Composite Materials and Structures

P. K.Sinha Published by: Composite Centre of Excellence, AR & DB, Department of Aerospace Engineering I.I.T. Kharagpur Dedicated To: My Family Members Including Students PREFACE Natural Composites such as stones, woods and bones as structural materials, had been used by mankind since prehistoric days. There is little denying the fact that these materials not only guided the process of development of human civilisation during its early phase, but also moulded the course of history in many occasions. The twentieth century heralded myraids of man-made composite materials that are stronger, stiffer and at the same time lightweight. The advent of some of these advanced composite has totally revolutionised the concept of material development and made significant impact on the structural engineering technology, The major lead has been provided by the aerospace industry, but the present situation is such that all engineering industries are vying with each other to make the best use of composite. Whether it is a sunglass frame, a helmet, a bullet - proof vest, an artificial tooth, a canine hip joint, a hokey stick, a violin, a shell roof, a microwave antenna, a drive shaft, an automotive bumper, a submarine hull, an aircraft wing or a satellite solar panel substrate - everywhere in all structural forms there lie the imprints of composites. It is therefore obvious that in current years, composites should attract the attention of engineers and scientists and deserve to form a part of the engineering curriculum in several engineering disciplines such as the aerospace, civil, mechanical, marine and materials engineering. Currently there exists an extensive published literature, specially in the form of handbooks, monographs and conference proceedings that deal with many aspects of composites, several technical journals are regularly publishing research articles since the last 2 -3 decades. A couple of journals are also exclusively devoted to composite materials and structures. There exist a few reference books as well to aid students in following courses taught in the subject in academic institutions. But most of these published materials are beyond the reach of the general student community due to several practical reasons, Hence the major thrust of the present effort is to develop a text book for engineering students in final year undergraduate and postgraduate levels in several engineering disciplines with the principal focus on materials and structures, Materials structures are very closely related - one cannot conceive a structure without a material to give the former a form to resist forces, Besides, composite materials pose special problems. The design of a composite material always precedes that of a composite structure. A composite material can be effectively tailored to suit a particular design need. It is obligatory that a composite structural designer should have a thorough knowledge about the process and development of composite materials and their characteristic behaviour. In the present book an equal emphasis is therefore directed to provide due coverage of both materials and structures. The contents are accordingly organised to provide a smooth transition from acquiring the basic knowledge of composite materials to deeper understanding of design and analysis of composite structural components. The book is structured primarily keeping engineering students in mind, but it will also be a useful reference and guide to practicing engineers as well as to those who do not necessarily desire to become specialists in the field. It is worth pointing that the book should also be of considerable interest to researchers as several new results and advanced ideas are included, which may deserve further work. The interest of Allied Publishers Limited for publishing of the present book is greatly appreciated. The author is indebted to his current research students, P.K. Aditya, D.K. Maiti, B Maruthi Rao and T V R Chowdary for providing computational and other supports. The author gratefully acknowledges the support from Dr. K.G. Narayanan, Mr. M. Ramakrishnan, Dr. A.K. Rath, Dr. K.G. Satyanarayana, Dr.Baldev Raj, Dr. Y.R. Mahajan and Dr. P.D. Mangalgiri by providing a couple of interesting photograps, Thanks are due to Mr. A. Choudhury for carefully typing the manuscript and to Mr. S. Mukhopadhyay for neatly drafting the illustrations. Finally, the author is appreciative of the partial financial support received from the CEP/QIP, IIT, Kharagpur for preparation of this manuscript. June 4, 1995 Prof. P.K. Sinha Department of Aerospace Engineering I.I.T., Kharagpur - 721302 The manuscript of the Book, Composite Materials and Structures was written by my Husband (Late) Prof. P. K. Sinha for possible publication. Due to several unavoidable reasons, he could not publish the book during his life time. His intention of writing this book was to help the student community and researchers in the field of composite materials and structures. He expressed his wish during his last visit to the department to publish the book through departmental home page as e-book so that students and researchers can access the book easily. Considering his noble wish, I requested Head of the Department, Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur to pursue the matter and release the book as e-book from the webpage of the Department and accordingly department got the necessary approval from the higher authority. I gratefully acknowledge the effort made by Head of the Department, Department of Aerospace Engineering to publish the book through webpage of the department. The manuscript of the book was initially typed in Chi-writer and later it was converted to Microsoft word. The effort of Dr. V. V. Subba Rao is gratefully acknowledged to upgrade the version of the manuscript to Microsoft word. The correction made by Dr. N. V. S. Naidu in the manuscript dictated by (Late) Prof. Sinha is highly appreciated. I am indebted to Dr. D. K. Maiti to pursue the matter and Mr. K. Bera to convert the manuscript to web form. I also gratefully acknowledge all those who directly or indirectly helped the author to bring the book in this form. The corrections as well as suggestions for improving the content will be highly appreciated. This is the only one book written by (Late) Prof. P. K. Sinha. July 5, 2006 Mrs. Anuradha Sinha Wife of (Late) Prof. P. K. Sinha Department of Aerospace Engineering I.I.T., Kharagpur - 721302 CHAPTER-1 INTRODUCTION 1.1 NATURAL AND MAN-MADE COMPOSITES 1.2 AEROSPACE APPLICATIONS 1.3 OTHER STRUCTURAL APPLICATIONS 1.3.1 Civil Engineering 1.3.2 Automotive Engineering 1.4 OTHER APPLICATIONS 1.5BIBLIGRAPHY 1.6 EXERCISE 1.1 NATURAL AND MAN-MADE COMPOSITES A composite is a material that is formed by combining two or more materials to achieve some superior properties. Almost all the materials which we see around us are composites. Some of them like woods, bones, stones, etc. are natural composites, as they are either grown in nature or developed by natural processes. Wood is a fibrous material consisting of thread-like hollow elongated organic cellulose that normally constitutes about 60-70% of wood of which approximately 30-40% is crystalline, insoluble in water, and the rest is amorphous and soluble in water. Cellulose fibres are flexible but possess high strength. The more closely packed cellulose provides higher density and higher strength. The walls of these hollow elongated cells are the primary load-bearing components of trees and plants. When the trees and plants are live, the load acting on a particular portion (e.g., a branch) directly influences the growth of cellulose in the cell walls located there and thereby reinforces that part of the branch, which experiences more forces. This self-strengthening mechanism is something unique that can also be observed in the case of live bones. Bones contain short and soft collagen fibres i.e., inorganic calcium carbonate fibres dispersed in a mineral matrix called apatite. The fibres usually grow and get oriented in the direction of load. Human and animal skeletons are the basic structural frameworks that support various types of static and dynamic loads. Tooth is a special type of bone consisting of a flexible core and the hard enamel surface. The compressive strength of tooth varies through the thickness. The outer enamel is the strongest with ultimate compressive strength as high as 700MPa. Tooth seems to have piezoelectric properties i.e., reinforcing cells are formed with the application of pressure. The most remarkable features of woods and bones are that the low density, strong and stiff fibres are embedded in a low density matrix resulting in a strong, stiff and lightweight composite (Table 1.1). It is therefore no wonder that early development of aero-planes should make use of woods as one of the primary structural materials, and about two hundred million years ago, huge flying amphibians, pterendons and pterosaurs, with wing spans of 8-15 m , could soar from the mountains like the present?day hang- gliders. Woods and bones in many respect, may be considered to be predecessors to modern man-made composites. Early men used rocks, woods and bones effectively in their struggle for existence against natural and various kinds of other forces. The primitive people utilized these materials to make weapons, tools and many utility-articles and also to build shelters. In the early stages they mainly utilized these materials in their original form. They gradually learnt to use them in a more efficient way by cutting and shaping them to more useful forms. Later on they utilized several other materials such as vegetable fibres, shells, clays as well as horns, teeth, skins and sinews of animals. Table 1.1 Typical mechanical properties of natural fibres and natural composites Materials Density Tensile modulus Tensile strength Kg/m 3 GPa MPa Fibres Cotton 1540 1.1 400 Flax 1550 1 780 Jute 850 35 600 Coir 1150 4 200 Pineapple leaf 1440 65 1200 Sisal 810 46 700 Banana 1350 15 650 Asbestos 3200 186 5860 Composites Bone 1870 28 140 Ivory 1850 17.5 220 Balsa 130 3.5 24 Spruce 470 11 90 Birch 650 16.5 137 Oak 690 13 90 Bamboo 900 20.6 193 Woods, stones and clays formed the primary structural materials for building shelters. Natural fibres like straws from grass plants and fibrous leaves were used as roofing materials. Stone axes, daggers, spears with wooden handles, wooden bows, fishing nets woven with vegetable fibers, jewelleries and decorative articles made out of horns, bones, teeth, semiprecious stones, minerals, etc. were but a few examples that illustrate how mankind, in early days, made use of those materials. The limitations experienced in using these materials led to search for better materials to obtain a more efficient material with better properties. This, in turn, laid the foundation for development of man-made composite materials. The most striking example of an early man-made composite is the straw- reinforced clay which molded the civilization since prehistoric times. Egyptians, several hundred years B.C., were known to reinforce the clay like deposits of the Nile Valley with grass plant fibres to make sun baked mud bricks that were used in making temple walls, tombs and houses. The watchtowers of the far western Great Wall of China were supposed to have been built with straw-reinforced bricks during the Han Dynasty (about 200 years B.C.). The natural fibre reinforced clay, even to-day continues to be one of the primary housing materials in the rural sectors of many third world countries. The other classic examples are the laminated wood furniture used by early Egyptians (1500 B.C.), in which high quality wood veneers are bonded to the surfaces of cheaper woods. The origin of paper which made use of plant fibres can be traced back to China (108 A.D.). The bows used by the warriors under the Mongolian Chief Djingiz Chan (1200 A.D.) were believed to be made with the adhesive bonded laminated composite consisting of buffalo or anti-lope horns, wood, silk and ox-neck tendons. These laminated composite bows could deliver arrows with an effective shoot in range of about 740 m. Potteries and hydraulic cement mortars are some of the earliest examples of ceramic composites. The cloissone ware of ancient China is also a striking example of wire reinforced ceramics. Fine metallic wires were first shaped into attractive designs which were then covered with colored clays and baked. In subsequent years, fine metallic wires of various types were cast with different metal and ceramic matrices and were utilized in diverse applications. Several other matrix materials such as natural gums and resins, rubbers, bitumen, shellac, etc. were also popular. Naturally occurring fibres such as those from plants (cotton, flux, hemp, etc.), animals (wool, fur and silk) and minerals (asbestos) were in much demand. The high value textiles woven with fine gold and silver threads received the patronage from the royalty and the rich all over the world. The intricate, artful gold thread embroidery reached its zenith during the Mughal period in the Indian subcontinent. The glass fibres were manufactured more than 2000 years ago in Rome and Mesopotamia and were abundantly used in decoration of flower vases and glass wares in those days. The twentieth century has noticed the birth and proliferation of a whole gamut of new materials that have further consolidated the foundation of modern composites. Numerous synthetic resins, metallic alloys and ceramic matrices with superior physical, thermal and mechanical properties have been developed. Fibres of very small diameter (<10?m) have been drawn from almost all materials. They are much stronger and stiffer than the same material in bulk form. The strength and stiffness properties have been found to increase dramatically, when whiskers (i.e., single crystal fibers) are grown from some of these materials. Figure1.1 illustrates the specific tensile strength and the specific tensile modulus properties are obtained by dividing the strength (M Pa) and modulus (G Pa) by either the density (kg m -3 ) or the specific gravity of the material. Because of the superior mechanical properties of fibers, the use of fibers as reinforcements started gaining momentum during the twentieth century. The aerospace industries took the lead in using fiber reinforced laminated plastics to replace several metallic parts. The fibres like glass, carbon, boron and Kevlar, and plastics such as phenolics, epoxies and polyesters caught the imagination of composite designers. One major advantage of using fibre reinforced plastics (FRP) instead of metals is that they invariably lead to a weight efficient design in view of their higher specific modulus and strength properties (Table 1.2). Composites, due to their heterogeneous composition, provide unlimited possibilities of deriving any characteristic material behavior. This unique flexibility in design tailoring plus other attributes like ease of manufacturing, especially molding to any shape with polymer composites, repairability, corrosion resistance, durability, adaptability, cost effectiveness, etc. have attracted the attention of many users in several engineering and other disciplines. Every industry is now vying with each other to make the best use of composites. One can now notice the application of composites in many disciplines starting from sports goods to space vehicles. This worldwide interest during the last four decades has led to the prolific advancement in the field of composite materials and structures. Several high performance polymers have now been developed. Substantial progress has been made in the development of stronger and stiffer fibres, metal and ceramic matrix composites, manufacturing and machining processes, quality control and nondestructive evaluation techniques, test methods as well as design and analysis methodology. The modern man-made composites have now firmly established as the future material and are destined to dominate the material scenario right through the twenty-first century. Table 1.2 Comparative mechanical properties of some man-made structural composites and metallic alloys Materials Specific Tensile Tensile Compressive Specific Specific Specific gravity modulus strength strength tensile tensile compressive modulus strength strength S E X t X c E/S X t /S X c /S G Pa M Pa M Pa G Pa M Pa M Pa Unidirectional Fibre Reinforced Plastics GFRP 2.0 40 1650 1400 20.00 825.0 700.0 CFRP 1.6 140 1450 1050 87.50 906.3 656.3 KFRP 1.5 90 1650 300 60.00 1100.0 200.0 Metals Steel 7.8 206 400-2500 400-2500 26.40 50-320 50- 320 Ti alloy 4.5 103 360-1400 360-1400 22.90 80-310 80- 310 Al alloy 2.8 69 55-700 55-700 24.60 20-250 20- 250 Mg alloy 1.8 47 150-300 150-300 25.00 83-166 83- 166 Beryllium 1.8 303 400 400 168-33 222 222 1.2 AEROSPACE APPLICATIONS One of the primary requirements of aerospace structural materials is that they should have low density and, at the same time, should be very stiff and strong. Early biplanes used wood for structural frameworks and fabrics for wing surfaces. The fuselage of World War I biplane fighter named Vieux Charles was built with wire braced wood framework. The monoplane, Le Monocoque, had an unusually smooth aerodynamic design. Its fuselage was made with laminated tulip wood, where one layer was placed along the length of the fuselage, the second in a right-hand spiral and the third in a left hand spiral around the fuselage. This laminated single shell wood construction provided highly polished, smooth surfaces. There was a significant reduction in the drag, and the plane could achieve a high speed of 108 mph. It won the Gordon Bennett speed race in Chicago in 1912. Almost all biplanes and monoplanes, with very few exceptions, were built of wood during the first quarter of the twentieth century. Lighter woods like balsa, poplar, spruce, tulip, etc. were more popular. The five-seater Lockheed Vega (first flight in 1927) also had highly polished, smooth, streamlined fuselage made of strips of spruce wood bonded together with resin. The Vegas were considered to be the precursor of the modern transport airplane and had the distinction of successfully completing many major flights such as crossing the American continent non-stop from Los Angels to New York, over-flying the Atlantic, encircling the globe and succeeding in several other long distant flights and races. Soaring planes, in those days, also had highly polished thin plywood fuselages. The thirties and forties noticed a gradual shift from wood to aluminium alloy construction. With the increase in the size and speed of airplanes, the strength and stiffness requirements for a given weight could not be met from wooden construction. Several new structural features, e.g., skin-stringer construction, shear webs, etc. were introduced. The aerospace grade aluminium alloys were made available. Two important airplanes Northrop Alpha and Boeing Monomail, which were forerunners in the development of several other aircraft, had aluminium alloy monocoque fuselage and a wood wing. These aircraft were introduced in 1930, although they were not the first to use metals. The switch over to light aluminium alloys in aircraft construction was no doubt, a major step in search for a lightweight design. The trend continued till fifties, by which almost all types of airplanes were of all-metal design. However, the limitations of aluminium alloys could be assessed as early as fifties with the speed of the aircraft increasing sharply (significantly more than the speed of sound), the demand for a more weight optimized performance, the fuel-efficient design and so on. The aluminium was stretched to its maximum limit. The search for newer and better materials was the only alternative. Continuous glass fibres, which were commercially available since thirties, are found to be very strong, durable, creaseless, non-flamable and insensitive to weathering. The glass fibres coated with resin can be easily moulded to any complex curved shape, especially that of a wing root and fuselage intersection and can be laid layer wise with fibres aligned in a desired direction as in the case of the three-layered wood fuselage of Le Monocoque. Fibreglass fabrics were successfully used in a series of Todai LBS gliders in Japan during the mid-fifties. Todai LBS-1 had spoilers made from fiberglass fabrics. Todai LBS-2 had a wood wing and a sandwich monocoque fuselage whose wall consisted of a balsa wood core sandwiched between two glass fibre reinforced composite face skins. The wing skin of another important glider, the Phoenix (first flight in 1957), developed in Germany was a sandwich with fiberglass-polyester faces and balsa wood core. The other successful glider SB-6, first flown in 1961, had a glass fibre-epoxy shell and a glass fibre composite-balsa sandwich box spar. The remarkable feature of all these gliders is that they exhibited superior flight performance and thus became the trend-setters in the use of glass fibre reinforced plastics. Glass fibres are strong, but not stiff enough to use them in high sped aircraft. The search for stiffer fibres to make fibre reinforced composite started in the fifties in several countries. The laboratory scale production of high-strength carbon fibres by Royal Aircraft Establishment, Farnborough, U.K. was reported in 1952. In USA, Union Carbide developed high-modulus continuous carbon fibres in 1958. High-strength graphite fibres were developed at the Government Industrial Research Institute of Osaka, Japan in 1959. Before the end of sixties the commercial production of carbon fibres (PAN based) started in full scale. Very high modulus boron fibres were also introduced during this time. High strength, low density organic fibres, Kevlar 49, were also marketed by Dupont, USA during early seventies. A host of synthetic resins, especially structural grade epoxy resins, were also commercially available. All these advanced materials provided the much- needed alternatives to less efficient aluminium alloy and fiberglass composites. The switch-over from the aluminium and GFRP to advanced composites in airframe construction was, however, very slow at the initial stage (Fig. 1.2). It started with the F- 14 fighter and the F-111 fighter bomber around 1972, but in the period of about two and a half decades, there were quite a few airplanes, in which almost all structures are made of composites (Table 1.3). Similar trend in material uses can be observed in the development of helicopters as well. As early as 1959-60, the Vetrol Company, now Boeing Helicopters, developed helicopter rotor blades with glass-epoxy faces and aluminium honeycomb core. In course of time several structural parts such as horizontal stabilizer, vertical pylon, tail cone, canopies, fuselage, floor board, rotor hub and landing gears were developed with various composites, which later culminated in the development of the all composite helicopter, Boeing Model 360 which was flight tested in 1987. The manufacturers of passenger aircraft soon realized the significance of using composites in the airframe structure. CFRP, KFRP and hybrid composites were extensively used throughout the wing, fuselage and tailplane sections of Boeing 767. Although the application of composites in civilian aircraft is relatively less, this trend is likely to change by the turn of the twenty-first century with the introduction of supersonic civil transports. Some of the future transport planes may fly at a very high speed (Mach 2-5). Significant advancement has been made in several high technology areas such as supersonic V/STOL flights, lightweight air superiority fighters with thrust vectoring, supersonic interceptors and bombers with high Mach number, advanced lightweight helicopters with tilt rotors, aerospace planes and hypersonic vehicles with multi-mode trans atmospheric cruise capability such as take-off and landing with a turbine engine, accelerating to Mach number 10-12 with a Scramjet and achieving an orbital velocity with a rocket engine. The composite materials will provide an increased number of choices to meet the tight weight budget and the critical performance level for all these advanced flight vehicles. Table 1.3 Advanced composites in selected aerospace applications Vehicles Components Composites Sailplanes SB-10 Middle portion of the wing CFRP SB-11, SB-12 Ventus, Nimbus, AS-W22 All composite CFRP Aeroplanes F-14 Stabilator BFRP F-15, F-16 Horizontal and Vertical tail skins BFRP Speed brakes CFRP A-4 Flap, Stabilator CFRP F-5 Leading edge CFRP Vulcan Airbrakes CFRP Mirage 2000 Rudder Boron/carbon/epoxy AV-8B Wing skin, Control surfaces, Front fuselage CFRP Rafale Wing structure CFRP Boeing 757 Control surface, Cowlings, Under &767 carriage doors, Fairings CFRP A 310-300 Fin box CFRP Lear Fan 2100 All composite CFRP Voyager All composite CFRP Starship All composite CFRP Airbus, Concorde, Delta 2000, Brake discs Carbon/carbon Falcon 900 DC-10 Aft pylon Boron/aluminium C-5A Wing box SiC/aluminium F-111 Fuselge segment Boron/aluminium Rockets and Space Vehicles Tactical Nose cone, Inlet Quartz/polyimide Missiles fairing, Fins Carbon/polyimide Polaris, Minuteman, Rocket cases KFRP, GFRP Poseidon, Trident Tomahawk Shaft for turbofan Borosic/titanium PSLV Upper stage solid motor case KFRP ARIANE Dual-launch structures, Fairings CFRP Hubble Space High-gain antenna boom Graphite/6061 Al Telescope INTELSAT Antennas, CFRP & KFRP Antenna support structure, Multiplexers, Solar array wings CFRP Viking Parabolic antenna reflector CFRP sandwich Boom CFRP Voyager Parabolic reflector, Subreflector CFRP support structure Dichroic subreflector KFRP INSAT, ARABSAT, ITALSAT Antenne reflectors CFRP OLYMPUS EUTELSAT Dual-grided reflectors CFRP & KFRP TDF-1 Solar array wing CFRP EURECA Micro-gravity spacecraft platform truss structure CFRP Space shuttle Main frame and rib-truss struts, Frame stabilizing braces, Nose landing Boron/aluminium gear and drag-brace struts The materials for the next-generation aeroengines will go a sea-through change in view of much hotter running engines to increase the thermal efficiency and enhance the thurst-to-weight ratio. It is envisaged that, for the future military aircraft, the thrust-to- weight ratio will double, while the fuel consumption will reduce by 50%. Metal-matrix composites (MMCs) and ceramic-matrix composites (CMCs), which are thermally stable and can withstand loads at high temperatures (Fig.1.3) will be of immense use in such applications. Carbon-carbon composites, which are ceramic composites can withstand load beyond 2000 0 C. The use of these advanced materials in aeroengines is likely to pick up in the first decade of the twentifirst century (Fig.1.4). Fan blades, compressor blades, vanes and shafts of several aeroengines are now either employing or contemplating to use in the near future metal matrix composites with boron, borsic, boron carbide, silicon carbides or tungsten fibres and aluminium, titanium, nickel and super alloy (e.g. NiCrAly or FeCrAly) matrices. S-glass, quartz And carbon fibre reinforced polyimides have recently been used on radomes and fins operating at high temperatures for short and long duration, because polyimides have high temperature strength retention properties compared to epoxies and phenolics. Carbon- carbon composites have been successfully employed in the brake discs of aircraft, rocket nozzles and several other components operating in extreme thermal environments. The material menu for rockets, missiles, satellite launch vehicles, satellites and other space vehicles is quite extensive and diverse. The trend is to design some of the upper stage structural components like payload structures, satellite frame works and central cylindrical shells, solar panel wings, solar booms, antennas, optical structures, thermal shields, fairings, motor cases and nozzles, propellant tanks, pressure vessels, etc. with composite materials to derive the maximum weight benefit. All space vehicles of recent origin have several composite structural systems. CFRP is the obvious choice because of its excellent thermo-mechanical properties, i.e., high specific stiffness and strength, higher thermal conductivity and lower coefficient of thermal expansion. The future large space stations are likely to be built with CFRP. Although BFRP has several positive features, it is mainly used for stiffening purposes. Both GFRP and KFRP are favoured for design of pressurized systems for their superiority in strength and cost- effectiveness. Beryllium, although not a composite, possesses highly favourable properties (Table 1.2) but it is sparingly used due to safety hazards, especially during fabrication. The examples of space applications of composites are too many. One of the early major application is the graphite-epoxy mesh grid off-set parabolic antenna reflector developed by Hughes Aircraft Company for the Canadian ANIK satellite which was launched in 1972. The European Remote Sensing Satellite ERS-I has several composite parts plus a large 10m long metallised graphite-epoxy radar antenna array. The Voyager spacecraft contains a large 3.7m diameter CFRP parabolic antenna reflector. The fairing of ARIANE 4 is a graphite composite stiffened shell structure of maximum 4m diameter and 8.6m height. A few other typical examples are listed earlier in Table 1.3. The composite application in the aerospace industries is a process of continuous development in which newer and more improved material systems are being utilized to meet the critical design and flight worthiness requirements. 1.3 OTHER STRUCTURAL APPLICATIONS 1.3.1 Civil Engineering The interest in the use of glass fibre reinforced polyesters in building structures started as early as sixties. The beautiful GFRP dome structure in Benghajj was constructed in 1968. The other inspiring example is the GFRP roof structure of Dubai Airport. This was built in 1972 and is comprised of clustered umbrella like hyperbolic paraboloids. Several GFRP shell structures were erected during seventies. Another striking example is the dome complex at Sharajah International Airport, which was constructed during early eighties. The primary advantage of using composites in shell structure is that any complex shell shape, either synclastic, anticlastic or combination of both, which is of architectural significance and aesthetic value, can be easily fabricated. The composite folded plate system and skeletal structures also became popular. The roof of Covent Garden Flower Market at Nine Elms, London covering an area of 1ha is an interesting example which was based on a modular construction. In this, pyramidal square modules were connected at their apices and bases to two-way skeletal grids. The modular construction technique helps to build a large roof structure which is normally encountered in the design of community halls, sports complexes, marketing centres, swimming pools, factory sheds, etc. Several other applications, where GFRP has been successfully used, include movable prefabricated houses, exterior wall panels, partition walls, canopies, stair cases and ladders, water tanks, pipes and drainages and led to its wide use in radomes and antenna towers. In one particular construction, the top 100 ft of a radar microwave link tower was built with GFRP and the guys were Kevlar fibres (also radio transparent) to reduce unwanted disturbances in air traffic control radar signals. Considering the future prospects of composites in civil structural application, ASCE Structural Plastics Research Council, as early as seventies endeavoured to develop design methods for structural plastics, both reinforced and unreinforced. However, the major deterrent for the popularity of composites in civil engineering structures is the material cost. But, in many applications, GFRP and KFRP may be cheaper considering the cumulative cost. The low structural weight will have direct bearing in lowering the cost of supporting skeletal structures and foundation. Moreover, ease of fabrication and erection, low handling and transportation cost, less wear and corrosion, simpler maintenance and repairing procedures, non-magnetic properties, integrity and durability as well as modular construction will cumulatively reduce the cost in the long run. The Living Environment house, developed by GE plastics in 1989, is an illustrative example of the multipurpose use of composites in a building. 1.3.2 Automotive Engineering Feasibility studies were carried out, since early seventies, to explore the possibilities of using composites in the exterior body panels, frameworks/chassis, bumpers, drive shafts, suspension systems, wheels, steering wheel columns and instrument panels of automotive vehicles. Ford Motor Co. experimented with the design and development of a composite rear floor pan for an Escort model using three different composites: a vinyl-ester-based SMC and XMC and a glass fibre reinforced prolypropylene sheet material. Analytical studies, static and dynamic tests, durability tests and noise tests demonstrated the feasibility of design and development of a highly curved composite automotive part. A composite GM heavy truck frame, developed by the Convair Division of General Dynamics in 1979, using graphite and Kevlar fibres (2:1 by parts) and epoxy resin (32% by wt) not only performed satisfactorily but reduce the weight by 62% in comparison to steel for the same strength and stiffness. The hybrid glass/carbon fibre composite drive shafts, introduced around 1982 in Mazdas, provided more weight savings, lower maintenance cost, reduced level of noise and vibration and higher efficiency compared to their metal counterparts. The more recent pickup truck GMT-400 (1988 model) carries a composite driveshaft that is pultruded around a 0.2cm thick and 10cm diameter aluminium tube. The composite driver shaft is 60% lighter than the original steel shaft and possesses superior dampening and torsional properties. Chevrolet Corvette models carry filament wound composite leaf springs (monoleaf) in both rear suspension (1081) and front suspension (1984). These springs were later introduced during 1985 on the GM Chevrolet Astro van and Safari van. Fibre glass reinforced polypropylene bumper beams were introduced on Chevrolet Corvette Ford and GM passenger cars (1987 models). Other important applications of composites were the rear axle for Volkswagen Auto-2000, Filament wound steering wheels for Audi models and composite wheels of Pontiac sports cars. Composites are recognized as the most appropriate materials for the corrosion resistant, lightweight, fast and fuel efficient modern automobiles, for which aerodynamics constitute the primary design considerations. All major automotive components like space frames, exterior and interior body panels, instrument panel assemblies, power plants, power trains, drive trains, brake and steering systems, etc. are now being fabricated with a wide variety of composites that include polymer, metal and ceramic matrix composites. The latter two composites will be of significance in heated engine components and brake pads. The pistons and connecting rods of modern diesel and IC engines are invariably made of composites with alumina fibres and aluminium or magnesium alloy matrices. The Ford`s probe V concept car is a classical example of multiple applications of composites in an automobile car. The present trend is to use composites even in the design of large size tankers, trailers, delivery vans and passenger vehicles. 1.4 OTHER APPLICATIONS Strong, stiff and light composites are also very attractive materials for marine applications. GFRPs are being used for the last 3-4 decades to build canoes, yatchs, speed boats and other workboats. The hull of a modern racing yatch, New Zealand, is of sandwich construction with CFRP faces. There is currently a growing interest to use composites, in a much larger scale, in ship industries. A new cabin construction material that is being tried in the Statendam-class ship building is a metallic honeycomb sandwich with resin-coated facing, that may lead to substantial weight saving. The Ulstein water jet has a long moulded inlet tract for better control of dimensional accuracy. The carbon/aluminium composite has been used for struts and foils of hydrofoils, and the silicon carbide/aluminium composite has been employed in pressure hulls and torpedo structures. The composites are also being increasingly used in the railway transportation systems to build lighter bogeys and compartments. The other important area of application of composites is concerned with fabrication of energy related devices such as wind-mill rotor blades and flywheels. The light artificial limbs and external bracing systems made of CFRP provide the required strength, stiffness and stability in addition to lightness. Carbon fibres are medically biocompatible. Composites made with carbon fibres and biocompatible metals and polymers have been found to be suitable for a number of applications in orthopaedics. A carbon?carbon composite hip joint with an aluminium oxide head has performed satisfactorily. Matrices such as polyethylene, polysulfone and polyaryletherketone reinforced with carbon fibres are also being used to produce orthopaedic implants. Composites also have extensive uses in electrical and electronic systems. The performance characteristics of CFRP antennas are excellent due to very low surface distortion. Composite antenna dishes are much lighter compared to metallic dishes. Leadless ceramic chip carriers are reinforced with Kevlar or Kevlar-glass coweave polyimides to reduce the incidence of solder joint microcracking due to stresses induced by thermal cycling. The stress level is reduced by matching the low coefficient of thermal expansion of ceramic chip carriers with that of tailored composites. Composites are, now-a-days, preferred to other materials in fabrication of several important sports accessories. A light CFRP golf shaft gives the optimum flexural and torsional strength and stiffness properties in terms of accuracy and the distance travelled by the ball. All graphite and graphite hybrid composite archery bows and arrows enhance arrow speed with a flattened trajectory and increased efficiency. The reduction in weight of a CFRP bobsleigh permits ballast to be added in the nose of the sleigh and thereby improves the aerodynamic characteristics due to the change in the position of the centre of gravity with respect to the centre of aerodynamic pressure. There are several other interesting composite leisure time items such as skis, tennis and badminton rackets, fishing rods, vaulting poles, hockey sticks, surf boards, and the list is likely to be endless in the twenty-first century. The day is not far when common utility goods will be made with composites. A few such examples are illustrated in Figs.1.5. 1.5 BIBLIGRAPHY 1. N.J.Hoff, Innovation in Aircraft Structures-Fifty years Ago and Today, AIAA Paper No. 84-0840,1984. 2. R.J. Schliekelmann, A Soft and Hard Future - A Look into Past and Future Developments of Structural Materials, AIAA International Annual Meeting on Global Technology 2000, Baltimore, 1980. 3. S.M. Lee (Ed.), International Encyclopedia of Composites, Vols.1-6, VCH Publishers, New York 1990-1991. 