE size would be large, while for densely distributed case a small RVE would suffice. Furthermore, for uniformly distrib- uted entities of the same geometry, a repeating unit cell containing a single entity can replace the RVE, while for the cases of nonuniform distribution of unequal entities, the RVE size will increase until a statistically homogeneous representation is attained. This implies that further increasing the RVE size will have no impact on the averages of the selected characteristics. As an example, if the selected characteristic is the affected matrix volume by a damage entity, as mentioned above, then the average value of this quantity will vary as the RVE size increases and will approach a constant value at a certain RVE size. The minimum RVE size beyond which no appreciable change in the considered average is found may be taken as the needed RVE. It is apparent that the RVE is not unique but is subject to the choice made for the particular formula- tion of the constitutive response of a continuum with damage. Consequently, there is no unique constitutive theory of a continuum with damage; however, the use of the concept of an internal state in a given theory requires specifying RVE in a consistent manner and assuring that the conditions for its existence are present. From the cases of damage mechanisms reviewed in Chapter 3 it can be noted that in composite laminates the damage tends to occur as sets of parallel cracks within the plies, each oriented along fibers in the given ply. It is therefore convenient to separate each set of ply cracks according to its orientation, referred to a fixed frame of reference, and assign it a damage mode number. Denoting damage mode by a = 1, 2, . . ., n, a damage mode tensor can be defined as D (a) ij = 1 V k a d ij _ _ k a ; (5:16) where k a is the number of damage entities in the ath mode, and V is the RVE volume. As noted above, if the ply cracks of a given orientation are uniformly spaced, then the RVE will reduce to the unit cell containing one crack. For nonuniform distribution of ply cracks, V must be large enough to provide a steady average of the damage mode tensor components. As defined by Eq. (5.16) the damage mode tensor will in general be asymmet- rical. Decomposing the influence vector a along directions normal and tangential to the damage entity surface S gives, a i = an i ÷ bm i ; (5:17) where n and m are unit normal and tangential vectors on S such that n i m i = 0. 142 Macro-damage mechanics Using Eq. (5.17) in Eq. (5.16) the damage entity tensor can be written in two parts as d ij = d 1 ij ÷ d 2 ij ; (5:18) where d 1 ij = _ S an i n j dS; d 2 ij = _ S bm i n j dS: (5:19) Here a and b are the magnitudes of the normal and tangential projections, respectively, of vector a i and vectors n i and m j are unit normal and tangential vectors, respectively. The damage mode tensor for a given mode can now be written as D (a) ij = D 1(a) ij ÷ D 2(a) ij ; (5:20) where D 1(a) ij = 1 V k a d 1 ij _ _ k a ; D 2(a) ij = 1 V k a d 2 ij _ _ k a : (5:21) This separation of the damage mode tensor in two parts allows the analysis to be simplified to avoid having to deal with asymmetric tensors. For instance, for damage entities consisting of flat cracks, the two parts of the damage mode tensor represent the two crack surface separation modes. If an assumption can be made that only the symmetric crack surface separation (known as mode I or crack opening mode in fracture mechanics) is significant, then the second term in Eq. (5.21) can be neglected. This will render the damage mode tensor symmetrical and it can then be written as D (a) ij = D 1(a) ij = 1 V k a _ S an i n j dS _ _ k a : (5:22) The consequence of this assumption was examined by Varna [20] for one class of laminates and it was found that not including the crack sliding displacement (CSD) for ply cracks inclined to the laminate symmetry directions resulted in errors in estimating the degradation of the average elastic properties of laminates. However, these errors were found to be small in absolute values while being significant in percentages. In fact for those ply crack orientations where CSD dominates, the cracks are difficult to initiate until high loads close to failure load are applied. Some damage mechanisms such as crystalline slip may require only the tangential part. For other situations the sliding between the crack faces can be negligible, e.g., for intralaminar cracks constrained by stiff plies, and fiber/ matrix debonds. Hence D ij 2(a) = 0 under such conditions. For cases where the 143 5.2 Continuum damage mechanics (CDM) of composite materials damage entity surfaces conduct tangential displacements only (e.g., CSD by flat cracks), it is possible to formulate the damage mode tensor as a symmetric tensor. One example of this is sliding of the fiber/matrix interface in ceramic matrix composites [16]. With stress, strain, and damage, all expressed as symmetric second-order tensors, a constitutive theory can now be formulated to have a convenient, usable form. Such a formulation is described next. 5.2.3 A thermodynamics framework for materials response Referring once again to Figure 5.4, a formulation of the constitutive response of a homogenized continuum with damage will now be discussed. In view of the observed behavior of common composite materials such as glass/epoxy and carbon/epoxy, only elastic response will be considered. Theoretical treatment of elastic response of solids is classical and can be found in textbooks. Incorpor- ating damage is, however, not a simple extension of the classical theory of elasticity. The CDM approach to be described here is based on thermodynam- ics and is naturally suited for thermomechanical response. It can be extended to incorporate nonmechanical effects, such as electrical and magnetic, as well as chemical. Every extension, however, comes with the price of having to deter- mine associated response coefficients (material constants) by a certain identifi- cation procedure. In the treatment presented here, the task of determining material constants is reduced by use of selected micromechanics. This way of combining micromechanics with CDM generates useful synergism, justifying the characterization of the combined approach as synergistic damage mechan- ics (SDM), to be discussed later. We begin with the conventional CDM frame- work first. At the foundation of CDM are the first and second laws of thermodynamics. Additionally, use is made of the concept of an internal state, which is identified here as the evolving microstructure depicted in Figure 5.4. As discussed before, this microstructure is homogenized into a damage field characterized by the set of damage mode tensors D ij (a) . The collectionof all variables resulting fromthermodynamics with internal state can now be placed in two categories: state variables and response functions (see Table 5.1). The thermodynamical response of a composite body, limited to small strains, is given by a set of five response functions: the Cauchy stress tensor s ij , Table 5.1 Thermodynamics variables for damage analysis Thermodynamic state variables Thermodynamic response variables 1. Strain tensor e ij = 1 2 u i; j ÷ u j;i _ _ 1. Cauchy stress tensor s ij 2. Temperature T 2. Specific Helmholtz free energy c 3. Temperature gradient g i = T ,i 3. Specific entropy 4. Damage tensors D ij (a) 4. Heat flux vector q i 5. Damage rate tensors _ D (a) ij 144 Macro-damage mechanics the specific Helmholtz free energy c, the specific entropy , the heat flux vector q i , and a set of damage rate tensors _ D (a) ij , a =1,2, ..., n. The thermodynamic state of the body is givenby the straintensor e ij = 1=2 ( ) u i; j ÷ u j;i _ _ , withdisplacement vector u i , absolute temperature T, temperature gradient g i = T ,i , and a set of damage tensors D ij (a) . Following Truesdell’s principle of equipresence, which states that all state variables should be present in all response functions unless thermodynamics or other relevant considerations preclude their dependency, we write s ij = s ij e kl ; T; g k ; D (a) kl _ _ ; c = c e kl ; T; g k ; D (a) kl _ _ ; = e kl ; T; g k ; D (a) kl _ _ ; q = q e kl ; T; g k ; D (a) kl _ _ ; _ D (a) kl = _ D (a) kl e kl ; T; g k ; D (b) kl _ _ : (5:23) The following balance laws should hold for the continuum body: v Balance of linear momentum: s ij;j ÷ rb j = r€ x j ; (5:24) where b j are the components of the body force per unit mass and r is the mass density. v Balance of angular momentum: s ij = s ji : (5:25) v Balance of energy: r _ u ÷ s ij _ e ij ÷ q i;i = rr; (5:26) where u is the specific internal energy per unit mass and r is the heat supply per unit mass. v Second law of thermodynamics in the form of the Clausius–Duhem inequality: s ij _ e ij ÷ r _ c ÷r _ T ÷ q i g i T _ 0; (5:27) where c = u ÷ T. Time differentiation of c in (5.23) gives, _ c = @c @e kl _ e kl ÷ @c @T _ T ÷ @c @g i _ g i ÷ @c @D (a) kl _ D (a) kl : (5:28) Substitution of Eq. (5.28) into Eq. (5.27) gives 145 5.2 Continuum damage mechanics (CDM) of composite materials s ij ÷ r @c @e kl _ _ _ e ij ÷ r ÷ @c @T _ _ _ T ÷ r @c @g i _ g i ÷r a @c @D (a) kl _ D (a) kl ÷ q i g i T _ 0; (5:29) where the summation sign is shown on a to include all damage modes. Now requiring (5.29) to hold for the independently varying strain, temperature, and temperature gradient gives the following results s ij = r @c @e kl ; (5:30) = ÷ @c @T ; (5:31) @c @g i = 0 : (5:32) Equation (5.32) states that the Helmholz free energy function does not depend on the temperature gradient and, consequently, Eqs. (5.30) and (5.31) eliminate this dependency from stress and entropy. The last two equations in the response function set (Eq. (5.23)) remain unaffected. Equations (5.30)–(5.32) lead to the following functional dependencies s ij = s ij e kl ; T; D (a) kl _ _ ; c = c e kl ; T; D (a) kl _ _ ; = e kl ; T; D (a) kl _ _ ; q = q e kl ; T; g k ; D (a) kl _ _ ; _ D (a) kl = _ D (a) kl e kl ; T; g k ; D (b) kl _ _ : (5:33) From Eq. (5.29) the following restriction (also known as internal dissipation inequality) results a R (a) kl _ D (a) kl ÷ q i g i T _ 0; (5:34) where R kl (a) are the thermodynamic forces conjugate to D kl (a) and are given by R (a) kl = ÷r @c @D (a) kl : (5:35) Each of these forces is analogous to the crack extension force (i.e., energy release rate) for a single crack. As an example, for a damage mode component, say D 11 of mode a = 1, the quantity R 11 (1) can be interpreted as the “force” causing an infinitesimal change in the internal state represented by D 11 (1) . Equation (5.34) 146 Macro-damage mechanics expresses the condition these forces must satisfy as damage evolves under thermo- mechanical impulses. The complete thermomechanical response for the composite body is governed by the set of functions Eq. (5.33) and the internal dissipation inequality Eq. (5.34). In view of the experimental data, which are mostly available for polymer matrix composites at room temperature, the thermomechanical framework will be developed further for the purely mechanical response. Thus, for isothermal condi- tions (T = 0, g i = 0) the set of response functions is reduced to the following s ij = s ij e kl ; D (a) kl _ _ ; c = c e kl ; D (a) kl _ _ ; _ D (a) kl = _ D (a) kl e kl ; g k ; D (b) kl _ _ ; a R (a) kl _ D (a) kl _ 0 : (5:36) Since s ij are derivable fromc, according to Eq. (5.30), it suffices to formulate c and _ D (a) kl for a purely mechanical response. Dealing with this scalar-valued function (c) as the sole response function for a given internal state of damage provides a favorable situation for further development of the theory. The form of the Helmholz free energy function can be chosen in different ways. A powerful way is possible by use of the theory of invariants for polynomial functions [21]. In the following it is illustrated for one case of orthotropic composites containing one damage mode. To derive the rate equations for s ij , we differentiate Eq. (5.30) with respect to time and use the functional dependency given in Eq. (5.36), which yields _ s ij = r @ 2 c @e ij @e kl _ e kl ÷ r a @ 2 c @e ij @D (a) mn _ D (a) mn : (5:37) Substitution of Eq. (5.35) gives _ s ij = r @ 2 c @e ij @e kl _ e kl ÷ a @R (a) mn @e ij _ D (a) mn ; (5:38) which may be rewritten as _ s ij = C ijkl _ e kl ÷ a K (a) ijmn _ D (a) mn ; (5:39) where C ijkl = r @ 2 c @e ij @e kl (5:40) and 147 5.2 Continuum damage mechanics (CDM) of composite materials K (a) ijmn = @R (a) mn @e ij = ÷r @ 2 c @e ij @D (a) mn : (5:41) The matrix C ijkl contains stiffness coefficients of the composite in its virgin mater- ial state, whereas coefficients of K ijmn (a) are functions determining the change of state caused by the internal dissipative mechanisms. The rate shown in Eq. (5.39) requires formulation of the scalar function c and the tensor components _ D (a) kl . The discussion of damage evolution is left for later; the next section will treat the stress–strain relations at a fixed damage state. 5.2.4 Stiffness–damage relationships The stiffness coefficient matrix C ijkl of the composite in a given state of damage is derivable from the Helmholtz free energy function c according to Eq. (5.40). The scalar valued function c can be written as a polynomial in its variables, i.e., c = c P e ij ; D (a) ij _ _ ; (5:42) where c P stands for the polynomial function. Let us now focus on the particular case of intralaminar cracking in composite laminates. Figure 5.5 shows an RVE illustrating one set of intralaminar cracks in an off-axis ply of a composite laminate. Although for clarity of illustration the cracking is shown only in one lamina, it is understood that in general it exists in multiple plies of the laminate. The thickness of the cracked plies is denoted by t c , s is the average crack spacing, t is the total laminate thickness, and W and L stand for the width and the length, respectively, of the RVE. The volume of the RVE, the surface area of a crack, S, and the influence vector magnitude, a, are specified as V = L:W:t; S = W:t c sin y ; a = kt c ; (5:43) where k, called the constraint parameter, is an unspecified constant of (assumed) proportionality between a and the crack size t c (also the cracked-ply thickness). Here, y is a positive quantity so that the surface area is always positive. Assuming a to be constant over the crack surface S, one gets from Eq. (5.22) D (a) ij = kt 2 c st sin y n i n j ; (5:44) where n i = sin y; cos y; 0 ( ). Expansion of the polynomial function in Eq. (5.42) can in general have infinite terms, which will obviously present an impractical situation. One way to restrict the functional form is by expanding the polynomial in terms that account for the 148 Macro-damage mechanics initial material symmetry. This is done in the polynomial invariant theory by using the so-called integrity bases [21]. Such bases have been developed for scalar functions of various vector and tensor variables. For the case of two symmetric second-order tensors, such as in Eq. (5.42), the integrity bases for orthotropic symmetry are given by Adkins [22]. Considering a single damage mode, a = 1, we have the following set of invariant terms e 11 ; e 22 ; e 33 ; e 2 23 ; e 2 31 ; e 2 12 ; e 23 e 31 e 12 ; D 11 ; D 22 ; D 33 ; D 2 23 ; D 2 31 ; D 2 12 ; D 23 D 31 D 12 ; e 23 D 23 ; e 31 D 31 ; e 12 D 12 ; e 31 e 12 D 23 ; e 12 e 23 D 31 ; e 23 e 31 D 12 ; e 23 D 31 D 12 ; e 31 D 12 D 23 ; e 12 D 23 D 31 : (5:45) For the sake of applying the constitutive theory to thin laminates where only in-plane strains are of interest, and for small strains, the expansion of the function c (Eq. (5.42)) can be restricted to no more than quadratic terms in strain compon- ents e 11 , e 22 , and e 12 To what extent the damage tensor components are to be taken in the expansion depends on the nature and amount of information that can Figure 5.5. A representative volume element illustrating intralaminar multiple cracking in a general off-axis ply of a composite laminate. 149 5.2 Continuum damage mechanics (CDM) of composite materials be acquired for evaluation of the material constants that will appear in the polynomial function. This issue will be discussed later. To begin with the simplest possible case, we will include only linear terms in D 11 , D 22 , and D 12 , which are the nonzero components for intralaminar cracks. Thus the set of invariant terms reduces to e 1 ; e 2 ; e 2 6 ; D 1 ; D 2 ; D 2 6 ; e 6 D 6 ; (5:46) where e 1 = e 11 ; e 2 = e 22 ; e 6 = e 12 ; D 1 = D 11 ; D 2 = D 22 ; D 6 = D 12 . The most general polynomial form for the Helmholtz free energy, restricted to second-order terms in the strain components and first-order terms in damage tensor components, is given by rc = P 0 ÷ c 1 e 2 1 ÷ c 2 e 2 2 ÷ c 3 e 2 6 ÷ c 4 e 1 e 2 _ _ ÷ c 5 e 2 1 D 1 ÷ c 6 e 2 1 D 2 _ _ ÷ c 7 e 2 2 D 1 ÷ c 8 e 2 2 D 2 _ _ ÷ c 9 e 2 6 D 1 ÷c 10 e 2 6 D 2 _ _ ÷ c 11 e 1 e 2 D 1 ÷ c 12 e 1 e 2 D 2 ¦ ¦ ÷ c 13 e 1 e 6 D 6 ÷ c 14 e 2 e 6 D 6 ¦ ¦ ÷ P 1 (e p ; D q ) ÷ P 2 (D q ); (5:47) where P 0 and c i , i = 1, 2, . . ., 14 are material constants, P 1 is a linear function of strain and damage tensor components, and P 2 is a linear function of damage tensor components. Setting the free energy to zero for unstrained and undamaged material, we have P 0 = 0, and assuming the unstrained material of any damaged state to be stress free, we get P 1 = 0. The stress components in the Voigt notation are now given by (from Eq. (5.30)) s p = r @c @e p ; (5:48) where p = 1, 2, 6. A differential in stress can now be written as ds p = r @c @e p @e q de q ÷ r @c @e p @D r dD r = C pq de q ÷ K pr dD r ; (5:49) where C pq = r @c @e p @e q (5:50) is the stiffness matrix when dD r = 0, i.e., at constant damage. This is illustrated for the uniaxial stress–strain response in Figure 5.6. As seen there, the elastic modulus at any point on the stress–strain curve is the secant modulus, not the tangent modulus. Combining Eqs. (5.47), (5.48), and (5.50), one obtains C pq = C 0 pq ÷ C (1) pq (5:51) 150 Macro-damage mechanics where C 0 pq = 2c 1 c 4 0 2c 2 0 Symm 2c 3 _ _ _ _ = E 0 x 1 ÷ n 0 xy n 0 yx n 0 xy E 0 y 1 ÷ n 0 xy n 0 yx 0 E 0 y 1 ÷ n 0 xy n 0 yx 0 Symm G 0 xy _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ (5:52) represents the orthotropic stiffness matrix for virgin composite material, in which E 0 x ; E 0 y ; n 0 xy ; G 0 xy are effective moduli for the undamaged laminate, and C (1) pq = 2c 5 D 1 ÷ 2c 6 D 2 c 11 D 1 ÷ c 12 D 2 c 13 D 6 2c 7 D 1 ÷ 2c 8 D 2 c 14 D 6 Symm 2c 9 D 1 ÷ 2c 10 D 2 _ ¸ _ _ ¸ _; (5:53) represents the stiffness change brought about by the damage entities of damage mode 1. It can be noted here that Eqs. (5.51)–(5.53) show linear dependence of the stiffness properties on damage tensor components. This is the consequence of including only linear terms in these components in the polynomial expansion of the free energy function, Eq. (5.47). Including higher-order terms will add additional constants c i , which will need to be evaluated. The evaluation procedure is described below, but it is remarked here that the formulation of constitutive response is in no way restricted only to linear dependence on the chosen damage measure. E 0 E s e Figure 5.6. Stress–strain curve of a composite with damage. The secant modulus E varies with the state of damage. 151 5.2 Continuum damage mechanics (CDM) of composite materials Case 1: Cracking in one off-axis orientation From Eqs. (5.51)–(5.53), it can be seen that the presence of damage entities, even for one orientation, removes the initial orthotropic symmetry. For the case of intralaminar cracks in one orientation, as illustrated in Figure 5.5, the nonzero components of the damage tensor D ij (1) are given from Eq. (5.44) by D 1 = D (1) 11 = kt 2 c st sin y; D 2 = D (1) 22 = kt 2 c st cos 2 y sin y ; D 6 = D (1) 12 = kt 2 c st cos y: (5:54) Inserting these into Eq. (5.51) and using Eq. (5.52) and Eq. (5.54) we obtain the stiffness matrix of the damaged composite laminate for a fixed state of damage as C pq = E 0 x 1 ÷ n 0 xy n 0 yx n 0 xy E 0 y 1 ÷ n 0 xy n 0 yx 0 E 0 y 1 ÷ n 0 xy n 0 yx 0 Symm G 0 xy _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ ÷ kt 2 c st sin y 2c 5 ÷ 2c 6 cot 2 y c 11 ÷ c 12 cot 2 y c 13 cot y 2c 7 ÷ 2c 8 cot 2 y c 14 cot y Symm 2c 9 ÷ 2c 10 cot 2 y _ ¸ ¸ _ _ ¸ ¸ _ : (5:55) Case 2: Cross-ply laminates For the special case of cross-ply laminates, y = 90 · , and hence C pq = E 0 x 1 ÷ n 0 xy n 0 yx n 0 xy E 0 y 1 ÷ n 0 xy n 0 yx 0 E 0 y 1 ÷ n 0 xy n 0 yx 0 Symm G 0 xy _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ ÷ kt 2 c st 2a 1 a 4 0 2a 2 0 Symm 2a 3 _ ¸ _ _ ¸ _ (5:56) where a 1 = c 5 ; a 2 = c 7 ; a 3 = c 9 ; and a 4 = c 11 . It can be observed that the ortho- tropic symmetry is retained for intralaminar cracking in cross-ply laminates. The engineering moduli can be derived from the following relationships 152 Macro-damage mechanics E x = C 11 C 22 ÷C 2 12 C 22 ; E y = C 11 C 22 ÷C 2 12 C 11 ; n xy = C 11 C 22 ; G xy = C 66 : (5:57) Thus, for cross-ply laminates with 90 · -ply cracks, E x = E 0 x 1 ÷ n 0 xy n 0 yx ÷ 2 kt 2 c st a 1 ÷ n 0 xy E 0 y 1 ÷ n 0 xy n 0 yx ÷ kt 2 c st a 4 _ _ 2 E 0 y 1 ÷ n 0 xy n 0 yx ÷ 2 kt 2 c sin y st a 2 ; E y = E 0 y 1 ÷ n 0 xy n 0 yx ÷ 2 kt 2 c st a 2 ÷ n 0 xy E 0 y 1 ÷ n 0 xy n 0 yx ÷ kt 2 c st a 4 _ _ 2 E 0 x 1 ÷n 0 xy n 0 yx ÷ 2 kt 2 c st a 1 ; n xy = n 0 xy E 0 y 1 ÷n 0 xy n 0 yx ÷ kt 2 c st a 4 E 0 y 1 ÷ n 0 xy n 0 yx ÷ 2 kt 2 c st a 2 ; G xy = G 0 xy ÷ 2 kt 2 c st a 3 : (5:58) Evaluation of material constants In the damage–stiffness relations Eq. (5.56) and Eq. (5.58), a i ,i = 1,2,3,4 are a set of four phenomenological constants, which need to be determined to predict stiffness degradation. As seen from Eq. (5.58), the shear modulus is uncoupled from the other three moduli and thus can be treated independently. These phenomeno- logical constants are material and laminate configuration dependent and can be evaluated for a selected laminate by using data generated either experimentally or by an analytical or a computational model. As an example, let the moduli of a given damaged cross-ply laminate at a fixed state of damage, s = s 1 , be given as E x ; E y ; G xy ; and n xy , then by using Eq. (5.56), we obtain a 1 = s 1 t 2kt 2 c E x 1 ÷ n xy n yx ÷ E 0 x 1 ÷ n 0 xy n 0 yx _ _ ; a 2 = s 1 t 2kt 2 c E y 1 ÷ n xy n yx ÷ E 0 y 1 ÷ n 0 xy n 0 yx _ _ ; a 3 = s 1 t 2kt 2 c G xy ÷ G 0 xy _ _ ; a 4 = s 1 t kt 2 c n xy E y 1 ÷ n xy n yx ÷ n 0 xy E 0 y 1 ÷ n 0 xy n 0 yx _ _ : (5:59) 153 5.2 Continuum damage mechanics (CDM) of composite materials In the above expressions it should be noted that while the values of a i are fixed for a given composite laminate (that has been homogenized), the parameter k depends on the ability of the cracks to perform surface displacements under an applied mechanical impulse. Thus this parameter may be viewed as a measure of the constraint to the crack surface separation imposed by the material surrounding the crack. One way to view this is by considering a crack of a given size embedded in an infinite isotropic material, in which case the crack surface separation is unconstrained and can be calculated by fracture mechanics methods. When the laminate geometry is finite and its symmetry is different from isotropic, the k parameter will take a value less than that for the infinite isotropic medium. This consideration allows us to assign k an undetermined value, say k 0 , for a reference laminate under reference loading conditions, and evaluate a change from this value for another crack orientation. This procedure will be discussed in more detail later. The CDM approach described here has been used to successfully predict degradation in the longitudinal and transverse moduli and the Poisson’s ratio for a variety of laminates, e.g., [0/90 3 ] s , [90 3 /0] s , and [0/±45] s as reported in [13–15, 17, 23]. Figure 5.7 shows the degradation of the longitudinal Young’s modulus in a [0/90 3 ] s glass/epoxy laminate with the applied tensile stress. The predictions agree with the observed values fairly well in the entire range of cracking. The predictions by the ply discount method are found to overestimate the total modulus reduction. The stress–strain curve constructed from the reduced modulus is shown in Figure 5.8. 0 0.5 0.6 0.7 0.8 0.9 1.0 E 1 /E 1 0 100 200 300 Gl./Ep. [0/90 3 ] s Calculated Measured Ply discount prediction s (MPa) x x x x x x x x x x x x x x Figure 5.7. Variation of the longitudinal Young’s modulus with applied stress for a glass/ epoxy [0/90 3 ] s laminate. Source: [23]. 154 Macro-damage mechanics 5.3 Synergistic damage mechanics (SDM) The observation that the k-parameter (hitherto referred to as constraint param- eter) may be viewed as a carrier of the local effects on damage entities within a RVE, while the a i -constants are material constants, led to a number of studies to explore prediction of elastic property changes due to damage in different modes. To be sure, the elastic properties are the averages over appropriate RVEs. At first it was found that from changes in E x and n xy due to transverse cracking in [0/90 3 ] s glass/epoxy laminates reported in [24] and assuming no changes in E y , changes in E x for the same glass/epoxy of [0/90] s configuration could be predicted with good accuracy. Also, in [0/±45] s laminates of the same glass/epoxy, the change in E x could be predicted by setting D 1 = D 2 (a good approximation, supported by crack density data). These results have been reported in [14]. Later, a systematic study of the effect of constraint on the constraint parameter was done by experimentally measuring the crack opening displacement (COD) in [±y/90 2 ] s laminates [25] for different y-values. By relating these values to the COD at y = 90 · normalized by a unit applied strain, the predictions of E x and n xy for different y could be made. Another study of the constraint effects was made by examining [0/±y 4 /0 1/2 ] s laminates, where the ply orientation y was varied. Once again, using experimentally measured COD for y = 90 · as the reference, the k-parameter for other ply orientations was evaluated from the COD values and E x and n xy for different y were then predicted [33]. While the experimental studies supported the idea of using the constraint parameter as a carrier of local constraints, the scatter in test data and the cost of 0 0.5 1.0 1.5 2.0 2.5 e (%) 0 50 100 150 200 250 s (MPa) Gl./Ep. [0/90 3 ] s Figure 5.8. Longitudinal stress–strain response for a glass/epoxy [0/90 3 ] s laminate. Source: [23]. 155 5.3 Synergistic damage mechanics (SDM) testing do not make the experimental approach attractive. Therefore, another systematic study of [0 m /±y n /0 m/2 ] s laminates was undertaken [28, 29] where com- putational micromechanics was employed instead of physical testing. An elaborate parametric study of the constraint parameter allowed developing a master curve for elastic property predictions. The most recent study [30] examines damage modes consisting of transverse ply cracks as well as inclined cracks of different orientations in [0 m /±y n /90 r ] s and [0 m /90 r /±y n ] s laminates. The reference value of k for y = 90 · , as well as changes in it for other ply orientations, are computed by finite element models of representative unit cells. Predictions made using k as a micromechanical carrier of local damage-induced perturbations are compared with available experimental data for [0/90/÷45/+45] s laminates. This approach of combining MIDM (local) and MADM (global) approaches is named synergistic damage mechanics (SDM). The following sections describe the use of SDM. First we take multidirectional laminates with cracking in two symmetric off-axis orien- tations, followed by the case of cracking in three orientations. 5.3.1 Two damage modes Let us consider a case where two modes of damage are active. In this case, the irreducible integrity bases for c P are given by, with a = 1 and 2, e 11 ; e 22 ; e 33 ; e 2 23 ; e 2 31 ; e 2 12 ; e 23 e 31 e 12 ; D (1) 11 ; D (1) 22 ; D (1) 33 ; D (1) 23 _ _ 2 ; D (1) 31 _ _ 2 ; D (1) 12 _ _ 2 ; D (1) 23 D (1) 31 D (1) 12 ; D (2) 11 ; D (2) 22 ; D (2) 33 ; D (2) 23 _ _ 2 ; D (2) 31 _ _ 2 ; D (2) 12 _ _ 2 ; D (2) 23 D (2) 31 D (2) 12 ; e 23 D (1) 23 ; e 31 D (1) 31 ; e 12 D (1) 12 ; e 23 D (2) 23 ; e 31 D (2) 31 ; e 12 D (2) 12 ; e 31 e 12 D (1) 23 ; e 12 e 23 D (1) 31 ; e 23 e 31 D (1) 12 ; e 31 e 12 D (2) 23 ; e 12 e 23 D (2) 31 ; e 23 e 31 D (2) 12 ; e 23 D (1) 31 D (1) 12 ; e 31 D (1) 12 D (1) 23 ; e 12 D (1) 23 D (1) 31 ; e 23 D (2) 31 D (2) 12 ; e 31 D (2) 12 D (2) 23 ; e 12 D (2) 23 D (2) 31 : (5:60) For a thin laminate loaded in its plane, the above set can be reduced by consider- ing only the in-plane strain and damage tensor components. Thus, the remaining integrity bases in the Voigt notation are given by e 1 ; e 2 ; e 2 6 ; D (1) 1 ; D (1) 2 ; D (1) 6 _ _ 2 ; D (2) 1 ; D (2) 2 ; D (2) 6 _ _ 2 ; e 6 D (1) 6 ; e 6 D (2) 6 : (5:61) 156 Macro-damage mechanics Using the above set of integrity bases, the most general polynomial form for rc, restricted to second-order terms in the strain components (assuming small strains) and first-order terms in damage tensor components (assuming small volume fraction or number density of damage entities in the RVE), is given by rc = P 0 ÷ c 1 e 2 1 ÷ c 2 e 2 2 ÷ c 3 e 2 6 ÷ c 4 e 1 e 2 _ _ ÷ e 2 1 c 5 D (1) 1 ÷ c 6 D (1) 2 ÷ c 7 D (2) 1 ÷ c 8 D (2) 2 _ _ ÷ e 2 2 c 9 D (1) 1 ÷ c 10 D (1) 2 ÷ c 11 D (2) 1 ÷ c 12 D (2) 2 _ _ ÷ e 2 6 c 13 D (1) 1 ÷ c 14 D (1) 2 ÷ c 15 D (2) 1 ÷ c 16 D (2) 2 _ _ ÷ e 1 e 2 c 17 D (1) 1 ÷ c 18 D (1) 2 ÷ c 19 D (2) 1 ÷ c 20 D (2) 2 _ _ ÷ e 1 e 6 c 21 D (1) 6 ÷ c 22 D (2) 6 _ _ ÷ e 2 e 6 c 23 D (1) 6 ÷ c 24 D (2) 6 _ _ ÷ P 1 e p ; D (1) q _ _ ÷ P 2 e p ; D (2) q _ _ ÷ P 3 D (1) q _ _ ÷ P 4 D (2) q _ _ ; (5:62) where P 0 and c i , i = 1, 2, . . ., 24 are material constants, P 1 and P 2 are linear functions of strain and damage tensor components, and P 3 and P 4 are linear functions only of the damage tensor components. Setting rc = 0 for unstrained and undamaged material, we have P 0 = 0; and assuming the unstrained material of any damaged state to be stress free, we get P 1 = P 2 = 0 on using Eq. (5.48). Considering the virgin material to be orthotropic and proceeding in a similar manner as above, we obtain the following relations for the stiffness matrix of the damaged laminate C pq = C 0 pq ÷ C (1) pq ÷ C (2) pq ; (5:63) where p, q = 1, 2, 6; C pq 0 is the stiffness coefficient matrix of the virgin laminate given by Eq. (5.52), and the changes in stiffness brought about by the individual damage modes are represented by C pq (1) and C pq (2) , which are given by C (1) pq = 2c 5 D (1) 1 ÷ 2c 6 D (1) 2 c 17 D (1) 1 ÷ c 18 D (1) 2 c 21 D (1) 6 2c 9 D (1) 1 ÷ 2c 10 D (1) 2 c 23 D (1) 6 Symm 2c 13 D (1) 1 ÷ 2c 14 D (1) 2 _ ¸ ¸ ¸ _ _ ¸ ¸ ¸ _ C (2) pq = 2c 7 D (2) 1 ÷ 2c 8 D (2) 2 c 19 D (2) 1 ÷ c 20 D (2) 2 c 22 D (2) 6 2c 11 D (2) 1 ÷ 2c 12 D (2) 2 c 24 D (2) 6 Symm 2c 15 D (2) 1 ÷ 2c 16 D (2) 2 _ ¸ ¸ ¸ _ _ ¸ ¸ ¸ _ : (5:64) 157 5.3 Synergistic damage mechanics (SDM) In general, for N damage modes, Eq. (5.63) can be written as C pq = C 0 pq ÷ N a=1 C (a) pq : (5:65) Let us now consider a special case of a general laminate undergoing damage in two symmetrically placed damage modes, such as 0 m =±y n =’ p _ ¸ s , with ’ restricted to angles that do not cause ply cracking. In such laminates, an in-plane tensile loading will produce an in-plane stress state in each off-axis ply consisting of normal stresses along and perpendicular to fibers in that ply and a shear stress in the plane of the ply. Depending on the values of y, ’, and ply properties, the stress perpendicular to the fibers could be tensile or compressive. Thus, on loading, an off-axis ply may or may not develop intralaminar cracks. When y = 90 · , the matrix will undergo multiple cracking in the transverse plies. For other cases of off-axis ply orientations, multiple cracking is typically observed to occur for angles from 50 · to 90 · under an axial tensile load. However, it has been observed that even in cases where these cracks do not initiate in the off-axis plies, the laminate moduli change with the applied load due to shear stress-induced damage within the plies. The damage state subsequent to ply cracking in the +y and ÷y plies can be represented by two damage mode tensors. For off-axis ply cracking, it is more convenient to rewrite the damage mode tensor defined in (5.44) in terms of the normal crack spacing, s y n = s y sin y, where s y is the crack spacing in the axial direction (see Figure 5.9) for a ply of orientation y. Accordingly, the damage mode tensors are given by D (a) ij = kt 2 c s y n t n i n j : (5:66) With reference to Figure 5.9(b) where the orientations of the two damage modes are shown, the damage mode elements are given by (a) (b) Figure 5.9. Damage characterization for two damage modes: (a) normal crack spacing s y n , and axial crack spacing s y in a cracked ply; (b) directions of normal vectors for cracks in +y- and –y-plies. 158 Macro-damage mechanics a = 1 : n (1) i = sin y; cos y; 0 ( ) ; D (1) 1 = k y ÷ t 2 c s y ÷ n t sin 2 y; D (1) 2 = k y ÷ t 2 c s y ÷ n t cos 2 y; D (1) 6 = k y ÷ t 2 c s y ÷ n t sin y cos y; a = 2 : n (2) i = sin y; ÷cos y; 0 ( ); D (2) 1 = k y ÷ t 2 c s y ÷ n t sin 2 y; D (2) 2 = k y ÷ t 2 c s y ÷ n t cos 2 y; D (2) 6 = ÷ k y ÷ t 2 c s y ÷ n t sin y cos y; (5:67) where the superscripts y + and y – indicate variables for the +y and –y plies, respectively. Assuming that the intensity and distribution of damage is the same in both +y and –y plies, we have k y ÷ = k y ÷ = k y ; s y ÷ n = s y ÷ n = s y n : (5:68) Substituting Eq. (5.67)–(5.68) into Eq. (5.64), we obtain C (1) 11 ÷ C (2) 11 = 2 k y t 2 c s y n t c 5 ÷ c 7 ( )sin 2 y ÷ c 6 ÷ c 8 ( )cos 2 y _ ¸ ; C (1) 22 ÷ C (2) 22 = 2 k y t 2 c s y n t c 9 ÷ c 11 ( )sin 2 y ÷ c 10 ÷ c 12 ( )cos 2 y _ ¸ ; C (1) 66 ÷ C (2) 66 = 2 k y t 2 c s y n t c 13 ÷ c 15 ( )sin 2 y ÷ c 14 ÷ c 16 ( )cos 2 y _ ¸ ; C (1) 12 ÷ C (2) 12 = k y t 2 c s y n t c 17 ÷ c 19 ( )sin 2 y ÷ c 18 ÷ c 20 ( )cos 2 y _ ¸ ; C (1) 16 ÷ C (2) 16 = k y t 2 c s y n t sin y cos y ÷c 21 ÷ c 22 [ [ = 0; C (1) 26 ÷ C (2) 26 = k y t 2 c s y n t sin y cos y ÷c 23 ÷ c 24 [ [ = 0 : (5:69) Thus, C (1) pq ÷ C (2) pq = 2a 1 D 1 ÷ 2b 1 D 2 a 4 D 1 ÷ b 4 D 2 0 2a 2 D 1 ÷ 2b 2 D 2 0 Symm 2a 3 D 1 ÷ 2b 3 D 2 _ ¸ _ _ ¸ _; (5:70) where the superscripts for denoting damage mode have beendroppedfor convenience, and a i and b i , i = 1, 2, 3, 4 are the two sets of four material constants, given by a 1 = c 5 ÷ c 7 ; a 2 = c 9 ÷ c 11 ; a 3 = c 13 ÷ c 15 ; a 4 = c 17 ÷ c 19 ; b 1 = c 6 ÷ c 8 ; b 2 = c 10 ÷ c 12 ; b 3 = c 14 ÷ c 16 ; b 4 = c 18 ÷ c 20 : (5:71) Here, a i and b i are functions of y. Denote 159 5.3 Synergistic damage mechanics (SDM) a 1 y ( ) = a 1 sin 2 y ÷ b 1 cos 2 y; a 2 y ( ) = a 2 sin 2 y ÷ b 2 cos 2 y; a 3 y ( ) = a 3 sin 2 y ÷ b 3 cos 2 y; a 4 y ( ) = a 4 sin 2 y ÷ b 4 cos 2 y : (5:72) Then, C (1) pq ÷ C (2) pq =D y 2a 1 y ( ) a 4 y ( ) 0 2a 2 y ( ) 0 Symm 2a 3 y ( ) _ ¸ _ _ ¸ _; (5:73) where D y = k y t 2 c s y n t : (5:74) Rewrite Eq. (5.72) as a i y ( ) = a i sin 2 y ÷ b i cos 2 y = a i sin 2 y 1 ÷ b i a i cot 2 y _ _ : (5:75) Consider for the moment the case when a i _ b i . Then, b i a i cot 2 y _ 1 for p 4 _ y _ p 2 : (5:76) Also, it can be expected that b i a i cot 2 y ¸ 1 for p 3 _ y _ p 2 ; i:e:; a i y ( ) ~ a i for p 3 _ y _ p 2 : (5:77) For this case, we have laminate stiffness matrix as C pq = E 0 x 1 ÷ n 0 xy n 0 yx n 0 xy E 0 y 1 ÷ n 0 xy n 0 yx 0 E 0 y 1 ÷ n 0 xy n 0 yx 0 Symm G 0 xy _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ _ ¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸ _ ÷ D y 2a 1 a 4 0 2a 2 0 Symm 2a 3 _ ¸ ¸ _ _ ¸ ¸ _ ; (5:78) which is of identical form to the expression for 90 · cracking in a cross-ply laminate, see Eq. (5.56). The moduli can be finally obtained using the relations in Eq. (5.57). The resultant expressions will be of the same form as in Eq. (5.58), except that kt c 2 /st is now replaced with D y . 160 Macro-damage mechanics The overall SDM procedure for this laminate is sketched in Figure 5.10. As illustrated, it combines micromechanics with CDM for complete evaluation of the structural response. Micromechanics involves analysis to determine CODs in cracked plies within a RVE (or unit cell, if applicable) of [0 m /±y n /0 m/2 ] s layup, from which the constraint effect is evaluated for different y and/or m. The constraint effect is carried over in the CDM formulation through the con- straint parameter. In a separate step, the damage constants a i are determined from experimental data for a reference laminate, which is chosen here to be [0/±90 8 /0 1/2 ] s . A cross-ply laminate of the same class as the ply layup and material in consideration is a good choice for the reference laminate because either the experimental data are often available or can be obtained by using any of the damage models for 90 · -ply cracking as described in the previous chapter. For evaluation of damage constants, expressions in Eq. (5.59) shall be used after replacing kt c 2 /st by D y . With the values of the damage constants and b = k y =k 90 known, the stiffness–damage relations given by Eq. (5.78) are employed to predict stiffness degradation with crack density. The overall structural behavior can be finally analyzed using the reduced stiffness properties for the laminate. For computation of average COD values, micromechanics can be performed using a suitable FE analysis. The 3-D FE model for [0 m /±y n /0 m/2 ] s layup, along COMPUTATIONAL MICROMECHANICS SYNERGISTIC DAMAGE MECHANICS STRUCTURAL ANALYSIS Analyze overall structural response to external loading using the reduced stiffness properties Use SDM to determine stiffness reduction in present laminate configuration [0 m / ±q n /0 m/2 ] s Structural scale: Meso Structural scale: Macro Structural scale: Micro – EXPERIMENTAL/ COMPUTATIONAL Determine COD and constraint parameter(s) ; = = + ( ( ) ( ( ) ∆u 2 ∆u 2 ∆u 2 n = κ q κ 90 ±q n ±q n +q n q b Evaluate damage constants using available data for reference laminate configuration [0/90 8 /0 1/2 ] s 90 8 ( ) ∆u 2 ( ) ∆u 2 Figure 5.10. Flowchart showing the multiscale synergistic methodology for analyzing damage behavior in a class of symmetric laminates with layup [0 m /±y n /0 m/2 ] s containing ply cracks in the +y and ÷y layers. 161 5.3 Synergistic damage mechanics (SDM) with boundary conditions and coordinate systems, is shown in Figure 5.11 [29, 31, 32]. As can be seen from Figure 5.12, the computed CODs averaged over the thickness of the cracked plies agree quite well with the experimental data over Figure 5.11. Representative unit cell for COD computation for laminates. Reprinted, with kind permission, from Int J Solids Struct, Vol. 45, C.V. Singh and R. Talreja, Analysis of multiple off-axis ply cracks in composite laminates, pp. 4574–89, copyright Elsevier (2008). 20 30 40 50 60 70 80 90 Ply orientation (q) 0 2 4 6 8 C O D ( µ m ) FEM Experiment Figure 5.12. Variation of average COD with change in ply orientation for a [0 m /±y n /0 m/2 ] s glass/epoxy laminate. Reprinted, with kind permission, from Int J Solids Struct, Vol. 45, C.V. Singh and R. Talreja, Analysis of multiple off-axis ply cracks in composite laminates, pp. 4574–89, copyright Elsevier (2008). 162 Macro-damage mechanics the whole range of ply orientations considered. This suggests that the 3-DFEanalysis is an accurate tool to evaluate the constraint parameter. The profiles of COD nor- malized with t c through the ply thickness are shown in Figure 5.13. As expected, for cross-ply laminates, the profile is symmetric about mid-plane of the cracked ply and consequently the maximum COD occurs at the mid-plane of the cracked layer. However, due to the difference in constraint from the surrounding material, this COD profile is different from an elliptical profile for a single crack in an infinite isotropic elastic medium subjected to a uniform far-field stress (see Figure 5.13 (b)). Thus, the magnitude of the average COD for a 90 · crack is different fromthat for an elliptical crack. For other ply orientations, the crack surface displacements are not symmetric about the mid-plane as shown in Figure 5.13 (c)–(d). This asymmetry increases as we go away fromcross-ply laminates (y =90 · ) and maximumCODdoes not occur midway through the thickness of the cracked y layers. The aspect ratio of COD profiles, g = Áu 2 ( ) max _ Áu 2 , varies from 1.33 for y = 90 · to 1.40 for y = 40 · (Figure 5.13 (a)), which is different from the aspect ratio of 1.273 ( = 4/p) for an (a) 1 0 3 u 2 / t c (b) 1 0 3 u 2 / t c (c) 1 0 3 u 2 / t c (d) 1 0 3 u 2 / t c Figure 5.13. COD profiles for cracked plies in [0/±y 4 /0 1/2 ] s laminates: (a) CODs averaged over +y- and –y-plies; (b) Crack profile for 90º-ply crack compared with an elliptic profile for an isotropic medium; (c), (d) COD profiles for +y and –y separately: (c) y = 70º, (d) y = 40º. Parts (c), (d) depict the asymmetry of opening displacements for off-axis laminates, especially at a ply orientation farther from y = 90 · . Reprinted, with kind permission, from Int J Solids Struct, Vol. 45, C.V. Singh and R. Talreja, Analysis of multiple off-axis ply cracks in composite laminates, pp. 4574–89, copyright Elsevier (2008). 163 5.3 Synergistic damage mechanics (SDM) elliptic profile. Here (Du 2 ) max represents the maximum COD and Áu 2 represents the CODaveragedover the thickness direction. Numerically, the COD, Du 2 , is computed by taking the difference of the node displacements on either side of the crack along a direction transverse to the local fiber direction (assuming the crack traverses parallel to the fiber). The elastic moduli E x and n xy predicted by the SDM approach are compared with the experimental data for y = 70 · and 55 · in Figure 5.14. The agreement with data is about the same as that obtained by using experimentally measured COD values to evaluate the constraint parameter. For y = 55 · , the stiffness reduction is caused by matrix cracking as well as shear-induced damage in off-axis plies, as discussed in [33], where a procedure for calculating the stiffness reduction due to shear damage was described. SDM predictions for this orientation thus include both effects. It is interesting to note that the stiffness predictions using experimentally measured COD values are subject to scatter, which is inherent in testing. Furthermore, the experimental procedure requires a specialized test set-up [33, 34], which is costly and takes a certain amount of training to operate. The 3-D FE computations on a unit cell, on the other hand, are easy to perform and can provide accurate values of CODs. The SDM approach is quite handy when the relative stiffness or thickness of cracked and supporting plies change. The difference in stiffness changes for 0 x E E x 0 x E E x (a) θ=70° (b) θ=70° n xy n xy 0 n xy n xy 0 θ=55° (d) (c) θ=55° Figure 5.14. Variation of longitudinal modulus and Poisson’s ratio for [0/±y 4 /0 1/2 ] s laminates: (a), (b) for y = 70 · ; (c), (d) for y = 55 · . Reprinted, with kind permission, from Int J Solids Struct, Vol. 45, C.V. Singh and R. Talreja, Analysis of multiple off-axis ply cracks in composite laminates, pp. 4574–89, copyright Elsevier (2008). 164 Macro-damage mechanics laminates with different ply thickness or axial stiffness are due to difference in the relative constraint from the supporting plies, and thus can be characterized by the changes in the constraint parameter. In [29] a parametric study was conducted to study the effect of changes in ply thicknesses (m and n) and axial stiffness ratio r = E ±y A _ E 90 A over CODs of [0 m /±y n /0 m/2 ] s laminates, where E A ±y and E A 90 represent the axial stiffness of ±y-plies and the 90 · -plies, respectively. Figure 5.15 shows the variation of average COD for changes in these parameters. The computed average CODs are given by the following parametric equation Áu 2 _ _ ±y n = U:f 1 y ( ):f 2 r ( ):f 3 m ( ):f 4 n ( ); (5:79) where U is the average COD for the reference laminate [0/90 8 /0 1/2 ] s , and f 1 y ( ) = sin 2 y; f 2 r ( ) = r ÷c 1 ; f 3 m ( ) = c 2 m ÷ c 3 ; f 4 n ( ) = c 4 n c 5 ; (5:80) are fitted functions where fitting constants (unitless) c 1 –c 5 are given as: c 1 = 0.0871; c 2 = 0.1038; c 3 = 0.8949; c 4 = 0.247; c 5 = 0.99 for [0 m /±y n /0 m/2 ] s glass/ epoxy laminates. The parametric study described above enables us to predict stiffness degrad- ation in off-axis laminates with different geometry and stiffness values. For example, one can consider a laminate with stiffer outer plies. The variation of engineering moduli E x and n xy for different stiffness ratios (r) for a [0/±70 4 /0 1/2 ] s laminate is shown in Figure 5.16(a). As expected, stiffer outer plies cause less severe degradation in the moduli. In contrast to changing the stiffness of outer plies, one can vary the number of constraining plies (m) or cracked plies (n). The effect of number of cracked plies on the change in stiffness moduli is shown in Figure 5.16(b). These results indicate that the cracking ply thickness, i.e., crack size, has significant effect on stiffness degradation while the thickness of the constraining plies as well as the change in axial stiffness ratio r have small effect. 5.3.2 Three damage modes We will now consider a special case of cracking in three off-axis plies, of which two are symmetrically opposite of each other, and the third is placed along the transverse direction. Thus cracking is in the +y, –y, and 90 · plies. For this case the stiffness matrix of the damaged laminate is given by C pq = C 0 pq ÷ C (1) pq ÷ C (2) pq ÷ C (3) pq ; (5:81) where C pq 0 is given by Eq. (5.52), and C pq (1) +C pq (2) is givenby Eq. (5.73). The components of the third damage mode (a = 3) corresponding to cracking in 90 · -ply are 165 5.3 Synergistic damage mechanics (SDM) (b) (a) (c) Figure 5.15. Variation of average COD for [0 m /±y n /0 m/2 ] s glass/epoxy laminates with (a) axial stiffness ratio, r (for m = 1, n = 4); (b) number of cracked plies, n; and (c) number of constraining plies, m. Reprinted, with kind permission, from Int J Solids Struct, Vol. 45, C.V. Singh and R. Talreja, Analysis of multiple off-axis ply cracks in composite laminates, pp. 4574–89, copyright Elsevier (2008). 166 Macro-damage mechanics D (3) 1 = k 90 t 2 90 s 90 t ; D (3) 2 = D (3) 6 = 0: (5:82) The integrity bases (Eq. (5.61)) have additional components for D (3) 1 . The free energy function thus gets the following terms added to Eq. (5.62) rc a = 3 ( ) = a / 1 e 2 1 D (3) 1 ÷ a / 2 e 2 2 D (3) 1 ÷ a / 3 e 2 6 D (3) 1 ÷ a / 4 e 1 e 2 D (3) 1 ; (5:83) where a / i ; i = 1; 2; 3; 4 are additional material constants. Substituting Eq. (5.83) into Eq. (5.48), we obtain C (3) pq =D (3) 1 2a / 1 a / 4 0 2a / 2 0 Symm 2a / 3 _ ¸ _ _ ¸ _; (5:84) where the contribution to the shear components is zero. It is important to emphasize here that the relative location of different damage modes in the whole laminate will cause different losses in stiffness due to damage in the laminate. To illustrate this let us consider two specific examples of laminates with damage modes consideredinthe present section, viz., +y, –y, and 90 · . Figure 5.17(a)–(b) shows representative unit cells for laminates with [0 m /±y n /90 r ] s (a) (c) (d) (b) 1.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 r =0.5 r =1 r =2 n=1 n=2 n=4 n=1 n=2 n=4 r =5 r =0.5 r =1 r =2 r =5 0.8 Crack density (1/mm) Crack density (1/mm) Crack density (1/mm) 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 1.2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.2 1 0.8 0.6 0.4 0.2 0 Crack density (1/mm) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.2 1 0.8 0.6 0.4 0.2 0 E x n xy n xy n xy E x 0 E x E x 0 0 n xy 0 Figure 5.16. Effect of change in axial stiffness ratio (r): (a), (b), and cracked ply thickness (n): (c), (d), on stiffness degradation in [0 m /±70 n /0 m/2 ] s glass/epoxy laminates. Reprinted, with kind permission, from Int J Solids Struct, Vol. 45, C.V. Singh and R. Talreja, Analysis of multiple off-axis ply cracks in composite laminates, pp. 4574–89, copyright Elsevier (2008). 167 5.3 Synergistic damage mechanics (SDM) and [0 m /90 r /–y n /+y n ] s configurations, respectively. The boundary conditions are shown for the 3-D FE computation of average COD values. The global laminate (X, Y, Z) and local crack plane (x 1 , x 2 , x 3 ) coordinate systems are also shown. Considering first the case of a [0 m /±y n /90 r ] s laminate, we note that ±y modes occur twice in the whole laminate, above and belowthe mid-plane of the laminate, whereas 90 · mode occurs only once. Thus, ÁC pq = C pq ÷ C 0 pq = N a=1 C (a) pq is given by ÁC pq = 2 C (1) pq ÷y ( ) ÷ C (2) pq ÷y ( ) _ _ ÷ C (3) pq 90 ( ) : (5:85) Collecting terms from Eq. (5.73) and Eq. (5.84), while assuming the a i ’s to be independent of y, we get ÁC pq = 2D y 2a 1 a 4 0 2a 2 0 Symm 2a 3 _ _ _ _ ÷ D 90 2a / 1 a / 4 0 2a / 2 0 Symm 2a / 3 _ _ _ _ ; (5:86) where, for the laminate configuration considered, D y = k y 2nt 0 ( ) 2 s y n t ; D 90 = k 90 2rt 0 ( ) 2 s 90 t ; (5:87) where t 0 denotes the thickness of a single ply. The special case when y = 90 · , and DC pq as given by Eq. (5.86) should be equal to that given by a single 90 · mode with crack size of (4n+2r) plies. Consider, for example, the DC 11 term. If we assume that the normal crack spacing is the same in all cracked plies, then DC 11 for a [0 m /±y n /90 r ] s laminate at y = 90 · from Eq. (5.86) is given by ÁC 11 = 2 C (1) 11 ÷ C (2) 11 _ _ ÷ C (3) 11 = 4D y [ y=90 a 1 90 ( ) ÷ 2D 90 a / 1 = 4k y [ y=90 2nt 0 ( ) 2 s 90 t a 1 ÷ 2k 90 2rt 0 ( ) 2 s 90 t a / 1 = 8t 2 0 s 90 t 2n 2 k y [ y=90 a 1 90 ( ) ÷ r 2 k 90 a / 1 _ ¸ : (5:88) (a) (symmetry about mid-plane) 0Њ layer +q layer (u) x=21 =u 0 –q layer y x z 90Њ layer (w) z =0 =0 (u) x=0 =0 (b) (symmetry about mid-plane) 0Њ layer +q layer (u) x=21 =u 0 (u) x=0 =0 –q layer y x z 90Њ layer (w) z=0 =0 Figure 5.17. Representative unit cell for (a) [0 m /±y n /90 r ] s and (b) [0 m /90 r /±y n ] s laminate configurations. 168 Macro-damage mechanics Since, for y = 90 · , [0 m /±y n /90 r ] s is equivalent to [0/90 2n+r ] s , we can consider their stiffness changes to be the same. Thus, DC 11 can also be written, using Eq. (5.84), as ÁC 11 = C (3) 11 = 2a / 1 D 1 = k 90 4n÷2r 4n ÷ 2r ( )t 0 [ [ 2 s 90 t :2a / 1 ; (5:89) where the sub-subscript “4n+2r” in k 90 4n÷2r represents the crack size for the 90 · mode in a [0/90 2n+r ] s laminate. Equating DC 11 from Eq. (5.88) and Eq. (5.89), we have 2n 2 k y [ y=90 a 1 90 ( ) ÷ r 2 k 90 a / 1 = k 90 4n÷2r 2n ÷ r ( )t 0 [ [ 2 a / 1 ; (5:90) i.e., a 1 90 ( ) = k 90 4n÷2r 2n ÷ r ( ) 2 ÷ r 2 k 90 2n 2 k y [ y=90 a / 1 : (5:91) Generalizing, we can write the interrelationship between two sets of constants as a i = k 90 4n÷2r 2n ÷ r ( ) 2 ÷ r 2 k 90 2n 2 k y [ y=90 a / i i = 1; 2; 3; 4: (5:92) Substituting Eq. (5.92) into Eq. (5.86), DC pq for a damaged [0 m /±y n /90 r ] s laminate is given by ÁC pq = D 2a / 1 a / 4 0 2a / 2 0 Symm 2a / 3 _ ¸ _ _ ¸ _; (5:93) where D = 4t 2 0 t 1 s y n k y k y [ y=90 2n ÷ r ( ) 2 k 90 4n÷2r ÷ r 2 k 90 _ _ ÷ r 2 k 90 s 90 _ _ ; (5:94) where the constraint parameters are given by k y = Áu y _ _ ±y 2n 2nt 0 ; k 90 4n÷2r = Áu y _ _ 90 4n÷2r 4n ÷ 2r ( )t 0 ; k 90 = Áu y _ _ 90 2r 2rt 0 : (5:95) Consider now the case of the [0 m /90 r /±y n ] s laminate configuration. It must be noted that, unlike [0 m /±y n /90 r ] s laminates, ±y damage modes in this case are centrally placed, thereby the corresponding equivalent crack size is 4nt 0 (with averaging over two +y and two –y layers). On the other hand, the crack size for the 90 · damage mode is rt 0 . The derived stiffness–damage relationships retain the form of Eq. (5.93). However, D in this case is given by D = 2t 2 0 t 1 s y n k y k y [ y=90 2 2n ÷ r ( ) 2 k 90 4n÷2r ÷ r 2 k 90 _ _ ÷ r 2 k 90 s 90 _ _ (5:96) and the corresponding constraint parameters are now given by 169 5.3 Synergistic damage mechanics (SDM) k y = Áu y _ _ ±y 2n 4nt 0 ; k 90 4n÷2r = Áu y _ _ 90 4n÷2r 4n ÷ 2r ( )t 0 ; k 90 = Áu y _ _ 90 r rt 0 : (5:97) The SDM procedure for this laminate configuration is quite similar to that described earlier for the [0 m /±y n /0 m/2 ] s layup in Figure 5.10. The damage constants are evaluated from Eq. (5.59) after replacing kt c 2 /st by D by using the stiffness degradation data for the reference laminate, which is chosen as [0/90 3 ] s in this case. Depending on the specific 90 · -ply placement, the corresponding constraint par- ameters are calculated using expressions in Eq. (5.95) or Eq. (5.97) by using the CODs computed from a suitable 3-D FE model. FE modeling details and COD computations are described in detail in [30, 32]. One important consideration in COD computation is the interaction between the families of cracks in different orientations which can modify the crack displacements and consequently the stiffness changes due to ply cracking [30]. This interaction can also modify the damage evolution as illustrated in the next chapter. In Figure 5.18, SDM predic- tions of stiffness moduli of [0/±70/90] s and [0/±55/90] s glass/epoxy laminates are compared with the independent calculations from 3-D FE analysis (see [30–32] for details on procedure to calculate stiffness changes using 3-D FE analysis). The SDM predictions for a quasi-isotropic laminate are shown in Figure 5.19. In the experiments it was observed that the laminates failed before the cracks in the plies could grow fully through the laminate width. The effect of partially grown cracks on stiffness changes is accounted for in the SDM analysis by reducing the crack density for layers by a “relative density factor,” defined as r r = Actual surface area for partial cracks Surface area for full cracks : (5:98) To find the actual surface area of partial cracks, the information regarding their actual length (along lamina width) is necessary. Since such data were not reported in the above experimental study [35], the results are presented for two assumed values of r r : 0.25 and 0.5. 5.4 Viscoelastic composites with ply cracking Polymers used as matrix materials in fiber-reinforced composites usually display viscoelastic behavior. However, the time-dependent behavior of the matrix does not translate fully in the composite and while it is isotropic in the polymer, the directionality of the fibers makes it anisotropic in the composite. Viscoelasticity models for composite materials, particularly for high temperature applications where the time dependency is more pronounced, are needed. The composite time- dependent response in most cases can be approximated by linear viscoelasticity. When ply cracking occurs the viscoelastic response can still be linear but with modified time-dependency. 170 Macro-damage mechanics θ=55Њ θ=55Њ θ=70Њ (a) θ=70Њ (b) Figure 5.18. Variation of longitudinal modulus and Poisson’s ratio for [0/±y/90] s laminates: (a) y = 70 · , (b) y = 55 · . The results are compared with independent FE calculations. SDM predictions are made for two cases: no interaction between ±y and 90 · cracks, and with maximum interaction between them. Reprinted, with kind permission, from Mech Mater, Vol. 41, C.V. Singh and R. Talreja, A synergistic damage mechanics approach for composite laminates with matrix cracks in multiple orientations, pp. 954–68, copyright Elsevier (2009). (a) (b) 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0 0.2 0.4 0.6 0.8 1 1.2 90Њ-Crack density (1/mm) 0 0.2 0.4 0.6 0.8 1 1.2 90Њ-Crack density (1/mm) +45Њ cracks +45Њ cracks –45Њ cracks –45Њ cracks [0/90/ ϯ45] s [0/90/ ϯ45] s [0/90] s [0/90] s E x E 0 x Test 1: [0/90/–45/+45] s Test 2: [0/90/–45/+45] s Test 3: [0/90/–45/+45] s Test 1: [0/90] s Test 2: [0/90] s Test 3: [0/90] s Test 4: [0/90] s Test 1: [0/90/–45/+45] s Test 2: [0/90/–45/+45] s Test 3: [0/90/–45/+45] s Test 1: [0/90] s Test 2: [0/90] s Test 3: [0/90] s n xy n xy 0 Figure 5.19. Stiffness reduction for quasi-isotropic [0/90/÷45/+45] s laminate compared with experimental data [35]: (a) longitudinal modulus, (b) Poisson’s ratio. Solid lines represent SDM predictions using assumed values of r r . Reprinted, with kind permission, from Mech Mater, Vol. 41, C.V. Singh and R. Talreja, A synergistic damage mechanics approach for composite laminates with matrix cracks in multiple orientations, pp. 954–68, copyright Elsevier (2009). 171 5.4 Viscoelastic composites with ply cracking One approach to treating the coupling of viscoelasticity and damage in com- posite materials is by internal variables based continuum models. Most models of this nature have been applied to particulate composites. One model that addresses composite laminates is by Schapery and Sicking [36]. This model extends the concept of work potential developed for elastic solids to viscoelastic composite laminates by using the pseudo variables defined by Schapery [37]. However, the Schapery–Sicking model makes certain assumptions that limit the usefulness of the model. It is assumed that the shear modulus and the transverse modulus of a ply are linearly related, and that the ply is transversely isotropic. Furthermore, modeling is done at the ply level, neglecting the constraint effects of the neighboring plies. The model developed later by Kumar and Talreja [38] does not suffer from these limitations. The CDM-based modeling of linear viscoelastic response of composites with damage in accordance with this model will be discussed below. Using the well-known Boltzman superposition principle the constitutive law for a viscoelastic material can be represented as s ij = _ t 0 C ijkl t ÷ t ( ) @e kl @t dt; (5:99) where C ijkl (t) is the relaxation modulus tensor. Taking the Laplace transform of both sides and using the convolution theorem, we obtain s ij = ~ C ijkl e kl ; (5:100) where the Laplace transform f s ( ) of a function f(t) is defined as f s ( ) = _ · 0 e ÷st f t ( ) dt; (5:101) and ~ C ijkl = sC ijkl : (5:102) The constitutive equations (5.100) in the Laplace domain are similar to the relationships of linear elasticity, except for the dependency on the trans- formation parameters. Additionally, the equilibrium equations and strain–dis- placement equations also retain their form in the Laplace domain. This correspondence between linear elasticity and linear viscoelasticity is the so- called Correspondence Principle [39], and it applies to solving boundary value problems as long as the boundary conditions do not change in time. Thus, if ply cracks are present in a composite laminate, then the Correspond- ence Principle can still be applied by viewing the cracked laminate as a homo- geneous solid in which the internal boundaries (cracks) retain the same boundary conditions in time. This implies that the cracks do not grow or heal in time. We shall therefore address the linear viscoelastic response of laminates 172 Macro-damage mechanics at a given (fixed) damage state. Specifically, we will express the relaxation modulus tensor in Eq. (5.99) as a function of damage. Following [38] a pseudo energy density function is defined in the Laplace domain in terms of transformed strain and damage variables, such that s ij = @ W e ij ; D (a) ij _ _ @e ij : (5:103) It is noted that the above equation is valid for a fixed ply crack density. The time- dependent deformation of the plies themselves will result in time-varying con- straint on the cracks, leading to the crack surface separation changing in time. For illustration, let us consider a cross-ply laminate with transverse cracks, Figure 5.5, with y = 90 · . In this case the damage mode tensor has a single nonzero component, which in the Laplace domain takes the form [38] D 11 = k s ( )t 2 90 s t t T : (5:104) where k s ( ) represents the constraint parameter in the Laplace domain, t 90 is the thickness of the cracked layer, t T is the total laminate thickness, and s t is the crack spacing. Using the same procedure as in Section 5.1.4, Kumar and Talreja [38] derived the following transformed constitutive relationships (in the Voigt notation) as s 1 s 2 s 6 _ ¸ _ ¸ _ _ ¸ _ ¸ _ = ~ C 11 ~ C 12 0 ~ C 12 ~ C 22 0 0 0 ~ C 66 _ ¸ ¸ _ _ ¸ ¸ _ e 1 e 2 e 6 _ ¸ _ ¸ _ _ ¸ _ ¸ _ ; (5:105) where ~ C pq = ~ C 0 pq ÷ ~ C (1) pq : (5:106) Here ~ C 0 pq is the transformed relaxation modulus of the cross-ply laminate without cracks and ~ C (1) pq , given by the following, is transverse cracking (one mode of damage, a = 1) ~ C (1) pq = D 11 2g 11 g 12 0 g 12 2g 22 0 0 2g 66 _ ¸ _ _ ¸ _; (5:107) where g 11 , g 12 , g 22 , and g 66 are material constants that appear as coefficient terms in the polynomial expansion of the pseudo strain energy density function W. The damage function D 11 is given by Eq. (5.104). The time-dependency of the crack opening displacement has been handled in two ways. In Kumar and Talreja [38] the function W was assumed to depend on 173 5.4 Viscoelastic composites with ply cracking the initial (t = 0) value of the damage tensor, thereby absorbing all time depend- ency in the presence of damage into the coefficient terms g 11 , etc. Later Varna et al. [41] chose to retain the time dependency in the damage tensor and dealt with the time-varying constraint to the COD explicitly. Taking the inverse Laplace transform of Eq. (5.106), one obtains C pq t ( ) = C 0 pq t ( ) ÷ 2r n t 90 t T k ij t ( ); (5:108) where r n = t 90 2s t (5:109) is the normalized crack density, and k 11 t ( ) = 2L ÷1 kg 11 =s ( ); k 12 t ( ) = L ÷1 kg 12 =s ( ); k 22 t ( ) = 2L ÷1 kg 22 =s ( ); k 66 t ( ) = 2L ÷1 kg 66 =s ( ) ; (5:110) where L –1 (*) represents the inverse Laplace transform. Note that k = k 0 , its value at time t = 0, if all time dependency is assumed to be in g 11 , etc. in the above equations. The functions k 11 , k 12 , and k 22 in Eq. (5.110) can be determined by procedures similar to those described above for elastic constants. Here these three unknown functions, which are decoupled from k 66 , can be evaluated from the time-varying differences of the axial modulus E x (t) and Poisson’s ratio n xy (t) from their initial values, and by assuming no change in the transverse modulus E y (t), as done in Kumar and Talreja [38]. The time-variation of the material constants can be determined from experimental data, if available, or by a micromechanics approxi- mation. The micromechanics can be analytical, if possible, or numerical, e.g., by a finite element model. In Kumar and Talreja [38] cross-ply laminates of given linear viscoelastic ply properties were considered for validation of the CDM approach described above. The functions k 11 (t), k 12 (t), and k 22 (t) were determined from the calculated visco- elastic response of a [0/90 2 ] s laminate at a fixed crack density of 0.4 cracks/mm. These functions were then used to predict the time variations of the axial modulus and Poisson’s ratio at other crack densities and for other cross-ply laminate configurations of the same material. The predictions were compared with inde- pendently calculated time variations of the properties by a finite element model and an analytical micromechanics model reported in Kumar and Talreja [40]. Predictions agreed well in all cases. Varna et al. [41] demonstrated the use of SDM for linear viscoelastic response predictions by explicitly treating the COD variations in time. The time dependency of COD, i.e., the constraint parameter k and its counterpart in the Laplace domain k, were calculated by a FE model. These were then inserted in 174 Macro-damage mechanics Eq. (5.110) to determine k 11 (t), k 12 (t), and k 22 (t). Predictions thus made agreed well with independently calculated viscoelastic response at different crack densities and different cross-ply laminate configurations. A parametric study was also performed in Varna et al. [41] to determine a master function for COD variation. That function has the following form k t ( ) = a ÷ b ÷ c t c 2t s ÷ 1 _ _ _ _ E 2 t ( ) E 1 t ( ) _ _ d ; (5:111) where a, b, c, and d are constants, E 1 and E 2 are the axial and transverse values, respectively, of the Young’s modulus of the ply material, and t s here is the thickness of the 0 · -plies in the cross-ply laminate. Note that if more generally a laminate of configuration [S/90 n ] s is used, then t s will be the thickness of the sublaminate denoted by S. For the nonlinear viscoelastic response of composites with damage another approach is needed since the Correspondence Principle does not apply. A CDM approach for this case was developed by Ahci and Talreja [42]. The material system in the experimental study performed there was a carbon fiber (T-650–35) fabric (8-harness satin, 3K tow size, with UC-309 sizing) in HFPE-II polyimide thermosetting resin. It was demonstrated that the viscoelastic response of the virgin material was linear within a range of stress and temperature, beyond which it became nonlinear. The microcracking within the fiber bundles further enhanced the nonlinearity of the viscoelastic response. Since the tests measured strain response under prescribed stress, the CDM formulation was appropriately modi- fied to consider stress as an independent variable. Thus the Gibbs free energy function G was formulated such that the strain response can be written as e ij = ÷ @G s kl ; D (a) mn ; T; g s _ _ @s kl ; (5:112) where the damage mode tensors D ij (a) are as defined by Eq. (5.20) and g s are the viscoelastic internal state variables. Note that the damage mode tensor is kept time independent by taking its initial value at time t = 0, analogous to the linear viscoelastic case in the CDM approach by Kumar and Talreja [38] treated above. Using polynomial expansion in terms of the integrity bases of the function G, similar to the procedures for the linear elastic and viscoelastic cases above, and by incorporating formulation of polymer viscosities available in the literature, the time-dependent in-plane strain response of an orthotropic composite with trans- verse cracks is derived as [42] e p = C E pq ÷ C D pq D ÷ C g pq g ÷ C E pq Dg _ _ s q ; (5:113) where p, q = 1, 2, and 6, and superscripts E, D and g to the compliance matrix C denote the elastic, damage, and viscous contributions, respectively, and the last term in the parenthesis stands for the interactive contribution between damage 175 5.4 Viscoelastic composites with ply cracking and viscosity. The damage variable D = D 11 , which is the only nonzero component of the damage mode tensor for transverse cracks. The material constants in the four matrices in Eq. (5.113) must be evaluated before these relationships can be used to predict the nonlinear viscoelastic response. The first matrix is simply the elastic response matrix and can be found by recording the instantaneous strain response to imposed stresses. The second matrix can be found experimentally in conditions where the viscoelasticity is negligible, and the procedure could be one of those discussed above for the case of damage in elastic composites. In the absence of damage, the third matrix in Eq. (5.113) represents the time-dependent response. Finally, the fourth matrix needs determining to see how damage enhances the viscoelastic deformation. A procedure for the complete evaluation of the material constants involved is thus far from trivial. An experimental procedure aided by finite element modeling was developed in Ahci and Talreja [42]. The resulting characterization of non- linear viscoelasticity and damage was used to illustrate the effects of temperature and stress levels as well as the nature of the damage–viscoelasticity coupling. 5.5 Summary This chapter has presented the basic concepts of continuum damage mechanics and illustrated its application to damage in composite materials. The classical field of continuum thermodynamics combined with internal variables characterization of damage provides a powerful tool to describe materials response affected by damage. In its conventional form the CDM relies on material parameter identification by experimental data. This has been a burden that has been increasingly challenging to carry because of the multiplicity of damage modes and the ensuing complexity of response changes. The extension of CDM developed by judiciously aiding it with selected micromechanics solutions (analytical or numerical) has eased the burden and thereby increased the attractiveness of CDM. This integrative approach of combining CDM and micromechanics has been named synergistic damage mechanics. This chapter has illustrated the application of SDM to multiple (up to three) damage modes in laminates. Results of parametric studies have also been presented to offer further insight into the consequences of multiple damage modes. Although most applications of polymer-based composites have been in struc- tures operating at temperatures safely below the glass transition temperature of the polymer, there are instances such as in jet engine casings where high tempera- tures are induced. Viscoelastic response of the composite in such cases takes place and must be addressed by itself and when damage occurs. The CDM and SDM methodologies have been presented for linear viscoelastic composites with damage. At high temperatures and with extensive damage the viscoelastic response can become nonlinear. This case has added complexity in identification of material constants. 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Varna, Physical interpretation of parameters in synergistic continuum damage mechanics model for laminates. Compos Sci Technol, 68:13 (2008), 2592–600. 21. A. J. M. Spencer, Theory of invariants. In Continuum Physics, ed. C. A. Eringen. (New York: Academic Press, 1971), pp. 239–353. 22. J. Adkins, Symmetry relations for orthotropic and transversely isotropic materials. Arch Ration Mech Anal, 4:1 (1960), 193–213. 177 References 23. R. Talreja, Transverse cracking and stiffness reduction in composite laminates. J Compos Mater, 19:4 (1985), 355–75. 24. A. L. Highsmith and K. L. Reifsnider, Stiffness-reduction mechanisms in composite laminates. In Damage in Composite Materials, ASTM STP 775, ed. K. L. Reifsnider. (Philadelphia, PA: ASTM, 1982), pp. 103–17. 25. J. Varna, N. V. Akshantala, and R. Talreja, Crack opening displacement and the associ- atedresponse of laminates withvarying constraints. Int J Damage Mech, 8:2 (1999), 174–93. 26. J. Varna, R. Joffe, and R. Talreja, A synergistic damage-mechanics analysis of trans- verse cracking in [±y/90 4 ] s laminates. Compos Sci Technol, 61:5 (2001), 657–65. 27. J. Varna, R. Joffe, and R. Talreja, Mixed micromechanics and continuum damage mechanics approach to transverse cracking in [S,90(n)](s) laminates. Mech Compos Mater, 37:2 (2001), 115–26. 28. C. V. Singh and R. Talreja, Damage mechanics of composite laminates with transverse matrix cracks in multiple orientations. In 48th AIAA SDM Conference, Honolulu, Hawaii, USA. (Reston, VA: AIAA, 2007). 29. C. V. Singh and R. Talreja, Analysis of multiple off-axis ply cracks in composite laminates. Int J Solids Struct, 45:16 (2008), 4574–89. 30. C. V. Singh and R. Talreja, A synergistic damage mechanics approach for composite laminates with matrix cracks in multiple orientations. Mech Mater, 41:8 (2009), 954–68. 31. S. Li, C. V. Singh, and R. Talreja, A representative volume element based on transla- tional symmetries for FE analysis of cracked laminates with two arrays of cracks. Int J Solids Struct, 46:7–8 (2009), 1793–804. 32. C. V. Singh, Multiscale modeling of damage in multidirectional composite laminates. Ph.D. thesis, Texas A&M University, College Station, TX (2008). 33. J. Varna, R. Joffe, N. V. Akshantala, and R. Talreja, Damage in composite laminates with off-axis plies. Compos Sci Technol, 59:14 (1999), 2139–47. 34. J. Varna, L. Berglund, R. Talreja, and A. Jakovics, A study of crack opening displace- ment of transverse cracks in cross-ply laminates. Int J Damage Mech, 2:3 (1993), 272–89. 35. J. Tong, F. J. Guild, S. L. Ogin, andP. A. Smith, Onmatrix crackgrowthinquasi-isotropic laminates – I. Experimental investigation. Compos Sci Technol, 57:11 (1997), 1527–35. 36. Schapery, R. A. and Sicking, D. L., On nonlinear constitutive equations for elastic and viscoelastic composites with growing damage. In Mechanical Behavior of Materials, ed. A. Bakker. (Delft, The Netherlands: Delft UniversityPress, 1995), pp. 45–76. 37. Shapery, R. A., On viscoelastic deformation and failure behavior of composite materials with distributed flaws. In Advances in Aerospace Structures and Materials, ASME-AD-01, ed. S. S. Wang and W. J. Renton. (Philadelphia, PA: ASTM, 1981), pp. 5–20. 38. R. S. Kumar and R. Talreja, A continuum damage model for linear viscoelastic composite materials. Mech Mater, 35:3–6 (2003), 463–80. 39. R. M. Christensen, Theory of Viscoelasticity: An Introduction, 2nd edn. (New York: Academic Press, 1982). 40. R. S. Kumar and R. Talreja, Linear viscoelastic behavior of matrix cracked cross-ply laminates. Mech Mater, 33:3 (2001), 139–54. 41. J. Varna, A. I. Krasnikovs, R. S. Kumar, and R. Talreja, A synergistic damage mech- anics approach to viscoelastic response of cracked cross-ply laminates. Int J Damage Mech, 13:4 (2004), 301–34. 42. E. Ahci and R. Talreja, Characterization of viscoelasticity and damage in high tem- perature polymer matrix composites. Compos Sci Technol, 66:14 (2006), 2506–19. 178 Macro-damage mechanics 6 Damage progression 6.1 Introduction As discussed in Chapter 1, material constitutive relationships are needed in order to conduct failure analysis of a structure subjected to service loading. Contrary to homogeneous materials such as metals whose stress–strain relationships can be specified a priori, composite materials suffer damage that can alter these relation- ships. Thus if strains are prescribed, the stress response can be expressed as s ij = C ijkl e kl ( )e kl ; (6:1) where the stiffness matrix C ijkl changes with strain as the composite material suffers damage. Determination of C ijkl as a function of the applied strain can be achieved by solving two related sub-problems: 1. Describe stiffness changes as a function of damage: In this step C ijkl is expressed in terms of some damage characteristic, such as the ply crack density, C ijkl = C ijkl r (a) _ _ ; (6:2) where r (a) is the crack density of damage mode a = 1, 2,. . ., N. As discussed in Chapter 4, micromechanics solutions for multiple damage modes are not always possible. Using the CDM formulation for multiple damage modes in linearized form we can express Eq. (6.2) as C ijkl = C 0 ijkl ÷ N a=1 C (a) ijkl r (a) _ _ ; (6:3) where C 0 ijkl is the stiffness tensor for undamaged laminate, and C (a) ijkl represents the stiffness changes brought about by damage mode a. 2. Describe the evolution of crack density as a function of applied loading: r (a) = r (a) e kl ( ): (6:4) Combining the solution to the two sub-problems above, we obtain C ijkl = C 0 ijkl ÷ N a=1 C DAM ijkl r (a) e mn ( ) _ _ : (6:5) The solution to the first sub-problem has been described in Chapters 4 and 5. This chapter concerns the second sub-problem. To take advantage of directional properties of plies, composite laminates are made from a mix of longitudinal, transverse, and angle plies. Since the unidirec- tional lamina has low strength in the transverse direction, it is prone to cracking along fibers. When the applied load is increased beyond the strain (or stress) at which initiation of cracking occurs, new cracks form in the cracked ply in between existing cracks. Initially these cracks are far apart and do not interact with each other. However, quickly they form a roughly periodic array of parallel cracks. Figure 6.1 shows the increase in density of transverse cracks (i.e., number of cracks per unit length normal to the crack plane) for typical configurations of cross-ply laminates when loaded under axial tension. Prediction of such curves has been an extensive subject of study. The approaches used to model crack initiation and progression (multiplication) can be divided into two categories: strength- based models and energy-based models. As the names suggest, the first category of models involves use of strength (failure) criteria, while energy balance concepts underlie the second category. 6.2 Experimental techniques Before discussing experimental observations and measurements in Section 6.3 below, a brief description of each of the techniques is given next. This is to help provide a background against which to interpret the observations. No attempt is made here to go into any depth on these techniques, as that would distract from the focus of this chapter. For further details references [2–4] are suggested. Over the past forty years many nondestructive evaluation (NDE) techniques have been developed to detect, monitor, and observe ply cracking damage in 40 30 20 10 0 n=4 [0/90 n /0],T300/934 Cracks/In. n=3 n=2 n=1 20 40 60 80 100 120 Ksi Figure 6.1. Transverse crack density vs. applied axial stress in [0 m /90 n ] s laminates [1]. Reprinted, with kind permission, from Composites Technology Review, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428. 180 Damage progression composite laminates. The quantities targeted for observations and measurements concerning ply cracking include the following: 1. Crack initiation strains (or stresses). 2. Increase in number of cracks with applied loading or with number of cycles in case of fatigue experiments. 3. Changes in stiffness properties with damage. 4. Crack opening displacements (COD) and crack profiles. 5. Final failure strains and damage leading to the final failure. Brief overviews of the main techniques are provided here. Edge replication Although direct optical microscopy can be used [5] for surface observation to evaluate damage, it requires in most cases specimens to be removed at regular intervals during loading. If feasible, anoptical microscope canbe mountedonthe testing machine itself, but it often requires the microscope to be brought close to the specimen surface for enough resolution. This limitation can be overcome by using edge replication instead. This is a simple and easy technique if the purpose is to monitor damage (count cracks) in composites, as was shown early by Stalnaker, Stinchcomb, and Masters [6, 7]. The approach involves the microscopic examination of surface replicas, which are prepared by pressing softened rubber tape (or tape with an adhesive, e.g., acetate) against the specimen edge. With this procedure it is possible to quickly obtain permanent records of the specimen edge while the specimen itself is held in loading. The popularity of the technique is evidenced by its utilization by many researchers in the field [2, 8–14]. Two examples of photomicrographs from edge replicas are shown in Figure 6.2. Acoustic emission In this technique, a sensor is used to monitor acoustic emission (AE) signals from stress waves generated due to some local failure in the material, such as formation of a ply crack. A major limitation of the approach is its inability to distinguish between damage types and to provide information concerning crack location and orientation [14]. However, if the damage mechanismis known based on prior experience with the material system, this technique can indicate the applied load at which the damage initiated. Using this to monitor damage and to take edge replicas accordingly has been found useful. A combination of ultrasonic polar backscattering scans and AE, which yields accurate crack location, has also been suggested [15, 16]. Figure 6.3 shows the variation of cumulative count of acoustic events and the corresponding stress–strain response of a ceramic composite subjected to tensile loading. X-ray radiography X-ray radiography is a useful technique in finding internal cracks that are not visible by optical microscopy. For clarity of observationa penetrant fluidthat absorbs X-rays is often necessary to provide sufficient contrast between exposed and unexposed regions. Pictures of the developed X-ray films can then be enlarged for further clarity. 181 6.2 Experimental techniques Transverse Cracking Transverse Cracking Longitudinal Cracking (a) (b) Figure 6.2. Photomicrographs of edge replicas showing details of damage development in fatigue loading of quasi-isotropic laminates: (a) [0/±45/90] s and (b) [0/90/±45] s laminate [2]. Reprinted, with kind permission, from Damage in Composite Materials, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428. 1200 (a) 900 600 300 0 0 20 40 Stress (MPa) C u m u l a t i v e A E c o u n t s 60 80 100 100 80 60 40 20 0 (b) 0 Strain (%) S t r e s s ( M P a ) 0.05 0.1 0.15 0.2 Figure 6.3. Variation of cumulative acoustic event (AE) counts (a), and stress–strain curve (b) for a ceramic composite loaded in tension [14]. Reprinted, with kind permission, from Damage Detection in Composite Materials, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428. 182 Damage progression The pictures, however, only provide 2-D images of cracks and therefore cannot be used to find the depth information needed to separate cracks of one layer from another. For that purpose penetrant enhanced X-ray stereo radiography can be used. The standard procedure is to make two X-ray radiographs of the object at slightly different orientations, which is usually done by rotating the X-ray source relative to the object. The spatial locations of cracks can be observed with a microscope but cannot be recorded. The approach has been successfully used to detect fiber fractures, delaminations and matrix cracks [17–19]. To enable imaging and finer precision detection of internal defects, the technique employed is X-ray tomography, which uses a medical scanner and yields a three-dimensional image of an object [20]. An X-ray radiograph of damage in a quasi-isotropic laminate is shown in Figure 6.4. Ultrasonic C-scan Ultrasonic C-scan is a nondestructive inspection technique in which a short pulse of ultrasonic energy is made incident on a sample. The attenuation of the pulse is influenced by voids, delaminations, state of resin cure, the fiber volume fraction, Figure 6.4. An X-ray radiograph of damage in a quasi-isotropic laminate [19]. Reprinted, with permission, from Effects of Defects in Composite Materials, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428. 