4. J.W. Weeton, D.M. Peters and K.L. Thomas (Eds.) Engineer`s Guide to Composite Materials, American Soceity of Metals, Metals Park, Ohio,1987. 1.6 EXERCISE 1. What are the special features of a structural composite? Compare between natural and man-made structural composites. 2. Why composites are favoured in engineering applications? Write a brief note on their uses in various engineering disciplines. CHAPTER-2 COMPOSITE MATERIALS 2.1 INTRODUCTION 2.2 REINFORCEMENTS 2.2.1 Fibres Glass Fibres Silica Fibres Boron Fibres Silicon Carbide and Boron Carbide Fibres Alumina Fibres Carbon Fibres Aramid Fibres 2.2.2 Particulates 2.3 POLYMERS AND POLYMER COMPOSITES 2.3.1 Thermoplastics 2.3.2 Thermosets 2.4 METALS AND METAL MATRIX COMPOSITES 2.5 CERAMIC MATRIX COMPOSITES 2.6 LAMINATE DESIGNATION 2.7 BIBLIOGRAPHY 2.8 EXERCISES 2.1 INTRODUCTION From this chapter onwards we restrict our attention primarily to man-made modern composites that are used in structural applications. The main constituents of structural composites are the reinforcements and the matrix. The reinforcements, which are stronger and stiffer, are dispersed in a comparatively less strong and stiff matrix material. The reinforcements share the major load and in some cases, especially when a composite consists of fibre reinforcements dispersed in a weak matrix (e.g., carbon/epoxy composite), the fibres carry almost all the load. The strength and stiffness of such composites are, therefore, controlled by the strength and stiffness of constituent fibres. The matrix also shares the load when there is not much difference between the strength and stiffness properties of reinforcements and matrices (e.g., SiC/Titanium composite). However, the primary task of a matrix is to act as a medium of load transfer between one reinforcement to the other. It also holds the reinforcements together. In that regard, the matrix plays a very vital role. Besides, the matrix may considerably influence the hygral, thermal, electrical, magnetic and several other properties of a composite. For example, to obtain a good conducting composite with SiC fibres one may choose an aluminium matrix rather than a titanium matrix. It may be noted that both the SiC fibres and the titanium matrix possess very poor thermal conductivities. The classifications of composites are commonly based on either the forms of reinforcements or the matrices used. There are two major forms of reinforcements: fibres (including whiskers) and particles (having various shapes and sizes). Accordingly, there are two broad classes of composites ? fibre reinforced composites and particle reinforced composites (or simply particulate composites). On the other hand, there are three important groups of matrices, namely, polymers, metals (and their alloys) and ceramics. The composites made using these matrices are classifies as polymer matrix composites (or polymer composites), metal matrix composites and ceramic matrix composites. Composites are also grouped in several other ways. One important class of composites is termed as laminar composites. They are also called laminated composites or laminates. A laminate usually consists of two or more layers of planar composites in which each layer (also called lamina or ply) may be of the same or different materials. Similarly, a sandwich laminate is a composite construction in which a metallic or composite core layer is sandwiched between two metallic or composite face layers. The composite face layers may also be in the form of laminates. Laminated and sandwich composite structures are very strong and stiff, and are commonly recommended for lightweight structural applications. 2.2 REINFORCEMENTS 2.2.1 Fibres Fibres constitute the main bulk of reinforcements that are used in making structural composites. A fibre is defined as a material that has the minimum 1/d ratio equal to 10:1, where 1 is the length of the fibre and d is its minimum lateral dimension. The lateral dimension d (which is the diameter in the case of a circular fibre) is assumed to be less than 254 ?m. The diameter of fibres used in structural composites normally varies from 5?m to 140?m. A filament is a continuous fibre with the l/d ratio equal to infinity. A whisker is a single crystal, but has the form of a fibre. Common low density fibres are manufactured from lighter materials especially those based on elements with low atomic number (e.g., H, Be, B, C, N, O, Al, Si, etc.). The cross-section of a fibre may be circular, for example as in the cases of glass, boron and Kevlar fibres, but some fibres may have regular prismatic cross-sections (e.g., whiskers) or arbitrary cross-sections (e.g., PAN, rayon and special pitch based carbon fibres). The irregularity in the cross-section may introduce anisotropy in the fibre. The typical microstructural morphology of common fibres are shown in Fig.2.1. From the micro-structure point of view, fibres can be either amorphous (glass), polycrystalline (carbon, boron, alumina, etc.) or single crystals (silicon carbide, alumina, beryllium and other whiskers). The strength and stiffness properties of a fibre are significantly higher compared to those of the bulk material from which the fibre is formed. Most of the common fibres are brittle in nature. The tensile strength of bulk brittle material is considerably lower than the theoretical strength, as it is controlled by the shape and size of a flaw that the bulk material may contain. As the diameter of a fibre is very small, a flaw, it may contain, must be smaller than the fibre diameter. The smaller flaw size, in turn, reduces the criticality of the flaw and thereby the tensile strength is enhanced. For example, the tensile strength of an ordinary glass (bulk) may be as low as 100-200 MPa, but that of a S-glass fibre may be as high as 5000 MPa. However, the tensile strength of a perfect glass fibre, based on intermolecular forces, is 10350 MPa. Further, the orientation of crystallites along the fibre direction also helps considerably in improving the strength properties. A whisker, being a single crystal, is not prone to crystal defects unlike polycrystalline fibres and provides very high strength and stiffness properties. The tensile strength and tensile modulus of a graphite whisker are as high as 25000 MPa and 1050 GPa, respectively. These values are quite significant compared to those of commercial fibres. The typical longitudinal tensile properties of a commercially available PAN based T300 fibre are 2415 MPa (strength) and 220 GPa (modulus). Typical thermomechanical and thermal properties of common fibres are listed in Tables 2.1 and 2.2, respectively. Both inorganic and organic fibres are used in making structural composites. Inorganic fibres (including ceramic fibres) such as glass, boron, carbon, silicon carbide, silica, alumina, etc. are most commonly used. The structural grade organic fibres are comparatively very few in number. Aramid fibres are the most popular organic fibres. Another recent addition is a high strength polyethylene fibre (Spectra 900) which has a very low density and excellent impact resistant properties. The carbon fibres may also be grouped with organic fibres, although they are more often considered as ceramic (inorganic) fibres. Inorganic fibres in general are strong, stiff, thermally stable and insensitive to moisture. They exhibit good fatigue resistant properties, but low energy absorption characteristics. Organic fibres, on the other hand, are cheaper, lighter and more flexible. They possess high strength and better impact resistant properties. Table 2.1 Typical mechanical properties of selected fibres Fibre Material Density kg/m 3 Tensile strength MPa Tensile modulus GPa Diameter ?m Glass 2550 3450-5000 69-84 7-14 Boron 2200-2700 2750-3600 400 50-200 Carbon 1500-2000 2000-5600 180-500 6-8 Kevlar 1390 2750-3000 80-130 10-12 Polyethelyne 970 2590 117 38 Silica (SiO 2 ) 2200 5800 72 35 Boron carbide (B 4 C) 2350 2690 425 102 Boron nitride 1910 1380 90 6.9 Silicon carbide (SiC) 2800 4500 480 10-12 TiB 2 4480 105 510 - TiC 4900 1540 450 - Zirconium oxide 4840 2070 345 - Borsic(SiC/B/W) 2770 2930 470 107-145 Alumina(Al 2 O 3 ) 3150 2070 210 17 Alumina FP 3710 1380 345 15-25 Steel 7800 4140 210 127 Tungsten 19300 3170 390 361 Beryllium 1830 1300 240 127 Molybdenum 1020 660 320 127 Quartz whisker 2200 4135 76 9 Fe whisker 7800 13,800 310 127 SiC whisker 3200 21,000 840 0.5-10 Al 2 O 3 whisker 4000 20,700 427 0.5-10 BeO whisker 2851 13,100 345 10-30 Fibre Material Density kg/m 3 Tensile strength MPa Tensile modulus GPa Diameter ?m B 4 C whisker 2519 13,790 483 - Si 3 N 4 whisker 3183 13,790 379 1-10 Graphite whisker 2100 20,800 1000 - Table 2.2: Typical thermal properties of selected fibres Fibre Melting point 0 C Heat Capacity kJ/(kg.K) Thermal conductivity W/(mK) Coefficient of thermal expansion 10 -6 m/mK Glass 840 0.71 13 5 Boron 2000 1.30 38 5 Carbon 3650 0.92 1003 -1.0 Kelvar 49 250 1.05 2.94 -4.0 SiC 2690 1.2 16 4.3 Steel 1575 0.5 29 13.3 Tungsten 3400 0.1 168 4.5 Beryllium 1280 1.9 150 11.5 Molybdenum 2620 0.3 145 4.9 Fe whisker 1540 0.5 29 13.3 Al 2 O 3 whisker 2040 0.6 24 7.7 Quartz Whisker 1650 0.963 10 0.54 Organic fibres as well as glass, silica, quartz and carbon fibres are commercially available in the form of strands, tows or yarns. A strand (or end) is a collection of filaments. A tow (or roving) consists of several ends or strands. A yarn is a twisted strand. Some twist is preferred for compactness and for making a composite with higher fibre content. However, an excessive twist should be avoided, as that may not permit the matrix to penetrate and wet all the fibres. These fibres are also used to make woven rovings and woven fabrics (clothes). The weave styles in the fabrics may be unidirectional (uniaxial), bidirectional (biaxial 2D and biaxial 3D) and multidirectional (multiaxial). In a uniaxial fabric wrap fibres are yarns that are laid along the roll. Fill/pick fibres, which constitute only a small percentage of the wrap fibres, are placed in the weft direction which is transverse to the roll direction. The weaving design methodology and weaving techniques, in most cases, adapt similar procedures that are followed in textile technology. A few typical weave styles are illustrated in Figs. 2.2 and 2.3. Commercial woven rovings are made with a simple plain weave style, whereas fabrics for carbon- carbon composites may adopt complex multidirectional weaving patterns. Hybrid fabrics may also be produced by mixing of various fibres in the warp and weft directions. Woven rovings and fabrics are quite often preimpregnated with resin to make prepregs that are convenient to use in the fabrication of composite parts. Glass Fibres Glass was first made by man in 3000 BC in Asia Minor. Continuous glass fibres were known to be used for decorative purposes in ancient times in Syria and Venice. The industrial manufacturing of glass fibres started in 1930`s for use in filters and insulations. Glass fibres currently comprise more than 90% of fibres used in polymer composites. There are five major types of glass used to make glass fibres. These are A glass (high alkali), C glass (chemical), D glass (low dielectric constant), E glass (electrical) and S glass (high strength), out of which the last two types, due to their superior mechanical properties, are most widely used in composite roofings, pressure vessels, containers, tanks, pipes, etc. E glass is a low alkali, aluminium borosilicate glass and is based on a mixture of alumina, boric acid, calcium carbonate and magnesia. S-glass is based on a mixture of silica, alumina and magnesia. For the manufacture of glass fibres, glass is premixed and formed into glass marbles or beads. The glass marbles or beads are then melt, and the molten glass is gravity fed, under a controlled temperature, through a platinum bushing containing a large number of very small orifices. The melt vitrifies within the bushing, and the filaments are simultaneously cooled and drawn rapidly to a small diameter. Figure 2.4 presents a schematic view of a fibre drawing process. The surfaces of drawn glass fibres are normally treated with appropriate sizing materials to promote adhesion with the resin matrix used, to facilitate weaving without causing mechanical damage to the fibre or to improve certain properties like, toughness and impact resistance. The cross-section of glass fibre is circular in nature and the diameter is usually in the range of 7-14 ?m. A glass fibre exhibits isotropic properties. Glass fibres are cheap, nonmagnetic, x-ray transparent, chemically inert, biocompatible, insensitive to moisture and temperature as well as possess high specific strength (strength to density ratio). However, a long duration loading under certain environmental conditions, may reduce the load carrying capacity of fibres by about 25%. This behaviour is known as static fatigue. Silica Fibres The silica (Si0 2 ) content of silica fibres ranges from 95 to 99.4% and is usually much higher compared to that of glass fibres. The glass fibres contain only 55 to 75% silica. The silica fibres are produced by treating glass fibres with acids so as to remove all impurities. A quartz fibre is a ultra-pure silica fibre. Quartz fibres are made from natural quartz crystals, in which the silica content is as high as 99.95%. There are a few other methods for producing high silica or quartz fibres. In one method, a polymer of silicon alkoxide is spun using a sol-gel process and subsequent heating of the fibre to 1000 0 c yields a 99.999% pure quartz fibre. Silica and quartz fibres have superior thermal properties compared to glass fibres. They have extremely low thermal conductivities and thermal expansion coefficients. They can withstand extreme changes in thermal environments. They can be heated to a very high temperature without causing any damage. These properties make them ideal materials for application in highly heated structures such as thermal shields, nose cones, rocket nozzles, exit cones, etc. Boron Fibres Boron fibres with consistent and good mechanical properties were first manufactured in the 1960`s. Boron is a multiphase fibre. A boron fibre is produced by depositing boron on a thin substrate by a chemical vapour deposition process. Substrates, which are thin filaments, usually made of tungsten or carbon. The substrate (of dia.8-12? m) is fed in a plating chamber containing a mixture of hydrogen and boron trichloride and is electrically heated (Fig. 2.5). The boron is deposited in an amorphous form on the surface of the substrate filament due to the chemical reaction as given by 2BCl 3 + 2H 2 → 2B + 6HCl The thickness of the deposited boron depends on the rate at which the substrate is passed through the plating chamber. The boron coated fibre is normally fed to successive plating chambers to increase the diameter of the fibre. The boron fibres are often treated chemically to remove surface defects, thermally to reduce residual stresses or by providing thin coatings (SiC, B 4 C or BN) to increase oxidization resistance and to make compatible with metal matrices. The boron fibre is marketed as a single filament. The boron filaments are now available in diameters 50?m, 100?m, 125?m,140?m and 200?m. The boron fibre with a tungsten substrate is costlier than that with a carbon substrate. A carbon substrate also reduces the density of the fibre. Boron fibres are usually impregnated with a resin to form tapes, as they are too stiff to weave. Boron fibres exhibit excellent stiffness properties, because of which they are used for stiffening of structural parts in aerospace applications. Their tensile strength is also quite good. The thermal properties are, however, in the intermediate range, although the melting point temperature is on the higher side. Silicon Carbide and Boron Carbide Fibres Like boron, there are several other multiphase fibres such as silicon carbide and boron carbide. Silicon carbides are also vapour deposited on tungsten or carbon filaments. The silicon carbide vapours are deposited when various chlorosilanes or their mixtures are used as reactants. A typical reaction is given as CH 3 S i Cl 3 → SiC + 3HCl The fibres are currently made in diameters of 100?m and 140?m. Silicon carbide fibres in general exhibit good high temperature characteristics. They are compatible with several lightweight alloys e.g., aluminium, nickel and titanium alloys. Silicon carbide on a carbon substrate has several other merits over its counterpart (silicon carbide on a tungsten substrate). It is cheaper and lighter. No reaction takes place between the deposited silicon carbide and the carbon substrate at a high temperature. The tensile strength is also on the higher side. The tensile strength and tensile modulus of a SiC whisker is 21000 MPa and 840GPa, respectively. SiC whiskers are grown by combining silicon and carbon at 1200-1500 0 C under special conditions. A boron carbide vapour deposited mantle on a tungsten filament substrate can be formed using mixture of boron trichloride and methane or carboranes. A boron carbide fibre can also be produced by heating carbon fibres in a chamber containing boron halide vapour. The melting point of a boron carbide fibre is 2450 0 C, and the fibre retains proper ties at a temperature higher than 1000 0 C. Alumina Fibres The commercial grade alumina fibre developed by Du Pont is known as alumina FP (polycrystalline alumina) fibre. Alumina FP fibres are compatible with both metal and resin matrices. These fibres possess a high melting point temperature of 2040 0 C. They also withstand temperatures up to 1000 0 C without loss of strength and stiffness properties. They exhibit high compressive strengths, when they are set in a matrix. Typical longitudinal compressive strengths of alumina FP/epoxy composites vary from 2275 to 2413 MPa. Alumina whiskers exhibit the tensile strength of 20700 MPa and the tensile modulus of 427 GPa. Carbon Fibres Carbon fibres are also commonly known as graphite fibres, although there are some basic differences between the two types. Graphitization takes place at a much higher temperature compared to the temperature at which carbonization takes place. The carbon content in the graphite fibre is also higher and is usually more than 99%. The manufacture of carbon fibres in the laboratory scale started in the early fifties. However, carbon fibres were made commercially available only during mid-sixties. They are made after oxidizing and carbonizing the organic textile fibre precursors at high temperatures. There are three common types of precursors: polyacrylonitrile (PAN), rayon and petroleum pitch. A typical fibre fabrication process based on a precursor is shown in Fig. 2.6. The molecular orientation already present in a precursor is formed along the fibre axis. Carbonization takes place at lower temperature (at about 1000 0 C). The carbonized fibre is then treated at higher temperature to facilitate the graphitization process. The degree of graphitization can be enhanced by raising the temperature further. High- modulus PAN and Rayon based graphite fibres need as high as 2500 0 C for proper graphitization. High strength PAN-based carbon fibres are treated at about 1500 0 C for required carbonization. The carbon fibre microstructural morphology changes considerably with the precursor used (Fig.2.1). This also affects the fibre-matrix interface characteristics. The fibres in general exhibit anisotropic behaviour. The average diameters of commercial fibres range from 6-8?m. Carbon fibres are produced in a variety of tensile strengths and tensile moduli. They are accordingly designated as ultrahigh, very high, high or intermediate modulus and high strength. The tensile strength and tensile modulus of carbon fibres may be as high as 5600 MPa and 500 GPa, respectively. Typical properties of some high modulus and high strength fibres are presented in Table 2.3. Carbon fibres have many other positive attributes, for which they are most popular in aerospace applications. They can withstand extremely high temperatures without loss of much strength and stiffness. The thermal conductivity is high and at the same time the coefficient of thermal expansion is almost negligible. These thermal characteristics make them outstanding candidate materials for high temperature applications. Further, carbon fibres are non-magnetic, x- ray transparent, chemically inert, bio-compatible and insensitive to moisture to a great extent. Carbon fibres are much costlier compared to glass and other organic fibres. Their application is, therefore, limited to strategic structural components, expensive sport goods and biological implants. Table 2.3: Typical properties of high performance carbon fibres Type Density Kg/m 3 Tensile strength MPa Tensile modulus GPa M40 1800 2740 390 M46 1900 2350 450 M50 1900 2450 490 T40 1800 5650 280 T75 1830 2620 545 T300 1770 3240 231 T400 1800 4500 250 T500 1800 5600 241 T800 1800 5600 290 Ultra high Modulus 1800 2240-2410 690-827 Aramid Fibres These are aromatic polyamide fibres. These are based on polymers formed by condensation of aromatic diacid derivatives with aromatic diamines. Kevlar is the trade mark for the commercially available aramid fibres marketed by Du Pont first in the early seventies. There are three types of Kevlar fibres: Kevlar, Kevlar 49 and Kevlar 29. Kevlar 49 and Kevlar 29 fibres posses the same strength, but Kevlar 29 has the two-third of the tensile modulus of Kevlar 49. Kevlar 29 is used for reinforcing rubber cordage and belting. Kevlar is similar to Kevlar 49, but is designed for tyre reinforcement. Kevlar 49 fibres are commonly termed as Kevlar fibres and find extensive uses in pressure vessels, motor cases and other structures where strength is the major design criterion. The diameter of Kevlar fibres range from 8-12?m. Kevlar fibres, being organic in composition, are susceptible to hygral and thermal environments. They are easily attacked by alkalis and acids. Each fibre is fibrillar in nature and consists of several long, stiff fibrils (aligned along the fibre axis) embedded in a softer matrix. Because of this microstructural morphology (Fig.2.1), Kevlar fibres are very weak in compression (due to buckling of fibrils), but exhibit good impact resistance. They are cheaper, non-magnetic, x-ray transparent and bio-compatible and resistant to flame, organic solvents, fuels and lubricants. 2.2.2 Particulates Particulates of various shapes and sizes are used as reinforcing particles. The shapes vary from a simple sphere (e.g., glass beads) to a complex polyhedron (e.g., crystals). The size ranges from a few microns to several hundred microns. Particles of various inorganic and organic materials are employed to make particulate composites. However, they should be compatible with the matrix system used. Materials like talc, clay, mica, calcium carbonate, calcium sulphate, calcium silicate, titanium oxide, wood dust, sand, silica, alumina, asbestos, glass beads, metal flakes, metal powder, carbon powder, ceramic grains and several polymeric particles are normally used. Besides strengthening the composite, particles also serve other purposes. They act as additives to modify the creep, impact, hygral, thermal, electrical, chemical and magnetic properties as well as wear resistance, flammability and such other properties of the composite. They may as well be utilized as fillers to change the matrix content and density of the composite. The strength, stiffness and other properties of the composite are dependent on the shape, size, distribution and blends of various particles in a given matrix and also on the particle-matrix interface condition. Depending on the composite`s end use, the volume content of the reinforcement may go up to 40-50%, or more. Short fibres are discontinuous fibres and may also be treated as particles with cylindrical shapes. Flakes/platelets are also commonly used. They are less expensive than short fibres, and can be aligned to obtain improved in plane directional properties compared to those of short fibres. Metal flakes (say, aluminium) can be used to improve the thermal and electrical conductivity of the composite, whereas mica flakes can be added to the matrix to increase the resistivity. Solid glass microspheres, silicate-base hollow microspheres and ceramic aluminosilicate macrospheres are used in reinforcing polymer matrices. The particle diameter for solid glass microspheres ranges from 5 to 50?m, whereas for hollow microspheres it ranges from 10 to 300?m. The spherical shape of these particles allows a uniform distribution of stresses through the matrix. The composite also behaves like an isotropic material. For a given volume, the surface required to wet with a resin is minimum for a sphere. This permits to provide a smooth surface finish to a product without increasing the resin content. One positive advantage with hollow microspheres is that the weight can be considerably reduced without compromising on the strength of a composite. The density of hollow microspheres ranges from 150 to 380 kg/m 3 and is considerably lower than that of the polymer matrix. The composite formed with hollow microspheres is, therefore, lighter than the matrix itself. Metal and ceramic matrices are also very commonly reinforced with particulates. Aluminium alloys reinforced with silicon carbide particles are found to exhibit higher strengths and stiffnesses. The tensile strength and Young`s modulus of the aluminium alloy AA2124-T6 matrix reinforced with the silicon carbide particles (volume content by 40%) are observed to be about 690 MPa and 150 GPa, respectively. Ceramic grains of borides, carbides, oxides, nitrides, silicides, etc. are dispersed in metal matrix to produce a host of particulate composites known as cermets. Some cermets, if properly made, may possess low density, but at the same time, may exhibit good thermomechanical properties. 2.3 POLYMERS AND POLYMER COMPOSITES Polymers (also known as plastics or resins) are far more popular than other two matrix materials, namely, metals and ceramics. Almost all reinforcements, inorganic and organic, can be used with polymers to produce a wide range of reinforced plastics or polymer composites. Polymers are particularly attractive due to several reasons. The densities of polymers are usually very low. Polymers are easily processable. The processing and curing temperature are normally in the lower range, and in some cases, the ambient temperature will suffice. This brings down the manufacturing cost substantially due to a low energy input. Further, polymers constitute a wide class of organic materials, each having a distinct characteristic feature. This makes them all the more attractive from the point of view of developing composites having different properties. Both thermoplastics and thermosets are employed in making reinforced plastics. Polyethelene, polystyrene, polyamides, nylon, polycarbonates, polysulfaones, etc. are common thermoplastics whereas thermosets are epoxy, phenolic, polyester, silicone, bismaleimide, polyimide, polybenzimidazole, etc. 2.3.1 Thermoplastics Although thermosets are commonly used in structural composites due to their higher strength and stiffness properties, there is a growing interest in recent years to use thermoplastics as well. The development of several high performance thermoplastics has been primarily responsible for this new trend. The main advantage with thermoplastic polymers is that they can be repeatedly formed by heat and pressure. A thermoplast is a collection of high molecular weight linear or branched molecules. It softens upon heating at temperature above the glass transition temperature, but regains its strength upon cooling. The increase in temperature activates the random motion of the atoms about their equilibrium positions and results in breakage of secondary bonds. The thermoplast softens and results in breakage of secondary bonds. The thermoplast softens and flows when pressure is also applied. When the temperature is lowered, new secondary bonds are formed and the polymer reverts to its original structure. The process of softening at higher temperature and regaining rigidity upon cooling is thus reversible in the case of a thermoplastic polymer. This characteristic behaviour helps it to be recast and reused several times. The repair of a damaged part also becomes simpler. The scrapage rate is also reduced. All these make thermoplasts very much cost effective. A thermoplastic polymer softens, but does not decompose unless the temperature is high enough to break the primary covalent bonds. Table 2.4 provides the typical thermo-mechanical properties of a couple of structural grade thermoplastic resins. These high performance resins are thermally stable at higher temperatures. They normally achieve high glass transition temperature Tg due Table 2.4 Typical properties of some high performance thermoplastics Properties Polyether ether ketones (PEEK) Polyamide- imide Polyether- imide Polysulfone Polyphenylene sulfide Density, kg/m 3 1300 1400 1270 1240 1340 Tensile strength, MPa 104 138 115 70 76 Tensile modulus, GPa 4.21 4.48 3.38 2.48 3.31 Poisson?s ratio 0.35 0.35 0.35 0.35 0.35 Coefficient of thermal expansion, 10 - 6 m/m K - 56 50 86.40 88 Maximum service temperature, K 630 - 490 490 565 to their relatively stiff, linear chains and high molecular weight. They are also strong and stiff and exhibit good creep resistant properties. They are relatively tougher and less sensitve to moisture. The structures of these resins are illustrated in Fig. 2.7. All resins are found to contain a high proportion of aromatic rings that are linked by a stable heteroatom or group (-CO-, SO 2 -, -O-, etc.). This provides a high degree of chain rigidity and thereby results in a higher Tg. In addition, the low aliphatic hydrogen (C-H) content enhances the thermal stability of all these resins at high temperatures. PEEK and polyphenylene sulfide are essentially crystalline polymers, and other resins shown in Table 2.4 are amorphous. PEEK has received considerable attention since its inception. The rigid rings, connected by fairly chemically inert groups (-o- and ?c-) make PEEK highly crystalline. The melting point and chemical resistance of PEEK are also considerably enhanced. PEEK has a Tg of 143 0 C and a melting point of 332 0 C. It is soluble only in concentrated sulphuric acid. The processing temperature ranges 300- 400 0 C. The moisture absorption limit is very low. The fracture toughness is comparatively higher. All these features of PEEK make it a highly attractive thermoplastic resin for application in reinforced composites. Graphite/PEEK composite prepregs are commercially available. Polysulfones reinforced with glass, aramid and carbon fibres have also found several applications. 2.3.2 Thermosets Thermoset polymers are formed from relatively low molecular weight precursor molecules. The polymerization process in a thermoset resin is irreversible. Once cured, they do not soften upon heating. They, however, decompose before softening upon further heating. Cross-linked and interlinked reactions lead to formation of chain molecules in two and three-dimension arrays. Because of three dimensional network of covalent bonds and cross links, thermosetting resins are listed in Table 2.5. At high temperature, the covalent bonds may break leading to destruction of the network structure and the polymer decomposes. Thermosetting resins vary widely with Tg values varying from 45-300 0 C and elongations ranging from 1% to more than 100%. Table 2.5 Typical properties of some thermosetting resins Properties Epoxies Polyesters Phenolics Polyimides Density, kg/m 3 1100-1400 1200 1200-1300 1400 Tensile strength, MPa 35-100 50-60 50-60 100-130 Tensile modulus, GPa 1.5-3.5 2-3 5-11 3-4 Poisson?s ratio 0.35 0.35 0.35 0.35 Coefficient of thermal expansion, 10 -6 m/mK 50-70 40-60 40-80 30-40 Service temperature, K 300-370 330-350 440-470 550-750 The most commonly used thermosets are epoxy, polyester and phenolic resins, among which polyster resins are most widely used in various common engineering goods and composite applications. However, epoxy resins constitute the major group of thermoset resins used in composite structures and adhesives, as they are stronger and stiffer. Phenolic resins are rich in carbon and possess good thermal properties and are normally used in high temperature applications especially as an ablative material in thermal protection systems. Silicone, bismaleimide, polyimide, polybenzimidazol, etc. are in fact, high temperature polymers that can perform at higher temperature ranging from 200- 450 0 C. The structures of some thermosetting resins are illustrated in Fig. 2.8. Epoxy resins in general possess good thermomechanical, electrical and chemical resistant properties. They are so called, because they contain two or more epoxide groups in the polymer before cross-linking. This epoxide group is a three membered cyclic ether O C C which reacts with several reagents. It is commonly found in glycidyl ethers and amines which are the major sources for epoxies in composite applications. The common epoxy is synthesized by condensing epichlorohydrin with bisphenol A in the presence of sodium hydroxide. Several other hydroxyl-containing compounds can replace bisphenol A. A wide variety or special purpose resins can thus be prepared. Some aerospace grade epoxy systems are based on an aromatic amine (glycidyl amines) instead of a phenol to increase the epoxy functionality leading to high cross- link density in the cured resin. Epoxy resins are cured using suitable curing agents or appropriate catalysts. The major curing agents are aliphatic amines, aromatic polyamines and polyanhydrides. Curing is the processs of reaction (ionic reaction, usually polyadditions) between the epoxide and the curing agent in which many epoxide groups are formed. Aliphatic amines are relatively strong bases and therefore react with aromatic amines to achieve cure at room temperature. The reaction is highly exothermic, and the pot life is shorter. This epoxy resin is useful for contact moulding, but not for prepregging and filament winding. Aromatic polyamines are normally solids and require high temperature (100-150 0 C) for mixing and curing. Anhydrides need higher thermal exposure (150-200 0 C) for a longer duration (8-16 hours) for proper curing. The reaction is low exothermic, but the pot life is longer. Both polyamines and anhydrides are suitable for prepreg manufacturing and filament winding. These epoxy resins are characterized by comparatively high thermal stability and chemical resistance. Catalysts can also be used along with curing agents to accelerate the curing process. Catalytic agents that are often used as curing agents to promote homopolymerisation of epoxide groups may be Lewis acids or bases. The commonly used catalytic curing agent is boron trifluoride blocked with ethyl amine (a typical Lewis acid). It is also used as a catalyst with aromatic amines to accelerate curing at a temperature of 150-200 0 C. Lewis bases are normally used as accelerators with anahydrides. Both Lewis acids and bases provide long pot lives. A polyester resin is comprised of an unsaturated backbone polymer dissolved in a reactive monomer. The polyester backbone polymer is formed by condensation of a mixture of diabasic acids (saturated and unsaturated) and one or more glycols. The components of the most commonly used polyester resin is phthalic anhydride (saturated acid), maleic anhydride (unsaturated acid) and propylene glycol. The backbone polymer is then diluted in styrene monomer (about 35% by weight). The solution is then blended with an inhibitor such as hydroquinone to prevent premature polymeisation. The process of curing is initiated by adding a source of free radicals (e.g., benzoyl peroxide or hydroperoxide and catalysts (e.g., organic peroxides such as cobalt naphthenate or alkyl mercaptans). Curing takes place in two stages: a soft gel is first formed and this is followed by a rapid polymerization with generation of heat. A higher proportion of unsaturated acid in the backbone polymer yields a more reactive resin, while with a higher quantity of saturated acid the reaction becomes less exothermic. During curing, the styrene monomer reacts with the unsaturated sites of the backbone polymer to form a three-dimensional cross-linked network. A small amount of wax is often added to the solution before curing to facilitate proper curing of the surface of a laminate. Wax, during curing, exudes to the surface to form a thin protective layer that reduces loss of styrene from the surface and prevents oxygen which inhibits reaction to come in contact with the radicals. Several types of polyester resins are commercially available. Vinyl-ester resins are high performance polyester resins, which are acrylic esters of epoxy resins dissolved in styrene monomer. Polyester resins can be reinforced with almost all types of reinforcements to make polyester composites. Polyester resins are cheaper and more versatile, but inferior to epoxy resins in some respects. Their use in advanced structural composites is therefore limited. However, they have been widely used in boat hulls, civil engineering structures, automobile industries and various engineering products and appliances. The commonly used phenolic (phenol-formaldehyde) resins are divided into two groups: resoles and novolacs. Resoles are one-stage resins which are synthesized with formaldehyde/phenol ratio greater than one (1.25:1) in presence of an alkaline catalyst. The polymerization process is not fully completed. It is stopped by cooling to obtain a reactive and soluble polymer which is stored at low temperature. The final polymerization process is initiated, during curing, by raising the temperature. The novolacs, on the other hand, are two-stage resins, made with an acid catalyst. The ratio of formaldehyde to phenol is kept about 0.8:1. In the first stage, the reaction is completed to yield an unreactive thermoplastic oligomer which is dehydrated and pulverized. A curing agent such as hexamethylenetetramine is added in the second stage, which decomposes due to heat and moisture during final curing to yield formaldehyde and ammonia. The ammonia also acts as a catalyst for curing. Resoles are used for prepregs and structural laminates. Novolacs are normally used as moulding compounds and friction products. Phenolic resins, as a whole, provide good dimensional stability as well as excellent chemical, thermal and creep resistance, and exhibit low inflammability. Phenolics char when exposed to the high temperatures and form a layer of carbon which in turn protects the underlying composite from being exposed to high temperature. This characteristic behaviour has made phenolic resins as candidate materials in making reinforced composites for high temperature applications and thermal shielding. Several high temperature thermosetting polymers are currently available, of which polyimides, bismaleimides, polybenzimidazole, silicone, etc. are of special interest to composite applications. Polyimides are made by polycondenstion of aromatic dianhydrides and aromatic diamines. The reaction between dianhydride and diamine (at a temperature lower than 100 0 C) first yields a soluble polyamic acid. Next cyclisation of the polyimides retain their usable properties at 300 0 C (continuous exposure) and can withstand an exposure of 500 0 C for a few minutes. Bismaleimides are addition polyimides. Silicone polymers are formed by intermolecular condensation of silanols which are produced from the halide or alkoxy intermediates. Both silicone and polybenzimidazole resins are normally used in the intermediate temperature range (200- 250 0 C). The reinforced plastics are very extensively used in engineering industries. Some important structural applications of fibre reinforced polymer composites are listed in Table 2.6. Table 2.6: Applications fibre reinforced polymer composites Composites Uses Aerospace CFRP Wing Skin, Front Fuselage, Control Surface Fin & Rudder, Access Doors, Under Carriage Doors, Engine Cowlings, Main Torsion Box, Fuel Tanks, Rotor Blades, Fuselage Structures and Floor Boards of Helicopters, Antenna Dishes, Solar Booms and Solar Arrays, etc. BFRP Horizontal and Vertical Tail, Stiffening Spars, Ribs and Longerons, etc. KFRP Nose Cones, Wing Root, Fairings, Cockpit and Fuselage of Helicopters, Motor Casings, Pressure Bottles, Propellant Tanks, Other Pressurised Systems, etc. GFRP Floor Boards, Interior Decorative Panels, Partitions, Cabin Baggage Racks and Several Similar Applications. Structural GFRP Folded Plates of Various Forms, Both Synclastic and Anticlastic Shells, Skeletal Structures, Walls and Panels, Doors, Windows, Ladders, Staircases, Chemical and Water Tanks, Cooling Towers, Bridge Decks, Antenna Dishes, etc. Marine and Mechanical GFRP Ship and Boat Hulls, Masts, Automobile Bodies, Frames and Bumpers, Bodies of Railway Bogeys, Drive shafts, Connecting Rods, Suspension Systems, Instrument Panels. Sports GFRP/CFRP Skis, Ski Poles, Fishing Rods, Golf Clubs, Tennis and Badminton Rackets, Hockey Sticks, Poles(Pole vault), Bicycle Frames, etc. 2.4 METALS AND METAL MATRIX COMPOSITES Polymer composites are used normally up to 180 0 C, but rarely beyond 350 0 C. The high temperature capabilities of inorganic reinforcements cannot be realized, when polymers are employed as matrix materials. Metal matrices, on the other hand, can widen the scope of using composites over a wide range of temperatures. Besides, metal matrix composites allow tailoring of several useful properties that are not achievable in conventional metallic alloys. High specific strength and stiffness, low thermal expansion, good thermal stability and improved wear resistance are some of the positive features of metal matrix composites. The metal composites also provide better transverse properties and higher toughness compared to polymer composites. Table 2.7 provides the list of some metal matrices and associated reinforcing materials. The reinforcements can be in the form of either particulates, or short fibres or continuous fibres. Cermets constitute an important group of metal matrix composites in which ceramic grains of sizes greater than 1 ?m are dispersed in the refractory metal matrix. A typical example is the titanium carbide cermet which comprises of 70% TiC particles and 30% nickel matrix and exhibits high specific strength and stiffness at very high temperatures. The thermo-mechanical properties of some common matrices are presented in Table 2.8. The aluminium matrices include several alloys such as AA 1100, AA 2014, AA6061, AA 7075, AA5052, etc. The composites with aluminium matrices are relatively lightweight, but their applications are limited to the lower temperature range Table 2.7: Metal matrices and reinforcements Matrix Reinforcements Aluminium and alloys C, Be, SiO 2 , B, SiC, Al 2 O 3 , Steel, B 4 C, Al 3 Ni, Mo, W, Z r O 2 Titanium and alloys B, SiC, Mo, SiO 2, B e ,ZrO 2 Nickel and alloys C, B e , Al 2 O 3, SiC, Si 3 N 4 , steel, W, Mo, B Magnesium alloys C, B, glass, Al 2 O 3 Molebdenum and alloys B, ZrO 2 Iron and Steel Fe, Steel, B, Al 2 O 3 , W, SiO 2 ,ZrO 2 Copper and alloys C,B, Al 2 O 3 , E-glass Table 2.8: Typical thermomechanical properties of some metal matrices Matrices Density kg/m 3 Tensile strength MPa Tensile modulus GPa Coefficient thermal expansion 10 -6 m/mk Thermal conduct- -vity W/(mk) Heat capa- -city KJ/ (kg.k) Melting point 0 C AA6061 2800 310 70 23.4 171 0.96 590 Nickel 8900 760 210 13.3 62 0.46 1440 Ti-6AL-4V 4400 1170 110 9.5 7 0.59 1650 Magnesium 1700 280 40 26 100 1.00 570 Steel 7800 2070 206 13.3 29 0.46 1460 Copper 8900 340 120 17.6 391 0.38 1080 because of its low melting point. Titanium and nickel can be used at a service temperature of up to 1000-1100 0 C. There are several systems such as engine components which are exposed to high level of temperature. Titanium and nickel composites are ideal for such situations, as they retain useful properties at 1000-1100 0 C. Ti-6AL-4V is the commonly used titanium matrix material. The other alloys of titanium include A-40Ti, A- 70Ti, etc. Nickel matrices are comprised of a series of Ni-Cr-W-Al-Ti alloys. Super alloys, NiCrAlY and FeCrAlY, are also used as matrices because of their high oxidation resistance properties. Molybdenum is a high temperature matrix and fibre material. Iron and steel matrices are cheaper and can be used at high temperatures, if the weight is not the major concern. Figure 2.9 exhibits a fractograph of the Al 2 O 3 fibre reinforced Mg alloy ZM21 composite. A scanning electron micrograph of the Al 2 O 3 fibre reinforced AA 2014 composite is shown in Fig. 2.10. The high temperature applications of metal matrix composites are listed in Table 2.9. The material cost is the major problem that currently limits their uses, otherwise most of the metallic structural parts can be replaced with metal matrix composite parts to gain advantages. Fig.2.9 Fig.2.10 Table 2.9: High temperature application of composites Composites Applications MMCs Aeroengine Blades, Combustion Chamber, Thrust Chamber, Nozzle Throat, Exit Nozzle, Engine Valves, Fins. CMCs Re-entry Thermal Shields and Tiles, Nozzle Throat, Nozzle Linings, Pump Seal Rings, Break Linings, Extrusion Dies, Valves, Turbochargers, Turbine Blades, Cutting Tools Carbon-Carbon Nose Cones of Re-entry Vehicles, Combustion and Thrust Chambers, Nozzle Throats, Exit Nozzles, Leading Edges of Re-entry Structures, Brake Discs. 2.5 CERAMIC MATRIX COMPOSITES Ceramic provide strength at high temperature well above 1500 0 C and have considerable oxidation resistance. They possess several desirable attributes like high elastic modulus, high Peierl`s yield stress, low thermal expansion, low thermal conductivity, high melting point, good chemical and weather resistance as well as excellent electromagnetic transparency. However, the major drawback of ceramics is that they exhibit limited plasticity. This low strain capability of ceramics is of major concern, as it, quite often, leads to catastrophic failure. For this reason ceramics are not considered as dependable structural materials. But such limitations may not exist with ceramic matrix composites, as suitable reinforcements may help them to achieve desirable mechanical properties including toughness. The ceramic matrices are usually glass, glass ceramics (lithium aluminosilicates), carbides (SiC), nitrides (SiN 4, BN), oxides (Al 2 O 3 , Zr 2 O 3 , Cr 2 O 3 , Y 2 O 3 , CaO, ThO 2 ) and borides (ZrB 2 , TiB 2 ). The reinforcements which are normally high temperature inorganic materials including ceramics, may be in the form of particles, flakes, whiskers and fibres. The commonly used fibres are carbon, silicon carbide, silica and alumina. The current resurgence in the research and development of ceramic matrix composites is due to their resistance to wear, creep, low and high cycle fatigue, corrosion and impact combined with high specific strength at high temperatures. The cutting rate of an alumina-SiC whisker cutting tool is ten times higher than that of conventional tools. The use of ceramic composites in aero-engine and automotive engine components can reduce the weight and thereby enhance the engine performance with higher thrust to weight ratios due to high specific strength at high temperatures. Automotive engines exhibit greater efficiency because of their low weight, better performance at high operating temperatures and longer life time due to excellent resistance to heat and wear. Several high temperature applications of ceramic matrix composites are presented in Table 2.9. Carbon-carbon composites are the most important class of ceramic matrix composites that can withstand temperatures as high as 3000 0 C. They consist of carbon fibres distributed in a carbon matrix. They are prepared by pyrolysis of polymer impregnated carbon fibre fabrics and preforms under pressure or by chemical vapour deposition of carbon or graphite. The polymers used are of three types: thermosets (furfurals, phenolics), thermoplastic pitches (coal tar based and petroleum based) and carbon-rich vapours (hydrocarbons such as methane, propane, acetylene, benzene). Phenolic resins are more commonly used in the manufacturing process of carbon-carbon composites. The phenolic resin impregnated carbon fibre preforms, on pyrolysis, converts the phenolic resin to a high proportion of amorphous carbon char. The composite material is found to be porous after the first pyrolysis. It is further impregnated with the phenolic resin and pyrolised, usually under vacuum and pressure, and the process is repeated several times to reduce the void content and realize the optimum density of the material. The major advantage of carbon-carbon composite is that various fabrics and shapes of preforms with multidirectional fibre alignments can be impregnated with resins and pyrolised to yield a wide class of one directional (1D), two directional (2D), three directional (3D) and multidirectional composite blocks of various shapes and sizes, which can be machined to produce the desired dimensions. Excellent wear resistance, higher coefficient of friction with the rise of temperature, high thermal conductivity, low thermal expansivity and high temperature resistance make them useful materials in high temperature applications. In absence of oxygen, carbon-carbon composites can withstand very high temperatures (3000 0 C or more) for prolonged periods. They are also used in prosthetics due to excellent biocompatibility. 2.6 LAMINATE DESIGNATION The structural applications of composites are mostly in the form of laminates. Laminates provide the inherent flexibility that a designer exploits to choose the right combination of materials and directional properties for an optimum design. A lamina is the basic building block in a laminate. A lamina may be made from a single material (metal, polymer or ceramic) or from a composite material. A composite lamina, in which all filaments are aligned along one direction parallel to each other, is called a unidirectional lamina. Some unidirectional laminae are illustrated in Fig. 2.11. Here the fibres (continuous) are oriented along a direction parallel to the x 1 axis. Note that the x 1 ' , x 2 ' axes are the material axes, and the x 1, x 2 axes are the reference axes. The orientation of the fibre with respect to the reference axis (i.e., x 1 axis) is known as the fibre angle and is denoted by ؠ(in degrees). A unidirectional lamina is designated with respect to the fibre angle خ For example, a 0 0 lamina corresponds to ؽ 0 0 , a 90 0 lamina corresponds to ؽ 90 0 and so on. Fig. 2.11 A laminate is designated by the manner laminae are stacked to form the laminate. For example, a (0 0 /?45 0 /90 0 ) laminate (Fig. 2.12) is one in which one 0 0 lamina is placed at the top, one 90 0 lamina is placed at the bottom and one +45 0 lamina and one -45 0 lamina are kept at the middle. Unless it is specified, it is normally assumed that all the laminae in a laminate possess the same thickness. A cross-ply laminate consists of only 0 0 and 90 0 laminae. An angle-ply laminate, on the other hand, contains only ?ؠlaminae. A laminate may be considered symmetric, antisymmetric or unsymmetric, in case there exists, with respect to the middle surface, any symmetry, antisymmetry or unsymmetry, respectively. Figures 2.13 and 2.14 illustrate several cross-ply and angle-ply laminates. It should be further noted that a [ (0 0 /90 0 ) n ] laminate is an antisymmetric cross-ply laminate consisting of n numbers of repeating two-layered (0 0 /90 0 ) cross-ply laminates. The total number of laminae in a [ (0 0 /90 0 ) n ] laminate is 2n. However, a [ (0 0 /90 0 ) ns ] laminate is a symmetric cross-ply laminate. It has symmetry about the midsurface of the laminate. The top half of the laminate contains n number of repeating of repeating two-layered (0 0 /90 0 ) cross-ply laminates. The bottom half consists of n number of two-layered (90 0 /0 0 ) cross- ply laminates so that the symmetry about the mid-surface is maintained. Note that the subscript `s` stands for symmetry and the number of laminae in a [ (0 0 /90 0 ) ns ] laminate is 4n. The laminates containing repeating (?ة angle-plies can also be identified in a similar way. A general unsymmetric laminate may contain 0 0 laminae, 90 0 laminae and/or ؠlaminae stacked in an arbitrary manner. For example, [ (0 0 ) 4 / (90 0 ) 2 ], [(90 0 ) 2 / (30 0 ) 2 ] and [0 0 /90 0 /30 0 /60 0 ] are all general unsymmetric laminates. An unsymmetry may also be introduced by stacking laminae made of different composites. A [0 0 c/90 0 g /0 0 k] laminate consists of a top layer of 0 0 carbon fibre reinforced composite, a middle layer of 90 0 glass fibre reinforced composite and bottom layer of 0 0 Kevlar fibre reinforced composite and is an unsymmetric cross-ply hybrid laminate. Various such hybrid laminates can be prepared for practical applications choosing various combinations of layers of metallic materials, polymer composites, metal-matrix composites and ceramic composites. The ? ARALL? is a hybrid laminate consisting of alternate layers of aramid/epoxy composite and aluminium alloys. Aramid epoxy composites are commonly combined with carbon epoxy composites to make carbon-kevlar hybrid composites to obtain a cost effective composite with superior compressive and impact resistant properties. Kevlar fibres are inexpensive compared to carbon fibres and are effective in resisting the impact forces. Carbon fibres, in turn, improve the compressive strength in the carbon-kevlar hybridization. 2.7 BIBLIOGRAPHY 1. J.W. Weeton, D.M. Peters and K.L. Thomas (Eds.), Engineer`s Guide to Composite Materials, American Society for Metals, Metals Park, Ohio, 1987. 2. S.M. Lee (Ed.), Encyclopedia of Composites, Vols. 1-4, VCH Publications, New York, 1990-1991. 3. R. Smith, Resin Systems, in Processing and Fabrication Technology, Delware Composites Design Encyclopedia, Vol.3 (Eds. M.G. Bader, W. Smith, A.B. Isham, J.A. Rolston and A.B. Metzner), Technomic Publishing Co., Inc. Lancaster, 1990, p.15. 4. B.C. Hoskin and A.A. Baker (Eds.), Composite Materials for Aircraft Structure, AIAA Education Series, American Institute of Aeronautics and Astronautics Inc., New York, 1986. 2.8 EXERCISES 1. Why fibres are preferred to other reinforcements ? What are whiskers ? Describe how fibres are fabricated using vapour deposition processes. 2. What are carbon-carbon composites and how they are produced ? Why they are recommended for high temperature applications ? 3. Write a note on metal matrix composites. 4. Write a note on ceramic matrix composites. 5. Describe the characteristics of a couple of thermosets and thermoplastics that are used for making composites for aerospace applications. 6. Why Kevlar fibres are recommended for strength and impact based structural designs? What are the basic differences between organic and other fibres ? CHAPTER - 3 COMPOSITE MANUFACTURING 3.1 INTRODUCTION 3.2 MOULDING PROCESS FOR POLYMER MATRIX COMPOSITES 3.2.1 Matched-die Mould Methods 3.2.2 Contact Mould Methods 3.2.3 Filament Winding 3.2.4 Pultrusion 3.3 FABRICATION PROCESSES FOR METAL MATRIX COMPOSITES 3.3.1 Diffusion Bonding 3.3.2 Powder Metallurgy Process 3.3.3 Casting 3.4 FABRICATION PROCESS FOR CERAMIC MATRIX COMPOSITES 3.5 MACHINING 3.6 JOINING 3.7 BIBLIOGRAPHY 3.8 EXERCISES 3.1 INTRODUCTION Manufacturing is a very broad discipline and encompasses several processes such as fabrication, machining and joining. The fabrication methodology of a composite part depends mainly on three factors: (i) the characteristics of constituent matrices and reinforcements, (ii) the shapes, sizes and engineering details of products and (iii) end uses. The composite products are too many and cover a very wide domain of applications ranging from an engine valve, or a printed circuit board laminate, or a large-size boat hull or to an aircraft wing. The fabrication technique varies from one product to the other. The matrix types (i.e., whether they are plastics, metals or ceramics) play a dominant role in the selection of a fabrication process. Similar process cannot be adapted to fabricate an engine blade made with fibre reinforced plastics and metal matrix composites. The process parameters may also have to be modified, even when one uses the same matrix type, but two different matrices. For example, the processing with phenolics requires additional heating, whereas epoxies can be processed under ambient conditions. Particulate reinforcements and short fibres are mixed with resin to produce either bulk moulding composite compounds (BMCs) or sheet moulding composite compounds (SMCs) which are then used as base materials to fabricate composite parts. One method commonly used with BMCs is the injection moulding in which the BMC is heated and then injected into the mould cavity. On the other hand, the comparable moulding method used for woven fibre fabrics is the resin injection moulding. The process parameters like temperature, injcection pressure and curing time vary from one method to the other. Moreover, a composite car body panel, though highly curved and complex in shape may be compression moulded, while a spar stiffened helicopter rotor blade may have to be fabricated using filament winding and other moulding methods. Further, the accuracy and sophistication required to fabricate an aircraft composite wing section may not be necessary while fabricating a composite bridge deck or a silo. The main purpose of this chapter is to outline briefly the basic features of common composite fabrication methods. No attempt is made to elaborate the actual fabrication procedure of a particular composite component, as this is beyond the scope of the present book. 3.2 MOULDING PROCESS FOR POLYMER MATRIX COMPOSITES Important moulding methods for fabrication of polymer matrix composite structural parts may be classified under matched die mould, contact mould (also called open mould), filament winding and pultrusion. There are two important stages in all moulding processes: laying and curing. The laying is the process in which moulding materials are laid on a mould in the mould cavity or on the mould surface that conforms to the shape of the part to be fabricated. The process of curing helps the resin to set, thereby providing the fabricated part a stable structural form. The moulding materials are obviously reinforced plastics, either in the form of separate resin and reinforcements, or in the form of composites like bulk moulding compound (BMC), sheet moulding compound (SMC) or prepregs. These composite forms of moulding materials eliminate the mess of using wet resins during the lay-up process. A bulk moulding composite compound is prepared by mixing chopped strands or particulate reinforcements with a pre-mixed resin (normally polyester resin) paste. Fillers, thickeners, catalysts and other additives are also blended. The final mix is, either in the bulk form or extruded in the form of a rope and then stored for future use in the matched- die compression moulding process. A sheet moulding compound is, on the other hand, fabricated in the form of a sheet. Chopped strands or other particulate reinforcements are sandwiched between two layers of polyester resin pastes coated on two polyethylene carrier films. This resin-reinforcement-resin sandwich covered on two sides by the carrier films is thoroughly compacted by forcing it through a series of rollers and stored as rolls. Sometimes, continuous rovings are also added in between two resin layers to improve directional properties. Carrier films are removed prior to using them in moulds. BMCs and SMCs can be used in several moulding processes. The temperature, pressure and curing time of BMCs and SMCs are dependent on the type of the compound and the shape of the finished part. Prepregs are prepared by pre-impregnating fibre fabrics with resin. The system is only partially cured. The final cure takes place during the moulding process. Prepegs can be used in all important moulding processes. However, high quality products are realized, when curing is done in an autoclave. Prepegs yield superior products having all kinds of shapes with uniform resin content and consistent quality. Matched-die Mould Methods Several moulding techniques fall under the matched-die moulding method. The common feature in all these techniques is that the mould consists of two parts that form a cavity in between them. The shape of the cavity corresponds to that of the part to be moulded. The moulds are usually fabricated with cast iron, steel and aluminium alloy. Fibre reinforced plastics or wooden moulds are also used in some cold moulding processes. The mating surfaces of the moulds are first polished, cleaned and coated with a release agent. Next a gel coat is applied. The gel coat is a special resin that sticks to the surface of the moulded part during curing and provides it with an excellent surface finish. Pigments and other additives are also added to the gel coat resin for colouring as well as improving its resistance to wear, corrosion, heat, flame, weathering, etc. The gel coat is not provided, when the moulded part is to be adhesively bonded to another part, as the coat may not allow proper bonding. Both thermosetting and thermoplastic resins are used in the moulding materials. Thermoplastics in general and some thermosets in particular need process temperature higher than the ambient temperature. This is met by heating either the mould or the moulding. Venting ports are provided in the moulds for escape of excess resin and volatile matters. Compression moulding is the most commonly used matched-die moulding method. It is employed in fabrication of automobile body panels, housings for electrical appliances and machines, covers, sinks and several other parts. Typical moulds are shown in Fig. 3.1. The moulds can have a single cavity or multiple cavities with complex curved shapes. Provision may exist to heat either or both the moulds. The pressure is applied by mounting the moulds in a mechanical or hydraulic press or by some external means. The precise application of pressure and temperature and their duration and cycles can be controlled. The process can also be easily automated. On application of pressure and temperature, the mould material softens and then flows and fills the mould cavity. Further, continuation of heat and pressure accelerates curing. The dimensions close to those of the desired finished part can be obtained in compression moulding. This reduces, to a great extent, subsequent trimming and machining. The moulding material may be a predetermined quantity of BMC, SMC, resin coated preforms/fabrics or prepregs. It is laid on the mould and then the moulds are closed. A barrier along the edge prevents the resin to flow out. The depth of the barrier also controls the thickness of the part. Heat and pressure are applied during curing. Once the curing is complete, the mould is opened and the part is removed. Some resins like polyesters and epoxies are highly exothermic and may not require external input of heat during curing. The moulding material with these resins can be cold pressed. Cold press moulding is relatively less expensive and is suitable, when a part is smaller in size and simpler in shape (flat or slightly curved panels). Cold stamping is also similar to compression moulding, but is normally used with thermoplastic sheets. The thermoplastic sheets are preheated, laid on the mould along with reinforcements and then cold pressed. An extension of this method to continuous production of fibre reinforced thermoplastic laminates is illustrated in Fig. 3.2. Alternate layers of fibre fabrics and thermoplastic films are fed through hot rollers that melt the resin and force it to penetrate and coat the individual fibres. The consolidated laminate is then passed through cold rollers which cure and harden the laminate. Conforming is also a matched-die moulding process specially developed to provide superior surface finish and durability to a composite part. A thermoformed thin thermoplastic layer is first placed on the mould (Fig. 3.3). The moulding composite material (BMC, SMC or resin-coated fabric) is then laid on the top of the thermoformed thermoplastic layer and hot pressed. The thermoplastic layer then sticks to the moulded part thereby providing it with a smooth surface having excellent properties. Various additives can be premixed in the thermoplastic layer to obtain desirable surface properties. Press moulding is again similar to the compression moulding process, but it is used to make flat, slightly curved and corrugated laminates. For production of good quality laminates with uniform resin content, a perforated release film (e.g., Teflon film with perforations every 50 mm) and a bleeder (e.g., glass cloth, jute cloth or absorbent paper) are placed on both sides of the composite part (Fig. 3.4). In some applications, a peel ply (e.g., nylon cloth) is also used to achieve the required surface finish. On the application of pressure the excess resin is squeezed out and passes through the pores of nylon cloth and perforations of Teflon film and gets absorbed by the bleeder. The pressure is applied normally at the time of gelation to avoid excess loss of resin and to allow uniform resin distribution. The uniform flow of excess resin out of the moulding can be achieved by applying vacuum to one of the surfaces (here in Fig. 3.4 either the top or the bottom surface) of the mould cavity. Injection moulding is a matched-die moulding process especially suitable for thermoplastic resin systems. Some thermosets can also be injection moulded. If the reinforcements are in the form of particles or very short fibres, they along with other additives, if any, can be premixed with resin. The mix is first heated in an injection chamber. The hot fluid mix is then forced into the closed mould cavity under high pressure and is allowed to cure. The cure part is removed after opening the mould. The method is suitable for fabrication of small to medium size parts such as valves, gears, instrument panels, etc. If the reinforcements are in the form of preforms or fibre fabrics, they are laid in the mould cavity and the fluid resin is then injected into the mould cavity. The process is known as resin injection moulding. The injection pressure helps the resin to infiltrate through the fibre lay-up. A vacuum is also applied to the mould cavity to facilitate the penetration and even distribution of resin. For larger mouldings such as boat hulls, resin is injected at several locations. Also, cold cure resins of low viscosity and long gel time are preferred, as the injection time is longer in such applications. In the reaction injection moulding, measured quantities of two liquid precursors such as a polyol and an isocyanate are mixed in a chamber and then injected into the hot mould cavity containing pre-laid reinforcements. Chopped strands and particulate reinforcements can also be blended with precursors prior to injection. The process is normally used for polymethane based systems and requires a relatively low injection pressure. It is highly suitable for mass production of composite parts. 3.2.2 Contact Mould Methods Contact mould methods are also known as open mould processes. In the open mould, there is only one mould (male or female) and as the name suggests the mould surface is open. A composite part is fabricated in contact with the open surface, and the shape of the open surface conforms to that of the moulded part (Fig. 3.5). The moulds are normally fabricated from cast iron, steel and aluminium alloy for application in hot processes, and fibre reinforced plastics and wood for cold processes. The mould surface is cleaned and polished prior to moulding and is filled up with coating of a release agent and a gel. The moulding materials are normally resin-coated woven rovings and fabrics, chopped strand mats and prepregs. These are laid on the mould by either the hand lay-up process or the spray-up process. In the hand lay-up process woven rovings, fabrics and/or chopped strand mats are placed layerwise on the release agent and gel coated mould surface. After laying of each layer, it is coated with resin using a brush or a spray gun. Some time gap is allowed for the applied resin coat to gel, before laying the next layer and applying resin to it. Squeegees or rollers are used for uniform distribution of resin and consolidation for the laminate. In the spray-up process, chopped strands or particulate reinforcements and the resin are sprayed separately to the mould surface. In some applications, continuous fibre strands are fed to a combined chopper and spray gun system by which chopped fibres and resin are sprayed simultaneously to the mould. After completion of the laying up process, the moulding is allowed to cure. Curing is done either at room temperature conditions or by heating the mould assembly in an oven. In the case of prepregs the hand lay-up process is employed. However prepregs do not require additional resin coating. For compact and voidless finished products of higher mechanical properties, other improved moulding processes such as vacuum bag moulding, pressure bag moulding processes such as vacuum bag moulding, pressure bag moulding and autoclave moulding are preferred. An improved open mould set-up with pre-curing lay-up details that are generally employed in a vacuum bag moulding, is shown in Fig. 3.6. The whole moulding system is covered with a flexible vacuum bagging film made of nylon or neoprene rubber. The edge of the bag is sealed using vacuum sealing compounds. Layers of vent cloth, perforated Teflon film, fiberglass bleeders, Teflon coated fiberglass fabrics and nylon peel ply, in that order, are kept between the vacuum bag and the composite lay- up. When vacuum is applied, these materials allow attainment of uniform vacuum throughout and provides path for escape of volatiles, trapped air and excess resin from the composite lay-up. The vacuum also induces uniform atmospheric pressure on the top surface of the mould assembly and thereby helps in uniform distribution of resin and futher consolidation of the laminate. It also provides better finished surfaces. The whole assembly can be put in an oven or an external heating arrangement can be made, if a high temperature curing is needed. In the case of pressure moulding, the vacuum bag is replaced by a pressure bag and the whole system is covered by a pressure plate (Fig. 3.7). The required pressure is then applied through an inlet pipe located at the cover plate. In this method, it is possible to apply pressure higher than one atmosphere. The higher pressure ensures proper consolidation and densification of the composite lay-up. However, the method cannot be applied to a male mould. Autoclave moulding is a highly sophisticated process in which controlled temperature and pressure can be applied. In addition, vacuum is also applied to suck volatile matters and entrapped air or gases. The whole assembly as shown in Fig. 3.6 is put inside an autoclave (Fig. 3.8) which is a pressurized cylindrical hot chamber. Curing takes place in presence of simultaneous pressure and temperature. After curing, the mould is taken out of the autoclave and the cured composite is laid with prepregs, as it permits controlled variation of prescribed temperature and pressure with respect to time. It yields highly densified and voidless quality products and is therefore greatly favoured in fabrication of all major aerospace components like aircraft wing parts, control surfaces, helicopter blades, filament wound rocket cases, pressure bottles, etc. 3.2.3 Filament Winding The filament winding process is employed for fabrication of a continuous fibre reinforced composite structure having an axis of revolution. Common examples of such structures are tubes, pipes, cylindrical tanks, pressure vessels, rocket motor cases, etc. Continuous fibre strands or rovings are first coated with resin in a resin bath and then fed through rollers to squeeze out excess resin and finally wound, under constant tension, around a collapsible mandrel. The outer diameter of the mandrel corresponds to the inner diameter of the part to be fabricated. The mandrel is usually made of steel. However, other materials like plastic foam and rubber are also used in fabrication of some mandrels. A steel mandrel can be so designed that it can be dismantled mechanically and removed part by part without damaging the filament wound composite part. Some foam mandrels can be chemically dissolved. An inflated rubber mandrel can be collapsed by deflating it. The mandrel is positioned, either horizontally (for helical winding) or vertically (for polar winding), on a carriage that moved back and forth along the direction parallel to the rotational axis. In addition to the translational (axial) motion induced by the carriage, the mandrel can also rotate about its own axis. Both rotational and axial motions of the mandrel can be properly controlled either manually or using an automatic system. The rotation of the mandrel about its axis of revolution facilitates winding of filaments on the outer surface of the mandrel. There are basically two types of filament winding patterns: helical winding and biaxial winding. In the helical winding (Fig. 3.9) a constant angle ؠ(known as helical angle) is maintained by controlling the rotational and axial motions of the mandrel. By reversing both axial and rotational motions, the filaments are wound with a minus helical angle, -خ Structural components having circular cylindrical shapes like tubes, pipes and cylinders are normally fabricated with alternating helical angle +ئnbsp; and ?خ When the filaments are wound at an angle =ؠ 90 0 , the winding is called hoop winding. Similarly, when ؽ 0 , it is termed as axial winding. In the biaxial winding pattern, there may exist a combination of either hoop winding and axial winding pattern commonly employed for filament winding of circular cylindrical closed end vessels (with or without small end openings), such as motor cases and pressure tanks, is known as polar winding or polar wrap (Fig.3.10). A combination of both polar and hoop winding is normally provided for proper strengthening in the circumferential direction. After winding is complete, the mandrel is removed from the carriage and placed in an oven, if required, for curing. Filament wound products for aerospace applications are normally cured in an autoclave. For such curing, vacuum bagging as described earlier, of the mandrel assembly has to be carried out before placing it in an autoclave. After curing, the mandrel is dismantled and the finished part is removed. Filament winding yields a component with a high degree of fibre volume fraction. Prepreg tapes can also be wound in similar ways. Filament winding can also be used, when the shape of the part is not a shell of revolution. The winding is first carried out on an inflated mandrel made of rubber. The mandrel is deflated, once winding is complete. The deflated filament wound component is placed in a closed match-die mould and the mandrel is inflated again to apply pressure from inside. The mould is simultaneously heated to facilitate curing. The cured finished part has the outer shape same as the inner shape of the mould. Several non-circular sections (e.g., box section and airfoil sections) can be fabricated using this technique. Braiding is another form of filament winding process and is employed in fabrication of bars, tubes, bends, etc. with both circular and non-circular hollow sections. It is carried out on an axially moving mandrel which is positioned through the central hole of a rotating ring. A number of spools containing continuous rovings are mounted on the ring. The fibres are pulled out from these spools and wound around the moving mandrel creating an inter-woven winding pattern which provides high interlaminar properties. 3.2.4 Pultrusion Pultusion, to some extent, is analogous to the metal extrusion process. In the pultrusion process continuous fibre reinforced structural sections can be produced by pulling the resin-coated filaments through a die unlike in the metal extrusion process, where hot metallic rods, bars and flats are pushed through a die to produce extruded parts. The pultrusion process is schematically shown in Fig. 3.11. Continuous fibre strands taken from a number of spools are sequentially pulled through a resin bath, a shaping guide and a hot die (or a cold die and an oven). The fibres are coated in the resin bath and the excess resin is squeezed out. The shaping guide provides a gradual change from a simpler to a more complex pre-formed shape close to that of the pultruded part. For example, to obtain a pultruded channel section, a flat form with no flanges is gradually changed to a channel section of desired dimensions. The final shape is realized when the preformed shape is pulled through a hot die and gets cured. Continuous strand mats and woven fabrics can also be pulled along with filament strands to provide better transverse properties to the pultruded sections. The die is a very critical component in the fabrication process. It is usually made of chromium plated steel and should have a highly smooth surface. A smooth surface inhibits sticking of the resin at the entry segment, where only the gelation of resin, but not curing has been initiated. Thermosets like epoxies and polyesters are normally used in the pultrusion process. Phenolics can also be used, but have to be preheated. Experimentation with thermoplastics has also been carried out. Pultrusion is a continuous process and therefore provides scope for automation. A pultursion machine ?SPACETRUDER?, designed and developed at Vikram Sarabhai Space Centre, Trivandrum, India is shown in Fig. 3.12. It can produce continuous lengths of FRP sections such as rounds, square bars, channels, angles, etc. using glass, carbon or aramid fibres and epoxy or polyester resins. 