183 6.2 Experimental techniques the condition of the fiber/matrix interface, and any foreign inclusions present. The technique is, however, limited by its inability to detect very small defects or cracks with their planes parallel to the wave direction [14]. In some cases a damaged laminate material may contain a large number of small and partially grown internal cracks. On increase in loading, these cracks can grow further and coalesce to form larger cracks. The common NDE techniques described above cannot provide accurate information on the formation, number, size, and progression of these internal cracks. For such cases a through-transmis- sion ultrasonic C-scan imaging with inclined focusing transducers in confocal configuration has been recently suggested [21, 22]. Other methods based on vibrations and lamb waves have also been suggested [23, 24]. Technique for COD measurement In addition to the observations of damage certain measurements also need to be made as dictated by the models. For instance, the COD of ply cracks is a quantity that enters the SDM approach described in Chapter 5. Since there are no standard methods for measuring this quantity, Varna et al. [26–27] developed a set-up for this purpose. To observe an individual ply crack, the set-up uses a miniature materials tester (MINIMAT) for loading a thin strip cut out from the cracked laminate. The open crack is observed under an optical microscope equipped with a video camera. The video signal transmitted to a TV monitor displays the crack profile at sufficient magnification (~ 2×10 3 ) to measure the COD. Figure 6.5 presents COD as a function of position along the cracked 90 · -ply thickness (z-direction) and compares it with the theoretical shape prediction obtained using Linear Elastic Fracture Mechanics (LEFM). Raman spectroscopy Recently Katerelos and coworkers [28] have developed an experimental technique based on Raman spectroscopy, which uses the property that the Raman vibrational wave numbers (frequencies) of certain chemical groups of commercial reinforcing fibers, such as aramid or carbon, are stress and strain dependent [16]. Thus, the wave number shift along an embedded fiber can be utilized to determine stress or strain. The technique can provide a high spatial resolution of ~ 1mm. However, for the approach to work the matrix needs to be translucent. Also, the data acquisition can take longer than milliseconds for a single measurement. Moreover, certain amorphous fibers such as glass have a weak Raman response. Aramid fibers, on the other hand, scatter the waves well. Therefore, a small amount of aramid fiber placed in strategic positions within a glass-fiber laminate can act as Raman sensors of stress and strain. The micromechanical strain mapping results are then used to derive the properties, i.e., the longitudinal modulus of elasticity and the magnitude of the residual strains caused by cracking [29]. The approach has been successfully applied to evaluate ply cracking damage evolution and resulting stiffness changes [30–32]. 184 Damage progression 6.3 Experimental observations Experimental studies on initiation and growth of intralaminar cracking in composite laminates have been extensive. Most of the work has focused on 90 · -ply cracking in cross-ply and quasi-isotropic laminates. Chapter 3 reviewed some of those observa- tions to illustrate the nature of composite damage. A book chapter by Nairn [33] is also recommended for a good overview of the topic. In the following we review quantitative data related to the initiation and progression of ply cracking in laminates. 6.3.1 Initiation of ply cracking The applied loading (stress or strain) at which ply cracking in laminates first occurs is of interest from materials selection as well as design points of view. A good laminate configuration (orientation, thickness, and sequence of plies) will delay initiation of ply cracking to as high a load as possible. Experimental observations have indicated that all laminate configuration parameters influence ply crack initiation. Most early studies examined ply cracking in cross-ply lamin- ates of glass-polyester or glass/epoxy under axial tension. The ply cracking in the 90 · -plies could in most cases be observed by looking at the specimen surface under an optical microscope at low magnification. The near-transparency of composites 25 20 15 [O2/908]s Hybrid CF/GF (Different sections) LEFM theory 10 5 0 0.0 0.2 0.4 0.6 0.8 1.0 Relative position in 90-layer C r a c k o p e n i n g d i s p l a c e m e n t u ( m m ) z—t 90 2t 90 Figure 6.5. Crack opening displacement measured using MINIMAT along the 90 · -layer thickness of a hybrid CF/GF cross-ply laminate at various sections through the specimen width. The solid line depicts predictions from LEFM. Source: J. Varna, L. Berglund, R. Talreja, and A. Jakovics, A Study of crack opening displacement of transverse cracks in cross-ply laminates, Int J Damage Mech, Vol. 2, pp. 272–89 (1993). 185 6.3 Experimental observations of glass fibers made this possible. In carbon fiber composites, however, one had to rely on edge observations until X-ray radiography allowed imaging of the interior cracks. The axial strain values at initiation of transverse cracking in cross-ply laminates of glass/epoxy and carbon/epoxy fall typically in the range 0.4–1.0%. As the off-axis ply angles decrease from 90 · the initiation strain increases. For instance, for 45 · -plies cracking barely initiates at 1.0% axial strain [34]. Figure 6.6 illustrates the effect of off-axis angle on crack initiation strain for (a) GFRP [35], and (b) CFRP [36]. Data for notched samples are also indicated to illustrate the effect of local stress enhancement. In Figure 6.6 the data for un-notched samples with and without polished edges illustrate the effect of machining-induced flaws in initiating cracking. Also of note is the range of strain and the lowest strain for GFRP versus CFRP. In the latter case, the lowest strain to crack initiation is higher due to the higher elastic modulus of carbon fibers. In both cases the initiation strain at low off-axis angles is limited by the fiber failure strain. The thickness of the cracking ply also has a large effect on e init . Garret and others [37–44] performeda systematic study of thickness effect on cracking in [0 m /90 n ] s laminates. They used glass-reinforced polyester [37, 38] and glass-reinforced epoxy [39–41] as the laminate materials andvariedthe thickness of the 90 · -plies while keeping the thickness of the 0 · -ply constant at 0.5 mm. Figure 6.7 shows the variation of e init with respect to the total thickness of the 90 · -plies. As the thickness of the 90 · -plies increases, e init decreases. When the thickness of the 90 · -plies is more than 0.4 mm, the cracks initiate instantly and span the entire cross section of the 90 · -plies. At the other extreme, if the thickness of the 90 · -plies is less than 0.1 mm, cracks may be entirely suppressed and the laminate eventually may fail due to other damage mechanisms, e.g., delamination and fiber breakage. The experiments on carbon/epoxy laminates GFRP [0/␪/0] laminates notched Angle (degrees) 40 50 60 70 80 90 100 unnotched polished unnotched 3 (a) 1.2 1 0.8 0.6 0.4 0.2 0 40 I n i t i a l c r a c k i n g s t r a i n ( % ) 60 Off-axis angle (␪) CFRP (IM7/8552) [0 2 /␪ 4 ] s laminates 80 100 Unnotched Notched (b) 2.5 2 1.5 1 0.5 0 L o n g i t u d i n a l c r a c k i n g s t r a i n ( % ) Figure 6.6. Crack initiation strains for angle ply laminates as a function of ply orientation: (a) GFRP laminates; (b) CFRP laminates. Part (a) reprinted, with kind permission, from Composites A, Vol. 28, L.E. Crocker, S.L. Ogin, P.A. Smithand P.S. Hill, Intra-laminar fracture in angle-ply laminates, pp. 839–46, copyright Elsevier (1997). Part (b) reprinted, with kind permission, fromSpringer Science+Business Media: J Materials Science, Intra-laminar cracking in CFRP laminates: observations and modeling, Vol. 41, 2006, pp. 6599–609, N. Balhi et al. 186 Damage progression show similar behavior [41, 45]. In general, for [0 m /90 n ] s laminates, complete crack suppression is expected only when m / n _10 although the suppression effects may be felt in the range 2 _ m / n < 10 [46]. This difference in initiation strain for different thicknesses is attributed to the relative constraint of the supporting 0 · -plies onto the crack opening in the 90 · -plies [47]. The actual ply layup in the laminate may also play a role in determining e init . For instance, [90 2 /0] s laminates develop cracks sooner than [0/90 2 ] s laminates [48]. It is because the 90 · -plies in [90/0] s laminates are on the outer surface and may not experience much support for crack suppression from the inner 0 · -plies. For multidirectional laminates, the situation is even more complex and e init will depend on the thickness ratio and the stiffness properties of the cracking and supporting plies. The laminate preparation and processing method also may affect crack initiation; filament-wound laminates may be more prone to cracking than those made from prepregs using an autoclave. 6.3.2 Crack growth and multiplication The growth of ply cracks is usually unstable in composites such as GFRP and CFRP. The initial transverse cracks are found to grow through the lamina thickness quickly but are usually arrested at interfaces with adjacent plies of different orientation, e.g., at the 90/0 interface for cross-ply laminates [38, 40] and at the 90/÷45 interface in [0/90/÷45/+45] s laminates [34]. Continued loading usually leads to formation of progressively more cracks between the already formed cracks. Most experimental studies point out that once the ply cracks have grown through the lamina thickness, they often grow unstably along the fiber direction through the laminate width, and are thus described as “tunneling cracks.” In some cases, however, cracks in plies other than 90 · may not grow fully before the laminate fails by delamination [34, 49–51]. Such partial cracks can be observed in the ±45 · -plies of quasi-isotropic [0/90/÷45/+45] s laminates [34]. 2 1 0 0 1 S t r a i n t o F i r s t M i c r o c r a c k ( % ) 2 Total Thickness of 90° Plies (mm) 3 4 5 Figure 6.7. The strain to initiate ply cracking in [0 m /90 n ] s glass/epoxy laminates as a function of the total thickness of 90 · -plies [33]. Experimental data from [38]. Reprinted, with kind permission, from Polymer Matrix Composites, J. A. Nairn, Matrix microcracking in composites, pp. 403–32, copyright Elsevier (2000). 187 6.3 Experimental observations Once the ply cracking has initiated, more and more ply cracks start appearing in between existing cracks, and the crack density rises quickly. As the crack spacing between adjacent cracks decreases, the cracks start interacting. Closely interacting cracks typically provide a “shielding effect,” which tends to reduce the stresses between two adjacent cracks. Therefore, on further loading, the rate of cracking reduces and finally approaches a saturation value. Thus a typical damage growth curve consists of three stages: crack initiation, rapid rise in crack density by multiplication, and reducing rate of crack density evolution until saturation (see Figure 6.11). Reifsnider and associates [10, 52] described this microcrack satur- ation as a material state and called it the characteristic damage state (CDS). They proposed that CDS is a well-defined laminate property and does not depend on load history, environment, or thermal or moisture stresses. However, later investi- gations by Akshantala and Talreja [53] have suggested that CDS may not be a single state under fatigue loading but may depend on the maximum stress applied. The progression from initial rapid rise until saturation of crack density depends on the ply material, the ply stacking sequence, and also the laminate fabrication process. For example, well-made carbon/epoxy laminates typically have a rapid rise in ply crack density. The saturation crack density is often found to inversely scale with cracking ply thickness, with thin plies accumulating a large number of cracks/mm [54, 55]. This is well illustrated in Figure 6.8 where experiments conducted on GFRP specimens with different ply thicknesses [54] showed that thin 90 · -plies can accumulate numerous cracks. The evolution of average crack density with respect to applied load for the same specimens is shown in Figure 6.9. It can be observed here that the saturation crack density is roughly proportional to 1/t 90 . The damage evolution curves in Figure 6.10 also show that thicker cross-ply laminates typically show lower crack density at saturation. The damage evolution in outer and inner 90 · -plies is also compared. The laminates with outer 90 · -plies are Figure 6.8. Transverse cracking in GFRP specimens with transverse-ply thickness of (a) 0.75 mm, (b) 1.5 mm, and (c) 2.6 mm, strained to 1.6%. Reprinted, with kind permission, from Springer Science+Business Media: J Materials Science, Multiple transverse fracture in 90 degrees cross-ply laminates of a glass fiber-reinforced polyester, Vol. 12, pp. 157–68, K. W. Garrett and J. E. Bailey. 188 Damage progression observed to have lower saturation crack densities. A typical damage evolution curve for ply cracking based on these studies is shown in Figure 6.11. 6.3.3 Crack shapes When cracks are widely spaced, the maximum principal stress occurs on the plane midway between existing cracks. Thus, at low crack density there is a tendency for new cracks to form midway and develop into a periodic array. However, at large crack density cracks interact causing the maximum principal stress to shift towards the 0/90 interface close to an existing crack. This may result in curved or oblique microcracks forming near the 0/90 interface [54, 56, 57]. These cracks make an angle of 40–60 · with existing straight cracks. Lundmark and Varna [58] have recently reported that curved microcracks form more readily in the low-temperature regime than at room temperature. In fact, at low temperatures complex crack trajectories are observed to form (see Figure 6.12). This may result in a highly damaged region in the laminate encompassing multiple crack types (Figure 6.13). 6.3.4 Effect of cracking The most direct effect of ply cracking is the reduction of the thermomechanical properties of the laminate, including changes in the effective values of Young’s moduli, Poisson’s ratios, and thermal expansion coefficients. The changes in stiffness properties can in turn lead to change in the behavior of the whole structure, e.g., the magnitude of its deflection and vibrational frequencies, some- times making the structure unable to carry out its intended design function. Even if it does not cause the structure to fail, substantial ply cracking may give rise to 1.2 3.2mm 2t 90 2mm 1.5mm 0.75mm 1 0.8 0.6 0.4 0.2 0 0 50 100 150 200 250 Applied stress (MPa) A v e r a g e c r a c k d e n s i t y ( 1 / m m ) Figure 6.9. Average crack density as a function of applied stress for GFRP cross-ply laminates with different transverse ply thicknesses (2t 90 ). The experimental data are from [54]. 189 6.3 Experimental observations Stage I Applied load (strain or stress) C r a c k d e n s i t y ( n o . o f c r a c k s / m m ) Stage II Stage III Crack initiation & propagation through laminate width Multiple crack formation Saturation of progressive cracking Figure 6.11. A typical damage evolution curve for ply cracking in laminates. 1.4 (a) 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0 200 400 [0/90 4 ] s [0/90 2 ] s [0/90] s 600 800 Stress (MPa) C r a c k d e n s i t y ( 1 / m m ) 1.0 (b) 0.8 0.6 0.4 0.2 0.0 0 100 200 300 [90/0/90] T [90/0] s [90/0 2 ] s [90/0 4 ] s 400 500 Stress (MPa) C r a c k d e n s i t y ( 1 / m m ) 600 700 800 900 1000 Figure 6.10. Damage evolution curves for (a) [0/90 m ] s and (b) [90/0 m ] s laminates. Reprinted, with kind permission, from Polymer Matrix Composites, J. A. Nairn, Matrix microcracking in composites, pp. 403–32, copyright Elsevier (2000). 190 Damage progression more deleterious forms of damage such as delamination and longitudinal splits, or provide pathways for the entry of moisture and corrosive liquids. 6.3.5 Loading and environmental effects Most experiments are performed using uniaxial tension, but ply cracks will also form under other loading conditions, such as fatigue, biaxial, or shear loading. Biaxial loading of [0 m /90 n ] s laminates may show cracks in both 0 · - and 90 · -plies. If the material response remains linearly elastic after cracking, then neglecting crack interaction effects between plies the effect of biaxial loading of [0 m /90 n ] s laminates may be seen as equivalent to two uniaxial loading cases, one on [0 m /90 n ] s and the other on [90 m /0 n ] s laminates. On thermal loading, the differential shrinkage between the 0 · - and 90 · -plies may also induce biaxial loading [59–65]. In general, Normal crack Crack type A Crack type B Crack type C Crack type D a a Partial crack 0° layer t 0 t 90 90° layer w w w Figure 6.12. Different crack types observed during tensile testing of a [0 2 /90 4 ] s CF/EP laminate at cryogenic temperature (÷150 · C). Reprinted, with kind permission, from Eng Fract Mech, Vol. 75, P. Lundmark and J.Varna, Damage evolution and characterization of crack types in CF/EP laminates loaded at low tempratures, pp. 2631–41, copyright (2008), with permission from Elsevier. Figure 6.13. A snapshot of a highly damaged region of a [0 2 /90 4 ] s CF/EP laminate at an applied stress of 343 MPa (0.66% strain) during tensile loading at cryogenic temperature (÷150 · C). Reprinted, with kind permission, fromEng Fract Mech, Vol. 75, P. Lundmark and J.Varna, Damage evolution and characterization of crack types in CF/EP laminates loaded at low tempratures, pp. 2631–41, copyright (2008), with permission from Elsevier. 191 6.3 Experimental observations thermal effects can be accounted for in the analysis and predictions of classical laminate theory. Bailey et al. [41] studied the effect of thermal stresses and Poisson’s contraction on ply cracking in CFRP and GPRP cross-ply laminates. Thermal residual stresses typically lower crack initiation strains. Thermal effects are larger in CFRPs than in GFRPs due to larger differences in both thermal expansion coefficients and Young’s moduli in directions parallel and perpendicular to fibers [41]. For instance, a [0/90] s laminate with a ply thickness of 0.5mm shows 0.322% thermal strains for CFRPs and 0.094% for GFRPs. The Poisson’s effect was found to be greater in GFRPs because of their larger failure strains. Poisson’s strains can sometimes be so high that they can induce transverse cracking of the 0 · -plies. The mismatch between the thermal expansions of different plies can also result in a high-density form of microcracking known as “stitch cracking.” Lavoie and Adolfsson [66] studied this type of cracking in [+y n /–y n /90 2n ] s laminates (see Figure 6.14). Stitch cracks appear to form instead of interply delamination at the tip of a 90 · crack in the adjacent constraining ply when the included angle between the two is greater than 50 · . Stitch cracking is also observed in case of fatigue loading [19]. Variation in intralaminar fracture toughness over tempera- ture can also change the initiation and evolution of ply cracking [67]. The residual stresses due to moisture can also affect the cracking process [63, 68]. This could be due to degradation of the matrix during hygrothermal aging [69, 70]. A combination of moisture and thermal residual stresses can cause significant stiffness reduction, especially at high crack density [71]. Figure 6.14. X-radiograph of thermal stress-induced matrix microcracks in a [+454/–454/ 908] s laminate. The 90 · -ply matrix cracks run from left to right, and trigger the formation of many short, stitch-like –45 · -ply matrix cracks. Long +45 · -ply matrix cracks appear. Reprinted, with kind permission, from J. A. Lavoie and E. Adolfsson, J Compos Mater, Vol 35, pp. 2077–97, copyright # 2001 by Sage Publications. 192 Damage progression If laminates are loaded in bending, ply cracks will form on the tension side [11, 72, 73], and for such loading analyses need to account for the resulting local stress states [11, 73–77]. 6.3.6 Cracking in multidirectional laminates Cross-ply laminates are not very common in practical applications. In fact, efficient use of composite laminates in a wide range of applications requires placing plies in multiple orientations. A common example is a class of laminates known as quasi-isotropic that can be constructed in different ways; a common form usually has a mix of plies with 0 · , 45 · , and 90 · orientations, exemplified by the [0/90/±45] s configuration. In the off-axis plies of such laminates, cracks usually initiate at much higher applied axial strains than in 90 · -plies and may sometimes show curved patterns due to inclined principal stress trajectories. Curved cracks and cracks near free edges typically promote delamination at the ply interface [78]. Experimental data show that the majority of off-axis cracks form at the edges of the test coupons and often may not grow across the thickness and width before the specimen fails by extensive delamination. Furthermore, in some cases, other damage modes such as delamination or fiber fracture can occur even before any cracking in the off-axis plies ensues. Johnson and Chang [79, 80] carried out extensive experiments on a variety of multidirec- tional laminate configurations and found that for laminates having a ply angle greater than 45 · , ply cracking is a dominant damage mode, with possible delami- nation at free edges. For instance, in [0/y/0] laminates with y _ 45 · the dominant damage was ply cracking [31, 35], while in [90/30/–30] s a combination of edge delamination and fiber fracture led to final failure. Crocker et al. [35] also found that in [0/45/0] laminates, delamination ensued as soon as ply cracks initiated in 45 · -plies. In case of multidirectional laminates containing 90 · -plies, an off-axis ply adjacent to a 90 · -ply shows numerous partial cracks, which may or may not join to form through cracks on increase of loading. The cracks in the contiguous ply almost always start from its interface with the 90 · -ply. Experiments by Yokozeki and coworkers [50, 51] on [0/y 2 /90] s laminates also point out that the angle of intersection between the 90 · - and y-plies and the thickness of the y-plies may have a significant impact on the initiation and growth of cracking in these plies. In more general crack systems in multiple orientations, another important consideration is the interaction between cracks in two adjacent plies. This inter- action can enhance cracking in a certain orientation thereby causing further stiffness degradation. For instance, in quasi-isotropic laminates 45 · cracks pro- mote enhanced cracking in the 90 · -plies. This intra-ply crack interaction is a complex function of the relative crack positions, orientations, crack sizes (ply thicknesses), and density of cracks in different orientations. This makes determin- ation of stresses in cracked laminates generally impossible to solve analytically, 193 6.3 Experimental observations and numerical computations such as 3-D FEM are then necessary. Figure 6.15 depicts a representative cell of a [0/90/y 1 /y 2 ] s laminate with cracks in multiple off- axis orientations. 6.4 Modeling approaches In the following we describe the main approaches to predicting the evolution of ply cracking in composite laminates. In the models discussed the ply cracks are assumed to be fully developed through the ply thickness as well as in the specimen width direction. In a given array of ply cracks, all cracks are assumed to be of the same size, shape, and orientation. The problem to address here is the crack multiplication process as a function of applied loading. As described above, there are two directions in which this problem can be pursued: one based on a strength criterion (i.e., point failure), and the other based on an energy criterion (i.e., surface formation). This section deals exclusively with cross-ply laminates; multi- directional laminates are considered later. The basic BVP to solve here is the same as that approached earlier in Chapter 4, see Figure 4.7. We will use the same symbols and notations (Section 4.4), unless specified otherwise. 6.4.1 Strength-based approaches According to these models, microcracks form when the local stress (or strain) state in a ply reaches a critical level [37–39, 81–88]. The most commonly used values of critical level are: failure strain (e 1T ) or failure stress (s tu ) of a ply in transverse tension. Other lamina failure criteria such as Tsai–Wu and Hashin’s criteria, described in Section 2.2.4, are also utilized. Since the stress state at the onset of transverse cracking is not generally uniform [45], these models fail to account for differences in crack initiation (as a material point failure process) and crack progression (as a surface growth process). Consequently, the ply thickness effect on transverse cracking cannot be properly treated by these criteria. One problem with strength-based criteria, and more generally with all proced- ures that require knowledge of the local stress states for failure assessment, is that Figure 6.15. A representative cell illustrating multiple cracking systems in a [0/90/y 1 /y 2 ] s laminate. 194 Damage progression determining the local stresses analytically is possible only in a few cases, mostly for cross-ply laminates. The larger problem, however, is the lack of agreement of strength-based predictions with the experimental data. Use of statistical strength concepts [83, 84, 86, 89, 90] may improve predictions, but require additional material data. In the following we provide a brief overview of the strength-based models. A basic shear lag analysis applied to determination of crack initiation strain and crack multiplication was carried out in [91]; it has been covered previously in Section 4.2. It is a one-dimensional analysis, and therefore does not provide accurate stress perturbation caused by cracking. The analysis of multiple cracking in a unidirectional fiber composite was later applied to the case of transverse cracking in cross-ply laminates by Garrett and Bailey [38] (see Section 4.4.1). According to the analysis, the load shed by the transverse plies in the crack plane and transferred back to the transverse plies over the distance y is given as [38] F = 2t 0 wÁs 0 1 ÷e ÷by _ ¸ ; (6:6) where t 0 is the thickness of the 0 · -ply, w is the specimen width, Ds 0 is the maximum additional stress on the longitudinal ply as a result of cracking in the transverse ply (occurring in the plane of crack and decaying exponentially away from the crack surface), and b 2 is the shear lag parameter, defined in Eq. (4.62), as b 2 = G 90 xz0 1 E 90 x0 ÷ 1 lE 0 x0 _ _ ; (6:7) where G 90 xz0 is the initial in-plane shear modulus of the 90 · -ply, l = t 0 / t 90 is the ply thickness ratio, and E 0 x0 and E 90 x0 are the longitudinal moduli for the 0 · - and 90 · -plies, respectively. The transverse ply will fail in tension when F = 2t 90 ws tu ; (6:8) where s tu is the transverse ply strength. The first crack is assumed to form in the middle of the specimen length at Ds 0 = (t 90 / t 0 ) s tu , i.e., at an applied load of s 0 = E c e tu , where E c is the longitudinal modulus for the composite and e tu is the transverse ply cracking strain. Next, the cracking process will cause second and third cracks (one above and one below the first crack) to form simultaneously. From Eq. (6.6) and Eq. (6.8) with y = l, where 2l is the crack spacing, Ds 0 will be Ás 0 = 1 l s tu 1 ÷e ÷bl : (6:9) These cracks will perturb the force transferred such that the new cracking process will occur at Ás 0 = 1 l s tu 1 ÷e ÷bl ÷2e ÷bl=2 : (6:10) 195 6.4 Modeling approaches Similarly, the (N+2)th crack formation will occur when Ás 0 = 1 l s tu 1 ÷e ÷ bl N ÷2e ÷ bl 2N : (6:11) The evolution of crack density thus can be generated using the above iterative scheme. Figure 6.16 shows the model prediction of crack spacing variation with increasing loading for a glass-polyester specimen of length 130 mm and transverse ply thickness of 3.2 mm. Although this model represents the general trends of the experimental results it generally underestimates the average crack spacing. This is supposedly due to the constant strength of the 90 · -ply resulting in new cracks midway between existing cracks. Later models tried to address this issue by representing 90 · -ply strength as a probabilistic function. Manders et al. [92] considered a two-parameter Weibull distribution of the strength along the length of the 90 · -ply and represented the risk of rupture per unit volume as p(s) = s s + _ _ v ; (6:12) where the constant s* is the scale parameter in terms of stress, and v is the shape parameter. The cumulative distribution function for failure is given as 5 4 3 2 1 0 0 100 Applied stress (MNm –2 ) C r a c k s p a c i n g ( m m ) 200 300 Figure 6.16. Prediction of crack spacing as a function of applied stress using the shear lag model [38]. Reprinted, with kind permission, from Springer Science+Business Media: J Materials Science, Multiple transverse fracture in 90 degrees cross-ply laminates of a glass fiber-reinforced polyester, Vol. 12, pp. 157–68, K. W. Garrett and J. E. Bailey. 196 Damage progression S V = 1 ÷exp ÷ _ V p(s) dV _ _ _ _ = 1 ÷exp ÷A _ L p(s) dy _ _ _ _ ; (6:13) where A is the area, and L is laminate length. Taking a logarithm of all the terms in Eq. (6.13) gives ln (1 ÷S V ) = ÷A _ L p(s) dy ~ ÷ApL: (6:14) The quantity Ap is obtained by plotting ln(1 ÷ S V ) against L. The Weibull param- eters s* and v are obtained by fitting the experimental crack density evolution with the model. To describe the stress s in the laminate, Manders et al. [92] used initial shear lag analysis, covered previously in Section 4.4.1. The model predictions of crack spacing variation as a function of applied strain are shown in Figure 6.17. A similar analysis was performed by Fukunaga et al. [83] who also used the Weibull distribution for 90 · -ply strength but instead used interlaminar shear lag analysis, also covered in Section 4.4.1. The effects of thermal residual stresses and Poisson’s contraction are also included in this analysis. According to their analysis the applied axial stress and crack spacing (2l) are related as s c = E c Q 22 1 ÷ Q 12 A 12 Q 22 A 22 _ _ s + L 0 t 1 2lt 90 d 1 _ _ 1=v ÷s 90 xx R _ _ ; (6:15) where s + = s 90 xx0 ln 2=L 0 t 1 w ( ) 1=v with L 0 , w, and t 1 being the length, width, and thickness of a single 90 · -ply, respectively, and d 1 = 1 l _ l 0 1 ÷ cosh bx= cosh bl ( ) [ [dx is a parameter that reflects the effect of stress nonuniformity on 90 · -ply strength. The corresponding strains can be calculated from the stress–strain relations for the cracked laminate derived in [83] as 0 C r a c k s p a c i n g ( m m ) 30 0.2 0.4 0.6 0.8 1.0 Strain (%) 1.2 1.4 1.6 1.8 2.0 2.2 20 10 0 Figure 6.17. Variation of average crack spacing for a [0/90/0] glass/epoxy composite laminate as predicted using the probabilistic shear lag model by Manders et al. [92] for. Solid and open circles represent experimental data from two specimens, while the solid line represents the model prediction. 197 6.4 Modeling approaches e c = s c E c 1 ÷ t 90 Q 22 t 0 Q 11 1 ÷ Q 12 A 12 Q 22 A 22 _ _ tanh al al _ _ ÷ t 90 s 90 xx R t 0 Q 11 tanh al al : (6:16) The crack initiation strain can be obtained by setting tan al / al = 0 (as l ÷·) in Eq. (6.16), and 2l = L 0 , d 1 = 1 in Eq. (6.15). Therefore, the applied strain at first cracking in the 90 · -ply is given by e 0 = 1 Q 22 1 ÷ Q 12 A 12 Q 22 A 22 _ _ s + t 1 t 90 _ _ 1=v ÷s 90 xx R _ _ : (6:17) These statistical descriptions of ply strength yielded good results for cross-ply laminates. But these models cannot account properly for effects of changes in ply thickness. Also, as mentioned before, shear lag analysis is a one-dimensional stress analysis and therefore cannot be accurate. More recently, 2-D shear lag analysis based on Steif’s parabolic displacement variation in 90 · -ply [93] have also been tried [55, 94–96]. A detailed discussion on the probabilistic concept will be covered later in Section 6.5. 6.4.2 Energy-based approaches The origin of energy-based approaches to crack extension lies in fracture mechanics (see Section 2.4 for linear elastic fracture mechanics). In the classical version of brittle fracture, the energy considerations lead to the condition that crack tip growth becomes unstable when the energy release rate is equal to or greater than the fracture toughness of the material containing the crack, i.e., G_G c . This material property is obtained by an independent test, which has been standardized in some way. For perfectly brittle fracture, i.e., when no other energy dissipating mechanism other than that expended in the crack surface formation exists, the fracture toughness equals twice the surface energy. In multiple cracking within a composite laminate, however, the situation is not as assumed in the brittle crack extension of a crack tip for which the criterion stated above applies. Here, a ply crack on extension is arrested at the ply interfaces and any further input of energy to the laminate supplied by external loads goes into the formation of more ply cracks. Although the individ- ual ply crack goes through the stages of through-thickness growth and growth in the fiber direction, in most cases the ply cracks form quickly and the analysis is therefore focused on their multiplication, i.e., an increase in crack density. Because of this, a modification of conventional fracture mechanics, called finite fracture mechanics [97], has been proposed. Thus, in contrast to conventional fracture mechanics, transverse cracking comprises events that involve a finite amount of new fracture area. For cross-ply laminates, finite fracture mechanics coupled with variational stress analysis has been used in several works [5, 33, 46, 97–99]. The criterion for finite surface formation under brittle fracture condition can be written as ÁÀ _ g ÁA; (6:18) 198 Damage progression where DG is the energy change (release) during crack surface increase by area DA and g is the surface energy per unit area of the new crack formed. In the following we describe the energy-based approaches combined with dif- ferent approximate stress analyses used. Shear lag analysis Among the early energy-based analyses for ply cracking is that proposed by Parvizi et al. [81]. They recognized the role of constraint of the outer 0 · -plies to transverse cracking in [0/90] s laminates and took account of it in the energy balance during cracking. Recalling from ACK theory, Section 4.2, a crack does not form in a specimen loaded in constant tension until ÁW _ ÁU S ÷U D ÷2g m V m ; (6:19) where V m is the matrix volume fraction and, defined per unit cross-sectional area of the composite, DW is the work done by applied stress, DU S is the increase in energy stored in the composite volume, U D is the energy lost by some dissipative processes during cracking (sliding friction between debonded fibers and matrix), and g m is the matrix surface energy per unit surface area. For the present case of cracking in a transverse ply, the above inequality becomes ÁW _ ÁU S ÷2g t t 90 h ; (6:20) where g t is the surface energy of the transverse ply for cracking in a direction parallel to the fibers, and h = t 0 + t 90 . For a linear elastic body the work of external forces equals half the stored strain energy, i.e., ÁU S = 1 2 ÁW: (6:21) Substituting Eq. (6.21) into Eq. (6.20), the cracking will occur when ÁW _ 4g t t 90 h : (6:22) Oncracking anadditional stress is thrownontoouter uncrackedplies, andthe laminate increases in length. The work done during this process can be derived as [81] ÁW = 2E 90 x0 E c e 2 tu lE 0 x0 b ; (6:23) where b is as defined earlier in Eq. (6.7). Combining Eqs. (6.22) and (6.23), the strain to initiate cracking in transverse ply is given by e 0 = e min tu = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2t 0 E 0 x0 g t b hE 90 x0 E c ¸ : (6:24) The model predictions for various transverse ply thicknesses are shown in Figure 6.18 against experimental data for glass/epoxy cross-ply laminates. 199 6.4 Modeling approaches A more sophisticated energy-based analysis of progressive ply cracking using one-dimensional stress analysis was performed by Laws and Dvorak [100]. (Refer to Section 4.4.1 for the shear lag analysis.) Consider a cracked cross-ply laminate in state 1 where cracks are a distance 2l apart, and the ligament AB between the cracks is as yet uncracked (Figure 6.19(a)). When the applied load reaches a critical value s c , a new crack appears in this segment at some location C (Figure 6.19(b)). Assuming that the load is kept fixed during formation of this additional crack, Laws and Dvorak calculated the energy released during cracking of lamin- ate of width w as ÁÀ = 2t 2 90 whE c bt 0 E 0 x0 E 90 x0 s 90 xxR ÷ E 90 x0 E 0 x0 s c _ _ 2 tanh bl 1 2t 90 ÷tanh bl 2 2t 90 ÷tanh bl t 90 _ _ ; (6:25) 0 2.5 1 2 Transverse ply thickness 2d (mm) T r a n s v e r s e c r a c k i n g s t r a i n e t u , e t m u i n ( % ) 3 4 2.0 1.5 1.0 0.5 0.0 Experimental e tu Theoretical e tu min Figure 6.18. Values of crack initiation strain e min tu as a function of ply thickness 2t 90 = 2d (from Eq. (6.24)) and experimental data for various ply thicknesses in glass fiber/epoxy cross-ply laminates. The horizontal line depicts the limiting values of e min tu for large inner-ply thicknesses [81]. Reprinted, with kind permission, from Springer Science+Business Media: J Mater Sci, Constrained cracking in glass fiber-reinforced epoxy cross-ply laminates, Vol. 13, 1978, pp. 195–201, A. Parvizi, K. W. Garrett and J. E. Bailey. A B z x State 1 (a) (b) State 2 A s=2l l 1 l 2 B C Figure 6.19. Progressive multiplication of ply cracks in transverse layer of cross-ply laminate: (a) state 1 with crack spacing, s = 2l; (b) state 2 with an additional crack in the ligament AB at location C. 200 Damage progression where b 2 = Kt 90 1=E 90 x0 _ _ ÷ 1=lE 0 x0 _ _ _ ¸ is the shear lag parameter (see Eq. (4.67)), and s 90 xxR is the thermal residual stress in the transverse ply. Substituting Eq. (6.25) in Eq. (6.18), and using DA = 2wt 90 , a new crack will form if t 90 hE c bt 0 E 0 x0 E 90 x0 s 90 xxR ÷ E 90 x 0 E 0 x0 s c _ _ 2 tanh bl 1 2t 90 ÷tanh bl 2 2t 90 ÷tanh bl t 90 _ _ _ g: (6:26) The first ply failure stress (the applied stress at crack initiation) will be given by the limit l ÷·, i.e., s fpf c = bt 0 E 0 x0 E c g t 90 hE 90 x0 _ _ 1=2 ÷ E c E 90 x0 s 90 xxR : (6:27) The thermal residual stress, s 90 xxR , and other parameters in the expression above are known from property data for an undamaged laminate. Therefore, Eq. (6.27) provides a relation among s fpf c , b, and g. Now s fpf c and g can be determined from experimental data. Therefore, Laws and Dvorak regard Eq. (6.27) as the relation that determines the shear lag parameter b. Once b is known, the applied stress needed to cause cracking at location C is given by s c (l 1 ) = s fpf c ÷ E c E 90 x0 s 90 xxR _ _ tanh bl 1 2t 90 ÷tanh bl 2 2t 90 ÷tanh bl t 90 _ _ ÷1=2 ÷ E c E 90 x0 s 90 xxR : (6:28) In a practical scenario, the location C is random due to spatial variation of the resistance to crack formation. Let p be the probability density function for the next crack to occur at a given location. In a laminate which already contains cracks with normalized crack density, r c = t 90 / l, the expected value of the applied stress to cause additional cracking is then E s c r c ( ) [ [ = _ 2l 0 p x ( )s c x ( ) dx: (6:29) Three possible choices for p(x) are: Case 1: The next crack occurs midway, so p x ( ) = d(x ÷l); (6:30) where d(x) is the Dirac delta function. Case 2: All locations are equally likely. Therefore, p x ( ) = 1 2l : (6:31) Case 3: p(x) is proportional to the stress at the location. For this case, p(x) is given by [100] as 201 6.4 Modeling approaches p x ( ) = s 90 xxR ÷ E 90 x0 E 0 x0 s c _ _ 1 ÷ cosh bx t 90 cosh bl t 90 _ ¸ ¸ _ _ ¸ ¸ _ : (6:32) For case 1, the solution is explicitly given by E s c r c ( ) [ [ = s fpf c ÷ E c E 90 x0 s 90 xxR _ _ 2 tanh b 2r c ÷tanh b r c _ _ ÷1=2 ÷ E c E 90 x0 s 90 xxR : (6:33) For cases 2 and 3, the integral in Eq. (6.29) must be evaluated numerically. The model predictions for these cases of p(x) are shown in Figure 6.20 (with g = 193 J/m 2 and b = 0.9). Based on comparison with the experimental data, Laws and Dvorak argue that fracture mechanics-based p(x) (case 3) is the most promising choice. With this choice, model predictions also compare well with another set of experimental data from [1] for graphite/epoxy laminates. Variational analysis Nairn [99] used the variational approach [101] for cracked cross-ply laminates, including thermal residual stresses, in conjunction with the energy release rate criterion to predict crack densities in cracked cross-ply laminates. His predictions showed good agreement with experiments when the critical energy release rate for matrix cracking, defined as such, was deduced from test data rather than evaluat- ing it independently. Another damage evolution model for cross-ply laminates is by Vinogradov and Hashin [97, 98, 102]. It uses variational analysis [101] for stress computation and finite fracture mechanics [97] for cracking criterion. Recalling from Section 4.6, Figure 6.20. Crack density evolution in a glass/epoxy [0/90] s laminate for three choices of probability distribution function p(x) [100]. The experimental data are from [8]. Note: [8] has additionally three data points at the high load end that fall away from model predictions. Reprinted, with kind permission, from N. Laws and G. J. Dvorak, J Compos Mater, Vol. 23, pp. 900–16, copyright #1988 by Sage Publications. 