3.3 FABRICATION PROCESSES FOR METAL MATRIX COMPOSITES Aluminium, magnesium, titanium and nickel alloys are commonly used as metal matrices, although several other matrix materials including super alloys have also been used. Both metal and ceramic reinforcements are employed. The choice of a particular matrix-reinforcement system is mainly controlled by the end use of the fabricated composite part. Several parameters influence the selection of a particular fabrication process. These are (i) types of matrices and reinforcements, (ii) the shape, size, orientation and distribution of reinforcements, (iii) the chemical, thermal and mechanical properties of reinforcements and matrices, (iv) shape, size and dimensional tolerances of the part and (v) finally the end use and cost-effectiveness. Compared to standard metallurgical processes, fabrication methods for metal matrix composites are much more complex and diverse. Some problems that are of major concern are the densification of the matrix while maintaining its purity, the control of reinforcement spacing and proper chemical bonding between the matrix and reinforcements. Based on the physical state of the matrix i.e., solid phase and liquid phase, fabrication processes can be grouped under solid phase processing and liquid phase processing. In the solid phase processing, the matrix is in the form of sheet, foil or powder. The diffusion bonding and power metallurgy processes are the two major solid phase processing techniques, while casting (also known as liquid metal infiltration) processes are related to liquid phase processing. Solid phase processing has certain advantages over liquid phase processing. The processing temperatures are lower, diffusion rates are slower and the reaction between reinforcements and the matrix is less severe. Secondary processes like forging, rolling, extrusion and superplastic forming are also important, as much care is needed to reduce damage to reinforcements. 3.3.1 Diffusion Bonding The diffusion bonding employs the matrix in the solid phase, in the form of sheet or foil. Composite laminates are produced by consolidating alternate layers of precursor wires or fibre mats and metal matrix sheets or foils under temperature and pressure (Fig.3.13). The precursor wires are collimated filaments held together with a fugitive organic binder. This is achieved either by winding binder-coated filaments onto a circular cylindrical mandrel or by spraying the binder on the filaments that are already wound on a mandrel. When the solvent is evaporated, the fibre-resin combination forms a rolled fibre mat on the mandrel surface. The binder resin in precursor wires and fibre mats decomposes at a high temperature without leaving any residue. Under temperature and pressure metal sheets or foils melt and diffuse through fibre layers to form a laminate. A multilayered laminate may have any desired stacking sequence. A monotape (i.e., a unidirectional lamina) in which a precursor layer or a fibre mat is sandwiched between two metal sheets or foils, forms the basic building block. Several complex composite components can be fabricated by stacking monotapes as per design requirements. The temperature, pressure and their duration are very critical for making good quality composites. Carbon fibres have been successfully combined with matrices like aluminium, magnesium, copper, tin, lead and silver to make a wide range of carbon fibre reinforced metal composites. A number of products ranging from flat plates to curved engine blades have been fabricated using the diffusion bonding technique. One interesting example is the 3.6m long high gain antenna boom that acted as a wave guide for the Hubble space telescope. The boom is made with diffusion bonded carbon fibre reinforced aluminium (AA 6061) composite and is of tubular cross-section with internal dimensional tolerances of ?0.15 mm. Several other composites, for example, boron, beryllium and steel fibres in aluminium alloy matrix have been manufactured using the diffusion bonding process. Large composite sheets can be produced by employing the vacuum hot rolling technique. 3.3.2 Powder Metallurgy Process Almost all metals and their alloys can be converted into powder form. Metal powders are commonly produced by atomization techniques. A stream of molten metal is disrupted either by impacting another fluid (gas or water) jet under high pressure or by applying mechanical forces and electrical fields leading to formation of fine liquid metal droplets which then solidify resulting in fine powder particles. The inert gases, argon and nitrogen, are used in the gas atomization, and the resulting powder particles are smooth and spherical with 50-100?m diameter. The impact of very high intensity gas pulse waves in supersonic and ultrasonic gas atomization can lower particle sizes to 10?m. The water jet impact produces irregular particles (75-200 ?m diameter). Both gas and water contaminate particles with oxygen. In another method, known as vacuum atomization, the liquid metal supersaturated with gas under pressure is suddenly allowed to expand in vacuum causing the liquid to atomise and produce spherical powders with diameters ranging between 40-150?m. In the rotating electrode technique, a prealloyed electrode is rotated at a high speed (about 250 rps) while it is melted by an arc or plasma beam. Spherical droplets of the molten material are ejected centrifugally and, on being cooled in an inert environment, produce high quality, spherical powders with 150-200?m diameter. The centrifugal atomization process combined with rapid solidification yields spherical powders less than 100?m diameter. Both the electrohydrodynamic atomization in which an electrical field is applied on the surface of a liquid metal to emit droplets and the spark erosion technique, where repetitive spark discharge between two electrodes immersed in a dielectric fluid produces metal vapour, thereby yielding very fine powders with diameter as low as 0.5 ?m or less. Powder metallurgy is a versatile process but its application to fabrication of metal matrix composites may not be straight-forward, especially because of the presence of reinforcement phase. There are quite a few composite fabrication techniques using continuous fibres of which two processes that use hot pressure bonding need special mention. In one process (known as powder cloth process), metal powder filled clothes are first produced by mixing metal matrix powders with an organic binder and then blending with a high purity Stoddard solution. On application of low heat, the Stoddard solution evaporates leaving behind a dough-like mixture which, on rolling, yields a metal powder cloth. Alternate layers of powder clothes and fibre mats, when hot press bonded, form a composite laminate (Fig.3.14). The binders usually burn out without leaving any residue. When the reinforcements are in the form of short fibres and particulates, metal matrix powder and reinforcements are thoroughly blended, and the blend is degassed to remove volatiles and then a composite ingot is formed by either hot pressing in vacuum or hot isostatic pressing. The composite ingot is subsequently used to fabricate structural components using secondary fabrication processes. Major problems are encountered in controlling the shape, size and distribution of reinforcements in the matrix. The alignment of short fibres, elongation of particulates i.e., a sphere changing to an oblate or nonspherical shape, and uniform dispersion or clustering are the common occurrences that influence the microstructure of the composite. Figure 3.15 shows an optical micrograph of SiC particulate (30% by volume)/AA2124 composite a) in the vacuum hot pressed condition showing the necklace structure of particulate reinforcement around the matrix particles and b) in the as extruded condition showing the elimination of necklace structure and improved distribution. In the thermal spray processes, metal powders, are deposited on the fibre substrates using either plasma spray or arc spray techniques and composites are subsequently produced by consolidating these metal matrix coated fibres under heat and pressure. The plasma spray technique is employed to deposit spherical metal powders that are injected in the plasma stream (the temperature is about 10,000K and the traveling speed is around Mach 3) within the throat of the gun. The powder particle size is very critical, because the powder should melt, but not vapourise before it reaches the substrate. The arc spray technique uses continuous metal matrix wires of 0.16-0.32 cm diameter instead of metal powders. Two wires of opposite charge are fed through an arc spray gun. The electric arc produced between the wire tips causes the tips to melt. An argon gas stream that passes through the gun and between the wires, carries with it droplets of molten metal and deposit them on the fibre substrates. Both plasma spray and arc spray techniques have been used to produce composite monotapes by winding continuous fibres on a mandrel and then spraying metal matrix powders on them. These monotapes are subsequently used to fabricate structural components using the diffusion bonding process. The powder metallurgy process has been used to produce composites such as boron, carbon and borsic fibres with aluminium alloy, SiC fibres with cromium alloys, boron and Al 2 O 3 fibres with titanium alloy, tungsten and molybdenum fibres with nickel alloy and several other composite systems. 3.3.3 Casting Casting or liquid infiltration is the process in which molten matrix is infilatrated into a stack of continuous fibre reinforcements or discontinuous reinforcements (short fibres and particulates) and is then allowed to solidify between the inter-reinforcement spaces. In the case of discontinuous reinforcements, they can also be pre-mixed with molten matrices prior to casting, using techniques such as mechanical agitation, mixing by injection with an injection gun, centrifugal dispersion and dispersion of pellets (formed by compressing the metal matrix and reinforcements) in a mildly agitated melt. This pre-mix or the composite slurry is used for subsequent casting. There are several casting methods that can be used to produce metal matrix composite components. Some important casting methods are sand and die castings, pressure die castings, centrifugal casting, squeeze casting and investment casting. In the sand and die casting process, the preferential concentration of discontinuous reinforcements, either at the top or at the bottom depending on their densities lower or higher than the metal matrices, takes place in view of the slow cooling rate of sand moulds. A more uniform dispersion or dispersoids can be achieved by agitating the mix, cooling the mould or employing a metal mould. The pressure die casting produces relatively void free composites and permits fabrication of large-size parts with intricate shapes. In the centrifugal casting, solidification takes place in a rotating mould. In this process, the centrifugal acceleration forces the heavier discontinuous reinforcements to concentrate near the outer periphery and the lighter ones lie closer to the axis of rotation. Squeeze casting (also known as liquid forging) is the process in which the molten matrix is infiltrated, under high pressure, onto a preheated stack of discontinuous reinforcements or fibre performs laid on a metal die. Solidification takes place also under pressure. Several critical components have been developed using squeeze casting. The Toyota piston, made of ceramic fibre and aluminium matrix, is one such example. In the investment casting, continuous fibre reinforcements are laid using usual filament winding or prepreg laying procedures. Composites are then produced by infiltrating the lay-up with a molten matrix under pressure or vacuum. Casting is the most commonly used process for manufacture of metal matrix composites. Figure 3.16 exhibits a graphite/aluminium composite ingot and its composite products. 3.4 FABRICATION PROCESS FOR CERAMIC MATRIX COMPOSITES Ceramic such as glass, glass-ceramics, borides, carbides, graphite, nitrides and silicates reinforced with both metallic and ceramic particles, whiskers and fibres provide enhanced strength and toughness even at high temperatures. Some ceramic composites, especially, carbon-carbon composites exhibit remarkable strength properties at a temperature as high as 2000 0 C or more. The fabrication processes for ceramic matrix composites are, in many ways, similar to those for metal matrix composites. As in the powder metallurgy processes for MMCs, short fibres and particulate reinforcements are mixed with ceramic powders and then hot pressed to produce CMC products. Common dispersion particles are SiC, TiC, BN and ZrO 2 . SiC whiskers are very commonly used to reinforce matrices such as glass, ZrO 2 , B 4 C, Al 2 O 3 , cordierite, Si 3 N 4 and several other ceramics. They are rod or needle shaped single crystal short fibres with diameter ranging from 0.1-5.0 ?m and length 5-200 ?m. SiC whiskers which apparently look like powders and ceramic powders are thoroughly mixed and then hot pressed to make composite. The proper mixing of SiC whiskers and ceramic powders is critical to produce composites with desirable properties. This is usually carried out using high shear mixing, ultrasonic dispersion, milling and several other mixing methods. The cutting tools for high nickel alloys employ SiC whisker reinforced alumina. They provide cutting rates up to ten times higher than conventional tools. Due to excellent wear resistance of SiC whisker reinforced composites at high temperatures, these materials find wide uses in dies for metal extrusion, heat engine valves, grit blast nozzles and other high temperature applications. The plasma spray techniques, as employed in metal matrix composites, has also been used in ceramic composites. Hot pressing and sintering of ceramic materials normally require high temperature and pressure at which reinforcements degrade due to chemical reactions on the reinforcement-matrix interface. A number of glass systems such as lithium aluminosilicate (LAS), magnesium aluminosilicate (MAS), barium magnesium aluminosilicate (BMAS), etc. have been hot pressed at relatively lower temperature without causing any damage to the reinforcement. Transfer moulding and injection moulding techniques have also been successfully carried out using glass systems. Infiltration of molten ceramics is also a common fabrication process for ceramic matrix composites. The high melting points of ceramics, however, may degrade the reinforcements. One way to circumvent this problem is to use polymer precursors that bring down the process temperature. However, during the conversion of a polymer precursor to the ceramic matrix, a lot of volatile matters escape causing shrinkage of the matrix. The matrix also becomes porous. The porosity can be reduced to a large extent by reimpregnation. Several precursor polymers have been studied to produce SiC and Si 3 N 4 matrices. In the sol-gel technique, gels are used to aid uniform infiltration of matrices. For example, tetrafunctional alkoxides are employed to infiltrate oxide matrices. This reduces the fibre damage to some extent due to lower viscosity of the gel-mixed slurry, but the shrinkage problem remains. Reaction sintering (also known as reaction bonding or reaction forming) eliminates some of the problems associated with hot press sintering and liquid infiltration, such as fibre damage, matrix shrinkage, porosity, etc. In this process, ceramic matrices are reaction formed. A typical example is the Si 3 N 4 matrix, which is reaction formed by nitriding Si powder. The SiC matrix has also been successfully reaction bonded. The reaction sintering process seems to have great potential, although the process is not yet fully developed. The main drawback is that the resulting composite may have excessive porosity. In the chemical vapour deposition process (more often called as chemical vapour infiltration), a ceramic matrix is chemically vapour deposited on the surfaces within a fibre preform. The preform is kept in a high temperature furnace (reactor). A carrier gas (H 2 , Ar, He, etc.) stream passes through a vessel containing gaseous reagents and carries their vapour into the reactor. In the reactor, the chemical reaction of gaseous reagents leads to the formation and deposition of ceramic matrix vapour on the heated surface of the preform. Other reaction powders diffuse out of the preform and are carried by the flowing gas stream out of the furnace. The deposition process continues, until all the inter fibre spaces are filled up resulting in a homogeneous and more or less void free composite. The main advantage of this process is that it causes minimum damage to the fibres, as the process temperatures and pressures are relatively lower compared to those in hot press sintering and liquid infiltration. Also this process permits fabrication of composite parts with irregular shapes. The deposition reaction may be of reduction, thermal decomposition or displacement type. A typical reaction such as CH 3 Si Cl 3 ? SiC + 3 HCL is responsible for deposition of SiC vapour. The reagents and vapour deposition temperatures for a few ceramic matrices are listed in Table 3.1. Plasma has also been used to assist chemical vapour deposition, for example, as in the case of SiO 2 matrix. The deposition process is carried out under any of these conditions: (i) maintaining a uniform temperature through out the preform i.e., isothermal condition, (ii) providing a thermal gradient through the thickness of the preform, (iii) isothermal, but forcing the flow of reactant gases into the preform, (iv) both with thermal gradient and forced flow and (v) with pulsed flow, i.e., cyclic evacuation and filling of the reactor chamber with reactants. Carbon-carbon composites are fabricated by either resin impregnation and subsequent carbonization or chemical deposition of carbon vapour. The latter process has Table 3.1: Typical ceramic matrices, reagents and vapour deposition temperatures Ceramic Matrix Reagents Deposition temperature, o C ZrB 2 ZrCl 4 , BCl 3 , H 2 1000-1500 HfB 2 HfCl 4 , BCl 3 , H 2 1000-1600 TiC TiCl 4 , CH 4 , H 2 900-1600 SiC CH 3 SiCl 3 , H 2 1000-1600 B 4 C BCl 3 , CH 4 , H 2 1200-1400 Si 3 N 4 SiCl 4 , NH 3, H 2 1000-1550 BN BCl 3 , NH 3 , H 2 1000-1300 Al 2 O 3 AlCl 3 , CO 2 , H 2 500-1100 Cr 2 O 3 Cr (CO) 6 , O 2 400-600 SiO 2 SiH 4 , CO 2 , H 2 200-600 * Y 2 O 3 YCl 3 , CO 2 , H 2 1200 ZrO 2 ZrCl 4 , CO 2 , H 2 800-1200 TiSi 2 TiCl 4 , SiCl 4 , H 2 1400 C CH 4 900-2250 * plasma assisted been found to yield superior carbon-carbon composites and has been used to produce aerospace components such as aircraft brake discs and engine exit nozzles, nose cones, rotors, combusters, etc. A hydrocarbon reagent, e.g., CH 4 may be used along with hydrogen, nitrogen or any inert gas for chemical deposition of carbon vapour on a carbon fibre substrate (perform). 3.5 MACHINING Machining is an important stage of the overall manufacturing process. It helps to realize finished products with specified dimensions, surface finish and tolerances. Conventional machining processes require direct contact between the cutting tool and the part to be machined. The quality of the machined part and the tool wear are two major concerns in these machining processes. The performance and integrity of the machined part depend primarily on the quality of machining. The removal of material in a metallic alloy is based on a shear process. The material removal in a composite may be totally different from that in metals and their alloys. The application of conventional metal cutting tools in composites leads to flaws such as fibre damage, delamination and cracking which are not encountered in metal cutting. The presence of these defects and tool wear has been observed while applying the machining processes such as shearing, abrasive cutting, grinding, profiling, punching and drilling on fibre reinforced composites. Even the use of high speed drilling has not been found to be very much effective in reducing the fibre damage or tool wear. Tool wear has always been a major problem in machining composite parts. It depends on the feed rate, cutting speed, cutting direction, temperature, and relative hardness properties of the tool material and the composite part to be machined as well as several other parameters. The highest tool wear is observed for a particular combination of feed rate and cutting speed. The tool wear rate is found to be inversely proportional to the hardness of the tool, when the tool is harder than the composite to be machined. Diamond tools normally exhibit a longer tool life. Tungsten-carbide tools and silicon-carbide grinding wheels are also useful. It is difficult to machine boron fibre composites because of extreme hardness of boron fibres. Some special problems arise during machining of aramid fibres. These can be tackled using specially designed tools which are commercially available. There are several non-conventional machining methods which avoid direct contact between the machine and the work piece, thereby eliminating the problem of tool wear and improving the quality of the machined component. A narrow laser beam of 0.1 mm diameter or less with a power in excess of 10 8 W/cm 2 can be directed to cut various composite materials. The CO 2 laser system with 10.6 ?m wavelength has been successfully used to machine fibre reinforced polymer composites. A ND:YAG (neodymium/yttriumaluminium-garnet) laser system with wavelength of 1.06?m and 200 pulses per second, can be used to cut metal composites. The water jet cutting process employs a jet cutting nozzle of 0.13 mm diameter and water pressure in excess of 350 MPa. In the abrasive water jet process, abrasive particles are added to the jet stream to facilitate the cutting process. Both these water jet methods are used in cutting both metallic and non-metallic composites. The electrical discharge machining process is based on erosion caused by an electrical spark developed between an electrode and the part to be machined in the presence of a dielectric fluid and has been effectively used to machine composite parts which are electrically conductive. Other non-conventional machining processes such as electrochemical machining, electron beam machining and ultrasonic machining also have good potential in composite applications. One of the primary disadvantage with all these non-conventional machining processes is that it may not be possible to induce shape changes, as is done by conventional processes. However, these unconventional processes, in most cases, lead to high quality cuts with minimum fibre damage, delamination and cracking as well as excellent surface finish. 3.6 JOINING A joint is an essential element in a structural system. However, it is the weakest link, and therefore the selection of a particular type of joint, its fabrication and/or assembly need careful consideration. There are two basic types of joints, bonded and mechanically fastened, which are also applicable to all composite systems. Bonded joints are preferred, because of their efficient load transfer which is primarily through shear in the bond layer. The bond shear stress as well as direct or induced peel stresses can be controlled by configuring and designing the joint properly. Some of the common bonded joint configurations are illustrated in Fig. 3.17. The mechanically fastened joints (bolted, riveted, etc.), on the other hand, are relatively inefficient because of high stress concentrations around bolt holes. The process of load transfer in a mechanically fastened joint, especially involving deformable bolts or pins and composites, is very complex. The modes of failure in a typical single-pin bolt joint are shown in Fig.3.18. The residual strength of such a joint is quite low and is usually less than a half of the composite laminate strength. However, mechanical fastening is a practical necessity in view of ease of repeated dismantling and assembly as well as repairability. Bonded joints are again of two types ? adhesive bonded and fusion bonded or welded. An adhesive bonded joint normally involves application of an adhesive layer on the overlapping surfaces of the adherends. The adhesive layer is usually a thermosetting resin. The bonded joints are, therefore, highly effective for composites with thermosetting resins. They are also good for metal and even ceramic joints, provided they are not exposed to temperatures higher than the tolerable limit of the adhesive. The surface treatment of adherends, prior to the application of the adhesice layer, is necessary to achieve better adhesion properties. But the thermosetting adhesive layer is not compatible with thermoplastic composites. Thermoplastic composite adherends need to be fusion bonded. In fusion bonding no new material is added. The process involves surface preparation, heating and melting of the bond surfaces, pressing for intermolecular diffusion and entanglement of the polymer chains and cooling for solidification. The bond surfaces are treated mechanically and/or chemically to remove all contaminations including loose particles, dirt, release agent, oil, etc. The heating and melting are carried out using any of the welding techniques employed in plastics. In the hot-plate welding process, the surfaces to be bonded are heated by making contact with a hot plate. The hot gas welding normally employs a hot gas. In the case of a butt joint, similar to that of a metal joint, a thermoplastic welding rod is also used to fill the joining gaps. The resistance implant welding rod is also used to fill the joining gaps. The resistance implant welding is an internal heating method and requires an electrical resistive element to be embedded on the bond surface. Heating is done by passing an electric current through the resistive element. There are several other ways of external and internal heating, namely induction welding, laser and infrared heating, dielectric and microwave heating, friction heating, vibration welding and ultrasonic welding. For joining metal matrix composite parts, in addition to adhesively bonded and mechanically fastened joints, several other joining methods that are used in joining metals and their alloys can be employed. These are brazing, soldering, diffusion welding, fusion welding, resistance welding, ultrasonic welding, laser welding and electronic beam welding. These techniques have been employed in several metal matrix composite systems, e.g., B/Al, C/Al, Borsic/Al, B/Ti, SiC/Ti, W/Ti, C/Mg, Al 2 O 3 /Mg and several others. The joint efficiencies are found to vary between 25-60%. However, higher efficiencies have also been noted in some cases, and a joint efficiency as high as 98% has been realized in the fusion welding of Al 2 O 3 fibres. The fusion bonding or welding of ceramic matrix composite parts with ceramic marix and metal matrix composite parts, has not yet been properly investigated. 3.7 BIBLIOGRAPHY 1. G. Lubin (Ed.), Handbook of Composites, Van Nostrand Reinhold Co., NY, 1982. 2. Mel M. Schwrtz (Ed.), Fabrications of Composite Materials, American Society for Metals, Metals park, Ohio, 1985. 3. Mel M. Schwarts (Ed.), Composite Materials Handbook, McGraw Hill Book Co., NY, 1984. 4. S.M. Lee (Ed.), International Encyclopedia of Composites, Vols. 1-6, VCH Publications, New York, 1990-1991. 5. J.W. Weeton, D.M. Fosters and K.L. Thomas (Eds.), Engineers Guide to Composite Materialals, American Society of Metals, Metals Park, Ohio, 1987. 6. N.P. Cheremisinof (Ed.), Handbook of ceramics and Composites, Vol. 1, Marcel Dekker, Inc., NY, 1990. 3.8 EXERCISES 1. Describe briefly various moulding processes of composites. 2. Describe an appropriate method of fabrication of an open-ended rectangular thin-walled box section with the wall consisting of 0 0 /90 0 /?45 0 /90 0 /0 0 laminations. 3. Write notes on (i) filament winding and (ii) pultrusion. 4. Describe the fabrication processes for metal matrix composites. 5. Describe the fabrication processes for ceramic matrix composites. 6. Discuss the joining techniques for MMCs and CMCs. CHAPTER - 4 COMPOSITE PROPERTIES - MICROMECHANICS 4.1 INTRODUCTION 4.2 UNIDIRECTIONAL COMPOSITES 4.2.1 Elastic Properties (Engineering Constants) 4.2.2 Strength Properties of Unidirectional Composites 4.2.3 Hygrothermal Properties 4.3 PARTICULATE AND SHORT FIBRE COMPOSITES 4.3.1 Particulate Composites 4.3.2 Short Fibre Composites 4.4 BIBLIOGRAPHY 4.5 EXERCISES 4.1 INTRODUCTION The mechanical and hygrothermal properties of composites are of paramount importance in the design and analysis of composite structures. The mechanical properties constitute primarily the moduli and strength properties. The hygrothermal properties are coefficient of expansion due to moisture (β), misture diffusion coefficient (d), coefficient of thermal expansion (α), thermal conductivity (k) and heat capacity (c). Micromechanical analyses concern with the theoretical prediction of these properties of constituent fibres and matrices as well as several other parameters like the shape, size and distribution of fibres, fibre misalignment, fibre-matrix interface properties, void content, fibre fracture, matrix cracking and so on. The studies in micromechanics utilize micro- models, as the fibre diameters usually vary in the microscopic scale between 5-140 ?m. The micro-models should simulate the microstructure of a realistic composite, but that usually makes the models highly complex. The problems involving such complex models are normally tackled utilizing advanced analytical methods as well as numerical analysis techniques(finite element and finite difference methods). Even in the case of a complex model, a simplified idealization with a reasonably good approximation of the real composite is desirable otherwise it may lead to nowhere. It is not intended in this chapter to present the complete theoretical basis of various micro-models used for the analytical prediction of all composite properties. The presentation is limited to only a few simpler cases so as to acquaint the reader of the background of the development in this area. Additional micromechanics relations for unidirectional composites, that may find use in design applications, are listed in Table 4.1. Typical properties of some of the common fibres and matrices are listed in Tables 4.2 and 4.3, respectively. The composite properties of a few composite systems derived using some of the relations presented in this chapter are listed in Table 4.4. Tables 4.1 through 4.4 are included at the end of this chapter. 4.2 UNIDIRECTIONAL COMPOSITES 4.2.1 Elastic Properties (Engineering Constants) The stress-strain relation provides the basic interface between a material and a structure. For a one dimensional isotropic, elastic body, the Hooke's law σ = E ∈ defines the stress-strain behaviour. Here E is a material constant and is usually referred as elastic constant (engineering constant) or Young's modulus. Besides E, the other conventional engineering constant for a two-dimensional or three-dimensional isotropic body is Poisson's ratio ν. The shear modulus G is not independent, but is related to E and ν as G = E/2(1+ ν). A composite material is essentially heterogeneous in nature, therefore the engineering constants, defined above, for an isotropic material are not valid. We consider here a three-dimensional block of a unidirectional composite (Fig. 4.1), in which fibres are aligned along the x' 1 axis. The elastic behaviour for such a three-dimensional body is orthotropic, and the engineering constants are 11 E′ , 22 E′ , 33 E′ (three Young's moduli along three principal material axes x' 1 , x' 2 , x' 3 ), ν' 12 , ν' 13 , ν' 23 , ν' 21 , ν' 31 , ν' 32 , (six Poisson's ratios) and G' 12 , G' 13 , G' 23 , (three shear moduli). Of these, the first nine engineering constants i.e., three Young`s moduli and six Poisson ratios are not independent. Due to symmetry of compliances (see Eq. 6.18) these are related as given by 33 32 22 23 33 31 11 13 22 13 11 12 ; ; Ε′ ′ · Ε′ ′ Ε′ ′ · Ε′ ′ Ε′ ′ · Ε′ ′ ν ν ν ν ν ν (4.1) Note that, 12 22 11 13 33 11 23 33 22 21 11 22 31 11 33 32 22 33 ' ' / ' , ' ' / ' , ' ' / ' ' ' / ' , ' ' / ' , ' ' / ' ν ν ν ν ν ν · −∈ ∈ · −∈ ∈ · −∈ ∈ · −∈ ∈ · −∈ ∈ · −∈ ∈ (4.2) Here ν' 12 and ν' 13 are usually referred as major Poisson ratios. The 'mechanics of materials approach' provides convenient means to determine the composite elastic properties. It is assumed that the composite is void free, the fibre- matrix bond is perfect, the fibres are of uniform size and shape and are spaced regularly, and the material behaviour is linear and elastic. Consider a two-dimensional unidirectional lamina (Fig. 4.2), in which we define a small volume element which represents not only the micro-level structural details but also the overall behaviour of the composite. A simple representative volume element consists of an isotropic fibre embedded in an isotropic matrix (Fig. 4.2b). This volume element is further simplified as shown in Fig.4.2c, in which the fibre is assumed to have a rectangular cross-section with the same thickness as the matrix. The width ratio is chosen to be the same as the fibre volume fraction of the composite itself. The objective is to derive the composite properties (E' 11 , E' 22 , ν' 12 , G 12 ) in terms of the moduli, Poisson`s ratios and volume fractions of the fibre and the matrix. Longitudinal modulus , E' 11 The micro-model (Fig. 4.2c) is subjected to a uniaxial tensile stress σ` 11 as shown in Fig. 4.3. It is assumed that plane sections remain plane after deformation. Hence, ∈ ' 11 = ∈ ' 11f = ∈ ' 11m = ΔL/L and σ' 11f = E' 11f ∈ ' 11 , σ' 11m = E' 11m ∈ ' 11 , σ' 11 = E' 11 ∈ ' 11 (4.3) Now, σ' 11 W = σ' 11f W f + σ' 11m W m (4.4) Substituting Eq. (4.3) into Eq. (4.4) and rearranging, we have E' 11 = E' 11f W f / W + E' 11m W m / W (4.5) Noting that the volume fractions of the fibre and the matrix are V f = W f /W and V m = W m / W respectively, Eq. 4.5 reduces to E' 11 = E' 11f V f + E' 11m V m (4.6) Equation 4.6 defines the composite property as the 'weighted' sum of constituent properties and is often termed as the 'rule of mixture'. Transverse modulus, E' 22 The tensile stress σ' 22 is applied along the x' 2 direction (Fig. 4.4) and the same is assumed to act both on the fibre and the matrix. The strain on the fibre and the matrix are ∈ ' 22 f = σ' 22 / E' 22f and ∈ ' 22m = σ' 22 / E' 22m (4.7) also ∈ ' 22 = ΔW/W and ΔW = ∈ ' 22f (V f W) + ∈ ' 22 m (V m W) So, ∈ ' 22 = ΔW/W = ∈ ' 22 f V f + ∈ ' 22 m V m or, m m f V V 22 22 22 22 22 22 Ε′ ′ + Ε′ ′ · Ε′ ′ σ σ σ or, m m f f V V 22 22 22 1 Ε′ + Ε′ · Ε′ or, f m m f m f V V 22 22 22 22 Ε′ + Ε′ Ε′ Ε′ (4.8) Major Poisson 's Ratio, ν ' 12 The micro-model is stressed as in the case of determination of E` 11 (Fig. 4.3). The transverse contraction is noted as ΔW and is contributed by both the fibre and matrix. Thus, ΔW = (ΔW) f + (ΔW) m or, ΔW = W V f ν ' 12f ∈ ' 11 + W V m ν ' 12m ∈ ' 11 (4.9) Now , 11 22 12 ∈′ ∈′ − · ′ ν (4.10) and ∈ ' 22 = - ΔW/W (4.11) Combining Eqs. 4.9 through 4.11, one obtains 11 12 ∈′ ∆ · ′ W W ν or, ν ' 12 = V f ν ' 12f + V m ν ' 12 m (4.12) Inplane Shear Modulus, G ' 12 The micro-model is now subjected to a shear stress σ ' 12 as shown in Fig. 4.5, and both and the fibre and the matrix are assumed to experience the same shear stress. ∈ ' 12f = σ ' 12 / G ' 12f and ∈ ' 12m = σ ' 12 / G ' 12m (4.13) Now, Δ = ∈ ' 12 W = W V f ∈ ' 12f + W V m ∈ ' 12m or, ∈ ' 12 = V f ∈ ' 12f + V m ∈ ' 12m (4.14) also, 12 12 12 G′ ′ · ∈′ σ (4.15) Substituting Eqs. 4.13 and 4.15 into Eq. 4.14 and eliminating σ ' 12 from both sides, we get m m f f G V G V G 12 12 12 1 ′ + ′ · ′ or, f m m f m f V G V G G G G 12 12 12 12 12 ′ + ′ ′ ′ · ′ (4.16) Note that, for an isotropic fiber E' 11f = E' 22f = E f , ν' 12f = ν f and ) 1 ( 2 12 f f f f G G ν + Ε · · ′ (4.17) and for an isotropic matrix E' 11m = E' 22 m = E m , ν' 12m = ν m and ) 1 ( 2 12 m m m m G G ν + Ε · · ′ (4.18) Equations 4.6 and 4.12 provide a reasonably accurate estimate of longitudinal modulus E? 11 and ν? 12 , respectively. However, the transverse modulus E? 22 and the shear modulus G? 12 , estimated using Eqs. 4.8 and 4.16, are not so accurate mainly due to the reason that the stresses in both the fibre and the matrix are assumed to be the same. The volume element considered in the above mechanics of materials approach does not adequately represent the micro structure of the composite. Advanced analytical methods employ better micro-models along with the realistic material behaviour and boundry conditions. The analytical method using a self-consistent field model provides a better estimation of composite properties in comparison to the ?mechanics of materials? approach. The model assumes the composite to be a concentric cylinder (Fig. 4.6) in which a transversely isotropic matrix. Although the assumed micro-model is simple, it permits formulation of the problem based on the theory of elasticity so that it is possible to achieve the stress and strain variations in a realistic manner, and the relations for the effective composite properties are then derived. These properties are expressed as follows: f m m f m m f f m m m f f m m m f f V G G V V G V V 23 23 23 2 12 12 11 11 11 ) ( ) ( ) ( 4 ′ Κ′ − Κ′ + Κ′ ′ + Κ′ ′ Κ′ Κ′ ′ − ′ + Ε′ + Ε′ · Ε′ ν ν (4.19) f m m f m m f f m m f m f m m m f f V G G V V G V V 23 23 23 12 12 12 12 13 12 ) ( ) ( ) )( ( ′ Κ′ − Κ′ + Κ′ ′ − Κ′ ′ Κ′ − Κ′ ′ − ′ + ′ + ′ · ′ · ′ ν ν ν ν ν ν (4.20) ] ) ( ) [( ] ) ( ) [( 12 12 12 12 12 12 12 12 12 13 12 f m f m f f m f m f m V G G G G V G G G G G G G ′ − ′ − ′ + ′ ′ − ′ + ′ + ′ ′ · ′ · ′ (4.21) ] ) )( 2 ( 2 ) ( [ ] ) ( 2 ) ( [ 23 23 23 23 23 23 23 23 23 23 23 23 23 23 23 f m f m m m f f m m f m f m m f f m m m V G G G G G G G V G G G G G G G G ′ − ′ ′ + Κ′ − ′ ′ + ′ + ′ Κ′ ′ − ′ Κ′ + ′ ′ + ′ + ′ Κ′ ′ · ′ (4.22) f m f m f f m m f m m f V G V G G ) ( ) ( ) ( ) ( 23 23 23 Κ′ − Κ′ − ′ + Κ′ ′ Κ′ − Κ′ + Κ′ ′ + Κ′ · Κ′ (4.23) where K' is the plane strain bulk modulus. ) 4 1 4 1 /( 1 11 2 12 23 33 22 Ε′ ′ + ′ + Κ′ · Ε′ · Ε′ ν G (4.24) in which K', G' 23 , ν' 12 2 and E' 11 are defined in Eqs. 4.19 through 4.23. Κ′ Ε′ Ε′ Κ′ ′ − Ε′ Ε′ − Κ′ Ε′ · ′ 11 22 2 12 22 11 11 23 2 4 2 ν ν (4.25) with E' 11 , E' 22 , K', and ν' 12 defined in the above relations. Note that for isotropic fibres and matrices, ; ; 23 13 12 33 22 11 f f f f f f f f ν ν ν ν · ′ · ′ · ′ Ε · Ε′ · Ε′ · Ε′ , ) 1 ( 2 23 13 12 f f f f f f G G G G ν + Ε · · ′ · ′ · ′ m m m m m m m m ν ν ν ν · ′ · ′ · ′ Ε · Ε′ · Ε′ · Ε′ 23 13 12 33 22 11 ; ; , ) 1 ( 2 23 13 12 m m m m m m G G G G ν + Ε · · ′ · ′ · ′ and ) 2 1 ( 3 ; ) 2 1 ( 3 m m m m f f f f ν ν − Ε · Κ · Κ′ − Ε · Κ · Κ′ (4.26) 4.2.2 Strength Properties of Unidirectional Composites The strength of a material is defined as the level of stress at which failure occurs. The strength is a material constant. Most of the isotropic structural materials possess only one constant i.e., the uniaxial tensile strength. The shear strength is normally related to the tensile strength. A brittle isotropic material may have different strength values in tension and compression and may be termed as a two-constant material. In contrast, a composite is a multi-constant material. Referring to Fig. 4.1, it may be stated that a unidirectional composite may possess three normal strengths X' 11 , X' 22 , X' 33 and three shear strengths X' 12 , X' 13 , X' 23 . A normal strength may have different values in tension and compression, as the compressive force usually induces premature failure due to buckling of fibres which have extremely high slenderness ratio. So there are a total of nine independent strength constants X' 11 t , X' 22 t , X' 33 t , X' 11 c , X' 