202 Damage progression where the stress calculations are described in detail, complementary energy change due to presence of N transverse cracks can be derived as [98] ÁÅ + = N n=1 ÁÅ + n = s 90 xx0 _ _ 2 t 2 90 C 22 N n=1 w r n ( ); (6:34) where C 22 = l ÷1 ( ) 3l 2 ÷12l ÷8 _ _ 1 60E 2 (see Eq. (4.151)), and w(r n ) = ÷ d 3 f n dx 3 ¸ ¸ ¸ ¸ r n = 2a 1 a 2 a 1 2 ÷a 2 2 _ _ cosh (2a 1 r n ) ÷cos(2a 2 r n ) a 1 sin (2a 2 r n ) ÷a 2 sinh (2a 1 r n ) ; (6:35) and the summation in Eq. (6.34) is over all blocks bounded by adjacent cracks. It is noted that r n = l n /t 90 is the normalized crack spacing and should not be confused with the crack density. For clarity of notations, the reader is referred to Section 4.6. Initially when the cracks are far apart (r n ÷ ·), the function achieves its asymptotic value given by w(·) = 2a 1 a 1 2 ÷a 2 2 _ _ : (6:36) Now consider the state of a laminate with N cracks. When a new crack appears, the energy release can be expressed as [97] ÁÀ n = Å + s N÷1 _ _ ÷Å + s N _ _ ; (6:37) where s N and s N+1 are the stress fields before and after formation of a new (N+1)th crack. The stresses include both mechanical and thermal effects (if present). Assuming that the new crack appears instantly, both stress fields are evaluated at the same external load. Equation (6.37) can be rewritten as ÁÀ n = Å + s N÷1 _ _ ÷Å + s 0 _ _ _ ¸ ÷ Å + s N _ _ ÷Å + s 0 _ _ _ ¸ ; (6:38) where s 0 is the stress field in the undamaged material at the same external load. Assuming Eq. (6.34) is a good approximation of the energy, the energy release due to the (N+1)th crack is given by ÁÀ n = s 90 xx0 _ _ 2 t 2 90 C 22 N÷1 i=1 w r N÷1 i _ _ ÷ s 90 xx0 _ _ 2 t 2 90 C 22 N i=1 w r N i _ _ ; (6:39) where s 90 xx0 is the stress in the undamaged 90 · -ply, r N i is the nondimensional crack spacing for the ith block for the Nth cracking step (going from N to N+1 cracks). Using the energy release rate criterion, Eq. (6.18), taking averages and express- ing variables as continuous variables, the energy released during cracking can be expressed as (see [98] for full derivation) g = ÷ s 90 xx0 _ _ 2 t 90 C 22 d d r w r _ _ r 2 : (6:40) 203 6.4 Modeling approaches Using the lower bound of the laminate longitudinal modulus (see Section 4.6) by Hashin [101] 1 E + x = 1 E 0 ÷k 2 1 t 90 h C 22 w r ; (6:41) where k 1 = s 90 xx 0 =s 0 xx 0 if the temperature change is absent. Using A = L=2 r, we finally obtain the cracking criterion as g = 1 2 s 90 xx0 _ _ 2 d dA 1 E + x _ _ V; (6:42) where V is the laminate volume. Equation (6.42) represents a particular homo- thermal case of the general fracture criterion derived by Hashin [103] given by g = 1 2 s @S + @A s ÷ @a + @A sT ÷ 1 2 @c + p @A T 2 T r _ _ V; (6:43) where S* is the effective elastic compliance tensor, a* is the effective thermal expansion tensor, c + p is the effective specific heat of a composite, and T r is the reference temperature. The damage evolution predictions using this approach are very accurate when the probabilistic distribution of the energy release rate is utilized. This will be discussed in the next section. Nairn’s original result [48, 99] was quite similar to the fracture criterion stated above. However, in place of the usual energy release rate (2g) he used the matrix fracture toughness (G m ) and suggested that it could be obtained through fitting experimental data for ply cracking. If we assume that the new crack forms midway between the existing cracks, Nairn’s fracture criter- ion is G m = s 2 c E 2 2 E 2 c ÷ ÁaT 2 C 2 00 _ _ t 90 C 22 2w r=2 ( ) ÷w r ( ) [ [; (6:44) where C 00 = (1/E 2 ) + (1/lE 1 ) (see Eq. (4.151)). An alternative formulation is to allow the new crack to form anywhere between two existing cracks. If the prob- ability of crack formation at any position is proportional to the tensile stress, the energy release rate is given by G m = s 2 c E 2 2 E 2 c ÷ ÁaT 2 C 2 00 _ _ t 90 C 22 w d ( ) ÷w r ÷d ( ) ÷w r ( ) [ [; (6:45) where w d ( ) ÷w(r ÷d) ÷w(r) [ [ = _ r=2 0 w(d) ÷w(r ÷d) [ [ 1 ÷f(r ÷2d) [ [dd _ r=2 0 1 ÷f(r ÷2d) [ [dd ÷w(d): (6:46) 204 Damage progression For cracking in [90 m /0 n ] s laminates, the expressions for energy release rate remain the same except that the constant C 22 is now given as C 22 = (l+1)(3+12l+8l 2 ) (1/60E 2 ) (see Eq. (4.164)). It is noted that Nairn’s analysis included residual thermal stresses, which mainly change the crack initiation strain (discussed later in detail). The parameter G m is evaluated by fitting the model to the experimental data. The model predictions for a [0/90 3 ] s glass/epoxy laminate with G m = 330 J/m 2 and a thermal residual stress of 13.6 MPa are shown in Figure 6.21. Plain-strain formulation McCartney [104–109] developed a model based on the Gibbs free energy, instead of complementary strain energy as described above. He used a plane strain formulation for the estimation of elastic moduli of the damaged laminate, which has been covered in Section 4.7. Consider the damage progression from a state of m cracks to n cracks in the 90 · -ply and assume that each crack formation occurs under conditions of fixed applied tractions. Based on energy considerations, crack formation will occur when ÁÀ ÷ÁG _ 0; (6:47) where DG is the change in Gibbs free energy, and DG is the energy absorbed in the volume V of laminate due to the formation of new cracks, given by ÁÀ = V À o n ( ) ÷À o m ( ) [ [; (6:48) where o denotes the damage parameter which characterizes the crack density in the laminate. The corresponding change in the Gibbs free energy can be written as 300 200 100 0 0.0 0.2 0.4 0.6 0.8 1.0 Crack density (1/mm) A p p l i e d s t r e s s ( M P a ) Figure 6.21. The applied stress as a function of crack density in a [0/90 3 ] s glass/epoxy composite. The squares are data from [8]. The solid line is the energy release rate analysis fit using Eq. (6.45) with G m = 330 J/m 2 and assuming that the initial level of the residual thermal stresses in the 90 · -plies was 13.6 MPa [99]. Reprinted, with kind permission, from J. A. Nairn, J Compos Mater, Vol. 23, pp. 1106–29, copyright # 1989 by Sage Publications. 205 6.4 Modeling approaches ÁG = _ V g o n ( ) ÷g o m ( ) [ [dV; (6:49) where g(o) represents the Gibbs free energy per unit volume. After some math- ematical treatment (see [109] for details) the cracking criterion becomes ^e o n ( ) ÷^e o m ( ) [ [ 2 1 ~ E o n ( ) ÷ 1 ~ E o m ( ) ÷m A o n ( ) g o n ( ) ( ) 2 ÷m A o m ( ) g o m ( ) ( ) 2 ÷2 À o n ( ) ÷À o m ( ) [ [ > 0; (6:50) where ^e o ( ); ~ E o ( ); m A o ( ); g o ( ), and G(o) denote axial strain, axial Young’s modu- lus, in-plane axial shear modulus, applied in-plane shear strain, and the energy absorption per unit volume for length 2L of laminate, respectively, for given damage state o. G(o) is here given by À o ( ) h (90) hL M j=1 d (90) j g (90) j ; (6:51) where 2h (90) and 2h represent the total thickness of 90 · -plies and whole laminate, respectively; M is the number of potential cracking sites in the 90 · -plies, which are ordered from the top to bottom of the plies, taken in increasing order from the center of the laminate to the outside; and 2g (90) j is the fracture energy for the jth potential cracking site. The expressions for other parameters can be found in [109]. The crack initiation strain can be obtained by setting o m = o 0 (undamaged state) in Eq. (6.50). COD-based models Following the work of Parvizi et al. [81] and Wang and Crossman [110] on energy release rates to study the formation of cracks in cross-ply laminates, Joffe and coworkers [111–114] considered fully developed cracks and developed a methodology to predict the multiplication of transverse cracks based on the virtual crack closure technique. The idea is to probe the region between two existing cracks and introduce a virtual crack. For the introduced virtual crack the work performed to close the crack surfaces is calculated and compared with the energy needed to create a crack, the critical energy release rate (G c ), at this position. A crack is taken to form when the work to close the crack exceeds G c . Consider a damaged [0 m /90 n ] s laminate with a periodic system of “N” self- similar cracks with spacing s = 2l in the 90 · -ply (Figure 6.19(a)). At applied laminate stress s 0 (and corresponding far-field stress s 90 x0 in the 90 · -layer) a new crack develops midway between two existing cracks and the total number of cracks becomes 2N with spacing l (Figure 6.19(b)). According to the crack closure concept the released energy due to these N new cracks is equal to the work needed 206 Damage progression to close them. If we denote this work by W 2N÷N and the work to close all cracks simultaneously by W 2N÷0 , energy balance requires W 2N÷0 = W 2N÷N ÷W N÷0 ; (6:52) where the work to close N cracks with spacing s is W N÷0 = N 2 1 2 _ t 90 ÷t 90 s 90 xx0 u(z) dz = 2Ns 90 xx 0 t 90 u n (l) = N s 90 xx0 _ _ 2 E 2 t 2 90 ~ u n (l); (6:53) where unit width is assumed and u(z), u n ; and ¯ u n represent the variation of the normal crack opening displacement (COD) along the thickness direction, its average value, and its average value normalized with respect to the remote stress and transverse modulus for the ply, respectively. u n ; and ¯ u n are thus defined as u n = 1 t 90 _ t 90 0 u(z) dz; ~ u n = u n s 90 xx0 =E 2 _ _ t 90 : (6:54) Similarly, the work to close 2N cracks with spacing l is given by W 2N÷0 = 4N s 90 xx0 _ _ 2 E 2 t 2 90 ~ u n (l=2): (6:55) Substituting Eqs. (6.53) and (6.55) into Eq. (6.52), the energy released by forma- tion of a crack midway between two existing cracks of spacing s is W 2N÷N = 2N s 90 xx0 _ _ 2 E 2 t 2 90 2~ u(l=2) ÷ ~ u(l) [ [: (6:56) The cracks form when this work is greater than or equal to the cumulative surface energy of newly created surfaces, i.e., W 2N÷N _ 2 N 2t 90 G c : (6:57) From Eqs. (6.56) and (6.57), the criterion for crack formation is s 90 xx0 _ _ 2 2E 2 t 90 2~ u(s=2) ÷ ~ u(s) [ [ _ G c : (6:58) To analyze cracking in an arbitrary position between two pre-existing cracks, a new crack is introduced in an arbitrary position between the cracks (Figure 6.19 (b)), which leads to a new damage state with one crack spacing equal to s 1 and the second one equal to s 2 = s – s 1 . The cracking criterion in this case is s 90 xx0 _ _ 2 2E 2 t 90 2~ u(l 1 =2) ÷ ~ u(l 1 ) ÷2~ u(l 2 =2) ÷ ~ u(l 2 ) [ [ _ G c : (6:59) 207 6.4 Modeling approaches The authors applied their analysis for the prediction of crack density evolution in glass/epoxy [±y/90 4 ] s laminates for the case of varying crack spacing. A Weibull distribution for G c was utilized and the CODs were calculated using FE analysis. The predictions are shown against experimental data in Figure 6.22. Adolfsson and Gudmundson [11] also developed an energy-based damage evolution approach using their stress analysis. Basic stress analysis using this approach is covered in Section 4.9.1, although they updated their analysis to include bending loads, which can be found in [11, 115]. The energy model for crack density evolution is based on changes in strain energy due to cracking. From [11], the strain energy per unit in-plane area of the damaged laminate with n plies may be written as w (c) = 1 2 e ÷a (c) ÁT ÷e (R) _ _ T C (c) e ÷a (c) ÁT ÷e (R) _ _ ÷ n k=1 h k ÁT; s k(R) _ _ ; (6:60) where bold-face letters represent matrices; C (c) , a (c) , and e (R) are the stiffnesses, thermal expansion coefficients and residual stress-induced eigenstrains, respect- ively, of the cracked laminate. Expressions for these crack density quantities were derived in [115]. In Eq. (6.60), DT is the temperature difference between the curing and service temperature and h k are the functions containing energy stored in the laminate due to interlaminar constraints and thermal residual stresses (their expressions are given in Appendix A of [11]). From the strain energy, the energy release rate for the ith cracked ply is derived as 0.8 [±q/90 4 ] s FEM model, energy approach 0.6 0.4 0.2 0.0 q=0 q=15 q=30 q=40 0 C r a c k d e n s i t y ρ ( c r / m m ) 50 100 Stress s 0 (MPa) 150 200 250 Figure 6.22. Evolution of crack density as a function of applied stress using energy model for [±y/90 4 ] s laminates. Symbols represent experimental data and lines represent the average of four runs using the energy model [113]. Reprinted, with kind permission, from Compos Sci Technol, Vol. 61, R. Joffe, A. Krasnikovs, and J. Varna, COD-based simulation of transverse cracking and stiffness reduction in [S/90n]s laminates, pp. 637–56, copyright Elsevier (2001). 208 Damage progression G i = ÷ @U @A i = ÷ @ Aw (c) _ _ @A i ; (6:61) where A i is the crack surface area in ply i. The area A i is given by A i = Ar i with normalized crack density r i = t i / l i , where l i is the average spacing of cracks in the ith ply and A is the laminate in-plane area. From Eqs. (6.60) and (6.61) the energy release rate G i for cracking in the ith ply is given by G i = @a (c) @r i ÁT ÷ @e (R) @r i _ _ T C (c) e ÷a (c) ÁT ÷e (R) _ _ ÷ 1 2 e ÷a (c) ÁT ÷e (R) _ _ T @C (c) @r i e ÷a (c) ÁT ÷e (R) _ _ ÷ N k=1 @h k @r i : (6:62) The above expression contains the derivatives of the effective thermoelastic properties of the damaged laminate with respect to the ply crack densities. Calculating these quantities is a more complex task than determining the proper- ties themselves. For this purpose, either FEM or the approximate analytical expressions given in Appendix A of [11] have to be utilized. The model predictions for the stress–strain response of graphite/epoxy cross-ply laminates are compared with experimental data in Figure 6.23. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Strain (%) 0 100 200 300 400 500 600 S t r e s s ( M P a ) (0/90/0) (0/90 2 /0) (0/90 4 /0) (0/90 8 /0) Simulations Figure 6.23. Stress–strain response of graphite/epoxy cross-ply laminates. Symbols represent experimental data whereas the solid lines represent predictions from the model by Adolfsson and Gudmundson [11]. Reprinted, with kind permission, from Int J Solids Struct, Vol. 36, E. Adolfsson and P. Gudmundson, Matrix crack initiation and progression in composite laminates subjected to bending and extension, pp. 3131–69, copyright Elsevier (1999). 209 6.4 Modeling approaches Qu–Hoiseth analysis The evaluation of moduli for damaged cross-ply laminates using the approach proposed by Qu and Hoiseth [116] is covered in Section 4.5. The cracking criterion for this model is derived as G c = 2e c 2 t 90 E c E 2 (E 1 ÷E 2 )r exp ÷ dr 2t 90 _ _ ÷exp ÷ dr t 90 _ _ _ _ ; (6:63) where G c is the in-plane mode I fracture toughness of transverse ply; e c is the applied strain; E c is the plain-strain Young’s modulus of the undamaged cross-ply laminate in the longitudinal direction; E 1 and E 2 are longitudinal and transverse moduli of the ply, respectively; d is the average crack opening displacement of the 90 · crack; and r = t 90 / l is the normalized crack density. The threshold strain at which the transverse matrix cracking initiates can be obtained by setting r ÷0 in Eq. (6.63) as e 0 = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G c E 1 ÷E 2 ( ) dE c E 2 ¸ : (6:64) The model predictions for a cross-ply laminate made of AS/3501–06 material for two values of G c are shown in Figure 6.24. The experimental data are from [117]. 6.4.3 Strength vs. energy criteria for multiple cracking There is a fundamental difference between strength- and energy-based criteria when applied to multiple ply cracking in laminates. Strength essentially represents the failure at a material point when a specified stress component or function of 0.0 C r a c k d e n s i t y 0.0 0.5 s c = E c e c Prediction with G c =130J/m 2 1.0 1.5 2.0 0.5 Applied stress 1.0 AS/3501–06 [0 2 /90 2 ] s 1.5 2.0 Experimental data Prediction with G c =104J/m 2 Figure 6.24. Comparison between model predictions and experimental data for the COD model of Qu and Hoiseth [116] for a cross-ply laminate, s c represents the applied stress on the composite. Reprinted, with kind permission, from Fatigue Frac Eng Mater, Vol. 24, J. Qu and K. Hoiseth, Evolution of transverse matrix cracking in cross-ply laminates, pp. 451–464, copyright Wiley (1998). 210 Damage progression stress components reaches a critical value. This approach is a legacy of homo- geneous materials such as metals and ceramics where yielding at a point (in a metal) is assumed to occur according to, e.g., the von Mises criterion, or brittle failure (in a ceramic) is assumed to occur when the maximum tensile principal stress reaches a critical value. In case of yielding there is no ambiguity in terms of the stress components attaining critical value at a point since this type of “failure” can spread spatially from one point to another as indicated by the contour of the yield criterion. However, “brittle failure” represents the instability of crack growth, and unless a crack exists its growth is meaningless. This difficulty has been conventionally overcome (or bypassed) by assuming that brittle failure according to a point-failure (strength) criterion is the concurrent formation and instability of a crack. In an unconstrained failure case, such as the brittle failure of an unre- inforced ceramic, this approximating assumption causes little difficulty since the initiation of cracking and its unstable growth are usually not far apart, i.e., they occur at roughly the same applied load. However, when the constraint to crack growth is imposed by the presence of reinforcements, or stiff elements in the matrix generally, then the formation of a crack and the instability of its growth are determined by different conditions. This fact was realized in [91] when it was found that the strength criterion was inadequate to predict the formation of multiple cracking in unidirectional brittle matrix composites. Energy considerations were then made by recognizing the dissipation of energy in crack surface formation. Multiple cracking in composite materials is an inherent feature of the failure process due to the presence of directed interfaces (fiber/matrix and interlami- nar) that impart mechanisms of stress transfer from the cracking elements to the noncracking elements, which in turn provide a constraint to the cracks. Thus, the inadequate incorporation of constrained cracking in a multiple cracking process is bound to induce error. One example, unfortunately not uncommon, is using solutions of crack opening displacements in an infinite medium in models of multiple cracking. 6.5 Randomness in ply cracking Physical observations of ply cracking indicate that in the early stages of the cracking process randomness exists in the location of cracks, their size, and how the evolu- tion (growth and multiplication) of cracking occurs. As the cracking process evolves, randomness tends to decrease, and as the process approaches saturation, uniformity in crack spacing results. The causes of randomness are many, most induced by the manufacturing process. For instance, the fiber volume fraction can vary spatially. Image analysis reported in [118] showed that in a T300/914 carbon/ epoxy composite with an average fiber volume fraction of 55.9% the local volume fraction ranged between 15 and 85%. Other common defects are voids and inclu- sions in the matrix, partially cured regions of matrix, broken fibers, fiber waviness, unbonded regions of interfaces at fiber surfaces, and between plies. 211 6.5 Randomness in ply cracking Several attempts exist in the literature to treat random variations of microstruc- ture. Silberschmidt [119, 120] suggested a lattice scheme which incorporates the effects of the initial microstructural randomness as well as a dispersed evolution of damage and its transition to spatially localized matrix cracking. The scheme involves mapping a dynamic matrix of stress-renormalizing coefficients onto the lattice of elements covering the cracked (90 · ) layer. Figure 6.25 shows the distri- bution of ply cracks at different times in load history based on the scheme for a [0 2 /90 4 ] s T300–934 laminate during fatigue loading. Another example of crack distribution for [0/90]s glass/epoxy laminates, tested under quasi-static tensile loading by Manders et al. [87], is shown in Figure 6.26, along with the average axial stress distribution along the laminate length, esti- mated by Berthelot and Le Corre [121]. Clearly the stress state is such that it is difficult to predict damage evolution using deterministic approaches. Probabilistic notions are often used to correct the cracking predictions for random effects, as stated earlier. Although some probabilistic fracture criteria have already been discussed, we shall focus here on the variational stress model, emphasizing details of probabilistic considerations. 80 70 60 50 40 30 20 10 0 0 2 4 6 (cm) (b) (a) 8 10 s 9x 0x − ( M P a ) Figure 6.26. Crack distribution (a) and corresponding variation (b) of the average longitudinal stress in the 90 · -ply along the laminate length, estimated for [0/90] s glass/epoxy laminates [121]. Reprinted from Compos Sci Technol, Vol. 60, J. M. Berthelot and J. F. Le Corre, Statistical analysis of the progression of transverse cracking and delamination in cross-ply laminates, pp. 2659–69, copyright (2000), with permission from Elsevier. Figure 6.25. Distribution of ply cracks in a [0 2 /90 4 ] s laminate at different times in the load history based on the lattice scheme by Silberschmidt [119, 120]: (a) 100 cycles; (b) 4×10 3 cycles; (c) 10 5 cycles; and (d) 2×10 5 cycles [120]. Reprinted, with kind permission, from Springer Science+Business Media: J Mater Sci, Effect of micro-randomness on macroscopic properties and fracture of laminates, Vol. 41, 2006, pp. 6768–76, V. V. Silberschmidt. 212 Damage progression A fracture mechanics-based stochastic model to predict progression of ply cracking was initially proposed by Wang and coworkers [1, 110, 122–125]. The authors postulated that cracking in transverse plies of cross-ply laminates is governed by a characteristic distribution of “effective” flaws, which are essentially inherent material microcracks that cannot be seen until grown to macroscopic dimensions. Thus these microcracks act individually as initiators of cracks that propagate to form fully grown transverse cracks. The distribution of flaw size f(a) and spacing f(S) along specimen length are assumed to follow the following normal probability distributions f (a) = 1 a ffiffiffiffiffiffi 2p _ exp ÷ (a ÷m a ) 2 2s 2 a _ _ ; f (S) = 1 S ffiffiffiffiffiffi 2p _ exp ÷ (S ÷m S ) 2 2s 2 S _ _ ; (6:65) where 2a is the average flaw size, S is the average distance between two adjacent flaws, and m a , m S, s a , and s S are fitting parameters. The “worst” of the flaws causes the first ply cracking. With increased loading, smaller flaws cause further trans- verse cracking. The first transverse crack forms when G s c ; a 0 ( ) = G c ; (6:66) where s c is the longitudinal stress applied to the composite, 2a 0 is the initial flaw size, and G c is the critical energy release rate, which is assumed to be constant along the laminate length. The propagation of the flaw will be stable if G s c ; a 0 ÷Áa ( )<G c ; (6:67) and unstable if G(s c ; a 0 ÷Áa) > G c : (6:68) Progressive cracking will ensue when there is enough energy available for multiple flaws to propagate into fully grown transverse cracks. For crack formation after the first crack the energy release rate for flaw propagation depends on its relative distance S from the existing crack and can be expressed as G(s c ; a) = R(S)G 0 (s c ; a); (6:69) where G 0 is the energy release rate when no crack is present, and R(S) is the energy retention factor, with a value between 0 and 1, accounting for the presence of a neighboring crack. Similarly, for a flaw to propagate between two existing trans- verse cracks, the energy release rate is G(s c ; a) = R(S L )G 0 (s c ; a)R(S R ) (6:70) where S L and S R are the distance of the flaw from the left and right cracks, respectively. Chou et al. [126] implemented the approach using a Monte Carlo scheme. The results showed a fair agreement with experimental data. Essentially, 213 6.5 Randomness in ply cracking this approach predicts the event when a micro-flaw develops into a fully grown transverse crack, thereby predicting the multiplication of ply cracks. However, the approach has not gained wide usage because it requires many unknown param- eters which are found by fitting to experimental data. As mentioned earlier, experimental observations show that the transverse cracks usually grow quickly through the 90 · -ply thickness as well as the specimen width. Therefore, more recent approaches do not try to predict crack propagation; rather they focus on the multiplication of cracks, i.e., an increase in crack density. To illustrate how a more recent fracture criterion can be modified to include probabilistic measures we follow the treatment of Vinogradov and Hashin [98]. Accordingly, the uncertainties in the cracking process can be categorized into two probabilistic notions: “geometrical” and “physical.” The “geometrical” uncertainty refers to the probability of a crack to appear at a certain location between two existing adjacent cracks, while the “physical” aspect deals with the variation of material resistance to crack formation. The geometrical aspect of probabilistic cracking can be introduced by considering the statistical variation of distance between adjacent cracks, i.e., r = _ · 0 rp r ( )dr; w = _ · 0 w r ( )p r ( ) dr; (6:71) where p(r) is the probability density function (PDF) of distances between adjacent cracks. The criterion for first crack formation can be found by substituting r ÷· in Eq. (6.40) and using Eq. (6.36), to obtain g = s 90 xx0 _ _ 2 t 90 C 22 w · ( ) = 2 s 90 xx0 _ _ 2 t 90 C 22 a a 2 ÷b 2 _ _ : (6:72) In fact the criterion in Eq. (6.72) is expected to predict the initial stage of the damage evolution curve. For any material block between two existing adjacent cracks, the cracking criterion in Eq. (6.40) can be rewritten as [98] g = s 90 xx0 _ _ 2 t 90 C 22 w r ÷x 2 _ _ ÷w r ÷x 2 _ _ ÷w r ( ) _ _ ; (6:73) where x denotes the nondimensional coordinate of the new crack location between the two existing cracks. Equation (6.73) is a local criterion for crack formation because it deals with the location of the next crack. The “physical” nature of damage evolution can be achieved by having a probabilistic variation of material property g. Thus, g = G(x): (6:74) The parameter G(x) can be thought of as local toughness of the material by arguing that it is easy to form a crack at a section which contains many flaws and has a weak interface. The variation of g is usually described using a Weibull distribution, i.e., the PDF of g can be expressed as 214 Damage progression p r (g) = g 0 g ÷g min g 0 _ _ ÷1 exp ÷ g ÷g min g 0 _ _ _ _ ; g _ g min ; (6:75) where g min is the minimum possible value of g, and and g 0 are parameters of the distribution, usually evaluated by fitting experimental data. For different laminate systems, i.e., for different mixes of 0 and 90 · plies, e.g., [0 n1 /90 m1 ] s and [0 n2 /90 m2 ] s laminates, the distribution parameters may not be the same. If the parameters for the first laminate configuration, 1 , g 01 , are known (through fitting of experimental data), the parameters 2 , g 02 for the second laminate configuration can be found from the following relations À 2 ÷ 2 2 _ _ ÷À 2 1 ÷ 2 2 _ _ À 2 1 ÷ 2 2 _ _ = m 2 m 1 À 2 ÷ 1 1 _ _ ÷À 2 1 ÷ 1 1 _ _ À 2 1 ÷ 1 1 _ _ ; g 02 = g 01 À 1 ÷ 1 1 _ _ À 1 ÷ 2 2 _ _; (6:76) where G(x) represents the standard gamma function for the random variable x. The derivation can be found in the original article [98]. The simulation procedure for this model can be summarized as follows: 1. Choose a ply material or a laminate configuration of a ply material. 2. Distribute random points for possible crack locations along the laminate length. 3. Generate a random value of g at each point according to the Weibull distribution. 4. Fit the model predictions to the experimental data for crack density evolution to deduce the parameters of the Weibull distribution. 5. Calculate Weibull parameters for other laminate systems using Eq. (6.76), and predict the crack density evolution for these laminates. Some examples of the numerical simulation results with the fitted and calculated parameters of the distribution are shown in Figure 6.27. A recent strength-based analysis by Berthelot and Le Corre [121] has revealed that the choice of probabilistic distribution should account for weakness areas in the material. This analysis for [0/90 2 ] s carbon/epoxy laminate shows that a prob- abilistic distribution of strength which accounts for weakness areas properly corrects the crack density evolution in the beginning stage (see Figure 6.28). A divergence between models with and without consideration of weakness areas is always observed at low crack densities. This is because, initially, cracking is preferred at weakness areas where the fracture toughness of the material is low 215 6.5 Randomness in ply cracking (due to inherent defects) as compared to its average value in the whole laminate. For glass/epoxy laminates, Berthelot and Le Corre found that delamination occurs at high crack densities and is the cause of data deviating from the model prediction. 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) 200 C r a c k d e n s i t y ( 1 / m m ) [0/90 3 ] s [0/90 2 ] s 400 600 Stress (MPa) 800 1000 1200 1400 Avimid K Polymer/IM6 0 1.2 (b) 100 C r a c k d e n s i t y ( 1 / m m ) [0/90 4 ] s [0/90 2 ] s [0/90] s 200 300 400 Stress (MPa) Hercules AS4/3501–6 500 600 700 800 1 0.8 0.6 0.4 0.2 0 Figure 6.27. Prediction of crack density evolution in cross-ply laminates using the Vinogradov and Hashin model [98] for two material systems: (a) Avimid K polymer/IM6, (b) Hercules AS4/3501-6. The experimental data are from [127]. Reprinted, with kind permission, from Int J Solids Struct, Vol. 42, V. Vinogradov and Z. Hashin, Probabilistic energy-based model for prediction of transverse cracking in cross-ply laminates, pp. 365– 392, copyright Elsevier (2005). 216 Damage progression 6.6 Damage evolution in multidirectional laminates Although many generic features of ply cracking are evident in cross-ply laminates, this class of laminates is used only in limited cases. Most applications require a mix of lamina orientations in the laminate configuration to generate properties to carry combinations of normal loads, bending moments and torsion. In a multidirectional laminate the ply cracking in any ply will generally take place under stresses normal to and parallel to the fibers, as well as in-plane shear. Experimental investigations [27, 34, 35, 50, 51, 128] have clarified some of the complexities relating to mode- mixity of crack growth and interactions between cracks within and among plies. Consider now a [0/90/y 1 /y 2 ] s laminate where ply cracks can appear in the 90 · -, y 1 -, and y 2 -plies, assuming loading in the 0 · -direction. Figure 6.29 illustrates the development of cracking in multiple orientations in such a laminate configuration. As indicated there, cracking initiates first in the 90 · -plies at an overall strain e 90 0 , and, on increasing the load, this cracking multiplies. At the strain e y 1 0 the y 1 -plies 1400 (a) 1200 1000 800 600 Experimental results Without weakness areas With weakness areas 400 C r a c k d e n s i t y ( m – 1 ) 200 0 250 300 350 400 Average stress (MPa) 450 500 550 1400 (b) 1200 Experimental results Without weakness areas With weakness areas 1000 800 C r a c k d e n s i t y ( m – 1 ) 600 400 200 0 250 300 350 400 Average stress (MPa) 450 500 550 Figure 6.28. Evolution of crack density as a function of applied stress for a [0/90 2 ] s carbon/ epoxy laminate [121]. The experimental data are from [1]. Reprinted, with kind permission, from Compos Sci Technol, Vol. 60, J. M. Berthelot and J. F. Le Corre, Statistical analysis of the progression of transverse cracking and delamination in cross-ply laminates, pp. 2659–69, copyright Elsevier (2000). 217 6.6 Damage evolution in multidirectional laminates begin cracking (assuming y 1 >y 2 ), and witha further increase in the imposed load, an interactive cracking process continues inthe 90 · - andy 1 -ply orientations. At straine y 2 0 cracking initiates in the y 2 -plies, and eventually all off-axis plies conduct interactive crack multiplication. The crack initiation strains and crack multiplication rates dependonthe constraint imposedby the neighboring plies tothe cracks ina givenply. For predicting the evolution of ply cracking the authors of this book have developed an energy-based approach, which is capable of dealing with cracking in off-axis plies of orthotropic laminates. In [129] the approach is described and applied to several ply cracking cases. A brief description of the approach follows. As illustratedinFigure 6.30, twodamage states are considered: state 1 withNparallel off-axis cracks spaced at distance s, and state 2 where the cracks have multiplied to 2N and attained the spacing s/2. Evolution of cracking damage is assumed when the work requiredingoing fromstate 1 tostate 2 (whichis the same as the workneededtoclose N cracks in going from state 2 to state 1) exceeds a critical value, i.e., if W 2N÷N _ N:G c : 1 sin y t c ; (6:77) where y is the off-axis angle and G c is the critical (threshold) value of the energy required for multiple ply crack formation within the given laminate (more discus- sion about this later). The work required to form N additional cracks in going from state 1 to state 2 (the same as the work required to close those cracks) is given by W 2N÷N = W 2N÷0 ÷W N÷0 ; (6:78) 90 q 1 q 2 0 0 0 Figure 6.29. Cracking process in a [0/90/y 1 /y 2 ] s half-laminate. State 1: N cracks, crack spacing = s State 2: 2N cracks, crack spacing = s/2 t q s s/2 x 3 x 2 Figure 6.30. Progressive multiplication of off-axis ply cracks in a multidirectional laminate [129]. Reprinted, with kind permission, from Int J Solids Struct, Vol. 47, C. V. Singh and R. Talreja, Evolution of ply cracks in multidirectional composite laminates, pp. 1338–49, copyright Elsevier (2010). 218 Damage progression where W N÷0 and W 2N÷0 represent the work required to close N cracks in state 1, and 2N cracks in state 2, respectively, and the two quantities are calculated as (see [130] for detailed derivation) W N÷0 = N 1 sin y t c ( ) 2 1 E 2 s y 20 _ _ 2 :~ u y n s ( ) ÷ s y 120 _ _ 2 ~ u y t s ( ) _ _ ; (6:79) W 2N÷0 = 2N 1 sin y t c ( ) 2 1 E 2 s y 20 _ _ 2 :~ u y n s 2 _ _ ÷ s y 120 _ _ 2 ~ u y t s 2 _ _ _ _ ; (6:80) where ~ u y n ; ~ u y t are the normalized average crack opening and sliding displacements (COD and CSD), respectively. These are given by ~ u y n = u y n t c s y 20 =E 2 _ _ = 1 t c s y 20 =E 2 _ _ _ t y =2 ÷t y =2 u n (z)dz; ~ u y t = u y t t c s y 120 =E 2 _ _ = 1 t c s y 120 =E 2 _ _ _ t y =2 ÷t y =2 u t (z) dz; (6:81) where u n and u t represent the relative opening and sliding displacement of the cracked surfaces, respectively, and overbars represent averages. For the special case of cracking in the 90 · -ply only, the sliding displacement is zero and hence the criterion for ply crack multiplication is written as t c : s y 20 _ _ 2 E 2 2:~ u y n s 2 _ _ ÷ ~ u y n s ( ) _ _ _ G Ic ; (6:82) where G Ic is the critical energy release rate in mode I (crack opening mode). This is the same relation as that derived for cracking in cross-ply laminates by Joffe et al. [113] except that they consider centrally placed cracked 90 · -plies in their model and normalize the average COD with half the ply thickness (t c /2). For cracking in a general off-axis ply, one can use a multi-mode criterion given as w I G Ic _ _ M ÷ w II G IIc _ _ N _ 1; (6:83) where w I = s y 20 _ _ 2 t c E 2 2:~ u y n s 2 _ _ ÷ ~ u y n s ( ) _ _ ; w II = s y 120 _ _ 2 t c E 2 2:~ u y t s 2 _ _ ÷ ~ u y t s ( ) _ _ ; (6:84) where G IIc is the critical energy release rate in mode II (crack sliding mode), and the exponents M and N depend on the material system, e.g, for a glass/epoxy system, M = 1, N = 2 [130]. In our work [129] we interpret the critical material parameters G Ic and G IIc not in the usual linear elastic fracture mechanics sense where they are defined as the resistance to advancement of the crack front at the point of unstable crack growth. 219 6.6 Damage evolution in multidirectional laminates Instead, we postulate that the work required to go from state 1 to state 2 involves a range of dissipative processes that all depend on the material condition in a cracking ply within the given laminate. The material parameter representing the dissipated energy per unit of ply crack surface is, therefore, not what is obtained in a standard fracture toughness test for determining G Ic or G IIc . To emphasize that the critical energy terms used here are not the usual fracture toughness values G Ic or G IIc , we will henceforth use the symbols W Ic and W IIc instead. These new quantities are not to be obtained by independent tests, but are to be evaluated by fitting model predictions (Eq. (6.83)) to the experimental data for a reference laminate. This way the values obtained will be representative of the energies associated with multiple cracking within a laminate. A reference laminate is chosen from the class of laminates (material etc.) for which predictions are to be made, and for which experimental data are readily available [129, 131]. Further- more, as described in [129], it is argued that a ply crack within a laminate cannot be formed unless sufficient energy is available to open its surfaces (i.e., in mode I cracking). In other words, a pure sliding action will not by itself generate the set of parallel cracks illustrated in Figure 6.30. This will imply that the second term in Eq. (6.84) is negligible in comparison to the first term. With these assumptions and approximations the predictions of crack density evolution agree well with experi- mental data [129]. The complete procedure to implement the described energy model for micro- crack initiation and evolution in an off-axis ply of a general symmetric laminate is outlined below. The procedure is in two parts: Part I: Estimate W Ic 1. From FE simulations, determine the variation of normalized COD and CSD (Eq. (6.81)) with crack spacing. 2. Assume a value for W Ic . Plot the damage evolution for the reference laminate as follows: (a) Divide the specimen length into small intervals of length, dX = t y /10 is chosen here. (b) Find the multiple crack initiation strain, Eq. (6.82) with the COD value calculated with a very large spacing (s ÷·). (c) Assume a small initial crack density, e.g., r initial = 1/50t y is chosen here. (d) Choose a random length interval and check for cracking. A new crack forms when the criterion set in Eq. (6.82) is satisfied. Increase the crack density and eliminate the crackedlengthinterval fromfurther considerationfor plycracking. (e) Choose another length interval and repeat the previous step until the fracture criterion is satisfied. (f) Increase the applied strain. Repeat steps (d) and (e) using this strain value. 3. Iterate step 2 by varying W Ic so that the resulting evolution curve fits the experimental data for the reference laminate. For example, for predicting the damage evolution in [0/±y 4 /0 1/2 ] s laminates, we chose [0/90 8 /0 1/2 ] s as the refer- ence laminate. 220 Damage progression Part II: Predict the damage evolution for other off-axis plies: 1. From FE simulations, determine the variation of COD and CSD (Eq. (6.81)) with crack spacing for a given off-axis laminate. 2. Using the value for W Ic obtained above, predict the damage evolution by following steps 2(a)–(f) described in Part I. The above semi-analytical model is coded in a MATLAB program. The input data include the following laminate properties: ply material (elastic moduli), ply thicknesses and orientations (i.e., laminate layup), and variation of COD with respect to crack density (which can be obtained from independent 3-D FE analysis). The energy model described above was applied to predict damage evolution in glass/epoxy [0/±y 4 /0 1/2 ] s , quasi-isotropic ([0/90/∓45] s ), and [0 m /90 n /±y p ] s laminates [129]. Figure 6.31 shows the variation of crack initiation strains with off-axis ply orientation (y) for [0/±y 4 /0 1/2 ] s laminates. As expected, the crack initiation strain increases as y decreases and it may exceed 1.5% if y < 45 · . In fact, the experiments by Varna et al. [27] for this laminate revealed that ply cracks did not form fully for y < 40 · . Using the procedure described above, W Ic is evaluated by fitting model predictions with experimental data for a chosen reference laminate [0/90 8 /0 1/2 ] s . The evolution of crack density against applied strain for these laminates are shown in Figure 6.32 and Figure 6.33 for y = 70 · and 55 · , respectively. For these laminates, direct application of Eq. (6.84) does not yield accurate predictions. The reason is that at low crack densities, the work term w I is almost constant (i.e., independent of crack spacing). How- ever, on development of sufficient cracks with a distribution in inter-crack spacing, w I depends on the crack spacing due to interactions between adjacent cracks. To account for this behavior, Liu and Nairn [127] suggested that the effective crack spacing be used in place of the average crack spacing s. Thus, w I is modified as 40 50 60 70 Ply Orientation (deg) 80 90 0 0.5 1 1.5 2 2.5 3 3.5 4 C r a c k i n i t i a t i o n s t r a i n ( % ) Model Experimental Data Figure 6.31. Variation of crack initiation strain with ply orientation for a glass/epoxy [0/±y 4 /0 1/2 ] s laminate. The experimental data are from [27]. 221 6.6 Damage evolution in multidirectional laminates w I = s y 20 _ _ 2 t c E 2 2:~ u y n fs 2 _ _ ÷ ~ u y n fs ( ) _ _ ; (6:85) where the parameter f is the average ratio of the crack interval in which a microcrack forms to the average crack spacing. For [0/±y 4 /0 1/2 ] s laminates, the predictions shown in Figure 6.32 and Figure 6.33 are made with f = 0.8. The same model was also used to predict damage evolution in a quasi-isotropic laminate. The value of W Ic for this case was obtained by fitting model predictions with experimental data for reference [0/90] s laminate. Since 45 · -plies in the quasi- isotropic laminates contained partially grown cracks, this was accounted for in the analysis by considering cracks in multiple orientations while calculating CODs from FE analysis. This showed that COD for the 90 · -ply increased resulting in Figure 6.32. Damage evolution in [0/±70 4 /0 1/2 ] s laminates. The experimental data are from [27]. The crack density is average of crack densities in +70 · and ÷70 · -plies. Reprinted, with kind permission, from Int J Solids Struct, Vol. 47, C. V. Singh and R. Talreja, Evolution of ply cracks in multidirectional composite laminates, pp. 1338–49, copyright Elsevier (2010). Figure 6.33. Damage evolution in [0/±55 4 /0 1/2 ] s laminates. The experimental data are from [27]. The crack density is the average of crack densities in +55 · and ÷55 · -plies. Reprinted, with kind permission, from Int J Solids Struct, Vol. 47, C. V. Singh and R. Talreja, Evolution of ply cracks in multidirectional composite laminates, pp. 1338–49, copyright Elsevier (2010). 222 Damage progression enhanced cracking in the transverse ply. The model predictions for crack density evolution in this case are compared with experimental data in Figure 6.34. No modifier for crack spacing was needed for this case, i.e., f = 1. A parametric study performed for [0 m /90 n /±y p ] s laminates reveals that inter- actions between the crack systems of different orientations may have a significant effect on damage evolution. If y is close to 90 · , this intra-mode interaction is higher because of the close proximity of the 90 · and y-crack planes. Ply cracks usually initiate in the 90 · -plies first and then in the 60 · -layers. Thus, initial simulation assumes only 90 · -cracks; whereas after the initiation of 60 · -cracks, a multi-mode scenario is used in FE modeling. The 60 · -cracks influence the damage progression in the 90 · -layer. Model predictions for y =60 · for different values of m, n, and p are shown in Figure 6.35(a)–(c) for the 90 · -, 60 · -, and +60 · -layers, respectively. For p = 2, the model predicts that cracks in the 60 · - and +60 · -layers will initiate earlier than in the 90 · -layer. If y < 45 · , this intra-mode interaction is not appreciable, and for this case the damage evolution in the 90 · -layer may not be affected at all. 6.7 Damage evolution under cyclic loading While the fatigue process in composite materials is treated in the next chapter, here we shall describe a modeling approach for the evolution of transverse cracking in cross-ply laminates under cyclic axial tension. This case because of its simplicity of geometry serves as a good illustration of the fundamental ideas in the fatigue of composites. The reader is urged to refer to two papers [53, 132] for full details of the treatment described below. Figure 6.34. Evolution of 90 · -crack density in [0/90] s and [0/90/±45] s laminates. The experimental data are from [34]. Reprinted, with kind permission, from Int J Solids Struct, Vol. 47, C. V. Singh and R. Talreja, Evolution of ply cracks in multidirectional composite laminates, pp. 1338–49, copyright Elsevier (2010). 223 6.7 Damage evolution under cyclic loading (a) (b) (c) Figure 6.35. Evolution of crack density in a [0 m /90 n /±60 p ] s laminate for varying ply thicknesses in (a) 90 · -layer, (b) ÷60 · -layer, and (c) +60 · -layer. Reprinted, with kind permission, from Int J Solids Struct, Vol. 47, C. V. Singh and R. Talreja, Evolution of ply cracks in multidirectional composite laminates, pp. 1338–49, copyright Elsevier (2010). The guiding principle in any fatigue analysis must be to address the question: what is the mechanism of irreversibility that causes the accumulation of damage from one load cycle to another? The common energy-dissipating mechanisms are plasticity, friction, and surface formation. For composite laminates that are modeled as layered elastic solids plasticity is not admissible. Frictional processes within the volume of such composites are possible between crack surfaces if the surfaces are in contact. Finally, new surface formation without plasticity is pos- sible by brittle fracture. As a case for illustration we shall consider a cross-ply laminate with ply cracks in the central 90 · -plies that formed under the first application of an axial tensile load. The problem posed is: if the load is removed and repeatedly reapplied to the previous maximum value, when would new cracks form between the pre-existing cracks? To begin the analysis we note that for the cracked cross-ply laminate a solution of good accuracy for the stress field in the region between cracks is available (e.g., [101]). This solution is valid for perfectly and linearly elastic (no plasticity) laminates. Also, the solution does not apply to partial cracks, i.e., cracks that are not fully extended in the thickness and width directions of the 90 · - plies. Thus to use this solution we must retain the symmetry and periodicity of the cracks assumed in obtaining the solution. This condition eliminates analysis of the cyclic growth of transverse cracks from partial to full extent. Obviously, a numer- ical analysis of this case is possible, but that would not provide the analytic fatigue damage model we intend to develop. With the analytical stress solution to the cracked cross-ply laminate at hand we see that unless some irreversible mechanism is included no change in the crack density can be predicted since any repetition of load in an elastic solid cannot change the stress field. Therefore, in order to have an analytical stress solution, i.e., keeping the symmetry of laminate geometry and periodicity of cracks, and to incorporate irreversibility, a novel idea was proposed in [132]. According to that, all irreversibility leading to damage accumulation is lumped into delamination surfaces emanating from the transverse crack fronts. Figure 6.36(a) illustrates the resulting model geometry of the cracked laminate while Figure 6.36(b) shows the repeating unit cell. As shown, the pre-existing trans- verse cracks are spaced at distance 2a and the delamination on either side of the cracks is of distance d. The idea behind the model is that the delamination grows under applied cyclic loading, the same way as a crack does, imparting changes to the stress field in the region between transverse ply cracks. In this way the model captures cycle-dependent irreversibility, thereby allowing the fatigue-induced multiplication of transverse ply cracks to be modeled. Although the formation of new cracks between pre-existing cracks can be modeled by different criteria, in [132] a maximum stress criterion for cracking is used, supported by previous work [133]. It can, however, be shown that if the delamination surfaces are traction free, then the maximum axial stress between the ply cracks goes down as the delamination length d increases. This suggests that the irreversibility captured in such delamination growth is inadequate for the purpose. As argued 225 6.7 Damage evolution under cyclic loading in [132], the delamination surfaces are indeed under compressive stress [134], making it plausible that frictional contact exists between those surfaces. The ensuing frictional sliding was modeled by an interfacial shear stress, which provided the needed increase in the axial stress for crack formation. A description of the model now follows. Referring to Figure 6.36(a) and (b), the stress analysis is performed by a variational approach along the lines in [101], conducting the minimization of complementary energy separately for region I and region II. First, an admissible stress system in the x–z plane of region II is expressed as s L(m) ij = s 0(m) ij ÷s (m) ij ; (6:86) where s L(m) ij and s 0(m) ij are the stress components in the cracked laminate and in the virgin laminate, respectively, and s (m) ij are the perturbations; m = 1 and 2 indicate the 90 · - and 0 · -plies, respectively. The axial perturbation stresses in the plies are assumed to have the following form s (1) xx = ÷s 1 f 1 x ( ); s (2) xx ÷s 2 f 2 x ( ) ÷A x ( ) z [ [; (6:87) where s 0(m) xx = s m has been used; f 1 (x), f 2 (x), and A(x) are unknown functions. Applying the equilibrium in the x direction and the interface iso-strain condition at z = t 1 , f 2 (x) and A(x) in Eq. (6.87) are eliminated, so the axial perturbation stresses can be expressed by the only unknown function, f 1 (x). After integrating equilibrium equations and using Eq. (6.87), all perturbation stress components Z 90Њ 0Њ X N xx Y X d a 0Њ 90Њ Z Region I Region II t 1 t 2 h (a) (b) Figure 6.36. Schematic of a cracked cross-ply laminate: (a) uniformly distributed transverse matrix cracks in 90 · -plies with the associated delamination of 0 · /90 · interface; (b) a unit cell between two matrix cracks with delaminated region (region I) and perfectly bonded region (region II). 226 Damage progression are expressed as functions of f 1 (x) by applying interface continuity conditions at z = t 1 and traction-free boundary conditions at z = h. The admissible stress system for a cracked laminate is then established based on the only unknown function, f 1 (x). The corresponding complementary energy functional for linear elastic materials in a volume V, with only traction boundary conditions in region II, can be written as [101]: U c = U 0 c ÷U / c = 1 2 _ V s ijkl s 0(m) ij s 0(m) kl dV ÷ 1 2 _ V s ijkl s (m) ij s (m) kl dV; (6:88) where S ijkl are the components of the compliance tensor. Since the virgin laminate stresses are constant, U 0 c is not of importance to the analysis. Substituting all perturbation stress components into Eq. (6.88) gives U / c = (s 1 ) 2 _ (a÷d) ÷(a÷d) t 1 C 00 f 2 1 ÷t 3 1 C 11 (f / 1 ) 2 ÷t 5 1 C 22 (f // 1 ) 2 ÷t 3 1 C 02 f 1 f // 1 _ _ dx; (6:89) where C ij are constants determined by the elastic constants and thickness of each layer, a is half the crack spacing, and d is the delamination length. Minimizing U / c after introducing the nondimensional variable x = x/t 1 , the following Euler– Lagrange differential equation in f 1 is obtained d 4 f 1 dx 4 ÷p d 2 f 1 dx 2 ÷qf 1 = 0; (6:90) where p and q are constants determined by C ij . Dependent on the material elastic property and geometry of the given laminate, two solutions for Eq. (6.90) exist f 1 x ( ) = A 1 cosh ax ( ) cos bx ( ) ÷A 2 sinh ax ( ) sin bx ( ); p 2 4 ÷q _ _ <0 A 1 cosh ax ( ) ÷A 2 cosh bx ( ); p 2 4 ÷q _ _ > 0 _ _ _ _ _ _ (6:91) The axial normal stress of interest at the mid-plane in the 90 · plies is obtained as s xx 0; z ( ) = s 1 1 ÷f 1 0 ( ) ( ); ÷t 1 <z<t 1 : (6:92) The two constants, A 1 and A 2 , are found using the traction continuity conditions at x =±(a – d) if the stress in region I is known. To obtain the stress state in region I, a very similar variational approach is applied. For an admissible stress system, the perturbation stresses are assumed first as s (1) xx = ÷s 1 c 1 (x); s (2) xx = ÷s 2 c 2 x ( ) ÷A ÷ z [ [: (6:93) A cubic variation of shear stress along the interface (z = t 1 ) is enforced to account for the effect of frictional sliding along delamination s (1) xx (x; t 1 ) = t a 3 (x ÷a)[x ÷(a ÷d)[ 2 ; (6:94) 227 6.7 Damage evolution under cyclic loading where t is an unknown. After integrating the equilibrium equation, and applying continuity of stress across the interface at z = t 1 , as well as traction-free boundary conditions at the crack surfaces, the admissible stress system is built up as functions of the only unknown, t. By minimizing the corresponding complemen- tary energy functional, dU c dt = 0; (6:95) the only unknown t is solved, so the stress field in region I is obtained. By applying the traction continuity conditions at the boundary between region I and region II, A 1 and A 2 in Eq. (6.91) are determined. Substituting A 1 and A 2 into Eq. (6.92), finally, the axial normal stress at the mid-plane in 90 · plies is obtained for the given unit cell with specified crack spacing and delamination length. A typical variation of the axial normal stress with delamination at a fixed crack density is shown in Figure 6.37. The following power-law relation is assumed to describe the growth of delami- nation under cyclic loading dl dN = B Át l _ _ m ; (6:96) where Át = t max ÷t min ( )=s max and l = l=t 1 , and where l is the delamination length, denoted as d in the above stress analysis. Integration of Eq. (6.96) yields the relationship between delamination length and the number of cycles, N. From Figure 6.37 we see that after the initial drop the axial normal stress increases as delamination grows. When the maximum stress criterion s xx (x = 0, ÷t 1 < z < t 1 ) = s c is satisfied, a new crack forms midway between the pre-existing cracks, and the corresponding crack spacing (or crack density) is updated. In this way, the multiplication of matrix crack under fatigue loading is actually controlled by the growth of delamination, and therefore determined by the number of cycles through integration of Eq. (6.96). The damage evolution as a function of the number 0.0004 0.0002 s x x ( x = 0 , z = 0 ) ( P a ) 3.75×10 7 4.25×10 7 4.5×10 7 4.75×10 7 5×10 7 5.25×10 7 0.0006 d (m) Figure 6.37. Typical variationof the axial normal stress with delamination at a fixed crack density. 228 Damage progression of cycles in cross-ply laminates is finally quantitatively described in the model. Figure 6.38 shows the variation of crack density as a function of the number of cycles under a given cyclic tension. 6.8 Summary Initiation of cracks within the plies of a laminate, and their growth and multipli- cation, are part of the field of damage evolution in composites that constitutes a key element in the performance assessment of structures made of these materials. This chapter has focused on various stress and failure analysis methods associ- ated with the prediction of initiation and progression of ply cracks. Since cracks form from material defects that can be considered random in their size and spatial distributions, statistical considerations have been included in the analyses. For the formation of cracks the criteria used are based either on strength or on the energy associated with fracture. Both criteria have been treated and compared. Experimental data, wherever available, have been used to assess the predictions. Most of the damage evolution work in the literature has been for transverse cracking in cross-ply laminates. While this cracking mode has been amply dealt with, more recent work on oblique cracks, i.e., cracks in off-axis plies of multi- directional laminates, has also been treated. While the fatigue of composite laminates is the focus of a separate chapter, where a broad treatment of the subject is given, a section on ply cracking under cyclic loading has been included here to illustrate how damage accumulation under repeated loads is modeled. 2.0 1.8 1.6 1.4 C r a c k d e n s i t y ( / m m ) 1.2 1.0 0 200 000 400 000 600 000 Model prediction Experiment Number of cycles 800 000 1000 000 Figure 6.38. Transverse crack evolution with cycles for a [0/90 2 ] s carbon/epoxy laminate at a maximum stress of 482.633 MPa and a stress ratio of 0.1. 229 6.8 Summary References 1. A. S. D. Wang, Fracture mechanics of sublaminate cracks in composite materials. In Composites Technology Review. (Philadelphia, PA: ASTM, 1984), pp. 45–62. 2. J. M. Masters and K. L. Reifsnider, An investigation of cumulative damage develop- ment in quasi-isotropic graphite/epoxy laminates. In Damage in Composite Materials, ASTM STP 775, ed. K. L. Reifsnider. 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Int J Solids Struct, 27:15 (1991), 1925–41. 236 Damage progression 7 Damage mechanisms and fatigue-life diagrams 7.1 Introduction The fatigue of composite materials presents a tremendous challenge when one considers the number and variety of parameters that can possibly affect the governing mechanisms. There is a considerable risk of the fatigue design becoming empirically based, and quite cost-ineffective, if rational guidelines based on phys- ical models cannot be developed. To help alleviate this problem, we will in this chapter develop a mechanisms-based framework for interpretating the fatigue behavior of composites, beginning with the baseline configuration of unidirec- tional fiber-reinforced plies and proceeding later to laminate configurations and other fiber architectures. The framework in the form of fatigue-life diagrams will allow assessment of the effects of constituent properties, such as fiber stiffness and matrix ductility, and provide guidelines for fatigue design as well as for developing mechanism-based life prediction models. After a review of the fatigue-life diagrams and their utility, we shall discuss the fatigue design methodologies, taking the examples of aircraft components and wind turbine blades. Finally, a mechanisms-based modeling of multi-axial fatigue will be discussed. 7.2 Fatigue-life diagrams The S-N, or Wo¨ hler diagram, originating from metal fatigue, is a familiar way to represent the resistance of a given material to the cyclic application of loads. It describes the observed fact that the material strength, given by the maximum stress sustained in the first application of load, reduces with repeated application of load, and is inversely dependent on the number of cycles applied. The strength value corresponding to a pre-selected large number of cycles, e.g., 10 6 , is custom- arily taken as the fatigue limit. In some cases, a “true” fatigue limit exists, representing the stress value below which a fatigue mechanism cannot be initiated, but in most cases, one uses the operational definition of no failure until the selected high number of cycles. In a composite material with two constituents – fibers and matrix – the useful- ness of the Wo¨ hler diagram may be questioned. The obvious problem is: how do we represent the roles of the two constituents in determining the fatigue-life, and what roles do the constituents play in determining it? In order to address these issues, a framework for conceptual interpretation of fatigue of composite materials was proposed by Talreja [1]. That framework will be discussed below, beginning with the case of on-axis tension–tension fatigue of unidirectional composites. 7.3 On-axis fatigue of unidirectional composites Consider a unidirectional composite that is subjected to cyclic tension in the fiber direction. Let the specimens of the composite be tested under load control, i.e., the load varies between set limits of maximum and minimum. We shall examine the mechanisms operating and the failure resulting under this condition. To do this, let us construct a plot with the horizontal axis as a logarithm of the number of cycles applied and the vertical axis as the maximum strain attained in the first load cycle as seen in Figure 7.1. The reasons for plotting strain and not stress are motivated by the following considerations. (1) The failure in the first cycle occurs when the composite strain equals the failure strain of fibers irrespective of the fiber volume fraction. A point with coordinates (log 1, e c ), where e c is the composite failure strain, can thus be plotted in the diagram. (2) The composite fatigue limit is governed by the matrix fatigue limit, as will be discussed later. The matrix within the composite is subjected to strain controlled fatigue, due to the fiber constraint, although the composite is undergoing load controlled testing. Thus, the composite fatigue limit is expressible in terms of strain. Since the two extreme values of the fatigue-life are given in terms of strain, there appears to be no compelling reason not to plot the strain between these limits. We note that for a given fiber type (e.g., glass or carbon) and matrix material the stress–strain relationship changes when the fiber volume fraction or fiber stiffness change. The significance of plotting the maximum strain in the first cycle lies in the fact that this strain value provides a good reference to the damage attained in the first cycle and that the subsequent damage and fatigue-life are likely to depend on this damage state. Figure 7.1. Schematic illustration of fatigue-life diagram for a unidirectional composite, showing three regions with different damage mechanisms. 238 Damage mechanisms and fatigue-life diagrams As first described in Talreja [1], the test data plotted on the axes (log N, e max ) can be viewed as falling in three regions. Region I is the horizontally extending scatter band (e.g., between 5% and 95% probability of failure) of the composite failure strain. This region represents a lack of degradation in strength (failure strain), i.e., the underlying predominant mechanism of fiber failure is nonprogressive. The arguments to support this postulate will be described below. Region II is the fatigue-life scatter band, which deviates from region I at a certain number of cycles and extends down to the fatigue limit. This region is governed by the progressive mechanism of fiber-bridged matrix cracking, as described below. Finally, region III is the region of no fatigue failure (in a selected large number of cycles, say 10 6 ) lying below the fatigue limit. Figure 7.2 depicts three scenarios, the first of which shows fiber breaks resulting from the first application of a high load that produces a maximum strain lying within the scatter band of composite failure. These fiber breaks are caused by the local fiber stress (or strain) exceeding the fiber strength at those (weak) points during the first application of load. Unloading and reapplying the load would change the local fiber stresses only if an irreversible (inelastic) deformation occurs. Assume now that the matrix surrounding a fiber break undergoes only small inelastic deformation due to the constraint of the stiff fibers. This is likely if the matrix is relatively brittle, such as an epoxy. The case of a matrix of high ductility (i.e., ability to flow easily) will be considered later. For small inelastic deformation in the matrix the repeated application of load would cause little change in the stresses on fibers surrounding a broken fiber site. The next fibers to break in a subsequent load cycle could appear near any of the previously broken fibers – but not necessarily near all previously broken fibers – when the local stresses exceed their local strengths, see the illustration at N = N 1 in Figure 7.2. The important thing to note is that, due to the small cycle-to-cycle changes in the local fiber stresses and randomness of the fiber strength, it is unlikely that the fiber breakage will be a progressive mechanism, in the sense that the number of fiber failures in a given location increases monotonically. Thus the final failure, which results from a core of (a few) fiber failures growing unstably (see illustration in Figure 7.2), could occur in any of Figure 7.2. Nonprogressive fiber breakage in region I in fatigue of unidirectional composites. 239 7.3 On-axis fatigue of unidirectional composites the potential failure sites, without preference. The consequence of this is that the composite failure under the specified loading condition can occur at any number of load cycles. This can be described as a nonprogressive failure mechanism, with no associated strength degradation. Thus the scatter band in region I is horizontal. The mechanism in region II is illustrated in Figure 7.3. At applied maximum load levels below the lower bound of the region I scatter band, the composite failure resulting from a cluster of neighboring fiber breaks is unlikely. Instead, with increasing load cycles, the matrix will undergo fatigue cracking. The matrix cracks will progress by failing fibers or by debonding and going around them. A typical fiber-bridged crack would then appear as illustrated in Figure 7.3. Among several such cracks the one to cause failure earliest would be the one that grows unstably first. Region III of the fatigue-life diagram can be viewed as the domain of no fatigue failure in a pre-selected large (> 10 6 ) number of load cycles. One possible scenario in this region is illustrated in Figure 7.4. As shown in the figure, fatigue cracks in the matrix are likely to develop, but they remain confined to the parts of the cross sections between fibers. The driving force for the cracks is insufficient to advance them by failing and/or debonding fibers. Another likely scenario in this region is fiber-bridged matrix cracking progressing at rates too low to cause failure in the pre-selected large number of cycles. A “true” fatigue will exist only if a mechanism of effectively arresting matrix crack growth is available. Fiber-bridged matrix crack Crack front (N=N 1 ) Crack front (N=N f ) Figure 7.3. Fiber bridged matrix cracking in region II in fatigue of unidirectional composites. Figure 7.4. Matrix cracking between fibers in region III in fatigue of unidirectional composites. 240 Damage mechanisms and fatigue-life diagrams 7.4 Effects of constituent properties The baseline fatigue-life diagram provides a good starting point for making an assessment of the roles of constituent properties in determining the fatigue response of composite materials. In the following we shall discuss some expected trends in the fatigue-life caused by changes to constituent properties. Fibers of different materials, e.g., glass, carbon, and SiC, have different axial moduli and fail at different axial tensile strains. Carbon fibers of different stiffness are commercially available and these fibers can fail at strains as low as 0.5%, or at strains exceeding 1.8%, depending on processing and surface treatment. The com- posite stiffness and failure strain in the fiber direction are mainly determined by the fiber properties in most composites of practical interest. The failure of fibers is subject to a significant scatter due to imperfections, in particular the surface defects. The composite failure is therefore also a statistical process, with added complexities of stress transfer between matrix and fibers, fiber debonding, fiber misalignment, stress concentration in fibers near a broken fiber, etc. For our considerations regarding the effect on composite fatigue of changing fibers and/or matrix proper- ties we can viewthe fatigue-life diagram and identify the following trends, as shown in Figure 7.5, where fiber-bridged cracking is the progressive mechanism. 1. As explained above, region I consists of the horizontally extended scatter band of composite failure strain, e c . This strain can be raised or lowered by the fiber failure strain. Thus, the placement of region I with respect to the fatigue limit e m (which is primarily a matrix property) is determined by the choice of fibers. This in turn means that the extent of region II (e c À e m ) is also determined by fibers. This provides a remarkable flexibility in view of the wide variety of fibers that are commercially available and are being developed. Note also that region II can be made to vanish (i.e., no fatigue!) by selecting fibers such that e c is less than or equal to e m , as we shall demonstrate later. 2. The progressive fatigue mechanism in region II of the fatigue-life diagram is influenced significantly by fibers. The closing tractions on the matrix cracks are Figure 7.5. Trends in fatigue-life diagrams due to constituent properties. 241 7.4 Effects of constituent properties supplied by the bridging fibers as they are strained. For the same stretching strain on the fibers, stiffer fibers would supply greater forces than would more compliant fibers. Thus, lower fatigue degradation rates would result in com- posites with stiffer fibers, leading to a rightward displacement of region II with increasing fiber stiffness, as indicated in Figure 7.5. 3. Consider a ductile matrix versus a brittle matrix, in relative terms. A crack in a ductile matrix would cause higher strains at the crack tips than in a brittle matrix. Thus, higher crack opening displacement will result in a ductile matrix than in a brittle one for the same crack length. This would mean higher strains in the bridging fibers for cracks in composites with more ductile matrices, resulting in earlier fiber failures than in the case of brittle matrices. In terms of fatigue-life this would translate into lower fatigue-life for a more ductile matrix, as indicated by the trend line in Figure 7.5. 4. The effect of constituent properties on the fatigue limit can be viewed in the following way. First, fatigue in the composite cannot occur unless cracks can be initiated in the matrix. Thus, the matrix fatigue limit in strain cycling is the baseline limit (as described above). This limit can be enhanced by fibers that provide a means of arresting (obstructing) matrix crack growth. However, stiffer fibers would be more effective in suppressing the crack growth, than would be more compliant fibers, by supplying higher closing pressure on the matrix crack planes. This would result in the trend in the fatigue limit indicated in Figure 7.5. The interpretation of fatigue behavior of unidirectional composites under cyclic on-axis tension will now be illustrated by considering test data. 7.5 Unidirectional composites loaded parallel to the fibers The fatigue-life diagram, described above, is developed by generic considerations of damage mechanisms. In what follows we will illustrate the diagram for specific material systems. 7.5.1 Polymer matrix composites (PMCs) The fatigue-life diagram will be used as a baseline diagram for interpretation of fatigue of polymer matrix composites. Fatigue of other material systems – metal matrix composites and ceramic matrix composites – then becomes easy to inter- pret by way of emphasizing the differences and similarities with respect to polymer matrix composites. To begin, consider the fatigue-life data shown in Figure 7.6 for a glass/epoxy composite of three different fiber volume fractions indicated in the figure. These data are plotted in the conventional manner with the applied cyclic stress 242 Damage mechanisms and fatigue-life diagrams amplitude as the vertical axis. The fatigue-life curves appear as distinct lines for each case of the fiber volume fraction. The data are replotted in Figure 7.7 using the maximum strain applied in the first cycle as the vertical axis, in accordance with the fatigue-life diagram. The data for the three volume fractions are found to fall together, as seen in the figure. The conceived fatigue-life diagram with the three regions has been superposed on the data. The horizontal scatter band of composite failure strain is drawn as region I and fatigue limit of the epoxy resin in strain control testing, reported by Dharan [2] to be at 0.6% strain, has been drawn as the assumed fatigue limit of the glass/epoxy composites. A scatter band has been placed about the fatigue-life data to indicate region II. This plot suggests that plotting fatigue-life against strain allows more meaningful interpretation, as argued above. It also provides confirmation of the proposition that the fatigue Figure 7.6. Stress/life data of a unidirectional glass/epoxy composite loaded in tension parallel to the fibers [2]. Reprinted, with kind permission, from Fatigue in Composite Materials, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428. 0 2 4 6 8 log N f V f 0.008 0.016 0.024 e c e m e m a x 0.50 0.33 0.16 Figure 7.7. Fatigue-life diagram of a unidirectional glass/epoxy composite loaded in tension parallel to the fibers [2]. Reprinted, with kind permission, from Fatigue in Composite Materials, copyright ASTM International, 100 Barr Harbor Drive,West Conshohocken, PA 19428. 243 7.5 Unidirectional composites loaded parallel to the fibers limit of the matrix is a good indication of the composite fatigue limit, when plotted in the strain coordinate. The existence of region I is not strongly indicated by the glass/epoxy data, for reasons that will become clear as we examine other data. Consider now a set of unidirectional carbon/epoxy composites where each com- posite has different type of carbon fibers and the same epoxy matrix. Figure 7.8 illustrates two examples of stress–strain behavior, one with relatively low stiffness fibers (Figure 7.8(a)) and the other with relatively high stiffness fibers (Figure 7.8(b)). The composite failure strain e c , assumed equal to the fiber failure strain, has been marked on the strain axis for both cases. Note that the failure strain is lower for higher fiber stiffness in accordance with observed behavior of carbon fibers. Also marked is the fatigue limit of epoxy, at the same value in both cases, assuming for simplicity that it is a matrix property not affected by fibers. The consequence of changing carbon fiber stiffness is thus essentially to increase or decrease the extent of region II. Imagine now that a certain carbon fiber is used that has such high stiffness that the composite failure strain e c <e m , the fatigue limit. This can also be caused by defective fibers of low strength. The consequence of composite failing at strains below the fatigue limit is that no fatigue progression would be possible. In other words, region II would not exist, as the fatigue limit would lie above region I. Thus, only region I (a horizontal scatter band) will appear in the fatigue-life diagram. Figure 7.9 illustrates this phenomenon by using fatigue-life data gathered by Sturgeon [3] for a carbon/epoxy composite with high stiffness fibers. The fatigue limit, assumed for epoxy to be 0.6% strain, has been marked in the figure along with the average composite failure strain at approximately 0.48% strain. The next set of test data, plotted in Figure 7.10, is from Awerbuch and Hahn [4] for carbon/epoxy with fibers of stiffness lower than those in the composite used by Sturgeon [3] just discussed. Here the data suggest no fatigue degradation when viewed without the fatigue-life diagram. However, when the scatter band of region s s e m e m e c e c e e Fiber (a) (b) Fiber Composite Composite Matrix Matrix Figure 7.8. Effect of fiber stiffness and strain to failure on composite stiffness in a unidirectional composite in longitudinal loading, where e c is the composite failure strain and e m is the matrix fatigue limit. (a) Composite with low stiffness fibers; (b) composite with high stiffness fibers. 244 Damage mechanisms and fatigue-life diagrams I and the fatigue limit of epoxy are placed in the diagram, the test data suggest a different interpretation. The progressive fatigue mechanism of region II is now seen to exist, albeit in a narrow range of strain. Another case of differing interpretation with and without the fatigue-life diagram is shown in Figure 7.11, where data from Sturgeon [5] for a lower stiffness carbon/epoxy are plotted. Contrary to Sturgeon’s assertion that no fatigue deg- radation exists for this material, the fatigue-life diagram suggests that, depending on where the fatigue limit lies, there would be a region II of progressive fatigue damage. In the figure the fatigue limit is indicated at 0.6% strain as a reference for comparison with other cases of carbon/epoxy composites. As the last case of carbon/epoxy fatigue we take the data supplied by P. T. Curtis (personal communication) plotted in Figure 7.12. Here the regions of the fatigue-life diagram appear distinctly, illustrating the power of interpret- ation offered by the diagram. The value of recognizing the existence of region I is 0 0.002 0.004 0.006 4 8 log N f e c e m e max Figure 7.9. Fatigue-life diagram of unidirectional composite reinforced by stiff fibers with low strain to failure (data from [3]). It is notable that there is no fatigue degradation, since the mechanism is nonprogressive fiber breakage only. 0 2 4 6 log N f 0.002 0.006 0.010 e max e c e m Figure 7.10. Fatigue-life diagram of unidirectional composite reinforced by medium stiffness fibers, showing a narrow range of strain where fatigue occurs [4]. Reprinted, with kind permission, from Fatigue of Filamentary Composite Materials, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428. 245 7.5 Unidirectional composites loaded parallel to the fibers particularly evident. Without this, the error introduced in assessing the fatigue-life would be significant. As further illustration of the fatigue-life diagram, Figures 7.13 and 7.14 show test data for Kevlar ® -epoxy and Kevlar ® -J-2 polymer, respectively. The J-2 0 0.01 0.02 4 8 log N e m e c e max Figure 7.11. Fatigue-life diagram of a unidirectional composite with low stiffness fibers (data from [5]), with a relatively wide range of strain where fatigue occurs. 0 0.6 0.9 1.2 1.5 1 2 3 4 5 6 7 log N f ε c e max (%) e m Figure 7.12. Fatigue-life diagram of a unidirectional carbon fiber/epoxy composite with a distinct region I (data courtesy of P. T. Curtis, RAE, UK). 0 0.0 0.5 1.0 1.5 2.0 1 2 4 3 5 6 7 log N f e max (%) e c e m Figure 7.13. Fatigue-life diagramof a unidirectional Kevlar fiber/epoxy composite (data from[6]). 246 Damage mechanisms and fatigue-life diagrams polymer is an amorphous polyamide thermoplastic [6]. The generality of the fatigue-life diagrams is that they lend themselves to description and interpretation of any kind of fatigue-life data for composites. A comparison of fatigue perform- ances between different material candidates can then be made. Commercial mater- ial systems of unknown constituent composition can thus be compared, and the best contender for an intended application can be identified. 7.5.1.1 Experimental studies of mechanisms The fatigue-life diagrams developed in [1] and described above for PMCs were based on deductive reasoning in the absence of any systematic investigation of such mechanisms. Since then an investigation was conducted by Gamstedt and Talreja [7], which, through direct observations of on-axis tension–tension fatigue of unidirectional carbon/epoxy, gathered evidence to support the diagrams. For instance, the presence of fiber-bridged cracking postulated in region II was 0 1 2 3 4 5 6 7 log N f 0.0 0.5 1.0 2.0 1.5 e max (%) Figure 7.14. Fatigue-life diagram of a unidirectional Kevlar fiber/J2 polymer composite (data from [6]). Figure 7.15. A crack on the surface of a carbon/epoxy unidirectional composite subjected to tension–tension fatigue. The crack tips are squeezed by the bridging fibers, as illustrated schematically by the bottom figure. Reprinted, with kind permission, from Springer Science +Business Media: J Mater Sci, Fatigue damage mechanism in unidirectional carbon fibre-reinforced plastics, Vol. 34, 1999, pp. 2535–46, E.K. Gamstedt and R. Talerja. 