22 c , X' 33 c , X' 12 , X' 13, X' 23 . Attempts made using micromechanical analyses to determine these strength constants, met with little success. This is primarily due to the reason that the micro- models used in these analyses are grossly unrealistic. In fact, it is extremely difficult to simulate the realistic composite, as the initial microstructure changes continuously with the increase of applied stress and propagation of failure in the form of fibre fracture, matrix cracking, fibre-matrix debond and so on at several points located randomly within the composite. The brittleness of the fibre and the matrix aggravates the situation. This is illustrated in Fig. 4.7. Note that l c is the ineffective length. The presence of a single surface flaw in a brittle fibre causes the fibre to fracture at A (Fig. 4.7a). This induces high shear stresses and causes the fibre-matrix debond along the fibre direction (Fig. 4.7b). Also when a fibre fractures, a redistribution of stresses in the vicinity results in the tensile fracture of the adjacent fibre due to stress concentration. This process leads to the propagation of the crack in the direction transverse to the propagation of the crack in the direction transverse to the fibres (Fig. 4.7c). In fact, the final failure of a composite is resulted due to the cumulative damage caused by several micro and macro-level failures. Longitudinal Tensile Strength, X' 11 t A simple relation can be derived for the longitudinal tensile composite strength X' 11 t using the 'rule of mixtures' and is expressed as X' 11 t = X' 11f V f + X' 11m V m (4.27) Here it is assumed that, at a particular level of stress, all fibres fracture at the same time and the failure occurs in the same plane. That this idealization is grossly unrealistic has already been argued in the preceding paragraph. Now, let us examine the validity of Eq. 4.27 for two composite systems: (i) a carbon/epoxy composite, in which the fibre failure strain is less than the matrix failure strain, i.e., ∈ ' 11fu < ∈ ' 11mu (Fig. 4.8a) and (ii) carbon/carbon composite when ∈ ' 11fu > ∈ ' 11mu (Fig.4.8b). In these cases, both fibres and matrices are brittle. In the case of carbon/epoxy composite, when V f is much higher than V m , the strength of the composite is primarily controlled by the fibre fracture. Once the fibres fail, very little resistance is offered by the matrix. So, the strength of the composite is given by X' 11 t = X' 11f V ' f + σ ' 11m V m (4.28) where σ ' 11m is the stress level in the matrix when the fibres fracture. On the other hand, when V f is low, there is a sufficient amount of matrix to resist the load after the failure of fibres. In that case, X' 11 t = X' 11m t V m (4.29) It is therefore obvious that there exists a limiting value of V f at which the final failure changes from the fibre failure mode to the matrix failure mode. One may argue in a similar way to identify the possible failure mechanisms in the case of a carbon/carbon composite also (Fig. 4.8b) as well as in the cases of other composites in which either fibres or matrices or both are ductile. But the fact remains that there is no single relation which is able to define the uniaxial tensile strength of a realistic composite. However, Rosen's model of cumulative damage, which is based on the Weibull distribution of the strength-length relationship, provides somewhat better estimation of X' 11 t , when the fibres and the matrix exhibit brittle behaviour. This model assumes that the composite consists of N fibres of original length L and the weaker fibres fracture due to the applied tensile stress (Fig. 4.9). The original length is then divided into M segments, where each segment (bundle or link) is of length 1 c . Thus the composite forms a chain of M bundles (links). When the number of fibres are very large (high V f ) the strength of each bundle or chain link assumes the same value, i.e., the strength of the composite becomes equal to the link strength. This is expressed as ) / 1 exp( ) 1 ( / 1 1 11 β αβ β − · ′ − c f X (4.30) where α and β are material constants and can be determined experimentally. The tensile strength of the composite is then determined using β αβ / 1 1 11 11 ) 1 ( − · ′ · ′ e V X V X c f f f t (4.31) Note that l c is called the ineffective length or critical fibre length and is determined using the shear lag stress. It is given by 2 f c i X d l X ′ · (4.32) where X' f is the tensile fracture strength of the fibre, d is the diameter of the fibre and X i is the fibre-matrix interfacial shear strength. The longitudinal compressive strength X' 11 c of a unidirectional composite is primarily affected by the buckling of fibres. In a simplified model, the fibres are treated as isotropic thin plates lying in the x' 1 x' 2 plane (Fig. 4.10) and are supported on an isotrpic elastic medium (matrix). Fibres may buckle in two modes-extension and shear. In the extension mode, the matrix along the length of the fibre experiences alternate expansion and contraction, whereas the matrix is subjected to shearing deformation in the shear mode. The compressive strength is then determined employing the strain energy method. For the extensional mode, 2 / 1 11 ) 3 / )( / ( 2 m m f f f m m f c V V V V X Ε Ε Ε Ε + · ′ (4.33) or, m f m m f f f c V V V X Ε >> Ε Ε Ε ≈ ′ , ) 3 / ( 2 2 / 1 11 (4.34) and for the shear mode 11 X' / C m m G V · (4.35) The transverse strength properties normally depend on the matrix properties. The transverse tensile strength X' 22 t may also depend on the fibre-matrix interface strength, as illustrated in Fig. 4.11. The experimental data for some composites confirm that the transverse tensile strength enhances with the improvement in the fibre-matrix interface bond. The actual fracture path, however, is a mixture of fibre-matrix debond, fibre splitting and matrix cracking. A realistic model should be based on the variation of statistical data for all these failure modes. Two simple relations, for the prediction of the transverse tensile strength X' 22 t and transverse compressive strength X' 22 c of a unidirectional composite, are presented as follows: t m f m f f t X V V X )] / 1 )( ( 1 [ 22 22 Ε′ Ε − − − · ′ (4.36) c m f m f f c X V V X )] / 1 )( ( 1 [ 22 22 Ε′ Ε − − − · ′ (4.37) These relations assume that the transverse strength of a composite primarily depends on the strength of the matrix. 4.2.3 Hygrothermal Properties Transport Properties The evaluation of transport properties like moisture diffusivity, heat conductivity, electric conductivity, dielectric constant and magnetic permeability of a unidirectional composite follows the similar procedure when one uses a self-consistent field model. The resulting relations are, therefore, identical for all transport properties. The procedure is, hence, illustrated considering only one case ? the diffusion of moisture through a unidirectional composite. Consider the concentric cylindrical model as shown in Fig. 4.6. Both the fibre and the matrix are assumed to be moisture permeable. For example, aramid fibres and polymer matrices are moisture permeable. For the diffusion of moisture along the fibre direction (x' 1 axis), the moisture diffusion equation assumes the form 2 11 2 1 C C d t x ∂ ∂ ′ · ′ ∂ ∂ (4.38) where C is the moisture concentration per unit volume, t is time and d' 11 is the longitudinal moisture diffusion coefficient of the composite. Steady state condition Equation 4.38 takes the form 2 2 1 0 C x ∂ · ′ ∂ (4.39) Assuming the boundary conditions (Fig. 4.6) to be as at x' 1 = 0, C = 0 and x' 1 = L, C = C 0 (4.40) the solution is derived as L x C C 1 0 ′ · (4.41) that satisfies Eqs. 4.39 and 4.40. The direction of moisture diffusion per unit area parallel to the x' 1 direction is defined as L C d x d dC d q c 0 11 1 11 11 ′ − · ′ ′ − · ′ (4.42) The total rate of moisture diffusing through the cross-section of the concentric cylinder is given by m f f f c q R R q R q R 11 2 2 11 2 11 2 ) ( ′ − Π + ′ Π · ′ Π (4.43) Note that L C d x d dC d q f f f 0 11 1 11 11 ′ − · ′ ′ − · ′ and L C d x d dC d q m m m 0 11 1 11 11 ′ − · ′ ′ − · ′ (4.44) where d' 11f and d' 11m are the longitudinal moisture diffusivities for the fibre and the matrix, respectively. Substituting Eqs. 4.42 and 4.44 in Eq. 4.43 and noting that R f 2 /R 2 = V f and (R 2 - R f 2 ) / R 2 = V m one obtains d' 11 = d ' 11f V f + d' 11m V m (4.45) when the fibres (e.g., glass, carbon, etc.) are impermeable to moisture d' 11 = d ' 11m V m (4.46) The transverse moisture diffusion coefficient d ' 22 can also be determined using a similar self-consistent field model and is, given as ] ) ( ) [( ] ) ( ) [( 22 22 22 22 22 22 22 22 22 22 f m f m f f m f m f m V d d d d V d d d d d d ′ − ′ − ′ + ′ ′ − ′ + ′ + ′ ′ · ′ (4.47) When fibres are impermeable to moisture, Eq. 4.47 reduces to f f m V V d d + − ′ · ′ 1 1 22 22 (4.48) The longitudinal and transverse thermal conductivities k' 11 and k' 22 of the unidirectional composite can be determined by replacing 'd ' with 'k' in Eqs.4.45 and 4.47, respectively. Note that, in that case, heat conduction takes place both through the fibre and the matrix. The other transport properties can also be derived in a similar way using Eqs. 4.45 and 4.47. Expansional Strains The longitudinal expansional strains (due to temperature or moisture) of a unidirectional composite can be determined using the simple 'mechanics of materials approach ' as discussed earlier. Consider the micro-model in Fig. 4.3. The total longitudinal strains, after accounting for the mechanical strain and the expansional strain, are given as e 11 11 11 11 ∈′ + Ε′ ′ · ∈′ σ and also ; 11 11 11 11 e f f f f ∈′ + Ε′ ′ · ∈′ σ and also ; 11 11 11 11 e m m m m ∈′ + Ε′ ′ · ∈′ σ (4.49) Solving Eqs. (4.49) one gets 11 11 11 11 ) ( Ε′ ∈′ − ∈′ · ′ e σ and m e m m m f e f f f 11 11 11 11 11 11 11 11 ) ( ; ) ( Ε′ ∈′ − ∈′ · ′ Ε′ ∈′ − ∈′ · ′ σ σ (4.50) Assuming free expansion ∈ ' 11 = ∈ ' 11 e , the first relation of Eqs. 4.50 yields σ ' 11 = 0 (4.51) Therefore, σ ' 11 W = σ ' 11f W f + σ ' 11m W m = 0 or, 11 11 11 11 11 11 ( ) ( ) 0 e e f f f f m m m m W W ′ ′ ′ ′ ′ ′ ∈ −∈ Ε + ∈ −∈ Ε · (4.52) Dividing Eq. (4.52) by W and noting that e m m f 11 11 11 11 ∈′ · ∈′ · ∈′ · ∈′ and W W V f f / · and V m = W m / W one obtains m m f f m m e m f f e f e V V V V 11 11 11 11 11 11 11 Ε′ + Ε′ Ε′ ∈′ + Ε′ ∈′ · ∈′ (4.53) Observing that the thermal expansional strain of a specimen of length L due to a rise of temperature ΔT is given by ∈ e = LαΔT, the longitudinal thermal expansion coefficient α' 11 of a unidirectional composite is derived from Eq. 4.53 as follows: m m f f m m m f f f V V V V 11 11 11 11 11 11 11 Ε′ + Ε′ Ε′ ′ + Ε′ ′ · ′ α α α (4.54) Similarly, the longitudinal moisture expansion coefficient β' 11 of a unidirectional composite is obtained from Eq. 4.54 replacing 'α' by 'β'. For the transverse expansional strain ∈ ' 22 e , the 'self-consistent field model ' approach is, however, preferred. The expression for ∈ ' 22 e can be derived as 1 1 ] 1 ¸ Ε′ + Ε′ Ε′ ∈′ + Ε′ ∈′ ′ + ′ · ∈′ m m f f m m e m f f e f m m f f e V V V V V V 11 11 11 11 11 11 12 12 22 ] [ ν ν (4.55) The transverse thermal expansion coefficient α' 22 is then derived from Eq, 4.55 in a similar way 1 1 ] 1 ¸ Ε′ + Ε′ Ε′ ′ + Ε′ ′ ′ + ′ · ′ · ′ m m f f m m m f f f m m f f V V V V V V 11 11 11 11 11 11 12 12 33 22 ] [ α α ν ν α α (4.56) The transverse moisture expansion coefficient β' 22 is obtained from Eq. (4.56) by replacing 'α' with 'β'. 4.3 PARTICULATE AND SHORT FIBRE COMPOSITES A unidirectional composite provides some sort of regularity in the microstructure, as the fibres are continuous and aligned in one direction. This helps to assure a simple micro-model with a constant strain or stress field and use the 'mechanics of materials ' approach to determine the composite properties. Such a simple analytical treatment with constant stress or constant strain field is not adequate in the case of particulate and short fibre composites. The microstructure is not uniform through the composite medium. The point to point variation of the microstructure is quite significant in many situations due to wide variations in the shape, size and properties of fillers and reinforcements and their orientation and distribution in the matrix phase. The discontinuous nature of some of these reinforcements adds to more complexities. There exist innumerable high stress zones around irregular shaped particulate reinforcements and at the tips of short fibres. The assumption of constant stress and strain fields is no more valid. Further complications arise due to the anisotropy caused by the alignment of short fibres and flake particulates. All these preclude a general treatment of the problem. A single composite micro- model, in no way can represent all composites of this category. Composites with different reinforcements may require different micro-models and analytical treatments. This is probably the main reason why the micromechanics analysis of this class of composites has not received much attention from researchers. There is also another important reason for the dearth of information in the area. In comparison to particulate and short fibre composites, unidirectional composites find extensive uses in structural components in several engineering disciplines. This has created more awareness and, in turn contributed to the growth of knowledge in the micromechanics of unidirectional composites, while the understanding of the micromechanical behaviour of particulate and short fibre composite still continues to remain at its nascent stage. 4.3.1 Particulate Composites The simplest mechanics of materials approach uses classical Voigt (constant strain) and Reuss (constant stress) models to estimate the elastic properties for an isotropic composite. With the Voigt model, the bulk modulus k and the shear modulus G are given as P= V f P f + V m P m , where P=K,G and E = 9 KG / (3K+G) ν = (3K-2G) / (6K+2G) (4.58) and with the Reuss model, the relations are G V V m m f f , , , 1 Κ Ε · Ρ Ρ + Ρ · Ρ (4.59) The properties predicted by Voigt model (highest) and Reuss model (lowest) are two extremes to the real values. Several improved analytical models are known to exist, but are not easily amenable to simple design uses. The Halpin-Tsai model, which is based on a semi-empirical approach, is popular and provides both upper and lower bounds that fall within the Voigt and Reuss limits. Simple relations that are developed based on an improved combining rule are found to provide a reasonably good estimate of the properties of an isotropic composite (P f > P m and 0 < ν f < 0.5). These are presented as follows: )}] /( ) )}{( 1 ( 2 1 1 { 1 [ )}] /( ) {( 1 [ 2 m f m f f f m f m f f m Ρ + Ρ Ρ − Ρ − + − Ρ + Ρ Ρ − Ρ + Ρ · Ρ ξ ν ν ξ ξν (4.60) with P = K,G. For bulk modulus, m m ν ν ξ + − · Κ 1 ) 2 1 ( 2 : and for shear modulus, G: m m ν ν ξ 10 8 5 7 − − · (4.61) Young 's modulus E and Poisson 's ratio ν are then determined from Eqs.4.60 and 4.61 using Eqs.4.58. The thermal expansion coefficient α is given by , _ ¸ ¸ Κ − Κ − Κ , _ ¸ ¸ Κ − Κ − Κ Κ + + · m m f f m f f m m f m m f f V V ν ν α α α α α 1 (4.62) where K is obtained using Eqs.4.60 and 4.61. 4.3.2 Short Fibre Composites A simple model assumes a randomly oriented short fibre composite as a quasi- isotropic micro-laminate in which each lamina consists of a group of short fibres oriented along a particular direction. P? is determined using the modified Halpin-Tsai relation as given by )}] /( ) {( 1 [ )}] /( ) {( 1 [ m f m f f m f m f f m Ρ + Ρ Ρ − Ρ − Ρ + Ρ Ρ − Ρ + Ρ · Ρ′ ξ ν ξ ξν (4.63) where for E ' 11 , longitudinal modulus, ξ = 2l / D E ' 22 , transverse modulus, ξ = 2 G ' 12 , inplane shear modulus, ξ =1 G ' 23 , transverse shear modulus, ξ = (3 ? 4 ν m ) -1 Note, that l and d are the length and the diameter of the short fibre, respectively. Both the matrix and the fibre are isotropic in nature. The Poisson 's ratio ν' 12 is estimated using the simple mixture rule. The longitudinal tensile strength is dependent on the critical fibre length l c (Eq. 4.32) and is given by 11 (1 ) t c f f m m l V V l σ ′ ′ Χ · Χ − + where σ ' m is the stress on the matrix when the fibre breaks. Table 4.1 Additional micromechanics relations for unidirectional composites 1. Volume fractions: V f + V m + V v =1 (1) For a void free composite, V v = 0; V f + V m = 1 2. Mass fractions M f + M m = 1 (2) 3. Void volume fraction ρ V v = 1 - ρ[ (M f / ρf ) + (M m / ρm) ] (3) 4. Composite density ρ= ρf V f + ρmV m (4) 5. Fibre volume fraction 1 (1 ) /[1 ( / )( 1)] f v f m f V V ρ ρ · − + − Μ (5) 6. Matrix volume fraction )] 1 1 )( / ( 1 /[ ) 1 ( − Μ + − · m f m v m V V ρ ρ (6) 7. Transverse modulus )] / 1 ( 1 /[ 22 33 22 f m f m V Ε′ Ε − − Ε · Ε′ · Ε′ (7) 8. Shear moduli )] / 1 ( 1 /[ 12 13 12 f m f m G G V G G G ′ − − · ′ · ′ (8a) )] / 1 ( 1 /[ 23 23 f m f m G G V G G ′ − − · ′ (8b) 9. Poisson's ratio ) / 2 ( 11 22 12 23 23 Ε′ Ε′ ′ − + ′ · ′ ν ν ν ν m m f f V V (9) 10. Longitudinal compression strength f f c V 11 11 Χ′ ≈ Χ′ (fibre crushing) (10a) 11 12 / 1 (1 / ) c m f m f X G V G G ′ ′ 1 ≈ − − ¸ ] (microbuckling) (10b) 11 12 10 2.5 c t m X X X ′ ′ ≈ + (10c) 11. Transverse thermal conductivity ) / 1 ( 1 ) 1 ( 22 33 22 f m f f m m f V V V Κ′ Κ − − Κ + Κ − · Κ′ · Κ′ (11a) 22 33 2 1 2 (1 2 / 1 ( / 4 tan 1 / 1 / m f mf f m mf mf f mf f V V V V − ′ ′ Κ · Κ · Κ − Π 1 − Κ Π Κ 1 + Π− Κ 1 + Κ Π −Κ Π ¸ ] (11b) where K mf = 2(K m /K f -1) for a cylindrical fibre. ; ) ( 5 . 0 ) ( 33 22 m m f m m m f f m V V Κ + Κ − Κ Κ Κ − Κ + Κ · Κ′ · Κ′ (for MMCs) (11c) 12. Transverse moisture diffusivity m f d V d d ) 1 ( 33 22 − · ′ · ′ (12a) Π − · ′ · ′ / 2 1 ( 33 22 f m V d d d ) 2 1 2 1 ( / ) 4 tan 1 / 1 ( / ) mf f m mf mf f mf f d V d d d V d V − 1 − Π 1 + Π− 1 + Π − Π ¸ ] (12b) where d mf =2 (d m /d f -1) for a cylindrical fibre. 13. Thermal expansion coefficients 1 ) 1 / ( ) / ( 11 + − Ε Ε + Ε Ε · ′ f m f m m m f f f V V V α α α (13a) m f m f f f f V V V α ν α α α ) / 1 )( 1 ( 11 11 22 33 22 Ε′ Ε′ + − + ′ · ′ · ′ (13b) ) ( ) 1 ( ) 1 ( 11 33 22 m m f f m m m f f f V V V V ν ν α ν α ν α α α + ′ + + + + · ′ · ′ (13c) ; ) )( ( 33 22 m m f f m f f m m f m f m m f f V V V V V V Ε + Ε − Ε − Ε + + · ′ · ′ α α ν ν α α α α (for MMCs) (13d) 14. Transverse moisture expansion coefficients 22 33 22 (1 ) (1 ) 1 (1 ) f f m m f f f m V V V V V β β β 1 − Ε ′ ′ 1 · · − + ′ Ε + − Ε 1 ¸ ] (14) 15. Heat capacity 1 ( ) f f f m m m C k C k C ρ ρ ρ · + (15) Table 4.2: Typical properties of some common fibres S. N0. Property Boron Carbon(T300) Kelvar- 49 S-Glass E-Glass Rayon (T50) 1. Fibre diameter, d ?m 140 8 12 9 9 8 2. Density, ρ f gm /cm 3 2.63 1.77 1.47 2.49 2.49 1.94 3. Longitudinal Modulus, E' 11f GPa 400 220 150 85 75 380 4. Transverse Modulus, E' 22f GPa 400 14 4.2 85 75 6.2 5. Longitudinal Shear modulus, G' 12f GPa 170 9 2.9 36 30 7.6 6. Transverse Shear modulus, G' 23f GPa 170 4.6 1.5 36 30 4.8 7. Longitudinal Poisson's ratio, ν' 12f 0.2 0.2 0.35 0.2 0.2 0.2 8. Transverse Poisson's ratio, ν' 23f 0.2 0.25 0.35 0.2 0.22 0.25 9. Heat capacity, C f kJ/ (kg k) 1.30 0.92 1.05 0.71 0.71 0.84 10. Longitudinal Heat conductivity k' 11f W/ (mk) 38.0 1003.0 2.94 36.30 13.0 1003.0 11. Transverse Heat conductivity k' 22f W/ (mk) 38.0 100.3 2.94 36.30 13.0 100.3 12. Longitudinal thermal Expansion coefficient, α' 11f 10 -6 m/m/K 5.0 1.0 -4.0 5.0 5.0 7.7 13. Transverse thermal expansion coefficient, α' 22f 10 -6 m/m/K 5.0 10.1 54 5.0 5.0 10.1 S. N0. Property Boron Carbon(T300) Kelvar- 49 S-Glass E-Glass Rayon (T50) 14. Longitudinal compressive strength, X' 11f t MPa 4140 2415 2760 4140 2760 1730 15. Longitudinal compressive strength, X' 11f c MPa 4830 1800 500 3450 2400 1380 16. Shear strength, X' 12f MPa 700 550 400 1050 690 350 Table 4.3 : Typical properties of some common matrices S No. Property Poly- imide Epoxy Phe- nolic Poly- ester Nylon 6061 Al Nickel Titan- ium 1. Density,ρ m gm /cm 3 1.22 1.3 1.2 1.2 1.14 2.8 8.9 4.4 2. Young's Modulus, E m GPa 3.45 3.45 11 3 3.45 70 210 110 3. Shear Modulus, G m GPa 1.28 1.28 4.07 1.11 1.28 26.12 81.40 44 4. Poisson's ratio, ν m 0.35 0.35 0.35 0.35 0.35 0.34 0.29 0.25 5. Heat capacity, C m kJ/(kgk) 1.05 0.96 1.30 1.15 1.67 0.96 0.46 0.39 6. Heat conductivity, k m W/(mk) 2.16 0.18 0.21 0.25 0.19 171 62.0 7.0 7. Thermal expansion coefficient, α m 10 -6 m/m/k 36.0 64.3 80.0 80.0 46.0 23.4 13.3 9.5 8. Moisture diffusivity, d m 10 -13 m 2 /s 0.39 1.637 1.20 1.80 1.10 0.0 0.0 0.0 9. Moisture expansion coefficient, β m m/m/C 0.33 0.38 0.38 0.50 0.45 0.0 0.0 0.0 10. Tensile strength, X t m MPa 120 90 60 60 81.4 310 760 1170 11. Compressive strength, X c m MPa 210 130 200 140 60.7 310 760 1170 12. Shear strength, X s m Mpa 90 60 80 50 66.2 180 440 675 Table 4.4 : Thermoelastic properties of three unidirectional composites (V f = 0.6) S. No Property Kelvar/ Epoxy T300/ Epoxy Boron/ polyimide Fomulae used 1. Density, ρ gm /cm 3 1.40 1.58 2.07 Eq.4 * 2. Longitudinal modulus, E' 11 GPa 91.38 133.38 241.38 Eq.4.6 * 3. Transverse modulus, E' 22 GPa 4.00 8.29 14.87 Eq.7 * 4. Poisson 's ratio, ν' 12 = ν' 13 0.35 0.26 0.26 Eq.4.12 * 5. Poisson 's ratio, ν' 23 0.484 0.424 0.394 Eq.9 * 6. Inplane shear modulus, G' 12 = G' 13 GPa 2.26 3.81 5.53 Eq.8a * 7. Transverse shear modulus, G ' 23 GPa 1.44 2.90 5.53 Eq.8b * 8. Longitudinal conductivity, k' 11 W/ (mk) 1.836 601.87 23.66 Eq.4.45 * 9. Transverse conductivity, k' 22 W/ 0.57 0.72 6.95 Eq.11b * (mk) 10. Heat capacity, c kJ/(kgk) 1.017 0.933 5.28 Eq.15 * 11. Longitudinal thermal expansion coefficient, α' 11 (x10 -6 ) m/m/k -2.48 1.99 5.28 Eq.4.54 * 12. Transverse thermal expansion coefficient, α' 22 (x10 -6 ) m/m/k 61.32 2.73 1.48 Eq.13b * * Eqs. Of Table 4.1 4.4 BIBLIOGRAPHY 1. J.M. Whitney and R.L. McCromechanical Materials Modeling, Delware Composites Design Encyclopedia, Vol.2, Technomic Publishing Co., Inc., Lancaster, 1990. 2. J.W. Weeton, D.M. Peters and K.L. Thomas (Eds.), Engineer's Guide to Composite Materials, American Society of Metals, Metals Park,Ohio, 1987. 3. G.S. Springer and S.W. Tsai, Thermal Conductivities of Unidirectional Material, J.Composotie Materials, 1,1967,166. 4. M. Taya and R. Arsenault, Metal Matrix Composite. Pergamon, Oxford, 1989. 5. R.A. Schapery, Thermal Expansion Coefficient of Composite Materials Based on Energy Principles, J.Composite Materials, 2, 1968,157. 6. R.M. Jones, Mechanics of Composite Materials, McGraw Hill Book Companym New York, 1975. 7. S.W. Tsai and H.T. Hahn, Introduction to Composite Materials, Technomic Publishing co., Inc., Lancaster, 1980. 8. B.D. Agarwal and L.J. Broutman, Analysis and Performance of Fiber Composites, Wiley-Interscience, NY, 1980. 9. J.C. Halpin, Primer on Composite Materials : Analysis, Technomic Publishing Co. Inc., Lancaster,1984. 10. Z. Hashin, Theory of Fibre Reinforced Materials, NASA CR-1974, 1972. 4.5 EXERCISES (Use material properties and formulae given in this chapter for numerical results) 1. Using simple rules of mixture, derive expressions for E' 11 , E' 22 , ν' 12 and G' 12 . 2. For a matrix of given weight, what should the weight of fibres so that the fibre volume fraction of the composite is 0.7 i.e., V f = 0.7. 3. For a boron/ polyimide composite (V f = 0.7) determine the values of E' 22 and G' 22 using various formulae and make a comparative analysis. 4. For a carbon/aluminium composite (V f = 0.5) determine the values of E' 11 , E' 22 , ν' 12 and G' 12 . 5. Determine d ' 11 and d ' 22 for a carbon/epoxy composites (V f = 0.7) and for Kevlar/epoxy composite (V f = 0.6 and d f = 5 d m ). 6. Determine k ' 11 and k ' 22 for a carbon polyimide and boron/polyimide composites (V f = 0.7). 7. Determine α' 11 and α' 22 for a boron/aluminium composite (V f = 0.5). 8. Determine β' 11 and β' 22 for a Kevlar/epoxy composite (V f = 0.7). Assume β' f = 0.5 β' m . CHAPTER - 5 TEST METHODS 5.1 INTRODUCTION 5.2 CHARACTERISATION OF PROPERTIES 5.2.1 Density 5.2.2 Fibre Volume Fraction 5.2.3 Fibre Tensile Properties 5.2.4 Matrix Tensile Properties 5.2.5 Tensile Properties of Unidirectional Lamina 5.2.6 Inplane Shear Properties 5.2.7 Compressive Properties of Unidirectional Lamina 5.2.8 Interlaminar Shear Properties 5.3 NDT METHODS 5.3.1 Acoustic Emission 5.3.2 Holographic Interferometry 5.3.3 Radiography 5.3.4 Thermography 5.3.5 Ultrasonics 5.4 BIBLIOGRAPHY 5.5 EXERCISES 5.1 INTRODUCTION Testing is a very broad and diverse discipline that concerns with the (i) characterization of physical, mechanical, hygral, thermal, electrical and environment resistant properties of a material, that are required as design input, (ii) quantification of inclusions, voids, cracks, delaminations and damage zones for design assessment, and (iii) testing and qualification for final product realization. Testing, at the final stage may also involve application of simulated service loads and exposure to accelerated environmental conditions. Testing, in fact, is a continuous process, it interacts with a product at every stage of its design and development and, in most cases, continues until the product ensures optimum performance requirements even after it is put to a certain period of continuous service. The testing of composites is much more involved than that of most other materials. The number of material parameters to be determined is quite large. Take the case of a three-dimensional unidirectional composite (Fig. 5.1). The mechanical parameters that are to be generated are three longitudinal moduli (E' 11 , E' 22 , E' 33 ), three Poisson 's ratios (ν ' 12 , ν ' 13 , ν ' 23 , ), three shear moduli (G' 12 , G' 13 , G ' 23 ) and nine strength constants ? six normal strengths (X' 11 , X' 22 , X ' 33 , in tension and compression) and three shear strengths (X' 12 , X' 13 , X' 23 ) and likewise ultimate tensile, compressive and shear strains. Besides, there are important parameters, namely moisture diffusivities (d ' 11 , d ' 22 , d ' 33 ), coefficients of moisture expansion (β' 11 , β' 22 , β' 33 ), maximum moisture content ) ( ∞ Μ , thermal conductivities (K' 11 , K' 22 , K' 33 ), coefficients of thermal expansion α ' 11 , α ' 22 , α ' 33 ), heat capacity ( C ) and several other parameters related to impact, fracture, fatigue, creep, viscoelasticity, plasticity, strain-hardening, etc. A specimen plays a vital role in realizing the desired test objective. The shape and size of a specimen and its test requirements vary from one test to the other. The heterogeneity and anisotropy of composites control the specimen design. A ' dog bone ' type specimen may be used for the tensile testing of a quasi-isotropic composite (particulate or randomly oriented short fibre composite) whereas a unidirectional composite requires a flat specimen with end tabs. A specimen to study the microstructure may need a Scanning Electron Microscope so as to obtain finer details of individual reinforcements, interfacial bonds, voids, micro-cracks, etc., on the other hand the testing of a proto-type aircraft composite wing component specimen may involve elaborate fixtures and instrumentations to simulate the load and measure the test data. All aspects of testing, namely, specimen preparation, design of test-rigs and fixtures, instrumentation and analysis of test data, are vital to the success of a test programme. The statistical allowables of the test data are normally based on A-Basis: The A parameter value is the value above which at least 99% population of values is expected to fall with a 95% confidence level. B-Basis: The B parameter value is the value above which at least 90% population of values is expected to fall with a 95% confidence level. S-Basis: The S parameter value is the minimum value specified by the governing specifications. Typical Basis: The typical parameter value is an average value. No statistical assurance is associated with this value. It is to be remarked that the test procedures to determine all the intrinsic composite properties are not yet fully developed. In this chapter, therefore, the salient features of a few established test methods for characterization of properties of fibres, matrices and unidirectional composites are described. All these test methods are applicable to polymer composites, but some of them may also be used for metal-matrix and ceramic-matrix composites. For specific details the relevant ASTM standard should be referred (See Table 5.1). Nondestructive testing (NDT) serves three major purposes. It assures quality control of materials and products during manufacture and assembly. It ensures the integrity of manufactured parts and their assemblies during service life. It generates a NDT database that forms the basis for evaluation and assessment of a component. NDT methods are, in general, indirect in nature and therefore require accuracy both in measurement and interpretation of data. NDT techniques have made enormous strides in recent years. Sophistication in electronic instruments, computerization, real time monitoring using video systems, as well as several other innovations and advances in data measurement and analysis techniques have revolutionized the NDT technology. Some of these advanced NDT methods are of great significance to the area of composites and composite structures, because some of the flaws have the thickness dimensions smaller than 100?m, say, in the case of a tight delamination or even closer to the fibre diameter as in the cases of a fibre break, matrix cracking, etc. Delaminations are most undesirable defects in composite structures. Under certain loading conditions, these defects grow faster and severely limit the integrity of a structure. They also act as pockets in which diffused moisture or water can accumulate thereby causing further degradation of the material and structure. The detection and quantification of delaminations, especially the tight ones, are of major concern to NDT personnel. Besides, there are several defects like cracks, voids, inclusions, debonding, disbanding, etc. which need to be evaluated using NDT techniques. The purpose here is not to make a critical assessment of the progress made in the area but to present a brief account of some important NDT methods that are the current trend-setters in composite applications. The reader should refer to the recently published literature for more information on the prospects and limitations of a particular method. However, one should keep it in mind that an NDT method is better learnt, when it is used in practice. 5.2 CHARACTERISATION OF PROPERTIES 5.2.1 Density (ASTM D792-75) The density for fibres and matrices is determined by weighing the specimen in air and then weighing it while suspended on a wire and immersed in water, and then noting down the difference in water. In case the specimen is likely to have the density lower than that of water, a sinker is attached to the wire to facilitate immersion. The density ρ is obtained from ) ( ) 9975 . 0 ( b w a a − + · ρ (5.1) where a is the weight of the specimen in air, b is the total weight of specimen and sinker completely immersed, while the wire is partially immersed and w is the weight of a fully immersed sinker but partially immersed wire. The density of a composite can also be determined in a similar way. 5.2.2 Fibre Volume Fraction (ASTM D 3355-74) The matrix phase in this method is first digested by burning in an oven or using a digesting liquid. The fibres remain unaffected. They are cleaned and then weighed. The fibre volume fraction f V is then determined using the relation. ) / /( ) / ( ρ ρ w w V f f f · (5.2) where w and ρ are the composite weight and composite density, respectively, and w f and ρ f are those for the fibre. 5.2.3 Fibre Tensile Properties (ASTM 3379-75) The tensile strength and Young's modulus of a high-modulus fibre is measured after mounting it on special slotted tabs and loading it at a constant strain rate (Fig. 5.2). The fibre strength X f and the fibre modulus E f are obtained from the load displacement plot. 2 4 . f L d u Ρ Ε · Π and 2 max 4 d f Π Ρ · Χ (5.3) where P,d and u are the load, fibre diameter and axial displacement, respectively. 5.2.4 Matrix Tensile Properties (ASTM D638-80) A 'dog bone' specimen (Fig. 5.3a) is commonly used for a polymer material, and the strength X m corresponds to the ultimate failure load. The Young's modulus E m and ultimate strain mu ∈ are measured using electrical resistance strain gauges located at the centre of the specimen to determine ν m . For thin polymer sheets (in case it is not possible to make thick sheets), a flat specimen (Fig. 5.3b) is recommended. The following relations are used: ; 1 11m m WT ∈ Ρ · Ε m m m 22 11 ∈ ∈′ − · ν and WT m max Ρ · Χ (5.4) 5.2.5 Tensile Properties of Unidirectional Lamina (ASTM D 3039-76) High modulus and high strength fibrous composites cause special problem of grip integrity. Wedge action frication grips are used to hold the specimen. The materials for the tabs should have lower modulus of elasticity and higher percentage of elongation. The tab thickness may range from 1.5 to 4 times the specimen thickness. Typical specimen dimensions are presented in Fig. 5.4. The longitudinal tensile properties u 11 12 11 11 , , ∈′ ′ Χ′ Ε′ ν as well as transverse tensile properties u 22 21 22 22 , , , ∈′ ′ Χ′ Ε′ ν for a unidirectional lamina can be determined following this test method. For longitudinal properties a 0 0 lamina with the width W= 12.7mm is employed, and for transverse properties a 90 0 lamina with W = 25.4mm is used. The specimens are loaded monotonically at a recommended rate of 0.02 cm/min. The applied loads as well as longitudinal and transverse strains are measured. The determinable characteristics are computed using stress-strain plots and simple relations as given below: For 0 0 specimen : WT WT max 11 11 11 22 12 11 11 ; ; 1 Ρ · Χ′ ∈′ ∈′ − · ′ ∈′ Ρ · Ε′ ν For 90 0 specimen: max 22 22 22 11 21 22 22 ; ; 1 Ρ · Χ′ ∈′ ∈′ − · ′ ∈′ Ρ · Ε′ ν WT /WT 5.2.6 Inplane Shear Properties (ASTM D 3518-82) Several test methods, namely picture frame, rail shear (single and double), tube torsion, plate twist and tension testing of an off-axis specimen, were developed to determine the inplane shear properties of a unidirectional lamina. All these methods (5.5) require complex test specimens or special test fixtures. The off-axis composite in which all fibres are oriented at an angle φ , provides a simple specimen configuration (Fig. 5.5a). The relation used to determine the inplane shear modulus is given by (see Eq. 6.41) ) 2 1 ( 1 11 12 12 2 2 22 4 11 4 11 Ε′ ′ − ′ + Ε′ + Ε′ · Ε ν G n m n m (5.6) from which G' 12 is determined; as 11 22 12 , and ν ′ ′ ′ Ε Ε are known from the earlier test (Eq. 5.5) and E 11 ) ( 11 ∈ Ρ · WT is determined from the tensile test of an off-axis specimen. However, the testing of an off-axis specimen requires greater care because of coupling of normal stress and shear stress. The coupling introduces bending moment and shear forces (Fig. 5.5b) at the ends where it is attached to the grips. This in turn requires complex end fixtures to loosen the end fixity. The other alternative is to test a long specimen so that the middle portion of the specimen remains unaffected. ASTM D 3518-82 specifies a simple test method in which a ?45 0 symmetric laminate specimen (Fig. 5.6) is subjected to a tensile load. The specimen details (including end tabs details) are those given in ASTM 3039-76. The width W of the specimen is 12.7mm. The applied load and both longitudinal and transverse strains are continuously recorded till failure. The shear stress and shear strain are computed at the different levels of the applied load from the relations 12 12 11 22 / 2WT and σ γ · Ρ · ∈ + ∈ (5.6a) and the corresponding shear stress-strain curves are plotted. The shear modulus and the shear strength are then obtained from max 12 12 12 12 2 G and WT σ γ Ρ ′ ′ · Χ · (5.7) 5.2.7 Compressive Properties of Unidirectional Lamina (ASTM D3410-87) The premature failures, namely, fibre buckling, fibre breaking, matrix shearing, etc. are commonly encountered in a compression test. The main problem here is to ensure that the specimen failure is by compression. The specimen gauge length should be sufficiently short to restrict the failure mode to a truly compressive one. Besides the specimen ends are likely to get damaged due to want of close contact at all points between the end faces of the specimen and the platens of the testing machine during loading. Further, the specimen centerline should be perfectly aligned so as not to induce any eccentricity with respect to the load path. All these require an appropriate specimen configuration and a complex loading fixture. Three standard methods are currently available to determine the true compressive properties. The details of the testing procedure, specimen configuration, loading fixture, etc. are presented in ASTM D 3410-87. Of these the most commonly used method employs a Celanese test fixture (Figs. 5.7 and 5.8). The test fixture consists of truncated conical collet type friction grips contained in matching cylindrical end fittings (tapered sleeves). The colinearity of the end fittings is maintained by a hollow cylinder that houses all fittings. A central opening in the central part of the hollow cylinder provides an access to the gauge length of the specimen. A spacer is used to separate the grips and allow them to be closed with a preload, without preloading the specimen. The assembled fixture with specimen (Fig. 5.9) is loaded between the flat platens of the testing machine. The hollow cylinder, however, does not carry any load during the test. The recommended loading rate is 0.017 mm/mm/s. The applied load P and strains (longitudinal and transverse) are measured at regular intervals. The compressive strength and longitudinal modulus, and Poisson's ratio for a longitudinal (0 0 ) specimen are computed from max 22 11 11 12 11 11 1 ; . ; . c c c c c c BT BT ν ′ Ρ ∈ Ρ ′ ′ Χ · Ε · · − ′ ′ ∈ ∈ (5.8) The properties for a transverse (90 0 ) specimen can also be determined in a similar way. 5.2.8 Interlaminar Shear Properties The shear moduli G' 13 and G' 23 , and shear strengths X' 13 and X' 23 (Fig. 5.1) are normally termed as interlaminar shear properties. There are no reliable methods for determination of all interlaminar shear properties. ASTM D 2344-84 specifies the determination of the interlaminar shear strength X' 13 only. The apparent shear strength determined using this method should be used only for quality control and specifications purposes, but not as design criteria. Both flat and ring short beam specimens (Fig. 5.10) can be used for which the span to thickness ratio is 4 for most composites, except glass fibre composite when it is 5. For the flat specimens, the corresponding length to thickness ratios are 6 and 7 so as to provide allowance for the support pins (3.2 mm φ ). The loading nose consists of a 6.35 mm φ dowel pin. The recommended crosshead speed is 1.3 mm/min. The apparent interlaminar shear strength is obtained using bh max 13 75 . 