247 7.5 Unidirectional composites loaded parallel to the fibers found as exemplified by an image of a surface replica shown in Figure 7.15. The crack opening profile shows squeezing of the crack tips by fibers. A mechanism for arresting matrix crack growth in region III was also dis- covered in [7]. Its evidence is displayed in Figure 7.16. As seen there, a matrix crack growing transverse to the applied load is arrested by extensive fiber/matrix debonding. Without the energy dissipation in the debonding process the crack will have a greater driving force for extension. In [8] the fatigue propagation of fiber-bridged cracks in unidirectional PMCs was analyzed. As expected, in comparison to an unreinforced matrix crack, the fatigue growth rate of fiber-bridged cracks was found to be lower, and even showing decelerating growth. In another study, Gamstedt [9] examined the role of fiber/ matrix interfaces prone to debonding on the composite fatigue under on-axis tension–tension loading. If the interfaces resist debonding, which may be viewed as the baseline (normal) case, the fatigue mechanisms described above in developing the fatigue-life diagram would apply. As debonding increases, the fiber strength variability becomes increasingly important in determining the damage progression, and in significant debonding cases, the fiber-bridged crack growth gives way to debonding-assisted fiber breakage as the progressive mechanism. The slope and extent of region II of the fatigue-life diagram is then determined by the debonding- assisted fiber breakage. The flat region I, which is a consequence of the unconnected fiber breakage, then diminishes, and can eventually become part of region II. Figure 7.17 illustrates the cases of little or no debonding vs. extensive debonding. The former case is typical of relatively strong interfaces in carbon/epoxy composites. When fiber breaks occur, initially at randomly located weak points, they remain isolated and unconnected due to the lack of debonding. As discussed above in Fiber/matrix debonding Matrix crack Figure 7.16. A surface replica image showing a matrix crack arrested by fiber/matrix debonding. Reprinted, with kind permission, from Springer Science+Business Media: J Mater Sci, Fatigue damage mechanism in unidirectional carbon fibre-reinforced plastics, Vol. 34, 1999, pp. 2535–46, E.K. Gamstedt and R. Talerja. 248 Damage mechanisms and fatigue-life diagrams describing region I, any of the initial breaks are likely to form a critical-size crack that could grow unstably, leading to failure. Thus there is no progressive mechanism and composite failure results essentially at any number of cycles, resulting in a horizontal scatter band. In the case of interfaces that are prone to debond, the influence of an initial fiber break extends to other fibers, depending on the rate at which the debond crack elongates, and a progressive fiber breakage results. The rate of this damage progression depends on the resistance to the debond crack growth given by the interface region and the likelihood of the debond cracks connecting with other cracks. The overall path to final failure may be described in terms of an average progression rate. Modeling of this progression is far from an easy task. Attempts at this were made by Gamstedt [9], who conducted numerical simulation of the fiber breakage process and clarified the roles of interfacial debonding and fiber strength variability. Figure 7.18 compares the fatigue-life diagrams of unidirectional carbon/epoxy and carbon/PEEK (polyetheretherketone) composites loaded in tension–tension loading. The former material typifies the strong fiber/matrix bonding case (left in Figure 7.17) while the latter is prone to debonding. The PEEK matrix is a semi-crystalline thermo- plastic that develops a so-called trans-crystalline structure extending from the fiber surface. This structure provides weak planes parallel to fibers for debonding to occur. Althoughotherwise a toughmaterial, PEEKbecomes a source of brittle cracking of the interfaces. The extensive debonding in carbon/PEEK removes region I of the fatigue- life diagram by inducing progressive fiber breakage, as explained above. (a) (b) Figure 7.17. Two cases of the role of fiber/matrix debonding in fatigue damage progression under on-axis tension–tension loading of unidirectional fiber-reinforced PMCs. Case 1 (a) illustrates unconnected fiber breaks when little or no debonding occurs while case 2 (b) is for extensive debonding that connects fiber breaks and causes the fiber breakage to be progressive. Reprinted, with kind permission, from Springer Science+Business Media: J Mater Sci, Fatigue damage mechanism in unidirectional carbon fibre-reinforced plastics, Vol. 34, 1999, pp. 2535–46, E.K. Gamstedt and R. Talerja. 249 7.5 Unidirectional composites loaded parallel to the fibers 7.5.2 Metal matrix composites (MMCs) The fatigue-life diagram discussed above should be viewed as a baseline diagram for polymer matrix composites, as the mechanisms considered for its construction were motivated by observations and conjectures related to this material system. We shall now discuss what changes in the diagram are plausible when the matrix material is a metal. Note that the role of fibers is in modifying the fatigue mechanisms taking place in the matrix, which is the material conducting irrevers- ible deformation (plasticity). The three regions of the fatigue-life diagram for MMCs treated by Talreja [10] will be discussed next. Region I: As discussed earlier, this region is manifested by fiber failures. For polymer matrix composites we argued that the fiber failures occurred in a manner that did not have significant progressiveness (accumulation of fiber failures in a localized zone), leading to the final (composite) failure from random sites at random number of cycles. An important factor in this deduction was insufficient irreversible deformation in the matrix to allow cycle-dependent stress enhancement and failure of fibers. This scenario will change when a relatively brittle polymer matrix is replaced by a more ductile metal matrix. The cyclic plastic deformation of a metal matrix around a broken fiber will redistribute stresses in the neighboring fibers, allowing an accumulative fiber failure process to occur. This will introduce a localized progressive degradation, which on reaching a critical level will cause composite failure. Thus, the hori- zontal scatter band of region I, characteristic of polymer matrix composites, will now have a downward slope. Region II: The mechanism of fiber-bridged matrix cracking is also expected to be the primary progressive mechanism for metal matrix composites. The 0 0.6 0.9 1.2 1.5 M a x . s t r a i n ( % ) 1 2 3 4 5 6 7 Log cycles Carbon/PEEK Carbon/epoxy Figure 7.18. Comparison of the fatigue-life diagram for unidirectional carbon/epoxy and carbon/PEEK composites loaded in tension–tension cycles along fibers. Note the absence of region I in carbon/PEEK. 250 Damage mechanisms and fatigue-life diagrams difference with respect to polymer matrix composites lies in the role of fiber/ matrix debonding. The fiber/matrix bond in metal matrix composites is generally stronger, leading to shorter extent of the interface failure. Furthermore, increased ductility of the matrix is expected to provide more crack opening displacement via crack tip blunting, causing increased failure of the bridging fibers. The matrix ductility effect will thus be as conjectured above in describing the trend in region II of the fatigue-life diagram as illustrated in Figure 7.5. Region III: This region is expected to be the same as in polymer matrix composites. The composite fatigue limit is also expected to be related to the matrix fatigue limit. Let us now examine some test data for metal matrix composites. Figure 7.19 plots fatigue-life at room temperature for a SCS6/Ti-15–3 unidirectional composite under cyclic tension at two R-ratios. Static failure strain is plotted on the vertical axis at the first cycle. As discussed above, region I displays some progressiveness by deviating from a horizontal scatter band. Region II shows greater degradation rates and tends to the fatigue limit gradually. Note that the matrix fatigue curve, plotted as a broken line, tends to approximately the same fatigue limit as the composite. The data for the same composite tested at a high temperature are shown in Figure 7.20. The general features of the fatigue-life diagram at room temperature are retained with different quantitative characteristics. Comparisons Figure 7.19. Fatigue-life diagram of SCS6/Ti-15–3 at room temperature. Figure 7.20. Fatigue-life diagram of SCS6/Ti-15–3 at high temperature (540–550 C). 251 7.5 Unidirectional composites loaded parallel to the fibers of the two regions are displayed in Figures 7.21 and 7.22. Figure 7.21 shows the data in region I at room and high temperatures on log–log scales. The best-fit lines for the data show that higher fatigue degradation (more progressiveness in the fiber failure mechanism) occurs at high temperature than at room temperature. Similar region II comparison at the two temperatures is shown in Figure 7.22. A leftward shift of region II, indicative of more fatigue degradation, is seen with an increase in temperature. This is likely due to higher matrix ductility (lower yield stress) at the high temperature, in accordance with the trends in the fatigue-life diagram described above. 7.5.3 Ceramic matrix composites (CMCs) In comparison with polymers and metals, ceramics have insignificant irreversi- bility of deformation. This suggests that there should be little incentive for fatigue mechanisms to occur in ceramics. However, when ceramics are reinforced by Figure 7.21. Region I for SCS6/Ti-15–3 at high (540–550 C) and low temperature (20 C). Figure 7.22. Region II for SCS6/Ti-15–3 at high (540–550 C) and low temperature (20 C). 252 Damage mechanisms and fatigue-life diagrams fibers with which they bond weakly, a dissipative mechanism becomes available from reversed (i.e., back-and-forth, not reversible) frictional sliding at interfaces. This mechanism is believed to play a major role in causing fatigue of CMCs. In the following a discussion of the fatigue-life diagram for unidirectional CMCs under cyclic axial tension is presented. Before getting into a discussion of fatigue of CMCs it would be useful to briefly reviewthe damage and accompanying stress–strain response observed under mono- tonic axial tension. Several works in the literature have reported such data, and we shall mainly draw upon Sørensen and Talreja [11] to illustrate some characteristics of interest. Figure 7.23 shows three surface replicas of a SiC/CAS (calcium alu- minosilicate) glass specimen taken at three applied axial strains at room tempera- ture. For reference, the failure strain is approximately 1.0%. Note that at 0.15% strain a few irregularly spaced cracks transverse to fibers are seen. At this low strain level the cracks are partially developed, i.e., they do not extend across the entire width of the specimen. As the applied strain increases, the cracks span the width fully and acquire more regular spacing. At some point the crack spacing approaches a minimum value, usually described as saturation, as seen in the replica at 0.8% strain. The stress–strain response in the axial as well as transverse directions is shown in Figure 7.24, alongside a plot of the AE(acoustic emission) events recorded during testing. The double stress–strain curves are due to strain gages used on both sides of the specimen to monitor axiality of loading. Figure 7.25 summarizes the different stages of damage progression, indicating the ranges in which they operate, based on microscopy observations and AE event counts. When considering fatigue, the first question to ask is: what happens in the material in the second and subsequent load cycles that is different from the Fiber-bridged matrix crack Fibers 500 µm e =0.15% e =0.5% e =0.8% Figure 7.23. Micrographs of surface replicas of a unidirectional CMC with increasing strain: 0Á15%, left; e = 0Á5%, middle; e = 0Á8% right. Source: [11]. 253 7.5 Unidirectional composites loaded parallel to the fibers damage caused, if any, in the first load cycle? The answer to this question lies in knowing the source of irreversibility in the material. For instance, if a material is cracked (singly or multiply) in the first load cycle, it would not be cracked further in the next and subsequent cycles unless an irreversible mechanism such as plasticity or friction is present. As stated earlier, ceramic materials have insignifi- cant irreversible (plastic) deformation to enable crack growth in cyclic loading. The likely source of irreversibility is frictional sliding at the interface. Keeping this in mind we shall examine below the existence and nature of the three regions of the fatigue-life diagram for CMCs (see Figure 7.26). 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Strain (%) 0 100 200 300 400 S t r e s s ( M P a ) 500 600 Monotonic tension IV Fiber fracture III Fiber bridging (Matrix fully cracked) IV III II I I No damage II Matrix crack development (initiation of matrix crack) D a m a g e d e v e l o p m e n t SiC f /CAS-II RT, 100 MPa/s s pl =285 MPa s mc =120 MPa Figure 7.25. Schematic drawing of damage mechanisms at different stages in monotonic tensile loading of a unidirectional CMC loaded in tension parallel to the fibers. 2.0 1.5 1.0 0.0 0 100 200 300 400 500 600 0.0 0.25 0.50 0.75 1.00 AE events (×10 6 ) e (%) e L e T s ( M P a ) Figure 7.24. Measured acoustic emission events and stress–strain response. Source: [11]. 254 Damage mechanisms and fatigue-life diagrams Region I: When a first load is applied up to a maximum value within the scatter band of the composite failure strain, the matrix cracks develop beyond their saturation stage (see Figure 7.25). This means that a set of fully grown fiber- bridged cracks exist at the end of the first load cycle. It is likely that a significant portion of the fibers will be broken due to the high load applied. When the second and subsequent cycles are applied, the fiber stress will increase due to the increased debond length caused by the irreversible frictional sliding, supplying the increased shear stress at the interfaces. Also, the cyclic grinding of the fiber surfaces by the debris generated by the interfacial wear will damage fibers and degrade their strength. All this will fail the fibers and when any of the cross sections going through the crack planes has insufficient number of intact fibers to bear the applied load, composite failure will occur. Now the critical question is whether this failure mechanism is progressive or nonprogressive. If failure were to come from only one cross section, i.e., if only one crack existed, or if one crack was somehow the favored one, then the accumulative fiber breakage will result in a progressive failure. However, that is not characteristic of the damage here. We have multiple cracks, and failure could potentially result from any one of those cracks. Although each fiber-bridged crack has progressive (accumula- tive) fiber failure, the rate of progression would likely be discontinuous and differ from that of other cracks. Thus the final failure would not necessarily result from the weakest of the cross sections produced by the first load. We postulate, therefore, that the number of cycles for composite failure in this case cannot be derived from a single rate equation, and consequently, the failure process is nonprogressive. Region II: Consider now the case of loading where the first cycle maximum load is in a range such that the matrix cracks are either partially developed or are Figure 7.26. Mechanisms in the three different regions of the fatigue-life diagram for unidirectional CMCs loaded in tension parallel to the fibers. 255 7.5 Unidirectional composites loaded parallel to the fibers fully developed but the bridging fibers are not broken extensively. The partial cracks will grow in subsequent cycles by debonding and/or breaking fibers at the crack tips (or fronts). The cyclic frictional sliding at the interfaces will debond the bridging fibers, causing crack surface separation to increase as the debond length increases and as more bridging fibers break. With cyclic stressing, there- fore, partial cracks will grow to their full extent. As more debonding occurs, stress will be transferred from fibers to the matrix increasingly, resulting in matrix cracking between two pre-existing cracks. A state of crack saturation will be approached eventually, unless composite failure results from failure of all fibers bridging any of the cracks. The damage development in this region of the fatigue-life diagram has well-defined progressiveness from partial cracking to full cracking until crack saturation. The terminal point is when at least one crack loses all its bridging fibers. This point may occur before crack saturation (toward the lower end of region II) or beyond crack saturation (toward the higher end of region II). The main difference between this region and region I is that in the latter saturation cracking with extensive fiber breakage already occurs in the first load application. Region III: We have defined this region to be a range of loading (maximum strain in the first cycle) in which damage may develop but does not lead to a critical state (failure) in a pre-selected large number of cycles (typically 10 6 ). In a unidirectional composite, failure (separation) will occur when all the bridging fibers of at least one crack fail, or, alternatively, broken fiber ends are interlinked by interfacial debonds and matrix cracks to form two separating surfaces. In either case, interfacial frictional sliding plays an important role by increasing debonding and thereby stressing fibers to failure. What has been found, however, is that the wear of the debonded interfaces (breakage of asperities resulting in smoothing) can make the frictional sliding less effective. This could slow down the cyclic development of damage to the extent of not reaching failure in the pre-specified number of cycles. Sørensen et al. [12] found that the temperature rise of SiC/CAS specimens caused by frictional heating at debonded interfaces increased sharply just before failure, while for specimens that did not fail in 10 8 cycles the temperature rise stopped at some cycles and then took a downward turn (Figure 7.27). These authors have challenged the notion of a “true” fatigue limit in CMCs, arguing that damage continues to develop even after 10 8 cycles, as long as interfacial debonding occurred in the first cycle. They contend that the condition for no damage development is no initiation of fiber-bridged crack, providing the true fatigue limit as the stress (or strain) at which such crack forms. The discussion concerning fatigue limit in CMCs is somewhat academic because the energy dissipating mechanism of frictional sliding at debonded interfaces depends on the loading frequency and specimen geometry. This is because the time rate at which conduction of the heat generated by frictional sliding occurs affects the rate per cycle of energy dissipation by frictional sliding. 256 Damage mechanisms and fatigue-life diagrams The fatigue data on CMCs reported in the literature are inadequate for a good illustration of the usefulness of the fatigue-life diagram. Since the testing was done with a traditional S-N curve in mind the data generated were too few in region I to validate its existence. Also, because of the material cost, the data generated were at too few load levels to have all three regions of the diagram appear with clarity. 7.6 Unidirectional composites loaded inclined to the fibers We consider the simple case of a single fiber orientation in a composite placed at an angle to the applied cyclic tension load as shown in Figure 7.28. As illustrated in the figure, cracks initiate along the fibers, either at interfaces or in the matrix. At first the cracks may initiate from defects in the matrix or at interfaces and grow with cycles through the thickness as well as along the fibers. At some stage in the evolution of this damage, a single crack may grow to the extent of attaining unstable growth in the next application of load and separate the composite in two pieces, as illustrated in the figure. In comparison to the on-axis loading case, i.e., when the fibers are aligned with the loading direction, the damage mechanism of the off-axis loading case is drastically different and dramatically simple. The single progressive mechanism depicted in Figure 7.28 will produce a continuous curve (and the associated scatter band) in the fatigue-life diagram starting at the first-cycle failure strain and ending asymptotically in the fatigue limit. For each off-axis angle the curve will be different. A schematic depiction of the fatigue-life diagram is shown in Figure 7.29, which illustrates the fatigue-life dependence on the off-axis angle. The on-axis fatigue-life diagram is shown for reference in the figure in broken lines. 1 MPa 0 10 20 30 40 10 2 10 4 10 6 10 8 Cycles N T e m p e r a t u r e r i s e ( K ) X X 220 MPa 212 MPa Run out s max =260 MPa s min =10 [0 16 ] SiC f / CAS II 200 Hz, RT Figure 7.27. Temperature increase due to frictional heating during cyclic loading of a unidirectional SIC/CAS II composite. 257 7.6 Unidirectional composites loaded inclined to the fibers As Figure 7.29 illustrates, region I (horizontal scatter band) of the fatigue-life diagram does not exist when the applied cyclic load is inclined to the fiber direction. The single scatter band starts at the composite failure strain and asymptotically ends at the fatigue limit. The failure strain as well as the fatigue limit strain depend on the off-axis angle, y. From the S-N data reported for glass/ epoxy in [13] the fatigue-life diagrams for a few angles are shown in Figure 7.30. The curves drawn by visual fit to the data show the trends depicted in Figure 7.29. The fatigue limits extracted from the data are shown plotted in Figure 7.31 where the fatigue limits deduced from [14] for angle-ply laminates of glass/epoxy are also plotted. The significant improvement in the angle-ply laminates over unidirectional composites for the same off-axis angle is discussed in the next section on fatigue of laminates. e max e c e c (0<q<90) (q=0) logN q Figure 7.29. Fatigue-life diagram for unidirectional composites under off-axis loading. The diagram in broken lines is for the on-axis loading case. Figure 7.28. Cracking in a unidirectional composite under off-axis cyclic tension (left) and failure from growth of a crack (right). 258 Damage mechanisms and fatigue-life diagrams 7.7 Fatigue of laminates Composite structures are built by placing fibers in different orientations to effectively carry multi-axial loading. We will here consider how the mechanisms of fatigue damage are affected by multi-directional fiber placement in a 0 q(degrees) 0.004 0.002 0.006 30 60 90 e f l Figure 7.31. The off-axis fatigue limit from data in Figure 7.30 (dashed line) is shown for comparison with the fatigue limit of angle-ply laminates of glass/epoxy taken from data reported in [14], plotted against the off-axis ply angle. 0 2 4 6 lg N e db e c e c e c e m e m a x e c 0.008 0.006 0.004 0.002 (q=60°) (q=30°) (q=10°) q/deg (q=5°) 5 10 30 60 Figure 7.30. Fatigue-life data for glass/epoxy unidirectional composites from [13] plotted as fatigue-life diagrams for different off-axis angles of tension–tension loading. The fatigue limit of on-axis loading is indicated at 0.6% strain for reference. Reprinted, with kind permission, from Z. Hashin and A. Rotem, J Compos Mater, Vol. 7, pp. 448–64, copyright # 1973 by Sage Publications. 259 7.7 Fatigue of laminates laminate subjected to a cyclic tensile load and how that translates into changing the basic fatigue-life diagram described above. 7.7.1 Angle-ply laminates These laminates have two fiber orientations placed symmetrically about a princi- pal direction, e.g., [Æy n ] s illustrated in Figure 7.32. When loaded in the axial, i.e., 0 direction, a y-ply in the laminate behaves differently from a unidirectional composite loaded off-axis at the same y-angle. First, for the same applied axial strain, the stress state in the ply within the laminate differs from that in the unidirectional composite. Second, the unidirectional composite fails when a crack appearing along its fibers grows unstably, while in the laminate the unstable growth of a ply crack does not cause laminate failure. Instead, the presence of the interface to the ply (or plies) in the other off-axis orientation leads to multiple ply cracking. The so-called shear lag process underlying the crack multiplication has been treated in detail in Chapter 4. Figure 7.32 depicts schematically the multiple cracking and the consequent (associated) delamination. As a ply crack front encounters the ply interface, the intense stress state ahead of the crack front debonds the interface, causing local ply separation (delamination). This delamina- tion accompanies each ply crack and under cyclic loading undergoes growth along the interfacial plane. While this occurs, the multiple cracking process within the plies can continue due to stress redistribution in the plies, if the renewed stress state is high enough. Under what conditions would the renewed stress state be critical for producing new cracks is a subject of damage evolution modeling and will be discussed in a subsequent section of this chapter. The key elements of the fatigue failure process in laminates are illustrated by the angle-ply fatigue, and may be summarized as: multiple ply cracking, delamination formation, and growth, possibly followed by more ply cracking, and delamination merger in ply interfaces, leading to separation of plies, which in turn can overload Matrix crack Delamination q q Figure 7.32. Multiple matrix cracks in off-axis ply of a laminate (left) and subsequent delamination caused by fatigue (right). For clarity, cracks and delamination are shown for one ply only; the other ply is indicated in broken lines. 260 Damage mechanisms and fatigue-life diagrams individual plies resulting in ply failure (separation). The laminate failure (separ- ation in parts) comes from the final event of fiber failures. Thus while an off-axis loaded unidirectional composite has essentially a two- stage fatigue process consisting of ply crack initiation and growth until instability, in laminates the fatigue process generally has several interactive cracking processes. One can lump them in two parts as the critical and sub-critical failures. The critical failure is the fiber breakage process, which itself can have its progres- sion. The sub-critical failure process consists of all events until fiber failures. The enhancement of the off-axis fatigue limit in angle-ply laminates shown by data in Figure 7.31 can now be explained by the presence of the sub-critical failure processes in laminates and its absence in unidirectional composites loaded at off-axis angles. The fatigue-life diagram for angle ply laminates is expected to lose region I, except at the y-angle approaching 0 . The fatigue limit will not be adversely affected until the sub-critical damage processes become insignificant, such as at y > 30 for the data in Figure 7.31, and at y = 90 the fatigue limit will be the same as for a unidirectional composite under transverse loading. It is possible that at small y-angles some enhancement in the fatigue limit results from the lamin- ation effect of inducing multiple cracking. Experimental data does not seem to be available, however, to confirm this. 7.7.2 Cross-ply laminates Cross-ply laminates, [0 n /90 m ] s , are orthogonal angle-ply laminates, on which loading is usually applied in one of the two fiber directions. Historically, this laminate was the first class of laminates for which multiple ply cracking was experimentally observed and reported in the literature in the late 1970s (see Chapter 3 for details). In those studies, monotonically increasing tensile load was applied along the 0 -direction, resulting in multiple ply cracks in the 90 -plies. Since multiple cracking is a basic feature of damage in composite materials, this configuration and loading combination has been, and continues to be, a test-bed for modeling studies of multiple cracking and its consequences on the material response. The observations of multiple cracking in cross-ply laminates under monotonic- ally increasing load do not, however, reveal the key feature underlying progres- siveness of damage under cyclic loads. Since most studies typically observed the cracking process on the free edges of a flat specimen, the details of damage accumulation were missed. The first study to examine the interior of a laminate with painstaking patience and using X-ray radiography, combined with stereo radiography, was by Jamison et al. [15]. The key X-ray picture, along with a schematic to depict interior details, is shown in Figure 7.33. Other than the transverse cracks, the details seen in Figure 7.33 are not found to develop sufficiently under monotonic loading. However, these details hold the key to the further progression of transverse cracking under cyclic loading. As we shall 261 7.7 Fatigue of laminates discuss below in the section on modeling, the irreversible changes needed from one load cycle to the next can be lumped into frictional sliding between the surfaces of the propagating delamination. In constructing the fatigue-life diagram of a cross-ply laminate under cyclic axial tension, one needs to ask certain basic questions. First, are conditions available for region I to be present? As discussed above in explaining the reasoning behind this region, a statistical nonprogressive fiber breakage mechanism must be available for this region to exist. The presence of 0 -plies in the laminate makes this mechanism plausible at applied axial strains near the average fiber breakage strain that leave the laminate intact at the first application of load. Second, where is the fatigue limit? To determine the fatigue limit, if not available from test data, a good estimate will be given by the strain at which transverse cracking initiates. The reasoning is simply that if no cracks initiate in the transverse plies, no progression of crack multiplication and subsequent damage are possible. The models for crack initiation strain have been discussed in Chapter 6. What remains to complete the fatigue-life diagram for cross-ply laminates is the location of region II. If the scatter-band of this region is assumed to be straight and sloping, a good approximation to begin with, then by fixing its lower end at the fatigue limit (at, say, 10 6 cycles), its slope or the point of its deviation from region I remain to be determined. A model that has been successfully determined will be discussed below in the section on fatigue-life prediction. Figure 7.34 shows data reported by Grimes [16] for a carbon/epoxy cross-ply laminate, plotted on the strain scale. The anticipated fatigue-life diagram, as discussed above, is superimposed on the data. The scatter-bands drawn are guided by the data, as insufficient data have been reported to estimate these bands from probability distributions. The location of the fatigue limit strain has been taken at the value of stress (converted to strain) reported by the author as the value where the first transverse cracking was found. Transverse ply cracks Transverse cracks Axial splits Axial splits Interior delaminations Interior delaminations Figure 7.33. X-ray radiograph of a carbon/epoxy cross-ply laminate taken after tension– tension fatigue shows transverse ply cracks, interior delamination, and axial splits. The accompanying schematic clarifies the interior details [15]. Reprinted, with kind permission, from Effects of Defects in Composite Materials, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428. 262 Damage mechanisms and fatigue-life diagrams 7.7.3 General multidirectional laminates For practical structures the composite architecture is designed to satisfy multiple requirements, such as resistance to bending and torsion as well as thermal expan- sion. A common composite architecture is a laminate consisting of plies, each with unidirectional fibers, stacked in a sequence such that the resulting structure has the required combination of properties. An example is [0/Æ45/90] s laminate, which is quasi-isotropic, i.e., it has directionally independent average elastic moduli in its mid-plane. Many other laminate configurations are possible, but often the number of ply orientations are kept to three, and the most used ones are 0 , 45 , À45 , and 90 . From a fatigue point of view the assessment of laminates is remarkably simple. First, irrespective of other ply orientations, the presence of 0 -plies provides region I, which lies as a scatter-band about the fiber failure strain, as in the unidirectional on-axis loading case as well as in the cross-ply laminate case. The fatigue limit for any laminate is determined by the first cracking mechanism. Therefore, if the 90 -ply orientation is present in a laminate, the strain at which the first transverse cracking occurs will determine the fatigue limit. This strain value is affected by the so-called ply constraint, i.e., the ratio of the transverse ply modulus to the axial modulus of the constraint-providing plies as well as the thickness ratio of the cracking plies to the constraining plies. For further discussion of the constraint effect see [17] and Chapter 3. The progressive fatigue damage, repre- sented by region II, appears as a sloping scatter band, starting at a low number of cycles (10 2 –10 3 ) and asymptotically approaching the fatigue limit at a high number of cycles (10 6 –10 7 ). What remains to determine now is the slope of the region II band (or line). Obviously, a life prediction model accounting for the (sub-critical) progressive fatigue damage would be needed to predict this slope. Let us examine some test data to get some insight into this slope. Figure 7.35 shows the fatigue-life diagram for a glass/epoxy [0/Æ45/90] s lamin- ate under tension–tension loading along the 0 -direction with data from Hahn and Kim [18]. To construct the diagram, the failure strain of fibers (same as that of the 0 0.004 0.008 2 4 6 log N e m a x e c e dl Debonding in 90˚ plies, delamination Figure 7.34. Experimental data verifying the anticipated fatigue-life diagram of cross-ply laminates, with data from [16]. Reprinted, with kind permission, from Composite Materials: Testing and Design, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428. 263 7.7 Fatigue of laminates laminate) is taken to place the scatter-band of region I. Since the failure strain data were not available to determine the failure probabilities, the scatter-band was drawn based on other similar data regarding fiber failure strain. The fatigue limit was placed at 0.46% strain based on information concerning the strain at which transverse cracking was observed. Region II has been placed around the fatigue- life test data. It is noted that the scatter-band of region II is approximately straight (or negligibly curved) and meets the region I band at 10 2 –10 3 cycles. The lower end of the region II band is at approximately 10 7 cycles. Another data set to consider is from [19] for a carbon/epoxy [0/Æ45/90] s laminate under tension–tension loading along the 0 -direction (see Figure 7.36). The procedure for constructing the fatigue-life diagram is the same as that just described related to Figure 7.35. From the two examples of fatigue-life diagrams in Figures 7.35 and 7.36 it is remarkable that the conceptual framework these diagrams represent is a powerful means of interpreting and estimating fatigue-life of composite laminates. Under the restriction of tension–tension cycling along a principal direction, usually a e m a x e c 0.0100 0.0075 0.0050 0.0025 0 2 4 6 lg N e d.1. =0.0046 Figure 7.36. Fatigue-life data for a carbon/epoxy [0/45/90/À45 2 /90/45/0] s laminate under tension–tension loading along the 0º-direction [19]. The fatigue-life diagram is superimposed on the data. Reprinted, with kind permission, from Fatigue of Filamentary Composite Materials, copyright ASTM International, 100 Barr Harbor Drive, West Conshohocken, PA 19428. 0 0.005 0.010 0.015 2 4 6 lg N e c e d.1. =0.0046 e m a x Figure 7.35. Fatigue-life data for a glass/epoxy [0/Æ45/90] s laminate under tension–tension loading along the 0 -direction (data from [18]). The fatigue-life diagram is superimposed on the data. 264 Damage mechanisms and fatigue-life diagrams symmetry axis of the given laminate, the diagram can be constructed by looking for simple indicators of its characteristic features. If, for example, the laminate has a 0 -ply along which the cyclic tension load is applied, then region I will exist, providing a flat scatter-band around the failure strain of the composite (which equals the failure strain of fibers). The next thing to look for is the ply orientation in the laminate that makes the largest acute angle with the loading axis. In the two examples above, that angle was 90 . The ply with this orientation starts the fatigue process by initiating cracks along its fibers. A good approximation of the fatigue limit is the axial strain at which this cracking occurs. Methods for estimating this strain have been discussed in Chapter 6. Once the upper limit (failure strain) and the lower (fatigue) limit of the laminate have been marked in the fatigue-life diagram, the progressive fatigue mechanism is captured between those limits. One can model the progressive mechanism and predict fatigue-life, as we shall discuss in the next section, or simply get a good approximation to it by drawing a straight line (or scatter-band) going from the point on the fatigue limit line at selected number of cycles (10 6 –10 7 ) to the region line (or scatter-band) at a low number of cycles. A good guideline for taking the low cycle number is 10 2 for glass/epoxy and 10 3 for carbon/epoxy. 7.8 Fatigue-life prediction Fatigue-life prediction of composite materials has suffered from concepts and methodologies developed for metal fatigue that are not quite relevant. Except in a few cases, such as delamination growth, the analysis of single crack initiation, propagation, and unstable growth to failure does not apply to composite fatigue. Thus, fracture mechanics by itself has little use in composite damage. The associ- ated methodologies of stress intensity factor threshold, the so-called Paris Law, residual strength, etc. do not help in assessing fatigue damage tolerance and durability of composite structures. The classical (pre-fracture mechanics) approaches to metal fatigue, typified by fatigue-life evaluation based on empirical S-N curves, are just as lacking of useful- ness for composite fatigue. The obvious fact that fatigue mechanisms are local, and therefore dependent on microstructure, should suggest that homogenizing a composite and describing fatigue in terms of average stresses cannot correctly describe the driving impetus for composite damage. As argued above in explaining the construction of the fatigue-life diagram, strain correctly describes the limiting conditions for fatigue. The remarkable success of the fatigue-life diagram in evaluating the roles of constituents and interfaces between them, as well as in clarifying the impact of material properties, should justify moving away from the pure empiricism of the S-N diagrams. The misconception generated by the plot- ting of strain in the fatigue-life diagram that it requires strain-controlled testing is indeed unfortunate. The test data are required to be generated in the completely traditional way, i.e., in load control, while conversion of the average stress to 265 7.8 Fatigue-life prediction the average strain is to be done for plotting the data in the fatigue-life diagram. The result measured in strain (such as strain corresponding to a given number of cycles) is to be converted back to stress for use in design. The power of the conceptual framework of the fatigue-life diagram also lies in guiding mechanisms-based modeling of life prediction. The diagram assures that the right mechanisms are addressed in modeling. Needless to say, mechanisms- based modeling is the only rational approach to fatigue in composites. The alternative of empiricism is unreliable and costly in the long run. In the following we shall illustrate mechanisms-based fatigue-life modeling for cross-ply laminates. So far this laminate configuration is the only one treated by the approach. Its success, as we shall see below, should provide incentive for treating more general laminates. 7.8.1 Cross-ply laminates In Section 7.7.2 above, we discussed the fatigue damage mechanisms and con- struction of the fatigue-life diagram for cross-ply laminates. To model the pro- gressive damage in region II let us begin by reviewing the quantification of the transverse cracking mechanism. Figure 7.37 shows the variation of the transverse crack density (number of cracks per unit axial length) with the number of cycles (on logarithm scale) for different levels of maximum load. The characteristic feature of the damage evolution is that the crack density increases exponentially at first, followed by the rate of increase decreasing and approaching the saturation state of zero rate. The level to which the crack density saturates appears to depend on the load level. The first step in the life prediction procedure would be to predict the observed transverse cracking behavior under fatigue. This requires predicting the crack density at the first application of the maximum load and, next, determining the increase in crack density with the number of cycles having the same load excursion. Transverse cracking under monotonic loading was treated in Chapter 6. Here we will focus on crack multiplication in repeated loading. Figure 7.38 depicts the crack multiplication process. On first application of the cyclic load, transverse cracks of a certain spacing form. On next and subsequent applications of the same load, another crack between the previously formed cracks would be possible for one of two (assumed) reasons: (1) a flaw exists in the transverse plies that initiates a new crack, which grows and becomes identical to the previous cracks, and (2) the stress state between the pre-existing cracks changes with load cycling and becomes favorable at certain cycles to produce a new crack. In each case we must identify the mechanism of irreversibility from one load cycle to the next in order to model damage accumulation leading to formation of a new crack. In the first assumed case, we would need some knowledge about the flaws, possibly induced by the manufacturing process. The size and spatial distributions 266 Damage mechanisms and fatigue-life diagrams of the flaws are likely to be random, requiring probability distributions for their description. The initiation of cracks from flaws and their propagation with load cycling need the presence of irreversibility, which is likely to come from the inelasticity in the matrix (fibers are usually elastic) and/or microcracking in the crack-tip region. In this scenario of transverse crack progression the modeling effort will involve statistical and numerical simulation. The second assumed case relies on stress enhancement in transverse plies in the region between cracks without resorting to flaws. In this case, the initiation of new transverse cracks can be assumed to occur when the axial normal stress in the transverse plies reaches a critical value. Figure 7.39 shows the distribution of the three stress components in the longitudinal section of the transverse plies as 0 0 10 20 30 40 50 1 2 3 4 5 6 7 Cycles (Log n) C r a c k d e n s i t y l ( i n – 1 ) 85% 66% 53% 35% 28% Figure 7.37. Density of transverse cracks in a [0/90 2 ] s laminate plotted against the log number of tension–tension cycles in the axial direction. The data points are for different maximum load values indicated as percentages of the ultimate tensile strength of the composite [20]. Reprinted, with kind permission, from Springer Science+Business Media: Appl Compos Mater, Vol. 3, 1996, pp. 391–406, X.X. Diao, L. Ye and Y.W. Mai. Figure 7.38. Transverse cracks produced by the first application of load (left); generation of a new crack midway between the pre-existing cracks on repeated application of the load (middle); and completely grown crack at certain number of load applications (right). 267 7.8 Fatigue-life prediction calculated by variational analysis [21, 22]. These stresses result from interaction between cracks. As seen in Figure 7.39, the axial normal stress attains a maximum midway between cracks and its value reduces from its pre-crack (constant) value. Thus, in the absence of flaws, a new crack can form midway between cracks if the maximum stress there exceeds a critical value. However, since this stress is lower than the critical value at which the previous cracks formed, new cracks can only form if the applied load is increased. In a cyclic load of constant amplitude, new cracks are therefore not possible, if the assumption of no flaws still holds. The conclusion has to be that in this scenario of transverse cracks in an elastic composite, damage progression under cyclic loads cannot be achieved. If the modeling of damage progression in cross-ply laminates is pursued without entering flaws in the model, then the only plausible place for irreversible mechan- isms is the interface between the 90 -plies and the 0 -plies. It can be argued that a transverse crack approaching this interface is bound to cause damage to the interface in some form or another because of the intense stress field that accom- panies the crack front. The most likely damage is cracking in the interfacial plane (delamination). Figure 7.40 depicts this scenario, showing delamination of length 2l at the crack fronts that are 2s distance apart. Two possible sub-scenarios are –40 –30 –20 –10 10 20 Crack Crack 30 40 60 –4 –3 (MPa) 0 º 9 0 º –2 1 2 3 4 x s xz (interfacial) (1) s zz (interfacial) (1) s xx (1) z x 50 –1 Figure 7.39. Axial distribution of stresses in the 90 -plies under an axial tensile load (from [22]). Of interest is the axial normal stress, which is assumed to be constant in the z-direction. Reprinted, with kind permission, from Damage Mechanics of Composite Materials, J.A. Nairn and S. Hu, Matrix microcracking, pp. 187–243, copyright Elsevier (1994). 268 Damage mechanisms and fatigue-life diagrams now possible, one where the delamination surfaces are traction free, and the other where frictional sliding between the delamination surfaces can take place. In [23] stress analysis was conducted for transverse cracks and delamination with a variational mechanics method to estimate the stresses in the 90 -plies between the transverse cracks. A cubic variation of the shear stress along the delamination length 2l was assumed and the calculated axial normal stress then showed an increase in its maximum value with l. Figure 7.41 shows the axial stress maximum value for traction-free delamination and when a shear stress acts between the delamination surfaces. As seen in Figure 7.41, if delamination surfaces are assumed traction free, then the right conditions would not be present for new cracks to form between cracks formed in the first load cycle. The incentive for crack formation would indeed decline as the delamination crack propagates under cyclic loading. On 2s 2l Figure 7.40. Transverse cracks of spacing 2s in a cross-ply laminate with delamination emanating from crack fronts and extending a distance l on either side of the transverse crack. The delamination growth is assumed to be caused by the cyclic axial tension applied to the laminate. 0 0 5 10 15 20 25 30 35 M a x . s t r e s s i n 9 0 d e g r e e p l y ( M P a ) 0.2 0.4 0.6 0.8 1 l / s s– l l s s – l l s 0 0 50 100 150 200 250 300 M a x . s t r e s s i n 9 0 d e g r e e p l y ( M P a ) 0.2 0.4 0.6 0.8 1 l / s Figure 7.41. The variation of the maximum axial normal stress between two transverse cracks is shown when delamination at crack fronts exists. The stress reduces with increase in the delamination half-length l when the delamination surfaces are traction free (left) and it increases when a shear stress acts between the delamination surfaces (right). 269 7.8 Fatigue-life prediction the contrary, if delamination surfaces are engaged by asperities and/or by compressive normal stress on the surfaces, then frictional sliding between the surfaces will result, giving rise to a shear stress. This stress alters the redistri- bution of stresses between the axial and transverse plies, resulting in the increase of the maximum axial normal stress as indicated in Figure 7.41. The number of cycles beyond the first cycle that will elevate this stress to a critical value for crack formation is then the cycles needed to increase the delamination length by fatigue growth. Based on the assumption of frictional sliding of the delamination surfaces, Akshantala and Talreja [24] developed a procedure by which the transverse crack density under cyclic loading could be determined. Figure 7.42 shows examples of the predicted crack density variation with the number of cycles. Note the crack density variation displays saturation to different levels depending on the maximum stress in accordance with the experimental data in Figure 7.37, except the initial exponential rise in the crack density. Since the stress analysis model is for inter- acting transverse cracks the initial exponential increase in noninteracting cracks cannot be predicted by the model. Having predicted the crack density increase with load cycles, Akshantala and Talreja [24] proceeded to use these data to predict fatigue-life. The assumption was made that in the progressive damage represented by region II of the fatigue-life diagram for cross-ply laminates (see Figure 7.34), a certain maximum crack density is attained at failure of the laminate. This crack density denoted f lies between the maximum achievable crack density c under static loading and the minimum crack density fpf at the initiation of multiple cracking, commonly called the first ply failure (a misnomer since the ply cracks 0.0 0.0 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 Number of cycles (10 6 ) C r a c k d e n s i t y ( / m m ) 350 MPa 400 MPa 500 MPa Figure 7.42. Transverse crack density in a cross-ply laminate predicted by Akshantala and Talreja [24] at different cyclic load levels. Note the tendency for saturation at different levels depending on the load level. Reprinted, with kind permission, from Mater Sci Eng A, Vol. 285, A micromechanics-based model for predicting fatigue-life of composite laminates, pp. 303–13, copyright Elsevier (2000). 270 Damage mechanisms and fatigue-life diagrams but does not fail). The variation of f with failure load cycles was assumed to be as depicted in Figure 7.43. The equation describing this variation, f = A log N f + B, has empirical constants A and B, which are determined by using the minimum and maximum values of crack densities and their corresponding number of cycles. The extreme values of the fatigue cycles are the beginning and end of the progressive damage, i.e., region II. In Figure 7.43, these values are shown as 10 2 and 10 6 cycles for illustration. As suggested at the end of Section 7.7, a good approximation of the beginning of region II is 10 2 cycles for glass/epoxy and 10 3 for carbon/epoxy composites. Note that the assumed process of fatigue failure is not that it comes from the transverse cracks of a certain density, but that the crack density increases from its first-load value to that crack density. The failure of the laminate must involve delamination growth and fiber failures. The transverse crack density is simply assumed to scale with fatigue-life in the assumed way. Figure 7.44 depicts the procedure for fatigue-life prediction using the calcu- lated crack density from the stress analysis model [23] and the assumed relationship of the crack density to failure cycles as displayed in Figure 7.43. At a given cyclic load, the fatigue-life is given by the minimum number of cycles until the crack density equals the limiting value. Thus, the graphical method for determining this value is to find the point of intersection of the crack density increase curve and the straight line describing the limiting crack density (see Figure 7.44). The number of cycles corresponding to the intersection point is then the fatigue-life at that load level. The fatigue-life prediction thus obtained is compared with the test data for a cross-ply lamin- ate in Figure 7.45. Cycles to failure (log N f ) 10 2 10 6 C r a c k d e n s i t y ( h f ) h c h fpf h f =A logN f +B A 1 e e c e fpf Figure 7.43. The assumed variation of the transverse crack density at failure in fatigue plotted against the fatigue-life of a cross-ply laminate. The upper limit to this crack density is the maximum saturation crack density under static load and the lower limit is the minimum crack density at initiation of multiple cracking. The composite strains corresponding to the two limits are also indicated. Reprinted, with kind permission, from Mater Sci Eng A, Vol. 285, A micromechanics-based model for predicting fatigue-life of composite laminates, pp. 303–13, copyright Elsevier (2000). 271 7.8 Fatigue-life prediction 10 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 1 10 2 10 3 10 4 10 5 10 6 C r a c k d e n s i t y ( / m m ) 0.6% 0.65% 0.7% 0.76% 0.9% 1.0% 1.2% h fpf h c h f =– 0.425 N f +3.35 Number of cycles Figure 7.44. The crack density data points represent the initial (first load) values and their increase with cyclic loading at different load levels (indicated by the corresponding first- cycle peak strains). The crack density relationship to failure cycles, schematically described in Figure 7.43, is also shown with the calculated end-values of crack density. Fatigue-life at a given load level is given by the cycles at intersection of the calculated crack density with its failure value. Cycles to failure 10 0 10 1 10 2 10 3 10 4 10 5 10 7 10 6 0.0 0.5 1.0 1.5 2.0 2.5 M a x i m u m a p p l i e d s t r a i n ( % ) Fatigue limit Experiment data Model Region II Region 1 Figure 7.45. The predicted fatigue-life of a glass/epoxy [0/90] s laminate compared with test data from [25]. 272 Damage mechanisms and fatigue-life diagrams 7.8.2 General laminates In Chapter 6 we treated the damage progression of a broad class of laminates consisting of plies in multiple orientations subjected to axial tensile loads. From the models described there one can calculate the load at initiation of multiple cracking and crack densities resulting at the first application of the maximum load in the given cyclic load. For further progression of damage under repeated application of that load one must ask: what is the mechanism responsible for irreversibility that accumulates damage with each load cycle? For cross-ply lamin- ates we addressed this question and described the model that incorporates the answer to the question. There is every reason to believe that in laminates with multiple off-axis ply orientations the delamination occurring at the off-axis crack fronts conducts frictional sliding and thereby provides the needed irreversibility (energy dissipation). For cross-ply laminates the stress analysis conducted in [21] for transverse cracks with delamination cannot simply be extended to multiple crack orientations. In fact even without delamination the stress analysis of cross- ply laminates is not easily extended to the multiple crack orientation case. 7.9 Summary This chapter has provided a systematic conceptual framework for interpretation of the fatigue process in composite materials. No effort has been made to give a comprehensive exposition of the vast literature on the subject. Instead, emphasis has been placed on understanding of the physical mechanisms underlying fatigue and incorporating this in a systematic way in the representation called fatigue- life diagrams. These diagrams provide a healthy departure from the empiricism dominant in the fatigue literature. They also generate bases for material selection and give useful guidelines for mechanisms-based modeling. The field of fatigue damage modeling and life prediction needs a great deal of further work. The challenges of addressing the mechanisms of fatigue damage accumulation have discouraged most research efforts that have instead taken the path of empirical approach. In fact even in the century old field of metal fatigue empirical approaches are common. The so-called Paris Law of cyclic crack growth is basically a curve fit to the observed data. However, its simplicity has attracted practical use but has impeded advances in fundamental understanding. It is therefore not surprising that the composites community has so far also opted for empiricism. It is worth noting that because of the large number of parameters in composite materials (constituent properties, ply orientations, woven and other complex architectures, etc.) the empirical path is highly inefficient. As demon- strated by the fatigue-life diagrams, focusing on the essential mechanisms allows one to transcend the apparent complexity of the composite microstructure and fiber architecture. 273 7.9 Summary In the next chapter we shall outline the areas in composite fatigue that must receive attention by the composites community and must be supported by govern- ment and industry. References 1. R. Talreja, Fatigue of composite materials: damage mechanisms and fatigue-life diagrams, Proc R Soc London A, 378 (1981), 461–75. 2. C. K. H. Dharan, Fatigue failure mechanisms in a unidirectionally reinforced compos- ite material. In Fatigue in Composite Materials, ASTM STP 569. (Philadelphia, PA: ASTM, 1975), pp. 171–88. 3. J. B. Sturgeon, Fatigue and creep testing of unidirectional carbon fiber reinforced plastics. In Proceedings of the 28th Annual Technical Conference of the Society of the Plastics Industry. (Washington, DC: Reinforced Plastics Division, 1973), pp. 12–13. 4. J. Awerbuch and H. T. Hahn, Fatigue and proof-testing of unidirectional graphite/ epoxy composite. In Fatigue of Filamentary Composite Materials, ASTM STP 636, ed. K. L. Reifsnider and K. L. Lauraitis. (Philadelphia, PA: ASTM, 1977), pp. 248–66. 5. J. B. Sturgeon, Fatigue Testing of Carbon Fibre Reinforced Plastics, Technical Report, Royal Aircraft Establishment, Farnborough (1975). 6. R. B. Croman, Tensile fatigue performance of thermoplastic resin composites reinforced with ordered Kevlar® aramid staple. In Proceedings of the Seventh Inter- national Conference on Composite Materials, vol. 2, ed., Y. Wu, Z. Gu, and R. Wu. (Oxford: International Academic Publishers, 1989), pp. 572–7. 7. E. K. Gamstedt and R. Talreja, Fatigue damage mechanisms in unidirectional carbon fibre-reinforced plastics. J Mater Sci, 34 (1999), 2535–46. 8. E. K. Gamstedt and S. O ¨ stlund, Fatigue propagation in fibre-bridged cracks in unidir- ectional polymer-matrix composites. Appl Compos Mater, 8 (2001), 385–410. 9. E. K. Gamstedt, Effects of debonding and fiber strength distribution on fatigue- damage propagation in carbon fibre-reinforced epoxy. J Appl Polym Sci, 76 (2000), 457–74. 10. R. Talreja, A conceptual framework for interpretation of MMC fatigue. Mater Sci Eng A, 200 (1995), 21–8. 11. B. F. Sørensen and R. Talreja, Analysis of damage in a ceramic matrix composite. Int J Damage Mech, 2 (1993), 246–71. 12. B. F. Sørensen, J. W. Holmes, and E. L. Vanswijgenhoven, Does a true fatigue limit exist for continuous fiber-reinforced ceramic matrix composites? J Amer Chem Soc, 85 (2002), 359–65. 13. Z. Hashin and A. Rotem, A fatigue failure criterion for fiber reinforced materials. J Compos Mater, 7 (1973), 448–64. 14. A. Rotemand Z. Hashin, Fatigue failure of angle ply laminates. AIAAJ, 14 (1976), 868–72. 15. R. D. Jamison, K. Schulte, K. L. Reifsnider, and W. W. Stinchcomb, Characterization and analysis of damage mechanisms in tension–tension fatigue of graphite/epoxy laminates. In Effects of Defects in Composite Materials, ASTM STP 836. (Philadelphia, PA: ASTM, 1984), pp. 21–55. 274 Damage mechanisms and fatigue-life diagrams 16. G. C. Grimes, Structural design significance of tension–tension fatigue data on com- posites. In Composite Materials: Testing and Design (Proc. 4th Conference), ASTM STP 617. (Philadelphia, PA: ASTM, 1977), pp. 106–19. 17. R. Talreja, Transverse cracking and stiffness reduction in composite laminates. J Compos Mater, 19 (1985), 355–75. 18. H. T. Hahn and Y. Kim, Fatigue behavior of composite laminate. J Compos Mater, 10 (1976), 156–80. 19. J. T. Ryder and E. K. Walker, Effect of compression on fatigue properties of a quasi- isotropic graphite/epoxy system. In Fatigue of Filamentary Composite Materials, ASTM STP 636. (Philadelphia, PA: ASTM, 1977), pp. 3–26. 20. X. X. Diao, L. Ye, and Y. W. Mai, Simulation of fatigue performance of cross-ply composite laminates. Appl Compos Mater, 3 (1996), 391–406. 21. Z. Hashin, Analysis of cracked laminates: a variational approach. Mech Mater, 4, (1985), 121–36. 22. J. A. Nairn and S. Hu, Matrix microcracking. In Damage Mechanics of Composite Materials, ed. R. Talreja. (Amsterdam: Elsevier Science, 1994), pp. 187–243. 23. N. V. Akshantala and R. Talreja, A mechanistic model for fatigue damage evolution in composite laminates. Mech Mater, 29 (1998), 123–40. 24. N. V. Akshantala and R. Talreja, A micromechanics based model for predicting fatigue-life of composite laminates. Mater Sci Eng A, 285 (2000), 303–13. 25. C. J. Jones, R. F. Dickson, T. Adam, H. Reiter, and B. Harris, The environmental fatigue behaviour of reinforced plastic. Proc R Soc London A, 396 (1984), 315–38. 275 References 8 Future directions In Chapter 1 we discussed the durability assessment of composite structures, the overall goal for the subject of this book. As outlined there in Figure 1.1, the mechanisms of damage and their effects on deformational response constitute the main thrust of the field of damage mechanics, which is at the core of durability assessment. After discussing the physical nature of damage observed experimen- tally in Chapter 3, the next two chapters treated the two main approaches in damage mechanics – micro-damage mechanics (MIDM) and macro-damage mechanics (MADM), both aimed at predicting deformational response at fixed damage. Damage evolution was treated in Chapter 6, while Chapter 7 was devoted to fatigue, a subject that requires special attention due to the conceptual difficul- ties it poses. In closing the book we wish in this chapter to review what has been achieved and what directions the field of damage and failure of composite materials should pursue to further advance toward durability assessment and beyond. 8.1 Computational structural analysis Obviously, complex structural geometries require computational structural analysis. The analytical modeling of damage initiation and evolution, and its effects on deformational response of composite laminates, discussed in previous chapters, were developed for idealized simple cases. Direct application of these models is limited to structures with simple geometry and loading conditions. For complex geometries, such as an airplane wing or a wind turbine blade, usually subjected to multi-axial mechanical loads, and possibly combined with thermal and moisture environments as well as manufacturing-induced residual stresses, computational approaches are inevitable. In industry, one often uses commercial software, e.g., ANSYS, ABAQUS, and NASTRAN, and the obvious need is to integrate damage and failure analyses into these codes. Efforts have been made to attempt some simple test cases where FE analysis of composites is combined with damage using failure criteria [1]. A series of World Wide Failure Exercises (WWFE) [2–4] have been conducted to compare several composite failure models with experimental data and provide guidance for their usage in composite design. While previous exercises focused on the ultimate failure, the ongoing WWFE-III offers opportunities to examine the initiation and progression of sub-critical events and their effect on the mechanical response. Yet no comprehensive code exists which can combine stress analysis with damage prediction. The current codes predict structural failure based on lamina-based failure criteria and not on the basis of sub-critical damage such as ply cracking. To improve this situation, the authors are currently developing a user subroutine (UMAT) in ABAQUS with an ability to dynamically update the stiffness properties of a region using the energy-based fracture criterion described in Chapter 6. The aim is to predict the initiation of ply cracking and its evolution, as well as the resultant stiffness properties, using the synergistic damage mechanics approach described in Chapter 5. While Figure 1.1 outlined the overall methodology for durability assessment, Figure 8.1 here illustrates the procedure for performing computational stress analysis of a composite structure in combination with damage analysis. At first, stress analysis is conducted assuming no damage is present in the whole structure. At this point the necessary data input includes the geometry of the structure, the configuration (layup, thicknesses of plies, etc.) and material of the composite PRELIMINARY STRESS ANALYSIS INPUT Geometry, laminate configuration, material properties and loading conditions Create geometric model Meshing Apply loading & service conditions Stress analysis DAMAGE ANALYSIS Identify regions where damage might have developed Predict damage initiation & evolution Evaluate stiffness properties of damaged regions UPDATED STRESS ANALYSIS OUTPUT Failure characteristics, stress-strain response, deformation behavior, life and durability Update stiffness of damaged regions Perform stress analysis of whole structure again Figure 8.1. Flowchart illustrating computational stress analysis of a composite structure in combination with damage analysis. 277 8.1 Computational structural analysis laminate, and the loading and service conditions. Then, based on the stress pattern, regions are identified which are potential sites for damage under the prescribed loading and service conditions. These regions include any stress con- centrators and could be based on previous design experience. In this step, possible damage mechanisms are also identified. For instance, the region close to a hole or cut-out in the structure is expected to develop damage in the form of ply cracks and delamination. Next, using the preliminary stress states, damage initiation and its evolution are predicted with the methods and analyses treated in earlier chapters. The damage prediction step may involve a substructural analysis and be carried out over multiple length scales. The multiscale modeling aspect will be discussed later in the present chapter. The stiffness properties of damaged regions are then updated using the analytical predictions, and a new stress analysis is performed at the same applied load levels. On increased loading, the same iterative process is repeated until the analysis predicts failure based on the design criteria. Depending on the purpose and need, the output of the computational structural analysis may include detailed failure characteristics, overall stress–strain response, deformation characteristics (deflections, vibrational frequencies, etc.), as well as damage tolerance and life prediction of the structure. The main challenge in any computational scheme is to properly incorporate damage and its effects. It is not uncommon to find works where damage with all its complexities is reduced to a parameter D, which is then arbitrarily assigned a value between 0 and 1, often based on computational convenience and seldom account- ing for the physics behind damage. Much progress is still needed in taking the theoretical advances in damage mechanics into computational design procedures. One of the main hurdles is properly conducting multiscale computational analysis of damage. In the following we shall address issues in multiscale modeling of damage, which present some fundamental challenges arising in the computational treatments of composite failure. 8.2 Multiscale modeling of damage The computational power available today has given impetus to the so-called multi- scale modeling, which follows an intuitive notion that the physical phenomena occurring at lower length scales determine the material response at higher scales. Fiber-reinforced composites and, more generally, heterogeneous solids appear to be natural candidates for this modeling idea. Thus, numerous studies have resulted to address deformation, damage, and failure of these materials and to some extent also their nonmechanical properties. Avast literature on multiscale modeling exists, and a recent collection of works [5] covers a wide range of such treatments. Since in a heterogeneous solid the microstructural entities (heterogeneities) are embedded in the matrix, a given property (e.g., mechanical or thermal response characteristic) must be defined at a scale much larger than the charac- teristic size of the heterogeneities. This leads to the suggestion that, at a chosen 278 Future directions scale, a representative volume element (RVE) exists such that by appropriate averaging over this volume the property of interest can be computed. The RVE idea is a necessary fundamental concept in any proper multiscale treatment. The reader is referred to texts in micromechanics such as that by Nemat-Nasser and Hori [6] for in-depth treatment of the RVE concept and the associated aver- aging schemes. Denoting the RVE scale as mesoscale, and the scale of microstructural entities as microscale, the three-level micro-meso-macro hierarchy forms the basis of a multiscale modeling treatment. In fact if the “microstructure” in a heterogeneous material contains nano-scale reinforcements (e.g., particles, fibers or tubes), then further differentiation of the lower-end scale to nano- and micro-levels can be made. While the hierarchical multiscale modeling is a viable approach for micro- structures that are “stationary” (Figure 8.3), i.e., they do not change their charac- teristic size under a loading impulse, it is not a given that the approach will remain valid when energy dissipative mechanisms permanently alter the microstructure. Our concern here is multiscale analysis of the energy dissipative mechanisms underlying damage in composite materials for the purpose of assessing structural integrity and durability, and for this case further examination of the multiscale approach seems warranted. In the context of computational structural analysis described above in Section 8.1 and in a previous review of this subject [7] we refer to Figure 8.2, Figure 8.2. Durability analysis procedure for composite structures. u 1 u 2 t t t u Figure 8.3. Heterogeneous solid in pristine state (left) and two damage states, type 1 (middle) and type 2 (right). 279 8.2 Multiscale modeling of damage which was also discussed in Chapter 1. Here we draw attention to the damage mechanics part of the scheme. As indicated, micro-meso-macro multiscale model- ing is at the core of this analysis. The three-scale differentiation when damage exists does not necessarily coincide with the multiscale differentiation adopted before damage. This critical point was treated in [8], from which the following discussion is drawn. 8.2.1 Length scales of damage Let us consider three RVEs depicted in Figure 8.3 to discuss characteristic scales of damage and their relation to the scales of microstructure. The RVE to the left in the figure is for pristine (undamaged) composite. When the bounding surface of this RVE is subjected to a prescribed traction t, the combined deformation of the matrix and the heterogeneities (shown symbolically as filled circles) contained within the RVE produces a displacement u of points on the bounding surface. This displacement field can be written as u = u 0 + du 0 , where u 0 represents the deformation of the matrix in the absence of fibers and du 0 is the perturbation in it caused by the heterogeneities in the absence of damage. The middle RVE in Figure 8.3 illustrates a damage scenario – let us call it type 1 damage – where some of the heterogeneities (e.g., inclusions) have partially or fully separated from the matrix. The interfacial cracks, thus formed, then perturb further the deform- ation field within the RVE if the traction t on the RVE surface is sufficient to activate the cracks (i.e., displace the crack surfaces). The displacement of points on the bounding surface is now u 1 . Finally, the RVE to the right in the figure depicts a damage scenario – type 2 damage – where the cracks within the RVE are restricted to the matrix and are geometrically unconnected to the heterogeneities. The displacement response to surface traction t on the RVE in this case is denoted u 2 . The perturbation field du 0 can, in principle, be determined by know- ledge of the properties of matrix and heterogeneities, as well as configuration variables such as size, shape, and spacing of heterogeneities. Depending on the model employed, one may use limited configuration information such as volume fraction of heterogeneities, or more enriched information such as statistical cor- relation functions that describe their relative size and placement. A thorough treatment of the morphological characterization of microstructures is found in [9], while [10] gives an extensive review of models that aim at carrying the morphological information to RVE level averages. The models in general have multi-level features and often rely on homogeniza- tion concepts. In the absence of damage, the models may be characterized as “stationary” microstructure models, i.e., the microstructure configuration remains unchanged under application of RVE surface traction t. In calculating the RVE surface displacement response to the prescribed traction t in the presence of damage, one can take two alternative approaches. One approach is to homogenize the matrix and stationary microstructure and embed the damage entities (e.g., cracks) in the homogenized composite. The displacement response can then be 280 Future directions written as u i = u + d i u, where d i u results from the perturbation in the displace- ment field of the homogenized composite caused by damage (i = 1 or 2 for damage type 1 or type 2, respectively). This approach is common and has many versions, a familiar one being the Mori–Tanaka estimation procedure [11]. It is noted that the Mori–Tanaka procedure is a homogenization method for station- ary microstructures, i.e., for undamaged composites. For further reference, we shall call it the homogenized microstructure (HM) approach. The other approach is to retain the discrete nature of the stationary microstructure in estimating the RVE displacement response. In this approach, called here the discrete microstruc- ture (DM) approach, the displacement response can be written as u i = u 0 + d i u 0 , where d i u 0 results from combined perturbation in the displacement response of the matrix caused by the stationary microstructure and damage, the subscript index i referring still to the damage type. The two approaches to estimating the RVE surface displacement, just described, are approximate and will generally yield different results. In the HM approach, the explicit association of a damage entity to the microstructure is lost, since the microstructure in which the damage entity resides has been homogenized, while in the DM approach the heterogeneities as well as damage entities are explicitly present. Thus, in the HM approach, damage entities of type 1 and type 2 are both surrounded by the homogenized microstructure, thereby their association with the microstructure (as to how the microstructure affects their initiation) is lost. In this sense, the HM approach is insensitive to which of the two types of damage is treated. On the other hand, in conducting the DM approach a signifi- cant difference exists depending on whether damage of type 1 or type 2 is considered. Since the type 1 damage is geometrically tied to the stationary micro- structure, the perturbation caused by it in the local fields can be analyzed by viewing it as a modification to the perturbation induced by the heterogeneities. This would not be the case for type 2 damage, as it is unconnected to the heterogeneities but affected by them. In fact, for this reason, the HM approach would be preferable for type 2 damage. The considerations described above have been made for a specific purpose: to clarify the characteristic scales associated with the micro-level (heterogeneities) and their relation to the meso-level (RVE) scale. As noted, the micro–meso bridging in the absence of damage is clear and unambiguous. In fact, if the matrix has a heterogeneous structure itself, then knowing the characteristic scales of that substructure it would be possible to homogenize it with a multiscale (sub-micro to micro) approach. Thus, in general for stationary heterogeneities, the multiscale modeling is plausible and systematic, at least conceptually. This is far from the case when damage in the form of distributed internal surfaces exists. When these internal surfaces are geometrically connected to the microstructure, such as in type 1 damage, their characteristic micro-level scales can be deduced from those of the heterogeneities and the micro–meso bridging is then relatively simple. How- ever, few cases of damage in composite materials belong to this type, i.e., where the damage entities remain connected to the heterogeneities. Although this might 281 8.2 Multiscale modeling of damage occur in the initial stages of damage, the damage entities formed grow away from the heterogeneities. In the case of type 2 damage, where the damage entities have scales unrelated to those of the heterogeneities, the resulting complexities of scales question the viability of the hierarchical multiscale modeling, as discussed in more detail in the following. 8.2.2 Hierarchical multiscale modeling We shall now focus on the specific case of composite laminates in which individual plies are of a matrix material reinforced with unidirectional fibers. Thus the stationary microstructure here consists of fibers distributed in the matrix and corresponds to the picture on the left in Figure 8.3 for a heterogeneous solid in pristine state. The microstructural length scale, or micro-scale, is the size of a fiber (radius or diameter). The next scale in the hierarchy, the meso-scale, is the RVEsize, which depends on the distribution of fibers. Assuming all fibers are straight and parallel, then for the case of uniformly distributed fibers in the cross section a repeating unit of a fiber embedded in a surrounding matrix, i.e., a unit cell, replaces RVE. For nonuniform distribution of fibers, reference is made to treatments in [9] and [10]. Simpler estimates for ply properties can be found in any of the common texts on composite mechanics, e.g., [11]. Finally, the macro-scale for a composite laminate is a structural scale that depends on the structural geometry. When a composite laminate suffers damage under loading, a variety of length scales develop that may or may not be connected with the initial length scale of undamaged composite. Although this issue has been discussed above in general terms, we will treat it now specifically for composite laminates. 