0 Ρ · Χ′ (5.9) 5.3 NDT METHODS 5.3.1 Acoustic Emission Acoustic Emission (AE) is essentially a technique of listening to a material. Whenever there is a change of condition in the material during loading and other service conditions, e.g., the initiation and propagation of a crack, sound waves (transient elastic waves) are generated by the rapid release of energy and propagate through the medium which contains that crack. These sound waves can be detected using an AE sensor glued to the surface of the medium at a convenient location. In the early fifties Joseph Kaiser, a German scientist, conducted experiments with metals and wood using sensitive electronic instruments and listened to the sound emitted by these materials during the process of deformation. He noted a phenomenon, termed as 'Kaiser Effect', that a material that had emitted AE signals during earlier stressing, would exhibit AE signals again when the previous stress was exceeded. Since Kaiser's first experimentation, there has been an all round growth in the use of AE techniques in materials and structures including composites. AE sensors (piezoelectric transducers) are in principle high frequency microphones which first receive the sound waves and then convert them to electrical signals. These signals are very weak and therefore are amplified before they are passed to the signal conditioner where other electrical noises are filtered out. The filtered AE signals are then processed and analysed. A simple AE measurement system is schematically illustrated in Fig. 5.11. The electrical signals received by an AE sensor are processed by a wide variety of parameters : (i) count rate and total count of the number of signals which exceed a reference threshold, (ii) distribution of signal amplitude as a function of stress and time, (iii) energy of the detected signals and (iv) frequency content of the signals. AE can also be used to locate the crack or the signal source which emits AE signals. This requires the use of multiple transducers, and the source is located by the triangulation method, normally used to locate a seismic source. AE is an active NDT method and can be utilized for condition monitoring of composite parts and production control, as well as assessing severity of flaws and damages. It has been used extensively in composites not only to identify various failure modes, to define defects and to locate AE souces, but also to conduct real time monitoring during proof testing and in service. Each failure mode, namely, fibre breaking, matrix cracking, interfacial debond or delamination is found to exhibit distinct characteristic AE signals. But the identification of individual modes becomes extremely difficult when two or more failure modes occur simultaneously. The types of fibres and matrices, the anisotropy, the stacking sequence, structural boundaries, presence of micro-defects, etc. can considerably influence the AE signals and their propagation characteristics. All these problems need to be solved before AE can be routinely used as an NDT tool in development of composite materials and structures. 5.3.2 Holographic Interferometry The holographic technique was discovered by Nobel Laureate Dennis Gabor in 1947, but it gained prominence after the discovery of the helium-neon laser in 1962. In holography, the entire optical wavefront both with respect to amplitude and phase is recorded in a film and phase is recorded in a film is called 'hologram' (after the Greek word holos meaning 'whole'). A hologram preserves the three-dimensional character of an object for which the hologram has been made. A simple holographic set-up (Fig.5.12a) mounted on a vibration isolated table, uses a laser, the light from which is split into two waves by a beam splitter. One wavefront i.e., the reference wavefront after being reflected from a mirror system reaches directly a holographic film. The other wavefront, i.e., the object wavefront reaches the film after being reflected from the object. The two wavefronts create a complex interference pattern which is recorded on the holographic film. The interference lines represent points with the same displacement. The coherence of the laser light permits the interference of these two waves, although there exist relatively large differences in path length. The recorded holographic film, or the hologram when illuminated with the reference wave, the object wave is reconstructed and a three-dimensional view of the object can be observed behind the hologram (Fig. 5.12b). Several images which interfere with each other can be stored on the same film, which can be reconstructed when required. The holographic interferometry (HI) uses the technique of multiple exposure for application in NDT. The popular double exposure method, in which holograms of an object in two different states, e.g., stressed and unstressed, provide anomalies in the interference pattern which may in turn, reveal the existence of a flaw if any. The double exposure method is also known as frozen fringe HI. There are other HI methods which are used for specific purposes. The 'simulataneous method' or real time HI first creates a hologram of the object in a desired reference state, which is later used as a reference hologram with respect to which subsequent changes in the object position are recorded by filming the hologram image. The 'time averaging method' is used to record small amplitude oscillations of vibratory parts. The hologram of a vibrating body is first recorded on a film for a time interval longer than the period of oscillation and in the process a set of holograms are superimposed. The resulting hologram when reconstructed, reveals nodal lines as dark interference stripes. HI has a great potential for NDT applications. The capability of HI is enhanced considerably after the introduction of video and popular with the NDT personnel working in the field of composites and composite structures. The real time monitoring of a component using HI, especially during the service life, still poses a problem as it is not easy to isolate vibration in the production and in-service environment. The vibrational displacement of the object as small as one-quarter wavelength of the laser light may produce fringes on the hologram. The use of phase-locked holography may alleviate problems associated the low frequency environmental vibration. The phase-locked holography uses the diffuse reflection of an unexpanded beam shone on a small portion of the test object as the reference beam. Another important development in this area is the electronic shearogrphy in which no separate reference beam is used. In this case, the returning object beam is doubly imaged with a video system. One image is then found to be slightly shifted or sheared relative to the original one. This shearing fringe pattern can be isolated from the real fringes. 5.3.3 Radiography X-ray radiography is the most commonly used NDT technique in industrial applications. X-rays are independent of the magnetic and electrical properties of a material and hence can be used with all materials. Two major characteristics of X-ray radiographic NDT method are that X-rays are absorbed differentially by different media and they produce photochemical effects in photographic emulsions. The intensity of a transmitted X-ray beam, when it passes through a medium, is given by I = I 0 e -?h (5.10) where I is the intensity of the transmitted beam, I 0 is the intensity of the incident beam, ? is the absorption coefficient of the medium and h is the thickness through which the beam travels. The absorption coefficient ? depends not only on the material, but also on the wavelength of X-rays. Thus it is observed from Eq 5.10 that the X-ray beam attenuates when it passes through a material. The attenuation depends on the absorption coefficient and the thickness of the material. If there exists any defect, say, a void, in the material, the void (? =0) does not absorb X-rays. So the intensities of the X-ray beams passing through the material thickness with and without a void will be different. When these transmitted beams are allowed to strike a photographic film, they create a contrast on the exposed film or radiograph (i.e., more the intensity, darker the film appears) from the knowledge of which the existence of a void can be predicted. It is also possible to determine the thickness and composition of a material by examining differences in the exposed film. Fig. 5.13 illustrates the principle how the radiograph is produced when a stepped specimen containing a hole is exposed to X-rays. Normally voids of small sizes (closer to fibre dimensions), and cracks and delaminations that exist normal to the X-ray beam are not easily detectable. However, inclusions, cracks, delaminations and other material defects and damages that are aligned parallel to the X-ray beam can be readily revealed. The X-ray radiography has also been applied to investigate the microstructural details of damages using low energy X-rays as well as using an X-ray opaque penetrant (e.g., tetrabomo-ethane or zinc iodide). The penetrant, however, should not react chemically with the constituents of the composite medium. The development of radiography with microfocus (in which electrons are focused on a small area by means of a magnetic field) opens up new vistas for locating smaller details closer to fibre dimensions (10?m). Figure 5.14 shows the microfocal radiograph of a carbon composite panel with defects such as Teflon pieces, steel mesh, steel wire and steel balls. Microfocus radiography combined with real time image processing can be conveniently applied to investigate the nucleation and growth of cracks, delaminations and damages in composite and honeycomb structures. A large portion of the attenuation of the X-ray beam, especially with low X-ray photon energies is due to Compton scattering. The X-ray backscatter imaging uses the process of recording and investigating scattered radiation from the object. The backscatter radiation provides quantitative information about variations in density due to presence of flaws, delaminations, etc. as well as change in materials. The method is found to be very useful for the inspection of laminated composite pressure vessels and motor cases, and very tight delaminations can be easily detected. Computerized tomography (CT) provides a three-dimensional image of the desired section of an object and therefore all minute details of the variations in the image slice are recorded. The image is called a tomogram (after the Greek word tomos means 'to cut'). A point source of X-rays or gamma radiation is collimated to a flat, fan shaped beam which penetrates the slice of an object under inspection. The intensity of the transmitted beam is recorded by a detector. The movements of the beam and the recorder can be synchronized when the beam is rotated about the object along with the recorder and thereby a three-dimensional scanning of the whole slice is carried out. CT is now widely used in medical diagnostics and offers a great potential for uses in composite materials and structures. Neutron radiography is another NDT technique which is finding applications in polymer composite structures. However, the major limitation of this method is that a transportable neutron source should be available at the site of inspection. 5.3.4 Thermography Thermography is also an effective NDT technique. It is basically a method of mapping and interpreting the contours of isotherms (equal temperature) over the surface of a body. A variation in the thermal field within the body occurs due to the presence of inhomogeneities, discontinuities and other defects which form hot or cold regions depending on their thermoelastic properties. These hot or cold regions exhibit sharp temperature gradients and can be located in the isothermal mapping. A thermal field within a body can be created externally by exposing it to a hot or cold source, or internally during the process of deformation when being loaded. A low level of mechanical vibration can raise the temperature in the regions containing discontinuities. A low temperature field may require spraying the body with liquid nitrogen. The thermal wave imaging technique employs a pulsed heat source to create pulsed thermal waves in the body. The thermal waves are then detected using acoustical or optical methods. Thermal patterns or isotherms are usually recorded employing an infrared electronic camera (Fig.5.15). These are then related to inhomogeneities or defects. Thermography has been successfully used to detect delaminations and other types of flaws in composites. Figure 5.16 shows a thermal image of a blister between the peel ply and the subsurface on a 2mm thick CFRP panel. The image was acquired by heating the rear side of the panel using a hot air gun and focusing the infrared camera on the front side. Thermography should find extensive uses in metal-matrix composites, as metals are, in general, good thermal conductors. The real time thermography permits scanning and imaging a large surface area in a shorter period of time. The vibrothermography, in which mechanical vibrations are employed to induce thermal gradients near the damage regions, combined with the real time recording using an infrared video camera has been used to investigate damages in composites. 5.3.5 Ultrasonics Ultrasonics is also a popular NDT technique for composites. The ultrasonic inspection in composites employs high frequency sound pulses usually in the megahertz range. Piezoelectric transducers are normally used to produce sound pulses. These sound pulses (ultrasonic signals) are allowed to propagate as a narrow beam through a material under examination. The sound waves attenuate based on the characteristics of the material (even if it is homogeneous) as given by the relation h e α − Ι · Ι 0 (5.11) where I is the intensity of the transmitted sound wave, I 0 is the original intensity of the sound wave, α is the attenuation constant and h is the distance travelled by the sound wave. The intensity of propagating signals attenuates further due to the presence of inhomogeneities (e.g., different materials, poor adhesive bonding, etc.) and discontinuities (e.g., delaminations, cracks, voids, etc.) in the material. The sound signals are scattered and /or reflected at the interfaces of these defects. The characteristics of these defects are predicted by investigating the reflected and / or transmitted signals. Fig. 5.17 illustrates how a reflected signal relates to the presence of a crack. In fact, more than 99% of an ultrasonic signal is reflected from a crack surface which is a material-air interface. This method of monitoring the reflected sound signal is called the 'pulse-echo' ultrasonic test. A piezoelectric transducer (probe) located at the top surface of a test specimen transmits a very short, high frequency pulse. The pulse is reflected from the crack top as well as from the bottom surface and is received by the same transducer (receiver). The variations in the amplitudes of reflected pulses when compared with that of the start pulse give the measure of attenuation. The depth at which the crack is located can be determined monitoring the time of arrival i.e., by relating the time axis with the sound path length. By mapping the surface and using angle probes the size and orientation of a crack can also be determined. The main advantage of the pulse-echo method is that it requires access only from one side of a structure. The portable pulse-echo systems are very common in in-situ inspection. The pulse-echo system helps locating flaws at different depths. The pulse- echo C-scan can provide a map of all flaws located at different depths. The method is also very sensitive to foreign body inclusions. Even the existence of a piece of paper or a similar material contained within a laminate can be easily identified from the reflected signal strength. The transmitted ultrasonic signals can also be monitored by placing a probe (receiver) on the bottom surface of the specimen. This is called the 'through-transmission' ultrasonic test. The presence of a flaw will reduce the intensity of transmitted signals. The through transmission ultrasonic technique is relatively more popular in composite applications. The method also permits ease of automation. A 'through transmission' ultrasonic C-scan provides complete information of the quality of an inspected part. Delaminations, inclusions and other defects normal to the ultrasonic beam are easily identified in the C-scan mapping. High frequency transducers (5 to 10 MHz) are employed to locate small defects or cracks parallel to the beam. Short focus transducers with 15 MHz are also being used in the ultrasonic C-scan system. High frequency transducers provide the sharper image of a defect and therefore help identification. There are several other ultrasonic test techniques that are receiving sufficient attention in recent years. The 'ultrasonic polar backscatter' technique employs slightly angled beams. This helps detection of matrix cracking in oriented plies. The 'ultrasonic resonance' method makes use of the fact that the existence of a delamination reduces the normal surface stiffness of the material. A continuous ultrasonic wave is transmitted through the material, and the mechanical stiffness or impedance of the material is monitored. The reduced surface stiffness due to presence of a delamination decreases the surface loading on the ultrasonic probe and a shift in the phase, amplitude or resonant frequency is observed. The 'ultrasonic correlation' method enhances the sensitivity of ultrasonic signals (higher signal to noise ratio) by making use of a continuous wave cross-correlation technique. The method is very useful for highly attenuative composite materials. The 'acousto-ultrasonic or stress wave factor' test technique employs an ultrasonic transducer to send a simulated acoustic emission pulse through the test object. A defect or damage can modify the waveform which is monitored at a distance away from the source and is analysed. The 'ultrasonic microscopy' can image microstructural differences on the surface of a material. The reflection scanning acoustic microscope uses a very narrow high frequency (100 MHz to 1GHz) ultrasonic beam to scan the object line by line. Its limit of resolution is that of an optical microscope but the acoustic imaging, in some cases, provides additional information. One of the major disadvantages of the ultrasonic NDT method is that a coupling agent is needed between the probe and the specimen to transmit and receive ultrasound signals. Normally either the specimen is immersed in a water bath or a water jet is directed to the specimen. Other coupling agents are also used. The coupling agents may have a deleterious effect on the specimen material. Further, this also poses special problems, when the size of a part to be inspected becomes large. The other alternative in such situations is to make use of transducers with dry coupling. The transducer is coupled acoustically to the specimen via a plastic material which is attached to the tip of the transducer. Typical CRT patterns from artificially embedded defects (paper and Teflon) in a carbon-carbon composite material are illustrated in Fig. 5.18. Table 5.1 : ASTM standards for composite and related testing ASTM D618-91 : Conditioning of Plastics and Electrical Insulating Materials for testing. ASTM D792-75 : Specific Gravity and Density of Plastics by Displacement. ASTM D 1505-75 : Density of Plastics by the Density Gradient Technique. ASTM D 3355-74 : Fibre Content of Unidirectional Fibre/Polymer Composites (Also see ASTM D3171-76 and ASTM D 3553-76) ASTM D 3379-75 : Tensile Strength and Young's Modulus for High Modulus Single Filament Materials. ASTM D2324-76 : Tensile Properties of Glass Fibre Strands, Yarns and Rovings used in Reinforced Plastics ASTM D 4018-81 : Tensile Properties of Continuous Filament Carbon and Graphite Yarns, Strands, Rovings and Tows. ASTM D 638-91 : Tensile Properties of Plastics ASTM D 695-91 : Compressive Properties of Plastics ASTM D 3039-89 : Tensile Properties of Oriented Fibre Composites. ASTM D 3552-77 : Tensile Properties of Fibre Reinforced Metal Matrix Composites. ASTM D 2291-80 : Fabrication of Ring Test Specimens for Glass Resin Composites. ASTM D 2290-92 : Apparent Tensile Strength of Ring or Tubular Plastics and Reinforced Plastics. ASTM D 790-91 : Flexural Properties of Plastics and Electrical Insulating Materials. ASTM D 3518-82 : Inplane Shear Stress-Strain Response of Unidirectional Reinforced Plastics. ASTM D 3918-80 : Definitions of Terms Relating to Reinforced Pultruded Products. ASTM D 4475-85 : Apparent Horizontal Shear Strength of Pultruded Reinforced Plastics Rods by Short Beam Method. ASTM D 3914-84 : In-plane Shear Strength of Pultruded Glass- Reinforced Pultruded Plastic Rods. ASTM D 3916-84 : Tensile & Properties of Pultruded Glass Fibre Reinforced Plastic Rod. ASTM D 3914-80 : Inplane Shear Strength of Pultruded Glass- Reinforced Plastic Rod. ASTM D 3410-87 : Compressive Properties of Unidirectional or Crossply Fibre- Resin Composites. ASTM D 2344-84 : Apparent Interlaminar Shear Strength of Parallel Fibre Composites by Short-Beam Method. ASTM D 3479-76 : Tension Fatigue of Oriented Fibre, Resin Matrix Composites. ASTM D 671-78 : Flexural Fatigue of Plastics by Constant Amplitude- of-Force. ASTM D 2585-90 : Preparation and Tension Testing of Filament Wound Pressure Vessels. ASTM D 2105-90 : Longitudinal Tensile Properties of Fibreglass Reinforced Thermosetting Plastic Pipe and Resin Tube. ASTM D 897-78 : Tensile Properties of Adhesive Bonds. ASTM D 3876-79 : Inplane Shear Strength of Reinforced Plastics ASTM D 3846-85 : Inplane Shear Strength of Reinforced Thermosetting Plastics. ASTM D 1623-78 : Tensile / Tensile Adhesive Properties of Rigid Cellular Plastics ASTM D 1621-79 : Compressive Properties of Rigid Cellular Plastics. ASTM D 747-90 : Apparent Bending Modulus (Stiffness) of Plastics by Cantilever Beam Method. ASTM D 696-91 : Coefficient of Linear Expansion. ASTM D 648-88 : Deflection Temperature of Plastics under Flexural Load. ASTM D 3917-88 : Dimensional Tolerance of Thermosetting Glass- Reinforced Plastic. ASTM D 543-87 : Resistance of Plastics (incluting Cast/Hot-moulded/ Cold- moulded Resinous/Sheet Products) to 50 Chemical Reagents. ASTM E 162-90 : Surface Flammability of Materials using a Radiant Heat Energy Source. ASTM D 2843-88 : Density of Smoke from the Burning or Decomposition of Plastics. 5.4 BIBLIOGRAPHY 1. Annual Book of ASTM standards, American Society for Testing and Materials, Philadelphia, 1992. 2. R.B. Pipes, R.A. Blake, Jr., J.W. Gillespie, Jr. and L.A. Carlsson, Test Methods, Delware Composite Design Encyclopedia, Vol.6, Technomic Publication Co., Inc., Lancaster, 1990. 3. K.G. Boving (Ed.), NDE Handbook, Butterworth, London, 1989. 4. T.S. Jones and H. Berger, Nondestructive Evaluation Methods for Composites, International Encyclopedia of Composites (Ed. S. M. Lee),Vol.4m VCH, N.Y., 1990, p.37. 5. E.G. Henneke, Nondestructive Evaluation of Advanced Composite Materials, Proc. Indo-US workshop in Composite for Aerospace Application, Bangalore, India, 1990, p.41. 6. I.G. Scott and C.M. Scala, A Review of Nondestructive Testing of Composite Materials, Non-destructive Testing International, 15, 1982, p.75. 7. J.H. Williams and S.S. Lee, Acoustic Emission Monitoring of Fibre Composite Materials and Structures, J Composite Materials, 12, 1978, p.348. 8. J.B. Abbiss, M J Marchant and A C Marchant, Recent Application of Coherent Optics in Aerospace Research, Optical Engineering, 15, 1976, p.202. 9. D.W. Oplinger, B.S. Parker and F.P. Chiag, Edge-Effect Studies in Fibre Reinforced Laminates, Experimental Mechanics 14, 1974, p.347. 10. B.G. Martin, Analysis of Radiographic Techniques for Measuring Resin Content in Graphite Fiber Reinforced Epoxy Resin Composites, Materials Ebaluation, 35, 1977, p. 65. 11. D.J. Hagemaier and R H Faesbender, Nondestructive Testing of Advanced Composites, Materials Evaluation, 37, 1979, p. 43. 12. R.L. Crane, S. Allinikor and F. Chang, The use of Radiographically Opaue Fibers to aid the Inspection of Composites, Materials Evaluation, 36, 1978, p. 69. 13. F.H. Chang, D.E. Gordon, B.T. Rodini and R.H. McDaniel, Real-time Characteization of Damage: Growth in Gr/Ep Laminates, J. Composite Materials, 10, 1976, p.182. 14. R. Prakash, Nondestructive Testing of Composites, Composites, 11, 1980, p.217. 15. P.V. McLaughlin, E.V. McAssey and R C Deitrich, Nondestructive Examination of Fibre Composite Structures by Thermal Field Techniques, NDT International, 13, 1980, p. 58. 16. S.S. Russell and E.G. Henneke, Dynamic Effects during Vibrothermographic NDE of Composites, NDT International, 17, 1984, p.19. 17. P.Stanley and W.K. Chan, Quantitative Stress Analysis by Means of the Thermoelastic Effect, J Strain Analysis for Engg. Design, 20, 1985, p.129. 18. M. Holler, J.F. Williams, S. Dunn and R. Jones, Thermomechanical Analysis of Composite Specimens, Composite Structures, 11, 1989, p. 309. 19. D. Zhang and B.I. Sandor, Thermographic Analysis of Stress Concentrations in a Composite, Experimental Mechanics, 29, 1989., p.121. 20. A. Vany and K.J. Bowles, An Ultrasonic Acoustic Technique for Nondestructive Evalution of Fiber Composite Epoxy, Polymer Engineering and Science, 19, 1979,p.373. 21. J.C. Dake, Jr. (Ed.), Acousto-Ultrasonics: Theory and Application, Plenum Publ. Corp., N.Y., 1988. 22. R.L. Hollis, R. Hammer and M.Y. Al-Jaroudi, Subsurface Imaging of Glass Fibres in a Polycarbonate Composite by Acoustic Microscopy, J Materials Science, 19, 1984, p.1897. 5.5 EXERCISES 1. Prepare a list of various properties (physical, thermal, electrical, chemical, mechanical, etc.) that are important in design and development of composite materials and structures. 2. How do you determine the tensile properties of fibres and matrices ? 3. Which properties of a unidirectional composite can be determined using tensile tests? Describe the methods. 4. Establish the theoretical basis of the formula defined in Eq. 5.9. Why the data obtained using this test method are not used for design purposes? 5. Derive the formulae used for determination of inplane shear properties of a unidirectional composite. Discuss the relevant test methods. 6. Compare the advantages and disadvantages of various NDT methods used in composite testing. 7. Which NDT methods will you recommend if you have to detect i. A subsurface delamination ii. An inclusion iii. Distribution of voids iv. A crack v. Improper bonding vi. Criticality of a flaw. 8. Describe the uses of ultrasonic methods in NDT of composites. Can these techniques be used to determine composite moduli? CHAPTER - 6 MACROMECHANICAL BEHAVIOUR 6.1 INTRODUCTION 6.2 THREE-DIMENSIONAL MATERIAL ANISOTROPY 6.3 MATERIAL SYMMETRY 6.4 ELASTIC CONSTANTS AND COMPLIANCES IN TERMS OF ENGINEERING CONSTANTS 6.5 CYLINDRICAL ORTHOTROPY 6.6 TWO-DIMENSIONAL CASE: PLANE STRESS 6.7 TRANSFORMATION OF ELASTIC CONSTANTS AND COMPLIANCES 6.7.1 Three-Dimensional Case 6.7.2 Two-Dimensional Case 6.8 PARTICULATE AND SHORT FIBRE COMPOSITES 6.9 MULTIDIRECTIONAL FIBRE REINFORCED COMPOSITES 6.10 UNIDIRECTIONAL LAMINA 6.11 BIDIRECTIONAL LAMINA 6.12 GENERAL LAMINATES 6.13 LAMINATE HYGROTHERMAL STRAINS 6.14 STRENGTH CRITERIA FOR ORTHOTROPIC LAMINA 6.15 BIBLIOGTAPHY 6.16 EXERCISES 6.1 INTRODUCTION The heterogeneity in a composite material is introduced due to not only its bi- phase or in some cases multi-phase composition, but also laminations. This leads to a distinctly different stress strain behaviour in the case of laminates. The anisotropy caused due to fibre orientations and the resulting extension-shear and bending-twisting coupling as well as the extension-bending coupling developed due to unsymmetric lamination add to the complexities. A clear understanding of the constitutive equations of a composite laminate is thus desirable before these are used in analysis and design of composite structures. In this chapter, we first introduce to the readers the basic constitutive equations for a general three-dimensional anisotropic material with and without material symmetry, elastic constants and compliances and their relations to engineering constants, as well as transformation laws for elastic constants and compliances for both three and two-dimensional cases. We also discuss constitutive relations for several composite materials?particulate and short fibre composites, multidirectional fibre reinforced composites, unidirectional lamina and general laminates as well as lamina strength criteria. 6.2 THREE-DIMENSIONAL MATERIAL ANISOTROPY For a three-dimensional elastic anisotropic body (Fig. 6.1), the generalized Hook's law is expressed as 3 3 1 1 ij ijkl kl k l C σ · · · ∈ ∑∑ (i, j = 1,2,3) (6.1) where ij σ and kl ∈ are the stress and strain tensors, respectively, and ijkl C are the elastic constants. Here the indices i, j, k and l can assume values of 1, 2 and 3. This implies that there may exist 3 4 = 81 independent elastic constants. However, it is known from the theory of elasticity, that both stress tensor ij σ and strain tensor kl ∈ are symmetric. As ij σ = ji σ , ijkl C = jikl C and as kl ∈ = lk ∈ , ijkl C = iflk C (6.2) Thus, ijkl C = jikl C = ijlk C = jilk C (6.3) This results in reduction of possible independent elastic constants to thirty-six. Further, if there exists a strain energy U such that kl ij ijkl C U ∈ ∈ · 2 1 (6.4) with the property that ij ij U σ · ∈ ∂ ∂ , then ijkl C = klij C (6.5) Equation 6.5 in conjunction with Eq. 6.3 finally reduce the total number of independent elastic constants from thirty-six to twenty-one only. Such an anisotropic material with twenty-one independent elastic constants is termed as triclinic. Now, using the following contracted single index notations 1 1 11 11 2 2 22 22 3 3 33 33 4 4 23 23 23 23 5 5 13 13 13 13 6 6 12 12 12 12 ( ) 2 ( ) ( ) 2 ( ) ( ) 2 ( ) and σ σ σ σ σ σ σ σ τ γ σ σ τ γ σ σ τ γ ∈ ∈ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ∈ ∈ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ∈ ∈ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ · · ' ; ' ; ' ; ' ; ∈ · ∈ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ∈ · ∈ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ∈ · ∈ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ (6.6) the constitutive relations for the general case of material anisotropy are expressed as 1 11 12 13 14 15 16 1 2 22 23 24 25 26 2 3 33 34 35 36 3 4 44 45 46 4 5 55 56 5 6 66 6 symmetric C C C C C C C C C C C C C C C C C C C C C σ σ σ σ σ σ ∈ ¹ ¹ 1 ¹ ¹ ¹ ¹ 1 ¹ ¹ ∈ ¹ ¹ 1 ¹ ¹ ¹ ¹ 1 ¹ ¹ ∈ ¹ ¹ ¹ ¹ · · 1 ' ; ' ; ∈ 1 ¹ ¹ ¹ ¹ 1 ¹ ¹ ¹ ¹ ∈ 1 ¹ ¹ ¹ ¹ ∈ 1 ¹ ¹ ¹ ¹ ¹ ¹ ¸ ] ¹ ¹ (6.7) or, } ]{ [ } { 1 j ij C ∈ · σ ; i, j = 1, 2,?.,6 (6.8) Here, [ ij C ] is the elastic constant matrix. Conversely, { i ∈ } = [S ij ] { j σ } ; i, j =1, 2,?..,6 (6.9) where [S ij ] is the compliance matrix. Note that [S ij ] = [C ij ] -1 (6.10) Also, [ ij C ] =[ ji C ] and [S ij ] = [S ji ] due to symmetry. 6.3 MATERIAL SYMMETRY There may exist several situations when the distribution and orientation of reinforcements may give rise to special cases of material property symmetry. When there is one plane of material property symmetry (say, the plane of symmetry is x 3 = 0, i.e., the rotation of 180 degree around the x 3 axis yields an equivalent material), the elastic constant matrix [ ij C ] is modified as 11 12 13 16 22 23 26 33 36 44 45 55 66 0 0 0 0 0 0 0 0 ( 0) 3 ij C C C C C C C C C C C C one plane of symmetry symmetry C X C 1 1 1 1 1 · 1 ¸ ] 1 1 1 · 1 ¸ ] (6.11) Thus there are thirteen independent elastic constants, and the material is monoclinic. The compliance matrix [S ij ] for a monoclinic material may accordingly be written from Eq. 6.11 by replacing 'C ' with 'S '. If there are three mutually orthogonal planes of symmetry, the material behaviour is orthotropic. The elastic constant matrix ] [ ij C is then expressed as ] [ ij C orthotropic = 11 12 13 22 23 33 44 55 66 0 0 0 0 0 0 0 0 0 0 0 0 C C C C C C C symmetry C C 1 1 1 1 1 1 1 1 1 ¸ ] (6.12) Thus there are nine independent elastic constants. Correspondingly there exist nine independent compliances. Two special cases of symmetry, square symmetry and hexagonal symmetry, may arise due to packing of fibres in some regular fashion. This results in further reduction of independent elastic constant. For instance, if the fibres are packed in a square array (Fig. 6.2) in the X 2 X 3 plane. Then [ ij C ] square array = 1 1 1 1 1 1 1 1 ] 1 ¸ 66 66 44 22 23 22 12 12 11 0 0 0 0 0 0 0 0 0 0 0 0 C C symmetry C C C C C C C (6.13) There exist now six independent elastic constants. Similarly, when the fibres are packed in hexagonal array (Fig. 6.3), 11 12 12 22 23 22 22 23 66 66 0 0 0 0 0 0 0 0 0 1 ( ) 0 2 0 ij C C C C C C C hexagonal array C C symmetry C C 1 1 1 1 1 1 1 1 1 · ¸ ] 1 − 1 1 1 1 1 1 1 ¸ ] (6.14) In the case of hexagonal symmetry, the number of independent elastic constants is reduced to five only. The material symmetry equivalent to the hexagonal symmetry, is also achieved, if the fibres are packed in a random fashion (Fig. 6.4) in the X 2 X 3 plane. This form of symmetry is usually termed as transverse isotropy. The [ ij C ] matrix due to the transverse isotropy is the same as that given in Eq. 6.14. The compliance matrices corresponding to Eqs. 6.12 through 6.14 can be accordingly written down. However,it may be noted that in the case of rectangular array (Fig. 6.5), C 12 ≠ C 13 , C 22 ≠ C 33 and C 55 ≠ C 66 (Eq. 6.13). Material Isotropy The material properties remain independent of directional change for an isotropic material. The elastic constant matrix [ ij C ] for a three dimensional isotropic material are expressed as 11 12 12 11 12 11 11 12 11 12 11 12 0 0 0 0 0 0 0 0 0 1 ( ) 0 2 1 ( ) 0 2 1 ( ) 2 ij C C C C C C C isotropy C C symmetry C C C C 1 1 1 1 1 1 1 1 1 · ¸ ] 1 − 1 1 1 − 1 1 1 − 1 ¸ ] (6.15) The compliance matrix [S ij ] for an isotropic material can be accordingly derived. 6.4 ELASTIC CONSTANTS AND COMPLIANCES IN TERMS OF ENGINEERING CONSTANTS The elastic constants or compliances are essentially material constants. Incidentally, the determination of all these elastic constants or compliances is not easy to accomplish by simple tests. The material constants that are normally determined through characterization experiments (see chapter 4) are termed as engineering constants. They can also be evaluated using the micromechanics material models (chapter 5). All nine independent compliances and therefore elastic constants listed in Eq. 6.12 are now expressed in terms of nine independent engineering constants. The stress- strain relations for a three-dimensional orthotropic material, in terms of engineering constants, can be written as follows: ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] 1 ¸ Ε Ε − Ε − Ε − Ε Ε − Ε − Ε − Ε · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ ∈ ∈ ∈ 6 5 4 3 2 1 12 13 23 33 22 23 11 13 33 32 22 11 12 33 31 22 21 11 6 5 4 3 2 1 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 σ σ σ σ σ σ ν ν ν ν ν ν G G symmetric G (6.16) We know that, in terms of compliances, the stress-strain relations are { } 1 { } [ ] ij j S orthotropic σ ∈ · (6.17) Comparing Eqs. 6.16 and 6.17, we can express the compliances in terms of engineering constants. 11 22 33 11 22 33 1 1 1 ; ; ; S S S · · · Ε Ε Ε ; 11 12 22 21 21 12 Ε − · Ε − · · ν ν S S ; 11 13 33 31 31 13 Ε − · Ε − · · ν ν S S (6.18) ; 22 23 33 32 32 23 Ε − · Ε − · · ν ν S S 12 66 13 55 23 44 1 ; 1 ; 1 G S G S G S · · · The elastic constants can then be derived by inversion of the compliance matrix i.e. [ ij C ] = [S ij ] -1 and are given as follows: ∗ Ε Ε Ε − Ε · ] ) / ( 1 [ 2 23 22 33 11 11 ν C ∗ Ε Ε + Ε · · ] [ 23 13 33 12 22 21 12 ν ν ν C C ∗ Ε + Ε · · ] [ 13 23 12 33 31 13 ν ν ν C C ∗ Ε Ε Ε − Ε · ] ) / ( 1 [ 2 13 11 33 22 22 ν C (6.19) ∗ Ε Ε Ε − Ε · ] ) / ( 1 [ 2 12 11 22 33 33 ν C 44 23 55 13 66 12 ; ; C G C G C G · · · where 1 2 2 2 33 11 12 23 13 13 33 11 23 33 22 12 22 11 ( ) 1 2( / ) ( / ) ( / ) ( / ) ν ν ν ν ν ν ∗ − Ε · − Ε Ε − Ε Ε − Ε Ε − Ε Ε (6.20) In terms of engineering constants, the elastic constants and compliances for an isotropic material are given by 11 12 (1 ) ; (1 )(1 2 ) (1 )(1 2 ) C C ν ν ν ν ν ν − Ε Ε · · + − + − (6.21) and 11 12 1 ; S S ν · · − Ε Ε 6.5 CYLINDRICAL ORTHOTROPY Consider cylindrical coordinates r, θ, z as illustrated in Fig. 6.6. Here the z-axis is assumed to coincide with the X 3 -axis. The stress and strain components are represented as ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ 6 5 4 3 2 1 σ σ σ σ σ σ τ τ τ σ σ σ θ θ θθ r rz z zz rr and ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ ∈ ∈ ∈ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ 6 5 4 3 2 1 θ θ θθ γ γ γ r rz z zz rr (6.22) The stress-strain relations, in terms of compliances, become ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ · 1 1 1 1 1 1 1 1 ] 1 ¸ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ ∈ ∈ ∈ 6 5 4 3 2 1 66 55 44 33 23 22 13 12 11 6 5 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 σ σ σ σ σ σ S S S symmetric S S S S S S (6.23) { } [ ]{ } i ij j S σ ∈ · where 11 22 33 1 1 1 ; ; ; ZZ S S S rr θθ · · · Ε Ε Ε ; 21 12 rr r r S S Ε − · Ε − · · θ θθ θ ν ν ; 31 13 rr rz zr zz S S Ε − · Ε − · · ν ν (6.24) ; 32 23 θθ θ θ ν ν Ε − · Ε − · · z zz z S S 44 55 66 1 1 1 ; ; z rz r S S S G G G θ θ · · · The elastic constant matrix [ ij C ] is obtained by inversion of the compliance matrix [S ij ] i.e., [ ij C ] = [S ij ] -1 or from Eq. 6.19 by replacing the indices 1,2,3 with r, θ, z respectively. 6.6 TWO-DIMENSIONAL CASE: PLANE STRESS For the case of plane stress (Fig. 6.7) σ 3 = σ 4 = σ 5 = 0 (6.25) The stress-strain relations, with two-dimensional anisotropy, are ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ 1 1 1 ] 1 ¸ · ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ 6 2 1 66 26 22 16 12 11 6 2 1 Q symm Q Q Q Q Q σ σ σ (6.26) or, }, ]{ [ } { j ij i Q ∈ · σ i, j = 1,2,6 (6.27) Where [Q ij ] are the reduced stiffnesses (elastic constants) for plane stress. Similarly, in terms of compliances, the stress-strain relations are ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ 1 1 1 ] 1 ¸ · ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ 6 2 1 66 26 22 16 12 11 6 2 1 σ σ σ S symm S S S S S (6.28) or, }, ]{ [ } { j ij i S σ · ∈ i, j =1,2,6 (6.29) For the case of two-dimensional orthotropy (Fig. 6.8) the stress-strain relations are ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ∈′ ∈′ ∈′ 1 1 1 ] 1 ¸ ′ ′ ′ ′ ′ · ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ′ ′ ′ 6 2 1 66 22 12 12 11 6 2 1 0 0 0 0 Q Q Q Q Q σ σ σ (6.30) and ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ′ ′ ′ 1 1 1 ] 1 ¸ ′ ′ ′ ′ ′ · ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ∈′ ∈′ ∈′ 6 2 1 66 22 12 12 11 6 2 1 0 0 0 0 σ σ σ S S S S S (6.31) with 11 22 11 22 12 21 12 21 12 22 21 11 12 21 66 12 12 21 12 21 ; 1 1 ; 1 1 Q Q Q Q Q G ν ν ν ν ν ν ν ν ν ν ′ ′ Ε Ε ′ ′ · · ′ ′ ′ ′ − − ′ ′ ′ ′ Ε Ε ′ ′ ′ ′ · · · · ′ ′ ′ ′ − − (6.32) and 21 12 11 22 12 21 66 11 22 22 11 12 1 1 1 ; ; ; S S S S S G ν ν ′ ′ ′ ′ · · · · − · − · ′ ′ ′ ′ ′ Ε Ε Ε Ε (6.33) Note that the engineering constants 11 Ε′ , 22 Ε′ , 12 ν ′ , (or 21 ν ′ ) and G' 12 are referred to the orthotropic axis system X' 1 X' 2 (i.e., the material axes). 