8.2.2.1 Damage in unidirectional composites: transverse loading case Figure 8.4 (left) shows a typical view of damage observed in the cross section of a unidirectional fiber composite of a polymeric matrix loaded in tension normal to Debonding induces matrix cracking Matrix cracking causes debonding (b) (a) Figure 8.4. A typically observed micrograph (left) of a cross section of a unidirectional fiber composite loaded in tension normal to fibers. To the right are depicted two plausible scenarios underlying the picture on the left. 282 Future directions fibers. The damage appears to be a mix of fiber/matrix debonding (type 1 damage) and matrix cracking (type 2 damage) described above. However, a closer study of fiber/matrix debonding [12–14] suggested that this failure mechanism could be a consequence of cavitation-induced brittle cracking in the matrix. Type 2 damage, on the other hand, is matrix flow-induced ductile cracking in this case. Thus, in polymer matrix composites, damage of type 1 and type 2 could both be different realizations of matrix failure with drastically different governing scales, as dis- cussed below. In Figure 8.4 (right) are shown two plausible scenarios of cracking that can underlie the observed damage in Figure 8.4 (left). The first one, marked (a), is for fiber/matrix debonding that progresses out of the fiber surface into the matrix. Such matrix cracks then coalesce forming a continuous fiber-to-fiber crack. The second scenario assumes formation of matrix cracks first, which on growing towards fibers induce fiber/matrix debonding. This sequence of cracking would also produce a continuous fiber-to-fiber crack. Which of the two scenarios holds depends on the transverse loading-induced local stress states, which in turn depend on the microstructure configuration, i.e., fiber volume fraction and distri- butions of fiber diameter, inter-fiber spacing, etc. A discussion of the effect of local stress states on matrix damage follows. A cross-sectional region of a transversely loaded composite is illustrated in Figure 8.5. For nonuniformly distributed fibers in the cross section, stress analysis studies conducted in conjunction with the work reported in [12] suggested that for points in the matrix close to the fiber surface, such as that indicated in the figure, the deformation of the matrix is nearly or fully dilatational, while for points in the matrix away from fibers, such as the other point indicated in the figure, the matrix deformation has a significant distortional component. The mix of dilatational and distortional deformation will depend on the constraint to deformation imposed by the presence of fibers. For instance, for points at nearly-touching fiber surfaces and in matrix regions that are squeezed between three fibers, the deformation will approach the dilatational state, while at points in the matrix sufficiently away from fibers such that the local stress perturbation induced by fibers is negligible, the deformation state will have a high degree of distortion. For points near a fiber surface where conditions for dilatational deformation are favorable, the same studies [12, 13] proposed that cavitation of the polymer matrix is induced by the hydrostatic tension (the insert in Figure 8.5(b)). The cavities thus formed expand stably at first and become unstable when the dilatation energy reaches a critical value. The unstable cavity growth in a region with lack of sufficient distortional energy results in brittle cracking, which finds its way into the fiber/matrix inter- face. The consequent debonding grows as an arc-shaped crack along the fiber surface (Figure 8.5(b)) and subsequently diverts into the matrix along a direction normal to the local maximum tensile stress (Figure 8.4(a)). Further progression of damage produces the view seen in Figure 8.4 (left). An alternative damage scenario is illustrated in Figure 8.5(c), where a point away from fibers undergoes significant distortional deformation, which eventually 283 8.2 Multiscale modeling of damage localizes in shear-intensive bands, leading to ductile cracking. As the crack growth advances to brittle regions near fibers, fiber/matrix debonding occurs (Figure 8.4(b)). As in the other damage scenario, further progression of damage can produce the same view seen in Figure 8.4 (left). Modeling of the two damage scenarios depicted above involves analyses at two different sets of characteristic scales. For the cavitation-induced brittle cracking resulting in fiber/matrix debonding (the first of the damage scenarios, Figure 8.5 (b)), the characteristic scale at which cavitation begins is the molecular scale of the polymer matrix. Once debonding occurs, the scale of damage is the diameter of the fiber on whose surface the arc-shaped debond crack grows. For the second damage scenario (Figure 8.5(c)), the sequence of deformation and failure mechan- isms can be long and complex depending on the polymer morphology and the stress triaxiality (defined in an appropriate way to describe the mix of deviatoric and hydrostatic stress). These mechanisms can be classified as brittle, quasi-brittle, or ductile to characterize the relative degree of material flow involved in the cracking process. The characteristic scales of damage will then vary accordingly. A large body of literature exists on models that address the deformation and (a) s Dilatational Distortional (c) Distortional s s s s s (b) Figure 8.5. (a) Points in a transversely loaded heterogeneous solid with dilatational and distortional deformation; (b) cavitation and subsequent cracking induced by hydrostatic stress state; and (c) cracking in matrix caused by distortional flow. 284 Future directions failure of polymers ranging from molecular level treatments to continuum mech- anics analyses incorporating the details of the mechanisms in various explicit and implicit ways. Review of this literature is not the purpose here, but in the context of the multiscale damage in heterogeneous solids it is important to note that the presence of heterogeneities alters the nature of the problem. While for an unre- inforced polymer a multiscale modeling effort must address all scales of morph- ology that are activated by the loading, in the case of a polymer composite, activities are enhanced at certain scales and subdued at others, depending on the local stress perturbation caused by the heterogeneities. Thus, the hierarchy of scales and their relative roles are generally different in unreinforced and in heterogeneous polymers. Although the discussion here has been focused on polymers as matrix materials, the inferences drawn will largely apply to reinforced ductile metals as well. 8.2.2.2 Cross-ply laminate: transverse ply cracking Transverse ply cracking in a cross-ply laminate is an extensively studied area and has been reviewed in previous chapters; here the focus of discussion will be the characteristic scales of damage. When a cross-ply laminate is loaded in tension along the longitudinal plies, the first event of damage is the transverse crack formation in one of the two ways discussed above. The crack grows across fibers first, and then along fibers, eventually spanning the thickness and width of the transverse ply. Up to this point, the scales associated with the damage are as discussed above for transverse loading-induced damage in unidirectional compos- ites. On encountering the ply interfaces the transverse crack fronts bring about interfacial stress perturbation, which traverses a certain distance along the longi- tudinal plies, whence the stresses return to the pre-cracking state, unless a perturb- ation by another crack intervenes, in which case another equilibrium stress state results. This so-called shear lag distance is the distance on either side of the transverse crack where reduction of the axial stress in the transverse ply prevents another transverse crack from forming, and increased loading is needed to pro- duce such a crack. This phenomenon of stress transfer from cracked transverse plies to the longitudinal plies is responsible for multiple cracking in the transverse plies. The conditions for single crack formation versus multiple cracking were first explained in the landmark paper now commonly known as the ACK theory [15]. From the viewpoint of the characteristic scales of damage, which is what is of concern here, the situation changes drastically when damage evolves from the phase of single transverse crack formation to multiple transverse cracking. The scales associated with single crack formation were discussed above. In multiple cracking the governing scale is that of the shear lag distance along the 0/90 ply interface. This distance is determined by the ply properties and the thickness of plies in the two orthogonal directions. In other words, it is the homogenized ply properties and laminate configuration that determine the scale of multiple cracking, while it was the ply constituent properties and reinforcement morph- ology (fiber volume fraction, fiber diameter, inter-fiber spacing, etc.) that governed 285 8.2 Multiscale modeling of damage formation of a single transverse crack. This fact does not seem to be fully appreciated by most multiscale modeling efforts, which tend to treat transverse cracking with the scales of heterogeneities (fiber size and spacing, and associated distributions). 8.2.3 Implication on multiscale modeling: Synergistic damage mechanics The two cases discussed above are intended to illustrate the complexity and richness of the damage phenomena in composite materials. More cases have been treated elsewhere [7, 8, 16, 17]. It seems clear from these studies that a hierarchical multiscale approach for undamaged composites cannot generally be extended to cover all cases of damage. This may suggest that multiscale modeling for the purpose of structural integrity and durability should be approached on its own rather than tying it to the hierarchical approach for undamaged heterogeneous solids. Efforts in this direction were proposed and labeled as “synergistic damage mechanics” (SDM) [18]. Since the publication of that work, a systematic demon- stration of the viability of the approach has been made [19–23]. In Section 5.2 the SDM approach was described in detail. Here, for the sake of completeness of the multiscale modeling discussion we include two figures from Chapter 5. Figure 8.6 (same as Figure 5.4) shows the two-step homogenization procedure involved in the characterization of damage. As depicted in the figure, the stationary microstructure is homogenized first and represented by appropriate constitutive relations. The evolving microstructure, consisting of damage entities, Stationary microstructure Evolving microstructure RVE for damage characterization Step 2 V a i n j Fully homogenized continuum Characterization of a damage entity P P Step 1 Homogenization of evolving microstructure Homogenization of stationary microstructure Continuum after homogenizing the stationary microstructures Figure 8.6. Two-step homogenization of a composite body with damage is depicted. The characterization of damage and the associated RVE are also illustrated. 286 Future directions is homogenized next and regarded as an internal structure embedded in the homogenized solid of stationary microstructure. The new continuum is now represented by a thermodynamics framework in which internal state variables characterize damage by a set of second-order tensors. The internal variables are defined by descriptors that require averaging over a RVE. The RVE size is then the meso scale, while the characteristic size of the damage entities is the micro scale. The size-scales of the stationary microstructure enter separately in the step 1 homogenization. In this scheme the micro-scale level is the single damage entity size and the meso-scale level is the RVE size, as described above. The micro-level descriptor is the damage entity tensor d ij described in Chapter 5, while its average over the RVE, denoted D ðaÞ ij for a selected damage mode a is the meso-scale descriptor, also described in Chapter 5. Because of the way the damage entity tensor is constructed, it is possible to explicitly incorporate micro-level information into this descriptor. In this way the influence of the surrounding heterogeneous solid, i.e., the “microstructure,” on the damage entity can be analyzed by a convenient means and thereby transmitted to the meso level. A parameter, called the “constraint parameter,” has been devised to effectively accomplish this task, either experimentally or by computational micromechanics. This has all been described in detail in Chapter 5 for single and multiple modes of damage. Figure 5.10 from Chapter 5 is reproduced here as Figure 8.7 for a convenient recollection of the SDM procedure. The multiscale modeling of damage and the accompanying SDM methodology are the appropriate way to treat the effect of damage on the composite material response and by extension to assess structural durability. The hierarchical multi- scale treatment of heterogeneous solids, although suited for estimating their overall response, cannot generally be extended to account for damage. 8.3 Cost-effective manufacturing and defect damage mechanics A composite structure, constructed by any practical manufacturing process, is rarely perfect. The defects induced during manufacturing can be in the fiber architecture, e.g., fiber misalignment, irregular fiber distribution in the cross section, and broken fibers; in the matrix, e.g., voids; and in the interfacial regions, e.g., debonding and delaminations. Such defects must be analyzed to determine their effects on the composite structural integrity and durability. The results of such analyses can be used in two ways: (a) to develop acceptance/rejection criteria for the manufactured part; and (b) to design the part to meet performance requirements accounting for the defects. The former is what industry largely practices today, if at all. A host of nondestructive inspection (NDI) techniques are potentially possible to detect defects. These techniques can be applied to implement thresholds for product quality, e.g., a maximum void volume fraction or a maximum delamination surface area. 287 8.3 Cost-effective manufacturing and defect damage mechanics The field of designing a composite part with known defects is far from mature. What need to be developed are accurate analyses of effects of real-life defects as well as a strategy for incorporating the results in a cost-effectiveness assessment of the manufacturing process. As is the case, most of the cost of a part lies in the manufacturing process. In the following, we shall first review a cost-effectiveness assessment procedure, clarifying the different elements that make up the proced- ure. We shall then focus on the mechanics of materials approaches for analysis of real-life defects. For illustration we shall review some recent results on (i) elastic property changes due to matrix voids, and (ii) effects on the propensity for crack extension induced by interlaminar voids. In closing we shall make recommenda- tion for future work. 8.3.1 Cost-effective manufacturing Figure 8.8 describes the interrelated elements involved in the process of assessing the cost-effectiveness of a composite structure with respect to its long-term per- formance. To begin, the manufacturing process selected for a given composite structure is described by material and geometry parameters, processing param- eters and their time variations, as well as descriptors of machining, tooling, joining, assembly, etc., as needed. The product resulting from the manufacturing process is characterized in terms of the “material state” and its corresponding COMPUTATIONAL MICROMECHANICS SYNERGISTIC DAMAGE MECHANICS STRUCTURAL ANALYSIS Analyze overall structural response to external loading using the reduced stiffness properties Use SDM to determine stiffness reduction in present laminate configuration [0 m / ±q n /0 m/2 ] s Structural scale: Meso Structural scale: Macro Structural scale: Micro – EXPERIMENTAL/ COMPUTATIONAL Determine COD and constraint parameter(s) ; = = + ( ( ) ( ( ) ∆u 2 ∆u 2 ∆u 2 n = κ q κ 90 ±q n ±q n +q n q b Evaluate damage constants using available data for reference laminate configuration [0/90 8 /0 1/2 ] s 90 8 ( ) ∆u 2 ( ) ∆u 2 Figure 8.7. Flowchart showing the multiscale synergistic methodology for analyzing damage behavior in a class of symmetric laminates with layup [0 m /Æy n /0 m/2 ] s containing ply cracks in the +y and Ày layers. 288 Future directions properties. The conventional material state description consists of constituent properties and their relative proportions, e.g., volume fractions, and of the fiber architecture, e.g., ply thickness, orientation, and stacking sequence in a laminate, or fabric type (e.g., 5- or 8-harness satin), thickness, and layup in a woven fabric composite, etc. In addition to this, the material state needs to be described by certain defect descriptors. As we shall see, homogenized descriptions of the constituents and defects do not suffice for the cost-effectiveness assessment of a manufacturing process. The defect descriptors needed would depend on the manufacturing process. Examples of such descriptors are distributions (or other statistics) of fiber mis- alignment angles, of void size and location, of fiber/matrix interfacial disbonds, of delamination size and location, etc. The appropriate set of defect descriptors, along with the conventional material state descriptors, makes up a complete characterization of the manufactured composite material. Depending on the service environment in which the composite structure is to function, i.e., the design requirements imposed on the structure, the cost-effectiveness assessment will consider the necessary material properties and their relationships with the material and defect descriptors. A cost/performance trade-off exercise, and any iteration on it, will result in an optimized cost-effective product. In most applications, where long-term performance is the critical design consideration, one needs to look at the degradation of initial (end-of-manufacturing) properties of interest under the service environment. Thus the cost/performance trade-off will consider the residual properties. A common approach is to consider only the initial (i.e., pre- service) properties even for the long-term case, assuming implicitly that the residual properties will relate to the defects (and the cost) the same way as the initial properties do. This assumption is in fact questionable since the initial properties may not show sensitivity to some of the material defects that may turn out to be significant in governing the long-term properties. There is a variety of manufacturing processes used for composite structures, e.g., autoclave molding, liquid compression molding, resin transfer molding, Figure 8.8. Procedure for the cost-effectiveness assessment of composite structures. 289 8.3 Cost-effective manufacturing and defect damage mechanics filament winding, chemical vapor deposition, etc. Each of these processes pro- duces defects in the manufactured part that are usually characteristic of that process. The machining, joining, and assembly methods used for composite struc- tures produce defects that are generally different from those produced during molding, winding, and vapor deposition. For instance, the defects in the inter- facial region between two parts will be different depending on whether the parts were co-cured or adhesively bonded. Significant differences in the fatigue lives of co-cured vs. bonded joints have been reported [24]. In recent years, many methods have been developed to observe manufacturing defects in composite materials and structures by nondestructive evaluation, based mainly on ultrasound and radiography [25] and to some extent on thermal wave imaging [26]. In the conventional approach these methods are utilized primarily for quality control of the manufactured product. The premise of the quality control is often that the presence of defects is undesirable. If defects of more than certain threshold values are found, then the part is rejected and one strives to improve the manufacturing process. This inevitably increases the cost and can result in the composite part not being competitive with a metallic alternative. It is important to realize that the presence of defects is not undesirable in all cases. In fact, if some defects are allowed, the part can be produced at a lower cost. Figure 8.9 illustrates the dependence of strength per dollar of manufacturing cost on the defect density. As seen in the figure, the strength decreases gradually with defect density for low densities, and drops rapidly at high densities, while the manufacturing cost increases rapidly at low defect densities and falls off as more defects are allowed. Thus the strength of the part achieved per dollar of manufacturing cost increases with defect density, up toa point, beyond which the benefit of allowing more defects decreases. It must be remembered, however, that this situation is typical of the static strength. The dependence of residual strength in long-termloading on the initial defect density may show different behavior. This aspect has not been investigated sufficiently. Figure 8.9 also suggests that we should get away from the accept/reject approach and advance to what may be called defect engineering. More specifically, we should engineer the components to have a certain amount of defects in order to bring down the manufacturing cost while still having the performance require- ments satisfied. To achieve this higher level of engineering we need certain cap- abilities to be in place. Referring again to Figure 8.8, the connection between manufacturing (box at top) and the material state achieved (box at left) requires a capability to predict the defect structure along with the material composition attained from the employed manufacturing process. Some attempts in this direc- tion have been made. As an example, see references [27–31] for prediction of voids in a liquid compression molding process. Our efforts are focused on the connection between defect structures and the mechanical properties, as well as their degradation in service. This type of activity may be viewed as an extension of damage mechanics, which in its conventional form deals with initiation and evolution of damage and the consequent changes in mechanical properties. Thus our starting point in the extended damage mechanics 290 Future directions is not a homogenized continuum, but a composite with fibers and matrix as constituents, and in addition, defects. The defects in our analyses are real-life defects with their geometry and distribution as given by actual observations. A broader strategy for durability assessment that includes analysis of defects was coined as defect damage mechanics [32]. It is discussed next. 8.3.2 Defect damage mechanics To illustrate the mechanics of damage incorporating defects, we take two examples in the following. 8.3.2.1 Autoclave processing voids The first example deals with voids in composite laminates manufactured by autoclave molding where we describe the observed characteristics of voids and their incorporation in the modeling (for more details, see [33]). Figure 8.10 (upper part) shows two cross-sectional views, parallel and normal to fibers, of a unidirectional carbon/epoxy composite made by the autoclave process. The voids seen are generally not spherical and are largely trapped between the prepreg layers. In the lower part of the figure are two cross sections, a short distance (1.2 mm) apart, showing the voids more closely. Figure 8.11 summarizes numerous observations and measurements reported in the literature [35–37] concerning voids in composites made by autoclave molding. The shape can be described as elongated cylinders of elliptical cross section capped at the ends. The volume fraction of the voids is found to be less than 3% for a well- controlled autoclave process, which is safely below the 5% value taken for rejec- tion of parts in the aerospace industry. The process by which voids form suggests that the voids must displace the fibers around them as they settle down in their equilibrium positions. Most continuum models that homogenize the composite and “embed” voids do not account for this fact. Such models essentially “replace” fibers, not “displace” them. Reference [33] accounted for the fiber displacement as schematically illustrated by Figure 8.12. Strength/dollar Strength Manufacturing cost Defect density Figure 8.9. Illustration of the dependence on defect density of strength, manufacturing cost, and strength per unit cost. 291 8.3 Cost-effective manufacturing and defect damage mechanics The predictions of the elastic moduli by the Huang–Talreja procedure [33] are compared with experimental data in Figure 8.13. Note the large change in the through-thickness modulus (E zz ) due to the voids. 8.3.2.2 Interlaminar voids We now consider the effect of the presence of voids in an interlaminar plane (layer) on the growth of a crack in that plane. As described above, most voids in 1.2 mm 2.5 mm 2.5 mm Figure 8.10. Observed voids in unidirectional carbon/epoxy composites made by autoclave process. Upper pictures [34]: cross section parallel to fibers (left) and across fibers (right). Lower pictures [35]: two cross sections 1.2 mm apart showing voids. Upper pictures reprinted, with kind permission, from K. J. Bowles and S. Frimpong, J Compos Mater, Vol. 26, pp. 1487–509, copyright # 1992 by Sage Publications. Lower pictures reprinted, with kind permission, from Springer Science+Business Media: Review of Progress in Quantitative Nondestructive Evaluation, A morphological study of porosity defects in graphite-epoxy composite, Vol. 6B, 1986, pp. 1175–84, D. K. Hsu and K. M. Uhl. Figure 8.11. Observed characteristics of voids in carbon/epoxy composites made by autoclave molding. 292 Future directions Figure 8.12. Modeling of voids accounting for displacement of fibers. Reprinted, with kind permission, from Compos Sci Technol, Vol. 65, H. Huang and R. Talreja, Effects of void geometry on elastic properties of unidirectional fiber-reinforced composites, pp. 1964–81, copyright Elsevier (2005). Figure 8.13. Elastic moduli predicted by [33] compared with experimental results. Data for E x and E y are from [37] and for E z from [36]. Reprinted, with kind permission, from Compos Sci Technol, Vol. 57, C. A. Wood and W. L. Bradley, Determination of the effect of seawater on the interfacial strength of an interlayer E-glass/graphite/epoxy composite by in situ observation of transverse cracking in an environmental SEM, pp. 1033–43, copyright Elsevier (1997). 293 8.3 Cost-effective manufacturing and defect damage mechanics layered composites tend to place themselves between layers when manufactured with a compression molding process. These voids can have different shapes, sizes, and spacing in the interlaminar plane. This plane is also a plane that is prone to cracking under service or may have pre-existing flaws due to insufficient adhesion. A convenient way to make assessment of the effects of voids on interlaminar fracture is to consider the geometry used for evaluation of interlaminar fracture toughness. Ricotta et al. [38] conducted a systematic study of the voids’ influence on crack growth by considering this geometry. In the following some results from that study are discussed to illustrate the effects. Figure 8.14 shows a woven fabric composite where voids are found in the resin- rich regions between the fiber bundles. These regions are likely to develop cracks under service environment such as fatigue or fail under in-plane compression, leading to delamination. A representation of the effect of such voids on crack growth is illustrated in the figure where a double cantilever beam (DCB) specimen with voids ahead of the crack tip is shown. This geometry for mode-1 crack growth has been systematically analyzed in [38] considering various parameters such as void shape (circular and elliptical), void size, and distance of void from of the crack tip. The approach described in detail in [38] consists essentially of first validating an analytical method by finite element analysis and then using the method to conduct a parametric study of the effects of voids. The method uses a beam-on- elastic-foundation analysis accounting for shear compliance and material orthotropic symmetry. The voids are simulated as regions without support from the elastic foundation. The strain energy release rate with voids present (G I,v ) is calculated for different cases. Figure 8.15 shows the effect of a single circular void of different radius R placed at different distance D from the crack tip. The G I,v normalized by the value without void (G I ) shows increasing enhancement as the void approaches the crack tip, and this enhancement increases with increasing void radius. Figure 8.16 shows a similar effect for elliptical voids. The effects of multiple circular voids on the energy release rate are shown in Figures 8.17 and 8.18. Figure 8.17 shows the effects of two and three voids of fixed radius and fixed mutual spacing located at different distances from the crack tip. In Figure 8.18 an interesting effect of void interaction is shown. As seen there, for multiple circular voids where the nearest void is kept at a fixed distance from the crack tip, while the mutual void spacing is varied, the energy release rate does not show a monotonic dependence on the void spacing. Instead, the void interaction increases the energy release rate up to a certain void spacing, beyond which the effect of having multiple voids decreases. Finally, Figure 8.19 illustrates how the energy release rate increases with crack propagation when a void exists ahead of the crack tip. The increase of the energy release rate with crack length when no void exists is plotted for refer- ence. Thus the enhancement of the energy release rate is seen as the crack tip approaches the void. 294 Future directions Figure 8.14. Voids in resin-rich areas between bundles in a woven fabric laminate and a DCB specimen representation of crack growth in the presence of the voids. 1.3 1.2 1.1 G I , v / G I 1 0 5 10 R=0.2 mm R=0.1 mm R=0.08 mm R=0.05 mm Distance from crack tip D (mm) 15 D 20 Figure 8.15. Effects of a circular void of radius R and of distance D from crack tip on the energy release rate. 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0 2 a/b=1 a/b=1.5 a/b=2 a/b=4 b=0.1 mm D 2b 2a Distance from crack tip D (mm) 4 6 G I , v / G I Figure 8.16. Effects of an elliptical void of different aspect ratio a/b, b = 0.1 mm, and of distance D from crack tip on the energy release rate [38]. Reprinted, with kind permission, from Compos Sci Technol, Vol. 68, M. Ricotta, M. Quaresimin and R. Talreja, Mode-I strain energy release rate in composite laminates in the presence of voids, pp. 2616–23, copyright Elsevier (2008). 295 8.3 Cost-effective manufacturing and defect damage mechanics 8.4 Final remarks The analyses and methods presented in this book have been mostly directed at composite materials having continuous fiber reinforcement in individual layers that are stacked to form laminates. These material systems with polymers as matrix materials and with stiff fibers such as carbon have spurred the development of lightweight structures in the aeronautics industry. Today, new aircraft such as Boeing 787 and Airbus 380 are products of those developments. It is arguable, however, how much of the advancement in damage modeling presented in this book is embedded in the design of these aircraft. While this situation is under- standable due to the stringent and costly airworthiness certification requirements, it is hoped that eventually the output of research efforts in damage and failure will transfer to designing safer and more cost-effective structures. 1 0 1 3 voids R =0.1 mm 2 voids R =0.1 mm 1 void R =0.1 mm 2 Distance from crack tip D (mm) 3 4 1.1 1.2 1.3 G I , v / G I Figure 8.17. Effect on the energy release rate of circular voids of radius R = 0.1 mm and of 2.0 mm mutual spacing placed ahead of the crack tip for varying distance D from the crack tip. 1.11 1.10 1.09 1.08 0 2 4 3 voids R = 0.1 mm 2 voids R = 0.1 mm 6 Distance from first void c (mm) 8 10 G I , v / G I Figure 8.18. Effect on the energy release rate of multiple voids of fixed radius R = 0.1 mm of varying mutual spacing c placed at a fixed distance from the crack tip. 296 Future directions The use of composites has over the years expanded beyond the aerospace applications to other areas of structures. Carbon fiber composites have experi- enced an explosive growth in recent decades with an annual growth rate ranging from 10 to 15%. The emerging applications of composite materials in automotive and wind energy sectors place different challenges on design of these materials than what has been the case in the aerospace industry. Although the affordability of aerospace vehicles, even in the defense industry, has been a consideration, cost- effectiveness is a prime factor in design of automotive and wind turbine structures. The defect damage mechanics discussed above is bound to be an integral part of the design process for these structures in the future. Incorporating this approach in computational design methodologies will be a crucial next step. For wind energy applications the key factor is long-term durability, other than low cost. The design life of these structures is currently at 20 years (earlier it was 30 years!). For fatigue this translates to 10 million load cycles, or more. Most composites fatigue testing has traditionally been done until 10 6 cycles, motivated by metal fatigue where steels typically have a fatigue limit, which is revealed by the S-N curve flattening out before this number of load cycles. For composite mater- ials the fatigue limit is not as easily determined. As discussed in Chapter 6, considerations of damage mechanisms are necessary to deduce this property. This is a significant challenge for a highly complex composite construction in wind turbine rotor blades. A much greater challenge is to determine the fatigue life at a large number of cycles under multiaxial loading conditions typical for these structures. The field of multiaxial fatigue in composites must be given the support it deserves. A large-scale activity that is comprehensive in its approach is needed. It must include testing and evaluation at scales where damage initiates, to scales of damage progression (crack multiplication), and failure criteria that properly represent the mechanisms. The activity so far has been limited in scope and mostly focused on empirical and semi-empirical approaches. Most 540 520 500 480 G I ( J / m 2 ) 460 440 420 0 1 2 D–a Circular void, R=0.1 mm Without void Crack propagation (mm) 3 4 Figure 8.19. Increase in the energy release rate as the crack tip approaches the void is illustrated. The lower curve shows the energy release rate for comparison when no void exists. 297 8.4 Final remarks work continues to emulate metal fatigue despite fundamental differences in the underlying mechanisms [39]. This book has not specifically dealt with nanoscale reinforcements and compos- ite systems. 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Int J Fatigue, 32 (2010), 2–16. 300 Future directions Author index Aboudi 124 Adkins 149 Adolfsson 192, 208, 209 Ahci 175, 176 Akshantala 188, 270 Aoki 70 Aveston 3, 4, 49, 58, 66, 74 Awerbuch 244 Bailey 4, 49, 74, 80, 82, 84, 188, 192, 195, 196, 200 Balhi 186 Benthem 115 Berglund 97 Berthelot 74, 212, 215, 216, 217 Bowles 292 Bradley 40, 293 Broutman 49 Budiansky 44, 85, 249 Chaboche 4, 136, 137, 138 Chang 193 Cooper 49, 58 Cox 74 Crocker 186, 193 Crossman 206 Curtis 245 Daniel 79, 82 Dharan 239 Diao 276 Dvorak 76, 82, 85, 200, 201, 202 Eshelby 85 Fenner 46 Flaggs 80, 82 Frimpong 292 Fukunaga 77, 78, 79, 82, 197 Galiotis 40, 46 Gamby 124 Gamstedt 38, 247, 248, 249 Garrett 4, 49, 80, 82, 84, 186, 188, 195, 200 Grimes 262 Gudmundson 111, 113, 114, 117, 155, 208, 209 Hahn 244, 263 Halpin 19 Hashin 19, 23, 57, 87, 91, 97, 101, 104, 113, 124, 194, 202, 204, 214, 216, 259 Highsmith 77, 78, 82 Hill 85, 186 Hoiseth 86, 210 Hong 79, 82, 124 Hori 3, 84, 279 Hsu 292 Hu 85, 98, 268 Huang 292 Ishikawa 70 Jamison 261 Joffe 104, 208, 219 Johnson 193 Ju 137 Kachanov 4, 114, 117, 134, 141 Katerelos 40, 184 Kelly 4, 49, 58, 66, 74 Kim 125, 263 Koiter 115 Krajcinovic 4, 208 Kumar 172, 173, 174, 175 Kuriakose 101 Lavoie 192 Laws 76, 82, 85, 200, 201, 202 Le Corre 212, 215, 216, 217 Lee 79, 82, 124 Lemaitre 4, 136, 138 Li 119, 121, 122 Lim 79, 82 Liu 47, 190, 221 Lundmark 117, 118, 119, 125, 189, 191 Mai 276 Manders 74, 82, 84, 196, 197, 212 McCartney 104, 109, 125, 205 Mori 19, 44, 281 Murakami 136, 137 Na 124 Nairn 83, 85, 97, 98, 110, 125, 185, 187, 202, 204, 205, 221, 268 Nath 46 Nemat-Nasser 3, 84, 279 Niu 43 Nuismer 79, 80, 81, 82 Ogin 79, 82, 186 Ohno 136, 137 Ostlund 113 Pagano 124 Parvizi 49, 74, 199 Parvizi-Majidi 67 Qu 86, 210 Queresimin 295 Ravichandran 47, 190 Rebiere 124 Reddy 119, 123 Reifsnider 77, 78, 82, 188 Rice 45 Ricotta 294, 295 Robotnov 135 Rotem 259 Sahu 49 Schapery 172 Shoeppner 124 Shtrikman 19 Sicking 172 Silberschmidt 212 Silwood 58 Singh 162, 171, 218, 222, 223, 224 Sirivedin 46 Sjogren 38 Smith 186 Sørensen 253, 256 Steif 79, 82, 198 Sturgeon 244, 245 Tada 115 Talreja 4, 6, 43, 50, 101, 141, 162, 171, 172, 173, 174, 175, 188, 218, 222, 223, 224, 238, 239, 247, 250, 253, 270, 292, 295 Tan 79, 80, 81, 82 Tanaka 19, 44, 281 Timoshenko 43 Tracey 45 Tsai 19, 23, 194 Uhl 292 Vakulenko 117, 141 Varna 40, 97, 102, 104, 117, 118, 119, 125, 143, 174, 189, 191, 208, 221 Vinogradov 202, 214, 216 Wang 67, 206, 213 Weibull 196, 208, 215 Wood 40, 293 Wu 23, 194 Yaniv 79 Ye 276 Yokozeki 70, 193 Zhang 124 302 Author index Subject index ACK theory 4, 58, 199 Acoustic emission 181, 253 Characteristic damage state (CDS) 47, 48, 188 Classical laminate plate theory (CLPT) 24 COD based methods 206 Complementary energy 12, 92, 93, 94, 101, 203, 227 Computational methods 119 Computational structural analysis 276 Constitutive relations/Constitutive response/ Constitutive relationships/Constitutive equations 11–15, 20, 26, 172 Constraint factor. See Constraint parameter Constraint parameter 148, 155, 163, 165, 169, 174, 287 Constraint. See ply constraint Continuum damage mechanics (CDM) 4, 57, 134, 137, 144, 154, 161, 174, 176 Correspondence Principle 172 Cost-effective manufacturing 288 Crack density 86, 173, 192, 202, 208, 214, 266, 270 Normalized crack density 201 Crack density evolution. See damage progression Crack initiation 180, 185, 201, 273, 276, 278 Crack initiation strain 198, 205, 221 Crack opening displacement (COD) 2, 3, 32, 102, 111, 117, 155, 161, 163, 164, 174, 181, 207, 219 Crack progression. See Damage progression Crack sliding displacement (CSD) 3, 143, 219 Crack spacing 148, 195 Normalized crack spacing 203 Crack size 148 Critical energy release rate. See energy release rate: critical Cyclic loading. See Fatigue Damage 36 Damage characterization 139 Damage development. See damage progression Damage entity tensor 141 Damage evolution. See damage progression Damage evolution curve. See damage progression curve Damage initiation. See Crack initiation Damage mechanics 2, 33, 36, 48, 50, 57, 276, 280, 290 Damage mechanisms 36, 37, 142, 143, 181, 242, 278 Damage modes 45, 46, 142–143, 156, 158, 165, 167, 175, 179, 193 Damage mode tensor 142–143, 158 Damage progression 46, 139, 179, 180, 184, 187, 188, 192, 194, 196, 204, 208, 212, 215, 217, 221, 222, 229, 253, 266, 268, 270, 273, 276, 283, 284, 294 Curve 189, 190 Energy based approaches 198, 210, 277 Strength based approaches 180, 194, 210 Damage state 173 Damage tensor 151, 157 Damage tolerance 265 Debonding (interfacial) 37–38, 48, 248, 249, 251, 256, 283, 284, 287 Defect damage mechanics 287, 291, 297 Delamination 33, 39–41, 48, 183, 187, 193, 216, 225, 260, 271, 273, 287 Distortional energy (Tsai-Hill) criterion 23 Durability 1, 37, 265, 279, 287 Assessment 277 Edge replication 181 Energy release rate 31, 203, 204, 208, 294 Critical 31, 206, 213, 219 Failure 36 Failure criteria 21, 180 Fatigue 223, 265, 276 Fatigue-life 246, 271 Fatigue-life prediction 265–271 Crossply laminates 266–270, 273 General laminates 273 Fatigue-life diagrams 6–7, 237–265, 273 Polymer matrix composites (PMC) 242, 250, 251, 283 Fatigue-life diagrams (cont.) Angleply laminates 260, 261 Ceramic matrix composites (CMC) 252 Crossply laminates 261–262 Metal matrix composites (MMC) 250, 251 Multidirectional laminates 263 Quasi-isotropic laminates 263 Unidirectional composite 257 Fatigue limit 237 Fiber breakage 42, 48, 183, 239, 249, 255 Fiber failure. See fiber breakage Fiber microbuckling 42, 43 Finite element method (FEM) 117, 119, 209, 276 Finite strip method 121 Fracture 36, 48 Frictional sliding 256, 268 See also Interfacial sliding Generalized Hooke’s law 12 Generalized plain strain analysis 104, 205 Hashin criterion 23 Interfacial sliding 39, 41 Interlaminar cracking. See delamination Intralaminar cracking. See ply cracking Laminate 18 Balanced 29 Crossply 29, 285 Multidirectional 193, 217 Quasi-isotropic 29, 193 Symmetric 29 Layerwise theory 123 Length scales of damage 280 Linear elastic fracture mechanics (LEFM) 29, 184 Macro damage mechanics (MADM) 5, 51, 57, 126, 134, 276 Matrix cracking. See ply cracking Maximum stress theory 22 Maximum strain theory 22 Mechanisms of damage. See damage mechanisms Microbuckling. See Fiber microbuckling Microcracking. See ply cracking Micro damage mechanics (MIDM) 4, 51, 57, 126, 134, 276 Micromechanics 3, 17, 44, 57, 144, 161, 176 Multiple cracking 63 Multiple matrix cracking. See ply cracking Multiscale modeling 278 Hierarchical 279, 282, 286 Particle cleavage 44 Periodic boundary conditions 120 Ply constraint 49, 187 Ply cracking 39, 48, 61, 64, 170, 183, 194, 267, 283 Principle of minimum potential energy 16 Principle of minimum complementary energy 17, 91 Principle of virtual work 16, 92 Raman spectroscopy 69, 184 Randomness in ply cracking 70, 210 Randomness. See randomness in ply cracking Reference Laminate 220, 221 Representative volume element RVE 2, 5, 69, 110, 121, 134, 139–148, 161, 278, 280, 281, 282, 287 Residual stresses 96, 192, 201, 208 RVE. See representative volume element Self-consistent method. See Self-consistent scheme Self-consistent scheme 84–86 Shear lag analysis 198 Shear lag methods. See Shear lag models Shear lag models 65, 73–84, 101–104 Shear lag parameter 76–82, 103, 195, 197, 201 Shear lag. See Shear lag models Shear lag theory. See Shear lag models Stiffness changes. See Stiffness degradation, Stiffness – damage relationships Stiffness degradation 83, 85, 102, 154, 170, 193 Stiffness-damage relationships 148, 152, 157, 169, 179, 184 Stiffness reduction 192 Strain energy density 12 Strength criteria. See Failure criteria and Damage progression: strength based approaches Structural integrity 1, 37 Synergistic damage mechanics (SDM) 5, 155, 156, 164, 170, 171, 174, 176, 184, 286 Thermal expansion coefficient 125 Thermal residual stress 202, 205 Thermal stress. See Thermal residual stress Tsai-Wu criterion 23 Ultrasonic C-scan 183 Unidirectional lamina (UDL) 18 Variational analysis. See variational methods Variational methods 87, 97, 202, 226 Virtual work. See Principle of virtual work Viscoelastic composites 170 Viscoelastic response 173, 174 Viscoelasticity 170 Void growth 44 X-ray radiography 181, 261 304 Subject index