6.7 TRANSFORMATION OF ELASTIC CONSTANTS AND COMPLIANCES If the elastic constants and compliances of a material are known with respect to a given co-ordinate system, then the corresponding values with respect to any other mutually perpendicular coordinates can be determined using laws of transformation. These are explained in Appendix A. 6.7.1 Three-Dimensional Case The transformation of elastic constants from the X' 1 X' 2 X' 3 coordinates to mutually orthogonal X 1 X 2 X 3 coordinates (Refer Fig. A.1 and Eq. A. 22) is given as follows: [ ] [ ] [ ][ ] T ij ij C T C T ∈ ∈ ′ · (6.34) where transformation matrix ] [ ∈ T is given by Eq. A.8. Note that the elements of ] [ ij C and ] [ ij C′ correspond to the X 1 X 2 X 3 and X' 1 X' 2 X' 3 coordinates, respectively. One can use Eq. 6.34 in the following form ] [ ij C′ = ] [ ∈ T -T ] [ ij C ] [ ∈ T -1 = [T σ ] ] [ ij C [T σ ] T (6.35) if the transformation is required from X 1 X 2 X 3 coordinates to theX' 1 X' 2 X' 3 coordinates. Note that [T σ ] is defined by Eq. A.13. Simalarly, ] ][ [ ] [ ] [ σ σ T S T S ij T ij ′ · (6.36) T ij ij T S T S ] ][ ][ [ ] [ ∈ ∈ · ′ (6.37) The corresponding elastic constants ] [ ij C and compliances [S ij ] due to special cases of material symmetry and transformation matrices ] [ ∈ T and [T σ ] due to specific orientation of axes are to be reduced from the general three dimensional cases, before transformation is sought from one axis system to the other. 6.7.2 Two-Dimensional Case If the elements of ] [ ij Q and ] [ ij Q′ refer to the X 1 X 2 and X' 1 X' 2 coordinates (see Eqs. 6.26 and 6.30 and Figs. 6.7 and 6.8), respectively, then transformation laws for reduced elastic constants are obtained as follows: ] ][ [ ] [ ] [ ∈ ∈ ′ · T Q T Q ij T ij (6.38) T ij ij T Q T Q ] ][ ][ [ ] [ σ σ · ′ (6.39) where ] [ ∈ T and ] [ σ T are defined by Eqs. A.18 and A.19. The compliance matrices are accordingly transformed using Eqs. 6.36 and 6.37. Accordingly, from Eqs. 6.38 and A.18 it can be shown that ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ′ ′ ′ ′ 1 1 1 1 1 1 1 1 ] 1 ¸ − − − − − − − − − + · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ 66 12 22 11 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 4 4 2 2 2 2 2 2 2 2 4 4 2 2 2 2 4 4 26 16 66 12 22 11 ) ( 2 ) ( 2 ) ( 2 4 4 2 4 2 Q Q Q Q mn n m mn n m n m mn n m mn n m mn mn n m n m n m n m n m n m n m n m n m n m n m m n n m n m n m Q Q Q Q Q Q (6.40) In a similar way from Eqs. 6,36 and A.19 one obtains ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ′ ′ ′ ′ 1 1 1 1 1 1 1 1 ] 1 ¸ − − − − − − − − − + · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ 66 12 22 11 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 4 4 2 2 2 2 2 2 2 2 4 4 2 2 2 2 4 4 26 16 66 12 22 11 ) ( 2 2 2 ) ( 2 2 2 ) ( 8 4 4 2 2 S S S S mn n m mn n m n m mn n m mn n m mn mn n m n m n m n m n m n m n m n m n m n m n m m n n m n m n m S S S S S S (6.41) Note that m = cos φ and n = sin φ and ] [ ij Q′ and ] [ ij S′ are defined by Eqs. 6.32 and 6.33 respectively, in terms of engineering constants 11 Ε′ , 22 Ε′ , 12 ν ′ (or 21 ν ′ ) and G' 12 corresponding to principal material directions. If transformation is required from one anisotropic material axis system (say X 1 X 2 X 3 ) to another anisotropic material axis system (say, 3 2 1 X X X ), then from Eqs. 6.38 and A. 18 we can ] ][ [ ] [ ] [ ∈ ∈ · T Q T Q ij T ij or, 1 1 1 ] 1 ¸ − − − 1 1 1 ] 1 ¸ 1 1 1 ] 1 ¸ − − − · 1 1 1 ] 1 ¸ 2 2 2 2 2 2 66 26 16 26 22 12 16 12 11 2 2 2 2 2 2 66 26 16 26 22 12 16 12 11 2 2 2 2 n m mn mn mn m n mn n m Q Q Q Q Q Q Q Q Q n m mn mn mn m n mn n m Q Q Q Q Q Q Q Q Q or, · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ 26 16 66 12 22 11 Q Q Q Q Q Q ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ 1 1 1 1 1 1 1 1 ] 1 ¸ − − − − − − − − − − − − − − − − − + − − 26 16 66 12 22 11 2 2 4 4 2 2 3 3 3 3 3 3 4 2 2 2 2 4 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 3 3 3 3 2 2 4 4 2 2 2 2 3 3 2 2 2 2 4 4 3 3 2 2 2 2 4 4 3 3 ) ( 2 3 3 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 ) ( 2 4 4 4 4 2 4 4 4 2 Q Q Q Q Q Q n m m n n m mn n m mn n m n m mn n n m n m m n m mn n m mn mn n m n m mn mn n m n m n m n m n m n m mn mn n m n m n m n m n m n m mn n m n m m n n m n m n m n m n m (6.42) Similarly, using Eqs. 6.36 and A.19 one can write ] ][ [ ] [ ] [ σ σ T S T S ij T ij · or, · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ 26 16 66 12 22 11 S S S S S S ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ 1 1 1 1 1 1 1 1 ] 1 ¸ − − − − − − − − − − − − − − − − − + − − 26 16 66 12 22 11 2 2 4 4 2 2 3 3 3 3 3 3 4 2 2 2 2 4 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 3 3 3 3 2 2 4 4 2 2 2 2 3 3 2 2 2 2 4 4 3 3 2 2 2 2 4 4 3 3 ) ( 2 2 2 3 3 ) ( 2 2 2 ) ( 4 ) ( 4 ) ( 8 4 4 2 2 2 2 2 2 S S S S S S n m m n n m mn n m mn n m n m mn n n m n m m n m mn n m mn mn n m n m mn mn n m n m n m n m n m n m mn mn n m n m n m n m n m n m mn n m n m m n n m n m n m n m n m (6.43) 6.8 PARTICULATE AND SHORT FIBRE COMPOSITES Particulate composites, where reinforcements are in the form of particles, platelets and flakes, and short fibre composites may exhibit a wide range of elastic material behaviour depending on the shapes, sizes, orientations and distributions of reinforcements in the matrix phase as well as elastic properties of the constituent materials. The matrix behaviour is normally isotropic. The composition of these composites are first established by examining their morphology and then proper stress-strain relations can be obtained from the equations developed in the preceding sections. It is also to be noted whether the composite body under consideration is three-dimensional or two dimensional in character. For example, the behaviour of a three-dimensional composite with a typical reinforcement packing shown in Fig. 6.9a is anisotropic in nature. Here the reinforcements are oriented in some regular fashion with respect to the reference axes X 1 X 2 X 3 . The stress strain relations } { i σ = ] [ ij C } { j ∈ for this type of composites are given by Eq 6.8 with elements of ] [ ij C listed in Eq. 6.7. When the reinforcements are arranged parallel to the axes (Fig. 6.9b), the composite behaviour is orthotropic and Eq. 6.12 defines the corresponding ] [ ij C . If the orientation and distribution of reinforcements are found to be random in the matrix phase, as shown in Fig. 6.9c, the composite is assumed to behave like an isotropic material. Consequently, the elastic constant matrix ] [ ij C is reduced to that given in Eq. 6.15. For two-dimensional anisotropic, orthotropic and isotropic cases, some possible reinforcement arrangements are illustrated in Fig. 6.10. The stress-strain relations, as presented in section 6.6 can be accordingly used for these cases. If transformations of elastic constants and compliances are required from one axes system to another, then one can use the transformation rules discussed in section 6.7. Fig. 6.10 6.9 MULTIDIRECTIONAL FIBRE REINFORCED COMPOSITES Composites exhibit strong directional properties, when reinforcements are in the form of continuous fibres. In a multidirectional composite, fibres can be placed in any desired direction in a three-dimensional space, along which better stiffness (or strength) is desired. The shear properties can be greatly improved by providing diagonal reinforcements. Carbon-carbon composites form an important class of multidirectional composites due to several variations in weave design and perform construction. Similar multi-directional composite systems can also be designed and developed with both metal- matrix and ceramic-matrix composites. A typical multi-directional (5D) composite is shown in Fig. 6.11a. There are three bundles of orthogonal fibres f 1 , f 2 , f 3 and two bundles of diagonal fibres f 4 , f 5 . We consider here an integrated multidirectional fibre reinforced composite moder which contains n number of unidirectional fibre composite blocks that are oriented in n arbitrary directions with respect to a three-dimensional reference axes X 1 X 2 X 3 . Each unit block may have different fobre volume fractions. This arrangement makes n number of material axis systems, and therefore yield n sets of direction cosines between n material axis systems and the reference axes X 1 X 2 X 3 . For example, Figure 6.11b represents the orientation of the material axis system for the ith block. The corresponding transformation matrices [ ] i T ∈ and i T ] [ σ can then be written down using Eqs. A.8 and A.13, respectively. The material behaviour for each block with respect to its axes is orthotropic. The elastic constants for the ith block are then given as i i ij C C C C C C C C C C C C C 1 1 1 1 1 1 1 1 ] 1 ¸ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ · ′ 66 55 44 33 23 13 23 22 12 13 12 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] [ (6.44) The effective elastic constants for the n-directional fibre reinforced composite are then determined by averaging the transformed properties as follows: i i ij T i n i ij T C T n C ] [ ] [ ] [ ( 1 ] [ 1 ∈ · ∈ ′ · ∑ (6.45) Note that the overall fibre volume fraction is given as i n i f f V n V ) ( 1 1 ∑ · · (6.46) 6.10 UNIDIRECTIONAL LAMINA A unidirectional lamina is a thin layer (ply) of composite and is normally treated as a two-dimensional problem. It contains parallel, continuous fibres and provides extremely high directional properties. It is the basic building unit of a laminate and finds very wide applications in composite structures specially in the form of laminates. Therefore, the knowledge of its elastic macromechanical behaviour is of utmost importance to composite structural designers. Figure 6.12a depicts a unidirectional lamina where parallel, continuous fibres, are aligned along the X' 1 axis (fibre axis or longitudinal direction). The X' 2 axis (transverse direction) is normal the fibre axis. The axes X' 1 X' 2 are referred as material axes. The material axes are oriented counter clockwise by angle φ with respect to the reference axes X 1 X 2 . The angle φ (also referred as fibre angle) is considered positive when measured counterclockwise from the X 1 axis. This type of unidirectional lamina is termed as ?off-axis? lamina. An off-axis lamina behaves like an anisotropic two-dimensional body, and the stress-strain relations, given by Eqs. 6.26 through 6.29, can be used for the present case. When the material axes coincide with the reference axes (i.e., φ =0), as shown in Fig. 6.12b, the lamina is termed as ?on-axis? lamina and its behaviour is orthotropic in nature. The stress-strain relations are defined by Eqs. 6.30 and 6.31. The engineering constants 11 Ε′ , 22 Ε′ , 12 ν ′ , (or 21 ν ′ ) and G' 12 are usually known, as these can be determined either by using micromechanics theories (chapter 4) or by characterization tests (chapter 5). Using these engineering constants, the reduced stiffnesses ] [ ij Q′ and compliances ] [ ij S′ are then determined for an orthotropic lamina with the help of Eqs. 6.32 and 6.33. The transformed reduced stiffnesses ] [ ij Q and ] [ ij S can now be evaluated employing Eqs. 6.40 and 6.41. The stiffness ] [ ij Q and compliances ] [ ij S for three composite systems are computed for various fibre orientations and are listed in Tables 6.1 and 6.2. Typical variations of transformed properties ] [ ij Q and ] [ ij S with change in the fibre angle φ are illustrated in Figs. 6.13 and 6.14. Such plots aid to the basic understanding of the stiffness behaviour of an off-axis lamina with different fibre orientations. Note that the case φ =0 corresponds to an on-axis lamina. 6.11 BIDIRECTIONAL LAMINA</A< b> A bidirectional lamina is one which contains parallel, continuous fibres aligned along mutually perpendicular directions, as shown in Fig. 6.15. A lamina reinforced with woven fabrics that have fibres in the mutually orthogonal warp and fill directions can also be treated as a bidirectional lamina. The effects of undulation (crimp) and other problems associated with different weaving patterns are however, neglected. In Fig. 6.15 the X 1 ' X 2 ' is referred as material axes. The amount of fibres in both directions need not necessarily be the same. In a hybrid lamina, even the fibres in two directions may vary, but when the material axes X 1 ' X 2 ' coincide with the reference axes X 1 X 2 (Fig. 6.15a), the material behaviour is orthotropic and the lamina may be termed as ?on-axis? bidirectional lamina. If the X 1 ' X 2 ' plane rotates by an angle φ with respect to the X 1 X 2 axes (Fig. 6.15a), then the oriented lamina behaves as an anisotropic material and it can be identified as an ?off- axis? bidirectional lamina can also be treated as a two-dimensional problem and its elastic properties can be determined in an usual manner as discussed in sections 6.6 and 6.10. It may be mentioned that the anisotropy and stiffness behaviour of a bidirectional lamina can be greatly controlled by varying the types of fibres (say, carbon fibre along the X 1 ' direction and glass fibre along the X 2 ' direction) and volume fractions of fibres (V f ) in both directions. When the fibres and V f are same in both directions, then E' 11 = E' 22 and the material behaviour is square symmetric. Note that a square symmetric material is different from an isotropic material. 6.12 GENERAL LAMINATES We consider here a general thin laminate of thickness h (Fig. 6.16). The X 3 axis is replaced here by the z axis for convenience. The laminate consists of n number of unidirectional and/or bidirectional laminae, where each lamina may be of different materials and thicknesses and have different fibre orientations ( φ ). A thin general laminate is essentially a two-dimensional problem, but cannot be treated as a two- dimensional plane stress problem as has been done for a unidirectional lamina. The existence of extension ?bending couling causes bending, even if the laminate is subjected to inplane loads only. Therefore, thin plate bending theories are employed in derivation of constitutive relations. We assume that Kirchhoff 's assumptions related to the thin plate bending theory are applicable in the present case. Let u 1 0 , u 2 0 and w are the mid-plane displacements, and w is constant through the thickness of the lamina. Then the mid-plane strains are given by 0 0 0 0 0 0 0 1 2 1 2 1 2 6 1 2 2 1 , and u u u u x x x x ∂ ∂ ∂ ∂ ∈ · ∈ · ∈ · + ∂ ∂ ∂ ∂ (6.47) and the curvatures, which are constant through the thickness of the laminate, are 2 2 2 1 2 6 2 2 1 2 1 2 , and 2 w w w k k k x x x x ∂ ∂ ∂ · − · − · − ∂ ∂ ∂ ∂ (6.48) The strains at any distance z are then given as 0 0 0 1 1 1 2 2 2 6 6 6 ( ) , ( ) and ( ) z zk z zk z zk ∈ ·∈ + ∈ ·∈ + ∈ ·∈ + (6.49) Now from Eq. 6.26, we have at any distance z 1 11 12 16 1 2 12 22 26 2 6 16 26 66 6 z z z Q Q Q Q Q Q Q Q Q σ σ σ ∈ ¹ ¹ 1 ¹ ¹ ¹ ¹ ¹ ¹ 1 · ∈ ' ; ' ; 1 ¹ ¹ ¹ ¹ 1 ∈ ¹ ¹ ¸ ] ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ + ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ 1 1 1 ] 1 ¸ · 6 2 1 0 6 0 2 0 1 66 26 16 26 22 12 16 12 11 k k k z Q Q Q Q Q Q Q Q Q z (6.50) The stress and moment resultants (Fig. 6.17) are evaluated per unit length of the laminate as follows: 1 1 / 2 2 2 / 2 6 6 h h N N dz N σ σ σ − ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ · ' ; ' ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ∫ and 1 1 / 2 2 2 / 2 6 6 h h M M z dz M σ σ σ − ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ · ' ; ' ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ∫ (6.51) Thus, [ ] 0 1 1 / 2 / 2 0 1 1 11 12 16 2 2 0 / 2 / 2 6 6 h h h h k N dz Q Q Q z k dz k σ − − ¹ ¹ ¹ ¹ ∈ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ · · ∈ + '' ; ' ;; ¹¹ ¹ ¹ ¹¹ ∈ ¹ ¹ ¹ ¹ ¹ ¹ ∫ ∫ [ ] [ ] 0 1 1 0 11 12 16 2 11 12 16 2 0 6 6 k A A A B B B k k ¹ ¹ ∈ ¹ ¹ ¹ ¹ ¹ ¹ · ∈ + ' ; ' ; ¹ ¹ ¹ ¹ ∈ ¹ ¹ ¹ ¹ where / 2 / 2 h h Qdz − Α · ∫ and / 2 / 2 h h B Qz dz − · ∫ [ ] 0 1 1 / 2 / 2 0 1 1 11 12 16 2 2 0 / 2 / 2 6 6 h h h h k M z dz Q Q Q z k z dz k σ − − ¹ ¹ ¹ ¹ ∈ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ · · ∈ + '' ; ' ;; ¹¹ ¹ ¹ ¹¹ ∈ ¹ ¹ ¹ ¹ ¹ ¹ ∫ ∫ [ ] [ ] ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ + ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ · 6 2 1 16 12 11 0 3 0 2 0 1 16 12 11 k k k D D D B B B where / 2 / 2 2 / 2 / 2 and h h h h B Qz dz D Qz dz − − · · ∫ ∫ Proceeding in a similar manner, all stress and moment resultants can be expressed as listed below: ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ 1 1 1 1 1 1 1 1 ] 1 ¸ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ Μ Μ Μ Ν Ν Ν 6 2 1 0 6 0 2 0 1 66 26 16 66 26 16 26 22 12 26 22 12 16 12 11 16 12 11 66 26 16 66 26 16 26 22 12 26 22 12 16 12 11 16 12 11 6 2 1 6 2 1 k k k D D D B B B D D D B B B D D D B B B B B B A A A B B B A A A B B B A A A (6.52) with (A ij , B ij , D ij ) = ∫ − 2 / 2 / h h Q ij (1, z, z 2 ) dz; i, j = 1, 2, 6 (6.53) Equation 6.52 represents the constitutive relations for a general laminate, and A ij , B ij , and D ij are the inplane, extension bending coupling and bending stiffnesses, respectively. Note that all these stiffnesses are derived for a unit length of the laminate. The elastic properties of each lamina are generally assumed to be constant through its thickness, as these laminae are considered to be thin. Then A ij , B ij , and D ij are approximated as ) ( ) ( 1 1 − · − · Α ∑ k k k ij n k ij z z Q 2 2 1 1 1 ( ) ( ) 2 n ij ij k k k k B Q z z − · · − ∑ (6.54) 3 3 1 1 1 ( ) ( ) 3 n ij ij k k k k D Q z z − · · − ∑ From Eq. 6.52, it is seen that there exist several types of mechanical coupling in a general laminate. These are grouped together as follows: Extension ? Shear : A 16 , A 26 Extension ? Bending : B 11 , B 12 , B 22 Extension ? Twisting : B 16 , B 26 Shear ? Bending : B 16 , B 26 Shear ? Twisting : B 66 Bending ? Twisting : D 16 , D 26 Biaxial ? Extension : A 12 Biaxial ? Bending : D 12 As stated earlier, the coupling terms B ij occur due to unsymmetry about the middle surface of a laminate. However, all terms containing suffices '16 ' and '26 ' are resulted due to anisotropy caused by the fibre orientation φ other than 0 0 and 90 0 . Those containing suffices '12 ' are due to Poisson's effect. Although a heneral unsymmetric laminate contains all coupling terms, there are several laminates where some of these may vanish. These are listed in Table 6.3. There are several important points that are to be noted here. The first two laminates (serial nos. 1 and 2) which are christened as ?off- axis laminate? and ?on-axis laminate?; respectively are essentially paralles ply laminates where all laminae in a laminate have the same fibre orientation and therefore are stacked parallel to each other. These are, in fact, similar to unidirectional laminae. For a symmetric balanced angle-ply laminate D 16 and D 26 do not vanish, although A 16 = A 26 = 0. The only coupling effect that appears in an anti-symmetric cross-ply laminate is the extension-bending coupling due to presence of B 11 and B 22 and note that B 22 = - B 11 . But the existence of B 16 and B 26 cause an antisymmetric angle-ply laminate to experience extension-twisting coupling. Note also that extension-bending coupling is predominant for an unsymmetric cross-ply laminate. The mechanical coupling, as discussed above, influences the deformation behaviour of a laminate to a great extent. This can be better understood by examining the deformed shapes of a couple of laminates as illustrated in Figs. 6.18 through 6.20. Here the dotted lines represent the undeformed shape and the firm lines, deformed shapes. Consider first a simple off-axis laminate (or unidirectional lamina), subjected to an inplane stress resultant N 1 (Fig.6.18a) and an out-of-plane moment resultant M 1 (Fig. 6.18b). We know from Eq. 6.52 and Table 6.1 (B ij =0) that 1 11 1 12 2 16 6 1 11 1 12 2 16 6 D k D k D k Ν · Α ∈ + Α ∈ + Α ∈ Μ · + + (6.55) Thus, as illustrated in Fig. 6.18a, it is noted that a simple tension causes not only extension and contraction, but also shearing of the laminate. While the extension and contraction are due to A 11 and A 12 , respectively and the inplane shear deformation is due to presence of A 16 . This characteristic behaviour is seen especially in an anisotropic (off- axis) laminate. The shear deformation vanishes, if A 16 = 0, as in the case of an orthotropic (on-axis) laminate (serial no.2 of Table 6.1). Similarly, as can be seen in Fig. 6.18b, a simple bending due to M 1 has resulted not only longitudinal bending (due to D 11 ) and transverse bending (due to D 12 ), but also twisting (due to D 16 ). Figure 6.19 describes the deformation behaviour of an antisymmetric cross-ply laminate. The extension-bending coupling due to B 11 and B 22 can be clearly observed. In Fig. 6.19a a simple inplane tension is found to introduce bending in the laminate. Conversely, a simple bending causes extension of the laminate, a shown in Fig. 6.19b. Figure 6.20 depicts the deformed shape of an antisymmetric angle-ply laminate. Here the extension-bending and bending-shear coupling effects due to B 16 and B 26 are presented. In a similar manner, the deformation characteristics of other types of laminates can be illustrated. The most important point that is to be focused here is that fibre orientation and lamina stacking sequence affect laminate stiffness properties, which, in turn, control the deformation behaviour of a laminate. Table 6.4 provides the stiffnesses [A ij ], [B ij ]and [D ij ] for various stacking sequences of carbon/epoxy composites. The [Q ij ] values given in Table 6.1 have been used to compute the above stiffnesses. 6.13 LAMINATE HYGROTHERMAL STRAINS The changes in moisture concentration and temperature introduce expansional strains in each lamina. The stress-strain relation of an off-axis lamina (Eq. 6.28) is then modified as follows ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ + ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ 1 1 1 ] 1 ¸ · ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ e e e S S S S S S S S S 6 2 1 6 2 1 66 26 16 26 22 12 16 12 11 6 2 1 σ σ σ (6.56) with 1 1 1 2 2 2 6 6 6 e H T e H T e H T ¹ ¹ ¹ ¹ ¹ ¹ ∈ ∈ ∈ ¹ ¹ ¹ ¹ ¹ ¹ ∈ · ∈ + ∈ ' ; ' ; ' ; ¹ ¹ ¹ ¹ ¹ ¹ ∈ ∈ ∈ ¹ ¹ ¹ ¹ ¹ ¹ (6.57) and 1 1 2 2 6 6 H H H C β β β ¹ ¹ ∈ ¹ ¹ ¹ ¹ ¹ ¹ ∈ · ∆ ' ; ' ; ¹ ¹ ¹ ¹ ∈ ¹ ¹ ¹ ¹ and 1 1 2 2 6 6 T T T T α α α ¹ ¹ ∈ ¹ ¹ ¹ ¹ ¹ ¹ ∈ · ∆ ' ; ' ; ¹ ¹ ¹ ¹ ∈ ¹ ¹ ¹ ¹ (6.58) where the superscripts e, H, T refer to expansion, moisture and temperature, respectively, ΔC and ΔT are the change in specific moisture concentration and temperature, respectively, and β's and α's are coefficients of moisture expansion and thermal expansion respectively. Note that the spatial distributions of moisture concentration and temperature are determined from solution of moisture diffusion and heat transfer problems. Expansional strains transform like mechanical strains (Appendix A) i.e., } ]{ [ } { ∈ · ∈′ ∈ T . Inversion of Eq. 6.56 yields (see also Eq. 6.26), at any distance z (Fig. 6.16), 1 11 12 16 1 1 2 12 22 26 2 2 6 16 26 66 6 6 e e e z Z z Q Q Q Q Q Q Q Q Q σ σ σ ¹ ¹ ∈ − ∈ ¹ ¹ 1 ¹ ¹ ¹ ¹ 1 · ∈ − ∈ ' ; ' ; 1 ¹ ¹ ¹ ¹ 1 ∈ − ∈ ¹ ¹ ¸ ] ¹ ¹ (6.59) Thus, for a general laminate Eq. 6.52 will be modified as ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ Μ Μ Μ Ν Ν Ν + ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ 1 1 1 1 1 1 1 1 ] 1 ¸ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ Μ Μ Μ Ν Ν Ν e e e e e e k k k D D D B B B D D D B B B D D D B B B B B B A A A B B B A A A B B B A A A 6 2 1 6 2 1 6 2 1 0 6 0 2 0 1 66 26 16 66 26 16 26 22 12 26 22 12 16 12 11 16 12 11 66 26 16 66 26 16 26 22 12 26 22 12 16 12 11 16 12 11 6 2 1 6 2 1 (6.60) where the expansional force resultants are dz Q Q Q Q Q Q Q Q Q z e e e z h h e e e ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ 1 1 1 ] 1 ¸ · ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ Ν Ν Ν ∫ − 6 2 1 2 / 2 / 66 26 16 26 22 12 16 12 11 6 2 1 (6.61) and the expansional moments are 1 11 12 16 1 / 2 2 12 22 26 2 / 2 6 16 26 66 6 e e h e e e e h z z Q Q Q Q Q Q z dz Q Q Q − ¹ ¹ ¹ ¹ Μ ∈ 1 ¹ ¹ ¹ ¹ 1 Μ · ∈ ' ; ' ; 1 ¹ ¹ ¹ ¹ 1 Μ ∈ ¸ ] ¹ ¹ ¹ ¹ ∫ (6.62) These expansional force resultants and moments may considerably influence the deformation behaviour of a laminate. 6.14 STRENGTH CRITERIA FOR ORTHOTROPIC LAMINA Isotropic materials do not have any preferential direction and in most cases tensile strength and compressive strength are equal. The shear strength is also dependent on the tensile strength. A strength criterion for an isotropic lamina is, therefore, based on stress components, σ 1 , σ 2 and σ 6 for a two-dimensional problem and a single strength constant i.e., ultimate strength X. An orthotropic lamina (Fig. 6.8), on the other hand, exhibits five independent strength constants e.g., tensile strength X' 11 t a dcompressive strength X' 11 c along the X' 1 direction; tensile strength X' 22 t and compressive strength X' 22 c along the X' 2 direction and inplane shear strength X' 12 . Hence a strength criterion for a two-dimensional orthotropic lamina should involve the stress components σ' 1 , σ' 2 and σ' 6 and strength constants X' 11 t , X' 11 c , X' 22 t X' 22 c and X ' 12 . We present here a few important strength criteria that are commonly used to evaluate the failure of an orthotropic lamina. Maximum Stress Criterian A lamina is assumed to fail, if any of the following relations is satisfied 1 11 2 22 ; t t σ σ ′ ′ ′ ′ ≥ Χ ≥ Χ , when 1 σ ′ and 2 σ ′ are tensile 1 11 2 22 ; c c σ σ ′ ′ ′ ′ ≥ Χ ≥ Χ , when 1 σ ′ and 2 σ ′ are compressive. (6.63) 12 6 Χ′ ≤ ′ σ It is assumed that inplane shear strengths are equal under positive or negative shear load. Maximum Strain Criterian A lamina fails, if any of the following is satisfied 1 1 2 2 ; , t t u u ′ ′ ′ ′ ∈ ≥∈ ∈ ≥∈ when 1 ∈′ and 2 ∈′ are tensile 1 1 2 2 ; , c c u u ′ ′ ′ ′ ∈ ≥∈ ∈ ≥ −∈ when 1 ∈′ and 2 ∈′ are compressive. (6.64) 6 6u ′ ′ ∈ ≥∈ Note that the addition of suffix 'u' in strain components indicates the corresponding ultimate strains. The ultimate shear strains are also assumed to be equal under positive or negative shear load. If a material behaves linearly elastic till failure, the ultimate strains can be related to ultimate strength constants as follows: 1 11 11 2 22 22 / ; / t t t t u u ′ ′ ′ ′ ′ ′ ∈ · Χ Ε ∈ · Χ Ε 1 11 11 2 22 22 / ; / c c c c u u ′ ′ ′ ′ ′ ′ ∈ · Χ Ε ∈ · Χ Ε (6.65) 12 12 6 / G u ′ Χ′ · ∈′ Tsai-Hill Criterion The general three-dimensional orthotropic strength criterion is given by 2 2 2 1 2 3 1 2 2 2 2 1 3 2 3 4 5 6 ( )( ) ( )( ) ( )( ) 2 2 2 2 ( ) 2 ( ) 2 ( ) G H F H F G H G F L M N σ σ σ σ σ σ σ σ σ σ σ σ ′ ′ ′ ′ ′ + + + + + − ′ ′ ′ ′ ′ ′ ′ − − + + + (6.66) Assuming that normal stresses 1 σ ′ , 2 σ ′ and 3 σ′ an dshear stress 6 σ ′ act independently and substituting 1 σ ′ = X' 11 , 2 σ ′ = X' 22 , 33 3 Χ′ · ′ σ and 12 6 Χ′ · ′ σ in the above strength criterion, we obtain 2 11 ) /( 1 Χ′ · + H G ; 2 22 ) /( 1 Χ′ · + H F ; ; ) /( 1 2 33 Χ′ · + G F 2 12 ) /( 1 2 Χ′ · N (6.67) Combining Eqs. 6.67 we get 2 2 2 11 22 33 1 1 1 2 ( ) ( ) ( ) H · + − ′ ′ ′ Χ Χ Χ 2 2 2 11 33 22 1 1 1 2 ( ) ( ) ( ) G · + − ′ ′ ′ Χ Χ Χ (6.68) 2 2 2 22 33 11 1 1 1 2 ( ) ( ) ( ) F · + − ′ ′ ′ Χ Χ Χ Assuming transverse symmetry X' 22 = X' 33 and two-dimensional plane stress case (σ 3 = σ 4 = σ 5 =0), Eq. 6.66 reduces to 1 ) ( 2 2 ) )( ( ) )( ( 2 6 2 1 2 2 2 1 · ′ + ′ ′ − ′ + + ′ + σ σ σ σ σ N H H F H G or, 1 ) ( 2 12 6 2 11 2 1 2 22 2 2 11 1 · 1 ] 1 ¸ Χ′ ′ + Χ′ ′ ′ − 1 ] 1 ¸ Χ′ ′ + 1 ] 1 ¸ Χ′ ′ σ σ σ σ σ (6.69) When 1 σ ′ , 2 σ ′ or both are tensile or compressive, Eq. 6.69 can be used by substituting the corresponding tensile or compressive strength constants in it. Thus, if 1 σ ′ is tensile, X' 11 = X' 11 t , and if 2 σ ′ is compressive, X' 22 = X' 22 c and so on. Tsai-Hill / Hoffman Criterion Tsai-Hill/Hoffman criterion accounts for unequal tensile and compressive strengths. For a three-dimensional state of stress in an orthotropic material, this criterion is given as 2 2 2 1 2 3 2 3 1 3 1 2 4 1 5 2 6 3 2 2 2 7 4 8 5 9 6 ( ) ( ) ( ) 1 C C C C C C C C C σ σ σ σ σ σ σ σ σ σ σ σ ′ ′ ′ ′ ′ ′ ′ ′ ′ − + − + − + + + ′ ′ ′ + + + · (6.70) If tensile 1 σ ′ acts only and 1 σ ′ =X' 11 t , then from Eq. 6.70 (C 2 + C 3 ) X' 11 t + C 4 = 1/X' 11 t (6.71) If compressive 1 σ ′ acts only and - 1 σ ′ = X' 11 c , then (C 2 + C 3 ) X' 11 c ? C 4 = 1/X' 11 c (6.72) From Eqs. 6.71 and 6.72, we obtain 4 2 3 11 11 11 11 1 1 1 ; ( ) t c t c C C C · − + · ′ ′ ′ ′ Χ Χ Χ Χ (6.73) Similarly, consideration of 2 σ ′ and 3 σ′ yields 5 1 3 22 22 22 22 1 1 1 ; ( ) t c t c C C C · − + · ′ ′ ′ ′ Χ Χ Χ Χ (6.74) 6 1 2 33 33 33 33 1 1 1 ; ( ) t c t c C C C · − + · ′ ′ ′ ′ Χ Χ Χ Χ (6.75) Now, assuming 33 22 Χ′ · Χ′ , we derive from the above the following relations for C 1 , C 2 and C 3 : 1 22 22 11 11 1 1 2 t c t c C · − ′ ′ ′ ′ Χ Χ Χ Χ (6.76) 2 3 11 11 1 2 t c C C · · ′ ′ Χ Χ (6.77) Further, applying 6 σ′ only and 6 12 X σ′ ′ · yields 9 2 12 1 ( ) C X · ′ (6.78) Now, considering a two dimensional state of plane stress condition ) 0 ( 5 4 3 · ′ · ′ · ′ σ σ σ and substituting the values of C 1 , C 2 , C 3 , C 4 , C 5 and C 9 from the above relations, the strength criterion takes the following form: 2 2 2 6 1 1 2 2 1 2 11 11 11 11 22 22 11 11 22 22 12 ( ) ( ) 1 1 1 1 ( ) ( ) ( ) 1 t c t c t c t c t c σ σ σ σ σ σ σ ′ ′ ′ ′ ′ ′ ′ − + + − + − + · ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ (6.79) Tsai-Wu Quadratic Interaction Criterion For an orthotropic material under a two-dimensional state of plane stress condition, this criterion assumes the form 2 2 2 11 1 12 1 2 22 2 66 6 16 1 6 26 2 6 1 1 2 2 6 6 ( ) 2 ( ) ( ) 2 2 1 F F F F F F F F F σ σ σ σ σ σ σ σ σ σ σ σ ′ ′ ′ ′ ′ ′ ′ + + + + ′ ′ ′ ′ ′ + + + + · (6.80) Considering that the positive or negative inplane shear stress 6 σ ′ should not affect the results, the terms F 16 1 σ ′ 6 σ′ , F 26 2 σ ′ 6 σ ′ and F 6 6 σ ′ should vanish. Hence Eq. 6.80 reduces to 2 2 2 11 1 12 1 2 22 2 66 6 1 1 2 2 ( ) 2 ( ) ( ) 1 F F F F F F σ σ σ σ σ σ σ ′ ′ ′ ′ ′ ′ ′ + + + + + · (6.81) Now applying independently tensile and compressive normal stresses 1 σ ′ and 2 σ ′ , and inplane shear stresses 6 σ ′ , and substitution of 1 σ ′ =X' 11 t - 1 σ ′ =X' 11 c , 2 σ ′ =X' 22 t , - 2 σ ′ = X' 22 c and 6 σ ′ =X' 12 in Eq. 6.81 yields 11 1 11 11 11 11 1 1 1 , t C t C F F · · − ′ ′ ′ ′ Χ Χ Χ Χ 22 2 22 22 22 22 1 1 1 , t C t C F F · · − ′ ′ ′ ′ Χ Χ Χ Χ 2 12 66 ) ( 1 Χ′ · F (6.82) Employing the von Mises plane stress analogy, the remaining interaction coefficient F 12 can be defined 12 1/ 2 11 11 22 22 1 2( ) t C t C F · ′ ′ ′ ′ Χ Χ Χ Χ (6.83) Combining Eqs. 6.81-6.83, the Tsai-Wu criterian takes the following form: 2 2 2 6 1 1 2 2 1/ 2 11 11 11 11 22 22 22 22 12 1 2 11 11 22 22 ( ) ( ) ( ) ( ) 1 1 1 1 ( ) ( ) 1 t c t c t c t c t c t c σ σ σ σ σ σ σ ′ ′ ′ ′ ′ − + + ′ ′ ′ ′ ′ ′ ′ ′ ′ Χ Χ Χ Χ Χ Χ Χ Χ Χ ′ ′ − + − · ′ ′ ′ ′ Χ Χ Χ Χ (6.84) It is to be mentioned that the Tsai-Wu criterion (Eq. 6.84) accounts for interaction of stress components as well as both tensile and compressive strength constants and shear strength and is considered as a reasonably accurate and consistent representation of failure of an orthotropic lamina under biaxial stresses. The Tsai-Hill criterion (Eq. 6.69) is also very popular with composite structural designers. Table 6.1: Stiffnesses ] [ ij Q′ and ] [ ij Q for three unidirectional composites (GPa) Material 11 Q′ 22 Q′ 12 Q′ 66 Q′ Kelvar/Epoxy 91.87 4.03 1.41 2.26 Carbon/Epoxy 133.94 8.32 2.16 3.81 Boron/Polyimide 242.39 14.93 3.88 5.53 Material φ (degree) 11 Q 22 Q 12 Q 66 Q 16 Q 26 Q Kelvar/ Epoxy 0 91.87 4.03 1.41 2.26 0.00 0.00 30 54.15 10.23 17.17 18.02 28.12 9.92 45 26.93 26.93 22.42 23.27 21.96 21.96 60 10.23 54.15 17.17 18.02 9.92 28.12 90 4.03 91.87 1.41 2.26 0.00 0.00 Carbon/ Epoxy 0 133.94 8.32 2.16 3.81 0.00 0.00 30 79.53 16.72 25.17 26.82 40.48 13.92 45 40.46 40.46 32.84 34.48 31.40 31.40 60 16.72 79.53 25.17 26.82 13.92 40.48 90 8.32 133.94 2.16 3.81 0.00 0.00 Boron/ Plyimide 0 242.39 14.93 3.88 5.53 0.00 0.00 30 142.88 29.15 46.53 48.17 73.87 24.62 45 71.80 71.80 60.74 62.39 56.87 56.87 60 29.15 142.88 46.53 48.17 24.62 73.87 90 14.93 242.39 3.88 5.53 0.00 0.00 Table 6.2: Compliance ] [ ij S′ and ] [ ij S for three unidirectional composites (TPa) -1 Material 11 S 22 S 12 S 66 S Kelvar/Epoxy 10.94 249.75 -3.83 443.49 Carbon/Epoxy 7.50 120.66 -1.95 262.47 Boron/Polyimide 4.14 67.27 -1.08 180.96 Material φ (degree) 11 S 22 S 12 S 66 S 16 S 66 S Kelvar/ Epoxy 0 10.94 249.75 -3.83 443.46 0.00 0.00 30 103.48 222.88 -36.66 312.13 -141.32 -65.50 45 174.12 174.12 -47.61 268.35 -119.40 -119.40 60 222.88 103.48 -36.66 312.13 -65.50 -141.32 90 249.75 10.94 -3.83 443.46 0.00 0.00 Carbon/ Epoxy 0 7.50 120.66 -1.95 262.47 0.00 0.00 30 60.24 116.82 -26.40 164.66 -77.23 -20.76 45 96.68 96.68 -34.55 132.05 -56.58 -56.58 60 116.82 60.24 -26.40 164.66 -20.76 -77.23 90 120.66 7.50 -1.95 262.47 0.00 0.00 Boron/ Plyimide 0 4.14 67.27 -1.08 180.96 0.00 0.00 30 40.06 71.63 -21.21 100.42 -50.59 -4.08 45 62.56 62.56 -27.93 73.57 -31.56 -31.56 60 71.63 40.06 -21.21 100.42 -4.08 -50.59 90 67.27 4.14 -1.08 180.96 0.00 0.00 Table 6.3 : Stiffnesses for various types of laminates Case Laminate type Elastic behaviour Stiffnesses I. Symmetric Laminates 1. Off-axis laminate anisotropic all B ij =0; A ij = h Q ij (all plies oriented and uncoupled D ij = (h 3 /12) Q ij at φ ) 2. On-axis laminate orthotropic all B ij =0; A ij =h Q ij (all plies oriented either and uncoupled and D ij = h 3 /12) Q ij 0 0 or 90 0 ) with Q 16 = Q 26 = 0 3. Symmetric cross-ply specially all B ij =0; A 16 = A 26 = (odd number of orthropic and D 16 = D 26 =0; rest of 0 0 / 90 0 / 0 0 , etc. plies) uncoupled A ij and D ij are finite 4. Symmetric angle-ply anisotropic and all B ij =0; all A ij and D ij (odd number of φ /- φ / φ , uncoupled are finite etc. plies) 5. Symmetric balanced angle anisotropic and all B ij =0; A 16 = A 26 =0 rest ply ( φ /- φ /- φ / φ , etc. plies) uncoupled of A ij and D ij are finite. II. Unsymmetric Laminates 6. Antisymmetric cross-ply orthotropic and A 16 = A 26 = B 16 = B 26 = (even number of partly coupled B 12 = B 66 = D 16 = D 26 =0 0 0 / 90 0 / 0 0 /90 0 , etc. plies) rest of A ij ,B ij and D ij are finite with B 22 =-B 11 ; D 22 =-D 11 7. Antisymmetric angle-ply anisotropic and A 16 = A 26 =B 11 =B 22 (even number of partly coupled B 12 = B 66 = D 16 = D 26 =0 ( φ /- φ / φ /- φ , etc. plies) rest of A ij , B ij and D ij are finite. 8. Unsymmetric cross-ply orthotropic but A 16 = A 26 = B 16 = B 26 = (irregular stacking of coupled D 16 = D 26 =0; rest of 0 0 or 90 0 plies) A ij , B ij and D ij are finite. 9. General unsymmetric anisotropic and all A ij , B ij and D ij are laminate strongly coupled finite. Table 6.4: Stiffneses 1 ] 1 ¸ ij ij ij ij D B B A for carbon/epoxy composite laminates Laminate Thickness : 4mm Units : [A ij ], GPa-mm; [B ij ], GPa-mm 2 ; [D ij ], GPa-mm 3 1. 0 0 / 90 0 / 0 0 laminates 1 1 1 1 1 1 1 1 ] 1 ¸ 32 . 20 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 20 . 69 54 . 11 00 . 0 00 . 0 00 . 0 00 . 0 54 . 11 55 . 689 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 24 . 15 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 78 . 200 66 . 8 00 . 0 00 . 0 00 . 0 00 . 0 66 . 8 28 . 368 2. 45 0 / -45 0 / 45 0 laminate 1 1 1 1 1 1 1 1 ] 1 ¸ 92 . 183 09 . 155 09 . 155 00 . 0 00 . 0 00 . 0 09 . 155 78 . 215 14 . 175 00 . 0 00 . 0 00 . 0 09 . 155 14 . 175 78 . 215 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 94 . 137 87 . 41 87 . 41 00 . 0 00 . 0 00 . 0 87 . 41 83 . 161 35 . 131 00 . 0 00 . 0 00 . 0 87 . 41 35 . 131 83 . 161 3. 45 0 /-45 0 /45 0 / 45 0 laminate 1 1 1 1 1 1 1 1 ] 1 ¸ 92 . 183 09 . 125 09 . 125 00 . 0 00 . 0 00 . 0 62 . 125 78 . 215 14 . 175 00 . 0 00 . 0 00 . 0 62 . 125 14 . 175 78 . 215 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 94 . 137 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 83 . 161 35 . 131 00 . 0 00 . 0 00 . 0 00 . 0 35 . 131 83 . 161 4. 0 0 /90 0 / 0 0 / 90 0 laminate 284.83 8.66 0.00 125.62 0.00 0.00 8.66 284.53 0.00 0.00 125.62 0.00 0.00 0.00 15.24 0.00 0.00 0.00 125.62 0.00 0.00 379.37 11.54 0.00 0.00 125.62 0.00 11.54 379.37 0.00 0.00 0.00 0.00 0.00 0.00 20.32 − 1 1 1 1 1 − 1 1 1 ¸ ] 5. 45 0 /-45 0 /45 0 / -45 0 laminate 1 1 1 1 1 1 1 1 ] 1 ¸ − − − − − − − − 92 . 183 00 . 0 00 . 0 00 . 0 81 . 62 81 . 62 00 . 0 78 . 215 14 . 175 81 . 62 00 . 0 00 . 0 00 . 0 14 . 175 78 . 215 81 . 62 00 . 0 00 . 0 00 . 0 81 . 62 81 . 62 94 . 137 00 . 0 00 . 0 81 . 62 00 . 0 00 . 0 00 . 0 83 . 161 35 . 131 81 . 62 00 . 0 00 . 0 00 . 0 35 . 131 83 . 161 6. 0 0 /90 0 /0 0 / 0 0 laminate 1 1 1 1 1 1 1 1 ] 1 ¸ − − 92 . 20 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 00 . 0 26 . 86 54 . 11 00 . 0 81 . 62 00 . 0 00 . 0 54 . 11 49 . 672 00 . 0 00 . 0 81 . 62 00 . 0 00 . 0 00 . 0 24 . 15 00 . 0 00 . 0 00 . 0 81 . 62 00 . 0 00 . 0 91 . 158 66 . 8 00 . 0 00 . 0 81 . 62 00 . 0 66 . 8 15 . 410 6.15 BIBLIOGTAPHY 1. S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw Hill, N.Y., 1970. 2. Y.C. Fung, Foundations of Solid Mechanics, Englewood Cliffs, N.J., 1965. 3. S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic Body, MIR Publ. Moscow, 1981. 4. J.C, Halpin, Primer or Composite Materials: Analysis, Technomic Publ. Co., Inc. Lancaste, 1984. 5. R.M. Christensen, Mechanics of Composite Materials, Wiley Interscience, N.Y., 1979. 6. Z. Hashin and C.T. Herakovich (Eds.), Mechanics of Composite Materials- Recent Advances, Pergamon Press, N.Y.,1983. 7. S.W. Tsai and H.T. Hahn, Introduction to Composite Materials Technomic Publ. Co., Inc., Lancaster,1980. 8. J.M. Whitney, Structural Analysis of Laminted Composites, Technomic Publ. Co., Inc.,Lancaster, 1987. 9. J.R. Vinson and R.L. Sierakowski, The Behaviour of Structures Composed of Composite Materials, Kluwar Academic Publ., MA,1985. 10. S.W. Tsai, J.C. Halpin and N.J. Pangano (Eds.) Composite Materials Workshop, Technomic Publ. Co., Inc., Lancaster, 1968. 6.16 EXERCISES 1. State the generalized Hooke's law for a three-dimensional elastic anisotropic material and show that there are twenty-one independent elastic constants for a triclinic material. 2. Write down the elastic constant matrix for three-dimensional orthtropic, square symmetric, hexagonal symmetric and isotropic materials. 3. Distinguish between elastic constants and engineering constants. 4. For a two-dimensional orthotropic case, express ] [ ij Q′ and ] [ ij S′ in terms of engineering constants. 5. Derive expressions for ] [ ∈ T and ] [ σ T in terms of angle φ and show that ] ][ [ ] [ ] [ ∈ ∈ ′ · T Q T Q ij T ij and ] ][ [ ] [ ] [ σ σ T S T S ij T ij ′ · 6. Assume properties given in Table 4.4 for Kevlar/epoxy and carbon/epoxy/composites and determine [A ij ], [B ij ] and [D ij ] for a ] 0 / 45 / 0 [ 0 2 0 2 0 2 K C K t hybrid laminate (thickness 4 mm). 7. Make a critical assessment of various lamina failure theories. 8. Derive expressions for Tsai-Hill and Tsai-Wu strength criteria. CHAPTER - 7 LAMINATED COMPOSITE BEAMS AND PLATES 7.1 INTRODUCTION 7.2 THIN LAMINATED PLATE THEORY 7.3 BENDING OF LAMINATED PLATES 7.3.1 Specially Orthotropic Plate 7.3.2 Antisymmetric Cross-ply Laminated Plate 7.3.3 Antisymmetric Angle-Ply Laminated Plate 7.4 FREE VIBRATION AND BUCKLING 7.4.1 Specially Orthotropic Plate 7.4.2 Antisymmetric Cross-ply Laminated Plate 7.4.3 Antisymmetric Angle-Ply Laminated Plate 7.5 SHEAR BUCKLING OF COMPOSITE PLATE 7.6 RITZ METHOD 7.7 GALERKIN METHOD 7.8 THIN LAMINATED BEAM THEORY 7.9 PLY STRAIN, PLY STRESS AND FIRST PLY FAILURE 7.10 BIBLIOGRAPHY 7.11 EXERCISES 7.1 INTRODUCTION The formulae presented in this chapter are based on the classical bending theory of thin composite plates. The small deflection bending theory for a thin laminate composite beam is developed based on Bernoulli's assumptions for bending of an isotropic thin beam. The development of the classical bending theory for a thin laminated composite plate follows Kirchhoff's assumptions for the bending of an isotropic plate. Kirchhoff's main suppositions are as follows: 1. The material behaviour is linear and elastic. 2. The plate is initially flat. 3. The thickness of the plate is small compared to other dimensions. 4. The translational displacements ( 0 2 0 1 , u u and w) are small compared to the plate thickness, and the rotational displacements ( 0 0 1,1 2,2 and u u ) are very small compared to unity. 5. The normals to the undeformed middle plane are assumed to remain straight, normal and inextensional during the deformation so that transverse normal and shear strains ( 5 4 3 , ∈ ∈ ∈ and ) are neglected in deriving the plate kinematic relations. 6. The transverse normal stresses ) ( 3 σ are assumed to be small compared with other normal stress components 1 2 ( and ) σ σ . So that they may be neglected in the constitutive relations. The relations developed earlier in sections 6.12 and 6.13 are essentially based on the above Kirchhoff's basic assumptions. Some of these relations will be utilized in the present chapter to derive the governing equations for thin composite plates. It may be noted that Kirchhoff's assumptions are merely an extension of Bernoulli's from one- dimensional beam to two-dimensional plate problems. Hence a classical plate bending theory so developed can be reduced to a classical beam bending theory. Here, also, the governing plate equations are derived first, and the beam equations are subsequently obtained from the plate equations. 7.2 THIN LAMINATED PLATE THEORY Consider, a rectangular, thin laminated composite plate of length a, width b and thickness h as shown in Fig.7.1. The plate consists of a laminate having n number of laminae with different materials, fibre orientations and thicknesses. The plate is subjected to surface loads q's and m's per unit area of the plate as well as edge loads 1 2 6 , and N N Ν per unit length. The expansional strains, which may be caused due to moisture and temperature, are also included. Note that 0 2 0 1 , u u and w are mid-plane displacement components. It is assumed that Kirchhoff's assumptions for the small deflection bending theory of a thin plate are valid in the present case. One of these assumptions is related to transverse strains 3 4 2 5 1 ( ), ( ) and ( ) zz z z ∈ ·∈ ∈ ·∈ ∈ ·∈ which are neglected in derivation of plate kinematic relations i.e., stress-strain relations. Considering the dynamic equilibrium of an infinitesimally small element dx 1 dx 2 (Fig. 7.2) the following equations of motion are obtained 0 1,1 6, 2 1 1 ,1 N N q Pu Rw + + · − && && (7.1) 0 6,1 2, 2 2 2 , 2 N N q Pu Rw + + · − && && (7.2) 4,1 5,2 1 ,11 2 , 22 6 ,12 2 Q Q q N w N w N w Pw + + + + + · && (7.3) 0 1,1 6, 2 4 1 1 ,1 M M Q m Ru I w + · − + − && && (7.4) 0 2, 2 6,1 5 2 2 , 2 M M Q m Ru I w + · − + − && && (7.5) where commas are used to denote partial differentiation, and dots relate to differentiation with respect to time, t. Combining Eqs. 7.3 through 7.5, we obtain 1,11 6,12 2, 22 1 ,11 2 , 22 6 ,12 1,1 2, 2 0 0 1,1 2, 2 ,11 , 22 2 2 ( ) ( ) M M M q N w N w N w m m Pw R u u I w w + + + + + + + + · + + − + && && && && && (7.6) where / 2 / 2 h z h P dz ρ − · ∫ , / 2 / 2 h z h R z dz ρ − · ∫ and / 2 2 / 2 h z h I z dz ρ − · ∫ and z ρ is the density of the laminate at a distance z. Substituting Eqs. 6.60 in Eqs. 7.1, 7.2 and 7.6 and noting Eqs. 6.47 and 6.48,we obtain the following governing differential equations in terms of mid-plane displacements 0 2 0 1 , u u and w. 0 0 0 0 0 0 11 1,11 16 1,12 66 1, 22 16 2,11 12 66 2,12 26 2, 22 11 ,111 16 ,112 12 66 ,122 26 , 222 1,1 6, 2 0 1 1 ,1 2 ( ) 3 ( 2 ) e e A u A u A u A u A A u A u B w B w B B w B w q Pu Rw + + + + + + − − − + − − Ν − Ν + · − && && (7.7) 0 0 0 0 0 0 16 1,11 12 66 1,12 26 1, 22 66 2,11 26 2,12 22 2, 22 16 ,111 12 66 ,112 26 ,122 22 , 222 6,1 2, 2 0 2 2 , 2 ( ) 2 ( 2 ) 3 e e A u A A u A u A u A u A u B w B B w B w B w N N q Pu Rw + + + + + + − − + − − − − + · − && && (7.8) 11 ,1111 16 ,1112 12 66 ,1122 26 ,1222 22 , 2222 0 0 0 0 0 11 1,111 16 1,112 12 66 1,122 26 1, 222 16 2,111 0 0 0 12 66 2,112 26 2,122 22 2, 222 1,11 6,12 2,22 1,1 2, 2 1 4 2( 2 ) 4 3 ( 2 ) ( 2 ) 3 2 e e e D w D w D D w D w D w B u B u B B u B u B u B B u B u B u M M M q m m N w + + + + + − − − + − − − + − − + + + · + + ,11 6 ,12 2 , 22 1,11 2, 22 ,11 , 22 2 ( ) ( ) N w N w Pw R u u I w w + + − − + + + && && && && && (7.9) Note that the rotary inertia effects are usually neglected in development of a thin plate theory. The proper boundary conditions are chosen from the following combinations: 0 0 0 0 , , , or or or or n n n n t t nt nt n n n n nt t n n u u N N u u N N w w M M w w M Q Q · · · · · · · + · (7.10) where the subscript n is the direction normal to the edge of the plate, and relates to the tangential direction. Equations 7.7 through 7.9 can be further simplified using the stiffnesses listed in Table 6.3 for various kinds of laminates. For symmetric laminates, B ij = 0 and R = 0. Hence Eqs.7.7, 7.8 and 7.9 can be accordingly modified. Note that these equations become uncoupled. The modified forms of Eqs. 7.7 and 7.8 (with B ij = 0 and R = 0) represent the stretching behaviour of a symmetric laminated plate. The bending behaviour of a symmetric laminated plate is represented by the equation 11 ,1111 16 ,1112 12 66 ,1122 26 ,1222 22 , 2222 1,11 6,12 2, 22 1,1 2, 2 1 ,11 6 ,12 2 , 22 4 2( 2 ) 4 2 2 e e e D w D w D D w D w D w M M M q m m N w N w N w Pw + + + + + + + + · + + + + + − && (7.11) However, for a specially orthotropic plate with symmetric cross-ply lamination (D 16 = D 26 = 0), Eq. 7.11 is further reduced to 11 ,1111 12 66 ,1122 22 , 2222 1,11 6,12 2, 22 1,1 2, 2 1 ,11 6 ,12 2 , 22 2( 2 ) 2 2 e e e D w D D w D w M M M q m m N w N w N w Pw + + + + + + · + + + + + − && (7.12) For a homogeneous anisotropic parallel-ply laminate, where all plies have the same fibre orientation θ, noting that ij ij Q h D 12 3 · we obtain from Eq. 7.11. 11 ,1111 16 ,1112 12 66 ,1122 26 ,1222 22 , 2222 1,1 2, 2 1,11 6,12 2, 22 1 ,11 6 ,12 2 , 22 3 4 2( 2 ) 4 12 ( 2 2 ) e e e Q w Q w Q Q w Q w Q w q m m M M M N w N w N w h w h ρ + + + + + · + + − − + + + − && (7.13) In a similar way, for an orthotropic plate with either 0 0 or 90 0 fibre orientation, Eq. (7.13) is further simplified using Q 16 = Q 26 = 0. For a homogeneous isotropic plate, Q 11 = Q 22 = E/(1-ν 2 ), Q 12 = ν Q 1, Q 66 = E/[2(1+ ν)] and Q 16 = Q 26 = 0 and hence Eq. 7.13 reduces to ,1111 ,1122 , 2222 1, 1 2, 2 1,11 6,12 2, 22 1 ,11 6 ,12 2 , 22 1 2 ( 2 2 ) e e e w w w q m m M M M D N w N w N w h w ρ + + · + + − − + + + − && (7.14) where ) 1 ( 12 2 3 ν − Ε · h D and expansion moments M e ?s are accordingly derived. The solution of Eqs. 7.8 through 7.10 is difficult to achieve due to the presence of bending-extensional and other coupling terms as well as mixed order of differentiation with respect to x 1 and x 2 in each relation. The closed form solutions are restricted to a few simple laminate configurations, loading conditions, plate geometry and boundary conditions. The other analytical methods are based on the variational approaches such Rayleigh Ritz method and Galerkin method. 7.3 BENDING OF LAMINATED PLATES 7.3.1 Specially Orthotropic Plate Consider a rectangular, symmetric cross-ply laminated composite plate (Fig. 7.1) which is subjected to transverse load q only. Equation 7.12 reduces to 11 ,1111 12 66 ,1122 22 , 2222 2( 2 ) D w D D w D w q + + + · (7.15) All Edges Simply Supported Consider the simply supported conditions as given below 1 1 11 ,11 12 , 22 2 2 12 ,11 22 , 22 0, : 0 and ( ) 0 0, : 0 and ( ) 0 x a w M D w D w x b w M D w D w · · · − + · · · · − + · (7.16) We assume the Navier-type of solution. Let b x m a x m W w m n mn 2 1 1 1 sin sin π π ∑∑ ∞ · ∞ · · (7.17) that satisfies the boundary conditions vide Eq. 7.16. Further we assume that b x m a x m q q m n mn 2 1 1 1 sin sin π π ∑∑ ∞ · ∞ · · (7.18) Substitution of Eqs. 7.17 and 7.18 in Eq. 7.15 yields 4 4 2 2 4 11 12 66 22 / 2( 2 ) mn mn q W m m n n D D D D a a b b π · ¸ _ ¸ _ ¸ _ ¸ _ + + + ÷ ÷ ÷ ÷ ¸ , ¸ , ¸ , ¸ , (7.19) Note that, for an isotropic plate, 4 2 2 / mn mn q W m n D a b π · 1 ¸ _ ¸ _ + 1 ÷ ÷ ¸ , ¸ , 1 ¸ ] (7.20) The loading coefficient q mn is determined for a specified distribution of transverse load q (x 1 ,x 2 ) from the following integral: 1 2 1 2 1 2 0 4 ( , )sin sin a b mn b m x m x q q x x dx dx ab a b π π · ∫ ∫ (7.21) It can be shown that, for a uniformly distributed load q 0 , mn q q mn 2 0 16 π · (7.22) where m, n are odd integers. Hence for a specially orthotropic plate that is subjected to a transverse uniformly distributed load q 0 , the deflection w is given by 1 2 0 6 4 2 4 11 12 66 22 sin sin 16 2( 2 ) m n m x m x q a b w m mn n mn D D D D a ab b π π π ∞ ∞ · 1 ¸ _ ¸ _ ¸ _ + + + 1 ÷ ÷ ÷ ¸ , ¸ , ¸ , 1 ¸ ] ∑∑ (7.23) where m, n are odd integers. The moments M 1 , M 2 and M 6 and shear forces Q 4 and Q 5 can be obtained from the following relations: 1 11 ,11 12 , 22 ( ) M D w D w · − + 2 12 ,11 22 , 22 ( ) M D w D w · − + 6 66 ,12 2 M D w · − (7.24) 4 1,1 6, 2 Q M M · + 5 2, 2 6,1 Q M M · + Two Opposite Edges Simply Supported We now consider the simply supported conditions at a x , 0 1 · and either or both of the conditions at b x , 0 2 · (Fig. 7.1) may be simply supported, clamped or free. We can proceed with the Levy's type of solution. The solution of Eq.7.15 consists of two parts: homogeneous solution and particular solution. Thus, ) ( ) , ( ) , ( 1 2 1 2 1 x w x x w x x w p h + · (7.25) For this particular solution w p (x 1 ), the lateral load q(x 1 ) is assumed to have the same distribution in all sections parallel to the x 1 -axis and the plate is also considered infinitely long the x 2 -direction. Equation 7.15 takes the following form: 11 ,1111 1 ( ) D w q x · (7.26) Assume a x m W x w m m p 1 1 1 sin ) ( π ∑ ∞ · · (7.27) and a x m q x q m m 1 1 1 sin ) ( π ∑ ∞ · · (7.28) Substituting Eqs. 7.27 and 7.28 in Eq. 7.26, we obtain 4 11 4 4 m q D a W m m π · (7.29) Hence the particular solution is given by a x m m q D a x w m m p 1 1 4 11 4 4 1 sin ) ( π π ∑ ∞ · · (7.30) The homogeneous solution is obtained from the following form of Eq.7.15 11 ,1111 12 66 ,1122 22 , 2222 2( 2 ) 0 h h h D w D D w D w + + + · (7.31) Let us express w h (x 1 , x 2 ) by a x m x X x x w m m h 1 2 1 2 1 sin ) ( ) , ( π ∑ ∞ · · (7.32) Eq. 7.32 satisfies simply supported boundry conditions at a x , 0 1 · (Eq. 7.16). Substitution of Eq. 7.32 in Eq. 7.31 yields 4 4 2 2 1 11 12 66 , 22 22 , 2222 4 2 1 2( 2 ) sin 0 m m m m n x m m n D X D D X D X a a a π π ∞ · 1 ¸ _ ¸ _ − + + · 1 ÷ ÷ ¸ , ¸ , ¸ ] ∑ (7.33) 4 4 2 2 11 12 66 , 22 22 , 2222 4 2( 2 ) 0 m m m m m n D X D D X D X a a π ¸ _ ¸ _ − + + · ÷ ÷ ¸ , ¸ , (7.34) Let the solution be 2 exp m m X x a πλ · (7.35) Substituting Eq. 7.35 in Eq.7.34, the characteristic equation is obtained as follows: 11 2 66 12 4 22 ) 2 ( 2 D D D D + + − λ λ (7.36) the solution of which is given by [ ] } ) 2 {( 2 1 22 11 2 66 12 66 12 22 2 D D D D D D D − + t + · λ (7.37) The value of 2 λ , in general, is complex. Hence, the roots of Eq. 7.36 can be expressed in the form β α i t and β α i t − , where α and β are real and positive and are given as ) 2 ( 1 66 12 22 D D D + · α (7.38) } ) 2 ( { 1 22 11 2 66 12 22 D D D D D − + · β (7.39) Hence, the solution is 2 2 2 2 2 2 ( cos sin ) cosh ( cos sin )sinh m m m m m m x m x m x B a a a m x m x m x C D a a a πβ πβ πα πβ πβ πα Χ · Α + + + (7.40) Hence referring to Eqs. 7.25, 7.30 and 7.40, the final solution w(x 1 , x 2 ) to Eq. 7.15 can be expressed as follows: 2 2 2 1 2 2 2 2 1 4 1 4 4 1 11 ( , ) ( cos sin ) cosh ( cos sin )sinh sin sin m m m m m m m x m x m x w x x A B a a a m x m x m x m x C D a a a a q m x a D m a πβ πβ πα πβ πβ πα π π ∞ · · + ¸ 1 + + 1 ] + ∑ (7.41) The constant A m , B m ,Cm and Dm are determined from the relevant boundry conditions at b x , 0 2 · . Note that q m is determined from the loading distribution q(x 1 ) using the following integration. 1 1 1 0 2 ( )sin a m m x q q x dx a a π · ∫ (7.42) Hence for a uniformly distributed transverse load q 0 , 4 0 π m q q m · m = 1,3,5,? (7.43) The moments and shear forces are computed using Eqs. 7.24. 7.3.2 Antisymmetric Cross-ply Laminated Plate Now consider the rectangular plate, shown in Fig. 7.1, to be made up of cross-ply laminations of stacking sequence [0/90] n (refer case 6 of Table 6.3). Equations 7.7 through 7.9 then reduce to 0 0 0 11 1,11 66 1, 22 12 66 2,12 11 ,111 ( ) 0 A u A u A A u B w + + + − · 0 0 0 12 66 1,12 66 2,11 22 2, 22 11 , 222 ( ) 0 A A u A u A u B w + + + + · (7.44) 0 0 11 ,1111 , 2222 12 66 ,1122 11 1,111 2, 222 ( ) 2( 2 ) ( ) D w w D D w B u u q + + + − − · The simply supported boundary conditions considered here are 0 1 1 11 1,1 11 ,11 12 , 22 0, : 0; 0 x a w M B u D w D w · · · − − · 0 0 0 2 1 11 1,1 12 2, 2 11 ,11 0; 0 u N A u A u B w · · + − · (7.45) 0 2 2 11 2, 2 12 ,11 11 , 22 0, : 0; 0 x b w M B u D w D w · · · − − − · 0 0 0 1 2 12 1,1 11 2, 2 11 , 22 0; 0 u N A u A u B w · · + − · Assume the following displacement components b x n a x m A u m n mn 2 1 1 1 0 1 sin cos π π ∑∑ ∞ · ∞ · · b x n a x m B u m n mn 2 1 1 1 0 2 cos sin π π ∑∑ ∞ · ∞ · · (7.46) b x n a x m W w m n mn 2 1 1 1 cos sin π π ∑∑ ∞ · ∞ · · that satisfy the simply supported boundary conditions vide Eqs. 7.45. The transverse load q is represented by the double Fourier series in Eq.7.18. Now substitution of Eqs. 7.18 and 7.46 in Eqs 7.44 results in three simultaneous algebraic equations in terms of A mn , B mn and W mn . Solving these equations, we obtain [ ] mn mn mn q n n m m D m B b 4 4 66 12 2 2 2 11 4 66 3 11 3 3 ) ( η η π η Α + Α + Α + Α · Α ∗ [ ] mn mn mn q n n m m A D n B b B 4 4 66 2 2 2 11 4 66 12 3 11 3 4 ) ( η η π η Α + Α + Α + · ∗ (7.47) [ ] mn mn mn q n m n m n m D b W 2 2 2 2 66 12 2 2 11 2 66 2 2 66 2 11 4 4 4 ) ( ) )( ( η η η π η Α + Α − Α + Α Α + Α · ∗ where 2 2 2 2 2 2 2 2 2 2 4 4 4 11 66 66 11 12 66 11 2 2 2 2 2 2 2 4 4 4 4 4 4 12 66 11 11 12 66 8 8 8 66 ( )( ) ( ) ( ) 2( 2 ) ( ) 2( ) ( ) mn D m n m n m n D m n D D m n B A m n m n m n m n η η η η η η η η η ∗ 1 · Α + Α Α + Α − Α + Α + ¸ ] ¸ 1 + + − + + Α + Α ] ¸ 1 +Α + ] and η = a/b Using Eqs. 7.46 and 7.47, the stress and moment resultants (N 1 , N 2 , N 6 , M 1, M 2, and M 6 ) are derived from Eqs. 6.52, and the shear forces Q 4 and Q 5 are obtained from the last two relations of Eqs. 7.24. Figure 7.3 exhibits the maximum nondimensional deflections (at x 1 =a/2 and x 12 =b/2) for simply supported antisymmetric cross-ply laminated plates, which are plotted against the aspect ratio a/b. The deflections are considerably higher in the case of a two- layered (n=1) plate because of the extension-bending coupling (B 11 ). However, as the number of layers increases, the coupling effect reduces and the results approach to that of an orthotropic plate (n = ∞ ). 7.3.3 Antisymmetric Angle-Ply Laminated Plate Next consider a rectangular angle-ply laminated composite plate of stacking sequence [?؝ n (refer case 7 of Table 6.3) that is subjected to transverse load q. Equations 7.7 through 7.9 become 0 0 0 11 1,11 66 1, 22 12 66 2,12 16 ,112 26 , 222 ( ) 3 0 u A u B u B w B w Α + + Α + − − · 0 0 0 12 66 1,12 66 2,11 22 2, 22 16 ,111 26 ,122 ( ) 3 0 A u A u A u B w B w Α + + + − − · (7.48) 11 ,111 12 66 ,1122 22 , 2222 0 0 0 0 16 1,112 2,111 26 1, 222 2,112 2( 2 ) (3 ) ( 3 ) D w D D w D w B u u B u u q + + + − + − + · The following simply supported boundary conditions are assumed 0 0 1 1 16 1, 2 2,1 11 ,11 12 , 22 0, : 0; ( ) 0 x a w M B u u D w D w · · · + − − · 0 0 0 1 6 66 1, 2 2,1 16 ,11 26 , 22 0; ( ) 0 u N A u u B w B w · · + − − · (7.49) 0 0 2 2 26 1, 2 2,1 12 ,11 22 , 22 0, : 0; ( ) 0 x b w M B u u D w D w · · · + − − · 0 0 0 2 6 66 1, 2 2,1 16 ,11 26 , 22 0; ( ) 0 u N A u u B w B w · · + − − · The transverse load q is given by Eq. 7.18. The following displacement field b x n a x m A u m n mn 2 1 1 1 0 1 cos sin π π ∑∑ ∞ · ∞ · · b x n a x m B u m n mn 2 1 1 1 0 2 sin cos π π ∑∑ ∞ · ∞ · · (7.50) b x n a x m W w m n mn 2 1 1 1 sin sin π π ∑∑ ∞ · ∞ · · satisfies simply supported boundary conditions in Eqs. 7.49. Substituting Eqs. 7.18 and 7.50 in Eqs 7.48 and solving the resulting simultaneous algebraic equations we obtain 4 3 2 2 2 2 2 2 66 22 16 26 3 2 2 2 2 12 66 16 26 (3 ) ( ) 3 mn mn mn b n m n B m B n D m B m B n q η η η π η ∗ Α · Α + Α + ¸ 1 − Α + Α + ] 3 3 2 2 2 2 2 2 11 66 16 26 3 2 2 2 2 2 12 66 16 26 ( )( 3 ) ( )3 ) mn mn mn b m B m n B m B n D n B m B n q η η η π η η ∗ · Α + Α + ¸ 1 − Α + Α + ] 4 4 2 2 2 2 2 2 11 66 66 22 4 2 2 2 12 66 ( )( ) ( ) mn mn mn b w m n A m A n D n m q η η η π η ∗ · Α + Α + ¸ 1 − Α + Α ] (7.51) where 2 2 2 2 2 2 11 66 66 22 2 2 2 2 4 2 2 2 4 4 12 66 11 12 66 22 2 2 2 2 2 2 2 2 2 12 66 16 26 16 26 2 2 2 2 2 2 2 2 66 22 16 26 2 2 2 2 11 66 16 ( )( ) ( ) ( 2( 2 ) ) 2 ( )(3 )( 3 ) ( )(3 ) ( )( mn D m n m n m n D m D D m n D n m n B m B n B m B n n m n B m B n m m n B η η η η η η η η η η η η ∗ · Α + Α Α + Α ¸ 1 1 − Α + Α + + + ] ¸ ] + Α + Α + + − Α + Α + − Α + Α 2 2 2 2 26 3 ) m B n η + The values of maximum nondimensional deflections (at x 1 =x 2 =a/2) for simply supported antisymmetric angle-ply laminated square plates are plotted against the variation of ؠranging from 0 to 45 0 . A two-layered laminate is found to exhibit much higher deflections due to higher values of coupling terms B 16 and B 26 compared to the other cases. 7.4 FREE VIBRATION AND BUCKLING 7.4.1 Specially Orthotropic Plate Consider a rectangular specially orthotropic plate (Fig. 7.5) which is subjected to compressive loads N N − · 1 and PN N − · 2 per unit length along the edges. The plate is also assumed to be vibrating freely in the transverse direction. Equation 7.12 then becomes (note that 0 6 · N ) 11 ,111 12 66 ,1122 22 , 2222 ,11 , 22 2( 2 ) ( ) 0 D w D D w D w N w pw pw + + + + + + · && (7.52) All Edges Simply Supported The deflected shape w (x 1 ,x 2 , t) is assumed in the following form: t i mn mn e b x n a x m W w ω π π 2 1 sin sin · (7.53) that satisfies the simply supported boundary conditions in Eqs. 7.16. Substitution of Eq. 7.53 in Eq. 7.52 yields the frequency equation as follows 1 ] 1 ¸ + + + · + + Ω 4 2 2 2 22 66 22 12 4 4 22 11 2 2 2 2 ) 2 ( 2 ) ( n n m D D D D m D D pn m k mn mn η η η (7.54) where 22 4 4 2 2 D b p mn mn π ω · Ω , , 22 2 2 D Nb k mn π · b a · η and 1 2 / Ν · N p . Note that mn ω is the circular frequency. The non-dimensional frequency parameter mn Ω can be computed for a particular mode shape m and n for various values of aspect ratio (a/b), stiffness ratios and inplane loads. From Eq. 7.54, it is evident that a critical combination of compressive inplane biaxial loads can reduce the frequency to zero. When the frequency is zero, the inplane loads correspond to the buckling loads of the plate. Further, it may be noted that the tensile inplane loads will increase the frequency of the plate. Two Opposite Loaded Edges Simply Supported For a laminated plate, where the compressive edge load N N − · 1 acts along the simply supported edges x 1 =0,a and the unloaded edges x 2 =0, b may have any arbitrary boundary condition, a solution to Eq . 7.52 (note that 0 6 2 · Ν · N ) can be assumed to be in the form t i m e a x m x t x x w ω π 1 2 2 1 sin ) ( ) , , ( Χ · (7.55) Substituting Eq. 7.55 in Eq. 7.52 yields 2 22 , 2222 12 66 , 22 4 2 2 11 2( 2 ) 0 m m mn m m D D D a m m D N p a a π π π ω ¸ _ Χ − + Χ ÷ ¸ , 1 ¸ _ ¸ _ + − − Χ · 1 ÷ ÷ ¸ , ¸ , 1 ¸ ] (7.56) A solution to Eq. 7.56 can be obtained as follows b x m 2 exp λ · Χ (7.57) Substituting Eq. 7.57 in Eq. 7.56 we obtain 0 2 2 4 · + − C Bλ λ (7.58) where 2 2 66 11 2 22 22 2 D D m B D D π η ¸ _ · + ÷ ¸ , and 1 ] 1 ¸ Ω − − · 2 2 2 4 22 4 11 4 η η π m k D m D C with 2 2 4 2 2 4 22 22 Nb P b k and D D ω π π · Ω · Equation 7.58 has four roots i.e., 1 1 2 2 , , , i i λ α α α α · − − with 2 / 1 2 1 ] [ C B B − + · α and 2 / 1 2 2 ] [ C B B − + − · α Thus the solution is ) / sinh( ) / cosh( 2 1 2 1 b x B b x m m m α α + Α · Χ ) / sin( ) / cos( 2 2 2 2 b x D b x C m m α α + + (7.59) The coefficients A m , B m , C m and D m are determined from boundary conditions at x 2 =0, b. For example, let us consider the clamped edges along x 2 =0, b i.e., X 2 =0,b: w=w , 2 =0 (7.60) Combining Eqs. 7.59 and using Eqs. 7.60, we obtain the following homogeneous algebraic equations ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ · ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ 1 1 1 1 ] 1 ¸ − 0 0 0 0 cos sin cosh sinh sin cos sinh cosh 0 0 0 1 0 1 2 2 2 2 1 1 1 1 2 2 1 1 2 1 m m m m D C B A α α α α α α α α α α α α α α (7.61) The frequency equation is obtained from the condition that the determinant of the coefficients of A m , B m , C m and D m must vanish. This leads to 2 1 2 1 1 2 2 1 sin sinh ) / / ( ) cos cosh 1 ( 2 α α α α α α α α − · − (7.62) The critical value of N is computed by satisfying Eq. 7.62 for a particular m when the frequency becomes zero. 7.4.2 Antisymmetric Cross-ply Laminated Plate Consider the transverse force vibration of a simply supported rectangular antisymmetrically laminated cross-ply plate (see section 7.3.2), when subjected to compressive loads N N − · 1 , pN N − · 1 and 0 6 · N . Equations 7.44 hold good except the third equation where 'q' is replaced by ? ,11 , 22 [ ( ) ] N w pw pw − + + && ?. The following displacement field t i mn mn e b x n a x m A u ω π π 2 1 0 1 sin cos · t i mn mn e b x n a x m B u ω π π 2 1 0 2 cos sin · (7.63) t i mn mn e b x n a x m W w ω π π 2 1 sin sin · satisfy the boundary conditions in Eqs. 7.45. These, on substitution in Eqs. 7.44 modified as above, result in the following homogeneous algebraic equations: ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ · ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ 1 1 1 ] 1 ¸ − 0 0 0 ) ( mn mn mn mn mn mn mn mn mn mn mn mn W B A k J G J H F G F E λ (7.64) where 2 2 66 2 11 η n A m A E mn + · 12 66 ( ) mn F A A mnη · + b m B G mn η π 3 11 − · 2 2 2 66 11 mn m n η Η · Α + Α b n B J mn 2 3 11 η · 2 4 4 4 2 2 2 11 12 66 2 2 ( ) 2( 2 ) mn K D m n D D m n b π η η η 1 · + + + ¸ ] 2 2 2 2 2 2 ( ) p b N m p n ω η λ η π · + + and b a / · η . The frequency relation is derived by vanishing the determinant of the coefficient matrix of Eq. 7.64 and is given by 2 4 2 2 2 2 4 66 11 12 4 2 22 22 22 6 3 3 3 6 6 2 11 4 2 22 2 2 2 ( ) mn mn mn mn mn mn mn mn D D D m m m n k pn n D D D m H m n F n E B D E H F η η η η η η 1 ¸ _ ¸ _ ¸ _ Ω + + · + + + 1 ÷ ÷ ÷ 1 ¸ , ¸ , ¸ , ¸ ] ¸ _ ¸ _ + + − ÷ ÷ − ¸ ,¸ , (7.65) wher 2 mn Ω and k mn are defined in Eq. 7.54. The critical buckling load corresponds to the lowest value of k that satisfies Eq. 7.65 when the frequency is zero. The load parameters, k mn and nondimensional frequency parameters, mn Ω are plotted against the aspect ratio, a/b for simply supported antisymmetric cross-ply laminate as shown in Figs. 7.6 and 7.7, respectively. 7.4.3 Antisymmetric Angle-Ply Laminated Plate Now consider the transverse free vibration of a simply supported rectangular antisymmetric angle-ply laminated plate (vide section 7.3.3) which is subjected to compressive loads N N − · 1 , pN N − · 2 and 0 6 · N . The third equation in Eqs. 7.48 is modified substituting - ,11 , 22 [ ( ) ] N w pw p w + + && in place of q. The displacement field that satisfies the boundary conditions in Eqs. 7.49 is assumed as t i mn mn e b x n a x m A u ω π π 2 1 0 1 sin sin · t i mn mn e b x n a x m B u ω π π 2 1 0 2 sin cos · (7.66) t i mn mn e b x n a x m W w ω π π 2 1 sin sin · These displacement relations, when substituted in the modified Eqs. 7.48 yield the following homogeneous algebraic equations: ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ · ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ 1 1 1 ] 1 ¸ − 0 0 0 ) ( mn mn mn mn mn mn mn mn mn mn mn mn W B A k J G J H F G F E λ (7.67) where 2 2 66 2 11 η n A m A E mn + · 12 66 ( ) mn F A A mnη · + ) 3 ( 2 2 26 2 16 π π n B m B b n G mn + − · 2 2 22 2 66 η n m mn Α + Α · Η ) 3 ( 2 2 26 2 16 η π n B m B b m J mn + − · [ ] 4 4 22 2 2 2 66 12 4 11 2 2 2 ) 2 ( 2 η η η π n D n m D D m D b k mn + + + · 2 2 2 2 2 2 ( ) and / . p b N m pn a b ω η λ η η π · + + · The condition that the determinant of the coefficient matrix in Eq.7.67 vanishes, determines the frequency equation as follows: 2 4 2 2 2 2 4 66 11 12 4 2 22 22 22 2 2 2 2 2 2 16 26 16 26 2 2 2 22 22 2 2 1 3 3 mn mn mn mn mn D D D m m m n k pn n D D D L M m m m B B n n B B n N D D η η η η η η 1 ¸ _ ¸ _ Ω + + · + + + 1 ÷ ÷ 1 ¸ , ¸ , ¸ ] 1 ¸ _ ¸ _ − + + + 1 ÷ ÷ ¸ , ¸ , ¸ ] (7.68) where 2 2 2 2 2 2 11 66 16 26 2 2 2 2 2 12 66 16 26 ( )( 3 ) ( )(3 ) mn L m A n B m B n n B m B n η η η η · Α + + − Α + Α + 2 2 2 2 2 2 66 22 16 26 2 2 2 2 12 66 16 26 ( )(3 ) ( )( 3 ) mn M m A n B m B n m B m B n η η η · Α + + − Α + Α + 2 mn mn mn mn N E H F · − Figures 7.8 and 7.9 depict the variation of bucklin g load parameters and Figure 7.10 shows that of the frequency parameter mn Ω for simply supported antisymmetric angle-ply laminated square plates. 7.5 SHEAR BUCKLING OF COMPOSITE PLATE A closed form solution for the shear buckling of a finite composite plate does not exist. This is true also for an isotropic panel. Here the solution of a long specially orthotropic composite plate is considered. Consider the plate to be infinite along the x 1 direction and is subjected to uniform shear along the edges x 2 = ?b/2 (Fig. 7.11). In absence of inertia and other loads except the edge shear 6 Ν , Eq. 7.12 becomes 11 ,1111 12 66 ,1122 22 , 2222 6 ,12 2( 2 ) 2 0 D w D D w D w w + + + − Ν · (7.69) Assuming the solution to be of the form 1 2 2 ( ) exp k i k x w X x b · (7.70) where k is a longitudinal wave parameter and 1 − · i . Substituting Eq. 7.70 in Eq.7.69 we obtain 2 4 22 , 2222 12 66 ,22 11 6 ,2 2 2 2 2( 2 ) 2 . 0 k k k k k k k D D D D N i b b b ¸ _ ¸ _ ¸ _ Χ − + Χ + Χ − Χ · ÷ ÷ ÷ ¸ , ¸ , ¸ , (7.71) A solution to Eq.7.71 is assumed to be of the form 2 2 exp k i x b λ Χ · (7.72) which on substitution in Eq. 7.71 yields the following characteristic equation 0 2 1 ) 2 ( 2 4 11 2 2 2 66 12 4 22 · + Ν + + + k D k b k D D D λ λ λ (7.73) Equation 7.73 has four roots 1 2 3 4 , , and λ λ λ λ and the solution to Eq. 7.69 is written as follows: 3 2 1 2 2 2 4 2 2 / 2 / 2 / 2 / i x b i x b i x b i x b k Ae Be Ce De λ λ λ λ Χ · + + + (7.74) Combining Eqs. 7.70 and 7.74, the solution is obtained as 3 2 1 1 2 2 2 4 2 2 / 2 / 2 / 2 / 2 / ( ) i x b i k x b i x b i x b i x b w e Ae Be Ce De λ λ λ λ · + + + (7.75) The substitution of Eq. 7.75 in the specified boundary conditions at the edges x 2 = ?b/2 will result four homogeneous algebraic equations in terms of the four coefficients A, B, C and D. For a non-trivial solution, the determinant of this coefficient matrix must vanish. This condition yields the equation for the shear buckling problems. The critical buckling load corresponds to the minimum value of 6 Ν at particular value of k. 7.6 RITZ METHOD The Ritz method (also known as the Rayleigh-Ritz method), in most cases, leads to an approximate analytical solution unless the chosen displacement configurations satisfy all the kinematic boundary conditions and compatability conditions within the body. It is developed minimizing energy functional ) , , ( 0 2 0 1 w u u ∏ · ∏ on the basis of energy principles. The principle of minimum potential energy can be used for formulation of static bending and buckling problems. The free vibration problem is formulated using Hamilton 's principle. The total strain energy of a general laminated plate is given by (Fig. 7.1) / 2 1 1 2 2 6 6 1 2 0 0 / 2 1 ( ) 2 a b h z h U dx dx dz σ σ σ − · ∈ + ∈ + ∈ ∫ ∫ ∫ (7.76) Substituting Eq. 6.59 in Eq . 7.76, one obtains / 2 2 2 11 1 12 1 2 16 1 6 22 2 26 2 6 0 0 / 2 2 66 0 11 1 12 2 16 6 1 12 1 22 2 16 6 2 16 1 26 2 66 6 6 1 2 1 [ 2 2 2 2 ( ) ( ) ( ) ] a b h h e e e e e e e e e U Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q dx dx dz − · ∈ + ∈∈ + ∈∈ + ∈ + ∈ ∈ + ∈ − ∈ + ∈ + ∈ ∈ − ∈ + ∈ + ∈ ∈ − ∈ + ∈ + ∈ ∈ ∫ ∫ ∫ (7.77) Substituting Eqs.6.47 through 6.49 in Eq. 7.77, and using Eqs. 6.53, 6.61 and 6.62 we obtain 0 2 0 0 0 2 0 0 0 11 1,1 12 1,1 2, 2 22 2, 2 16 1,1 1, 2 2,1 0 0 0 0 0 0 0 2 0 26 2, 2 1, 2 2,1 66 1, 2 2,1 11 1,1 ,11 0 0 0 12 2, 2 ,11 1, 1 , 22 22 2, 2 , 22 16 1 [ ( ) 2 ( )( ) ( ) 2 ( )( ) 2 2 ( )( ) ( ) 2B {( )( ) 2B {( )( ) ( )( )} 2 ( )( ) 2 {( a b U u u u u u u u u u u u u u w u w u w u w w · Α + Α + Α + Α + + Α + + Α + − − + − Β − Β ∫ ∫ 0 0 0 , 22 1, 2 2,1 1,1 ,12 0 0 0 0 0 26 , 22 1, 2 2,1 2, 2 ,12 66 1, 2 2,1 ,12 2 2 11 ,11 12 ,11 , 22 22 , 22 16 ,11 ,12 2 0 0 26 , 22 ,12 66 ,12 1 1,1 2 2, 2 )( ) 2( )( )} 2 { ( ) 2( )( )} 4 ( )( ) ( ) 2 ( )( ) ( ) 4 ( )( ) 4 ( )( ) 4 ( ) e e u u u w w u u u w u u w D w D w w D w D w w D w w D w u u + + − Β + + − Β + + + + + + + − Ν + Ν − 0 0 6 1, 2 2,1 1 ,11 2 , 22 6 ,12 1 2 ( ) 2 ] e e e e u u w w w dx dx Ν + + Μ + Μ + Μ (7.78) The potential energy of external surface tractions and edge loads due to the deflections of the plate is given by 0 0 1 1 2 2 1 ,1 2 ,2 0 0 2 2 1 ,1 2 , 2 6 ,12 1 2 [( ) 1 { ( ) ( ) 2 ( )}] 2 a b V q u q u qw m w m w N w N w N w dx dx · − + + + + − + + ∫ ∫ (7.79) The kinetic energy of the laminated plate can be expressed as 0 2 0 2 2 0 0 1 2 1 ,1 2 , 2 0 0 2 2 ,1 , 2 1 2 1 [ {( ) ( ) ( ) 2 {( )( ) ( )( )} 2 {( ) ( ) }] a b u u w R u w u w I w w dx dx Τ · Ρ + + − + + + ∫ ∫ & & & & & & & & & (7.80) where P, R, and I are defined in Eqs. 7.6. The Ritz method can be utilized for seeking solution to a particular problem. The approximate displacement functions are chosen to be in the following form 0 1 1 1 2 1 ( , ) m i i i u a u x x · · ∑ 0 2 2 1 2 1 ( , ) m i i i u b u x x · · ∑ (7.81) 1 2 1 ( , ) r i i i w c w x x · · ∑ where the shape functions u 1i (x 1 , x 2 ), u 2i (x 1 , x 2 ) and w i (x 1 , x 2 ) are capable of defining the actual deflection surface and satisfy individually atleast the geometric boundary conditions, and a 1 , b 1 , and c 1 are undetermined constants. For the bending of a general laminated plate, the energy functions, ) , , ( 0 2 0 1 w u u ∏ is defined as U V Π · + (7.82) where U and V are represented in Eqs 7.78 and 7.79, respectively. The principle of minimum potential energy leads to the following conditions. 0, 0 and 0 i i i a b c ∂∏ ∂∏ ∂∏ · · · ∂ ∂ ∂ (7.83) that provide m + n + r simultaneous algebraic equations for the computation of m+n+r unknown coefficients a i , b i , and c i. The approximate solution is thus obtained by substituting these coefficients in the assumed displacement functions in Eqs 7.81. For the solution of plate buckling problem, only the edge loads are retained assuming N N − · 1 , N p N 2 2 − · , N p N 6 6 − · in the expression for the potential energy V in Eqs. 7.79 and 7.82. The application of conditions in Eq. 7.83 results a set of m+n+r algebraic homogeneous equations in terms of m+n +r coefficients a i , b i , and c i . The vanishing of the determinant of the coefficient matrix yields the buckling equation from which critical buckling loads are determined. For the free vibration problem, the displacement functions in Eqs. 7.81 can be modified to include the time dependence as follows: t i e x x U u ω ) , ( 2 1 1 0 1 · t i e x x U u ω ) , ( 2 1 2 0 2 · (7.84) t i e x x W w ω ) , ( 2 1 · where the U 1 (x 1 , x 2 ), U 2 (x 1 , x 2 ), and W(x 1 , x 2 ), correspond to the right-hand-side expressions in Eqs. 7.81. The energy functional П * includes the strain energy U and kinetic energy T. Hence П * = U+T. In the absence of surface forces and moments, edge loads and expansional stress resultants and moments, and substituting Eq. 7.84 in the above energy functional П * and carrying out the derivation of П * with respect to a i , b i , and c i and equating them to zero i.e. 0 i a ∗ ∂ ∏ · ∂ , 0 i b ∗ ∂∏ · ∂ and 0 i c ∗ ∂∏ · ∂ , we obtain a set of m+n+r coefficients a i , b i , and c i . The frequency equation is derived from the condition that the determinant of the coefficient matrix must vanish. 7.7 GALERKIN METHOD The Galerkin method utilizes the governing differential equations of the problem and the principle of virtual work to formulate the variational problem. Here the virtual work of internal forces is obtained directly from the differential equations without determining the strain energy. The Galerkin method appears to be more general than the Ritz method and can be very effectively utilized to solve diverse general laminated plate bending problems involving small and large deflection theories, linear and nonlinear vibration and stability of laminated plates and so on. Consider a general laminated plate (Fig. 7.1) to be in a state of static equilibrium under loads q 1, q 2 and q only Then the governing differential equations in Eqs. 7.7 through 7.9 can be expressed as follows: 0 ) , , ( 1 0 2 0 1 1 · − q w u u L 0 ) , , ( 2 0 2 0 1 2 · − q w u u L (7.85) 0 ) , , ( 0 2 0 1 3 · − q w u u L The equilibrium of the plate is obtained by integrating Eqs. 7.85 over the entire area of the plate. Note that, if required, the edge loads 1 2 6 , , N N N and expansional force resultants and moments can also be included in Eqs. 7.85. Assuming small arbitrary variations of the displacement functions 0 0 1 2 , and u u w δ δ δ and applying the principle of virtual work, we obtain the variational equations as follows: 0 0 0 1 1 2 1 1 1 2 0 0 [ ( , , ) ]( ) 0 a b L u u w q u dx dx δ − · ∫ ∫ 0 0 0 2 1 2 2 2 1 2 0 0 [ ( , , ) ]( ) 0 a b L u u w q u dx dx δ − · ∫ ∫ (7.86) 0 0 3 1 2 1 2 0 0 [ ( , , ) ]( ) 0 a b L u u w q w dx dx δ − · ∫ ∫ As in the Ritz method, we assume the approximate displacement functions in Eqs. 7.81, where the shape functions u 1i (x 1 ,x 2 ), u 2i (x 1 ,x 2 ) and w i (x 1 ,x 2 ), satisfy the displacement boundary conditions but not necessarily the forced boundary conditions, in which case the method leads to an approximate solution. Now, 0 1 1 1 2 1 ( , ) m i i i u u x x a δ δ · · ∑ 0 2 2 1 2 1 ( , ) m i i i u u x x b δ δ · · ∑ (7.87) 1 2 1 ( , ) m i i i w w x x c δ δ · · ∑ Substitution of Eq. 7.87 in Eq. 7.86 yields 0 0 1 1 2 1 1 1 2 1 2 1 0 0 [ ( , , ) ] ( , ) 0 a b n i i i a L u u w q u x x dx dx δ · − · ∑ ∫ ∫ 0 0 2 1 2 2 2 1 2 1 2 1 0 0 [ ( , , ) ] ( , ) 0 a b n i i i b L u u w q u x x dx dx δ · − · ∑ ∫ ∫ (7.88) 0 0 3 1 2 1 2 1 2 1 0 0 [ ( , , ) ] ( , ) 0 a b r i i i c L u u w q w x x dx dx δ · − · ∑ ∫ ∫ The variations of expansion coefficients , and i i i a b c δ δ δ are arbitary and not inter-related. This provides m+n+r equations. 0 0 1 1 2 1 1 1 2 1 2 0 0 [ ( , , ) ] ( , ) 0 a b i L u u w q u x x dx dx − · ∫ ∫ 0 0 2 1 2 2 2 1 2 1 2 0 0 [ ( , , ) ] ( , ) 0 a b i L u u w q u x x dx dx − · ∫ ∫ 0 0 3 1 2 1 2 1 2 0 0 [ ( , , ) ] ( , ) 0 a b i L u u w q w x x dx dx − · ∫ ∫ (7.89) to determine m+n+r unknown coefficients a i , b i , and c i . Note that, in a rigorous sense, the variational relations in Eqs. 7.86 are valid only, if the assumed displacement functions 0 0 1 2 , and u u w are the exact solutions of the problem. Thus, when these displacements are kinematically admissible and satisfy all the prescribed boundary conditions and compatibility conditions within the plate, the method leads to an exact solution. Equations 7.85 can be used for the buckling analysis of a laminated composite plate assuming q 1 = q 2 =0 and replacing ?q? with ? 1 ,11 2 , 22 6 ,12 2 N w N w N w + + ? where N N − · 1 , N p N 2 2 − · and N p N 6 6 − · . Following Eqs. 7.86 through 7.89 we obtain the variational relations of the following form: 0 0 1 1 2 1 1 2 1 2 0 0 [ ( , , )] ( , ) 0 a b i L u u w u x x dx dx · ∫ ∫ 0 0 2 1 2 2 1 2 1 2 0 0 [ ( , , )] ( , ) 0 a b i L u u w u x x dx dx · ∫ ∫ 0 0 3 1 2 ,11 2 , 22 6 ,12 1 2 1 2 0 0 [ ( , , ) ( 2 )] ( , ) 0 a b i L u u w N w p w p w w x x dx dx + + + · ∫ ∫ (7.90) Equations 7.90 are a set of m+n+r homogeneous algebraic equations in terms of m+n+r coefficients a i , b i , and c i . The condition that for a non-trivial solution, the determinant of the coefficient matrix should vanish yields the buckling equation. For the free vibration problem, the displacement functions are assumed to be of the form given in Eqs. 7.84. Considering only the transverse inertia in Eqs. 7.7 through 7.9 and neglecting surface forces and moments, edge loads and expansional stress resultants and moments, we obtain, following the procedure as in the case of buckling above, the variational relations of the form 1 1 2 1 1 2 1 2 0 0 [ ( , , )] ( , ) 0 a b i L U U W u x x dx dx · ∫ ∫ 2 1 2 2 1 2 1 2 0 0 [ ( , , )] ( , ) 0 a b i L U U W u x x dx dx · ∫ ∫ (7.91) 2 3 1 2 1 2 1 2 0 0 [ ( , , ) ] ( , ) 0 a b i L U U W P W w x x dx dx ω − · ∫ ∫ These are, again, a set of m+n+r homogeneous algebraic equations in terms of m+n+r coefficients a i , b i , and c i . The frequency equation is established from the condition that the determinant of the coefficient matrix must vanish so as to obtain a non-trivial solution. 7.8 THIN LAMINATED BEAM THEORY The small deflection bending theory of thin laminated composite beams can be developed based on Bernoulli's assumptions for bending of an isotropic beam. Note that Kirchhoff's assumptions are essentially an extension of Bernoulli's assumptions to a two- dimensional plate problem. Hence the governing laminated plate equations as developed in earlier sections can be reduced to one-dimensional laminated beam equations. Consider a thin laminated narrow beam of length L, unit width and thickness h (Fig. 7.12). The governing differential equations defined in Eqs. 77 through 7.9 reduce to the following two one-dimensional forms: 0 0 11 1,11 11 ,111 1,1 1 1 ,1 e A u B w N q Pu Rw − − + · − && && 0 0 11 ,1111 11 1,111 1,11 1,1 1 ,11 1,11 ,11 e D w B u M q m N w Pw Ru Iw − + · + + − − + && && && (7.92) Consider the bending of a laminated composite shown in Fig. 7.12 under actions of transverse load q(x 1 ) only. Equations (7.92) assume the form 0 11 1,11 11 ,111 0 A u B w − · 0 11 ,1111 11 1,111 D w B u q − · (7.93) Consider, for example, the following simply supported boundary conditions at x 1 = 0, L: 0 1 11 1,1 11 ,11 0; 0 w M B u D w · · − · and 0 1 11 1,1 11 ,11 0 N A u B w · − · (7.94) Assume the displacement functions to be of the forms L x m U u m m 1 1 0 1 cos π ∑ ∞ · · L x m W w m m 1 1 sin π ∑ ∞ · · (7.95) that satisfy the boundary conditions in Eqs. 7.94. Assume the transverse load q(x 1 ) as L x m q q m m 1 1 sin π ∑ ∞ · · (7.96) Substituting Eqs. 7.95 and 7.96 in Eqs. 7.93 and carrying out the algebraic manipulation, we obtain 11 3 2 11 11 11 ( ) m m B q U m A D B L π · ¸ _ − ÷ ¸ , and 11 4 2 11 11 11 ( ) m m A q W m D B L π · ¸ _ Α − ÷ ¸ , (7.97) where q m for a particular distribution of load q(x 1 ) is obtained from ∫ · L m dx L x m x q L q 0 1 1 1 sin ) ( 2 π (7.98) Equations 7.95 in conjunction with Eqs. 7.97 provide solution to the above beam bending problem. Note, that for a uniformly distributed transverse load q 0, 0 4 , 1, 3, 5,.... m q q m mπ · · Next consider the free transverse vibration and buckling of a simply supported laminated beam. The following governing differential equations are considered ( N N − · 1 ; see Eqs. 7.92): 0 11 1,11 11 ,111 0 A u B w − · 0 11 ,1111 11 1,111 1 ,11 0 D w B u N w Pw − + + · && (7.99) The displacement functions chosen are t i m e L x m U u ω π . cos 1 0 1 · t i m e L x m W w ω π . sin 1 · (7.100) that satisfy the boundary conditions defined by Eqs. 7.94. Substituting Eqs. 7.100 in Eqs. 7.99, we obtain two algebraic homogeneous equations in terms of coefficients U m and W m . For a non-trivial solution, the determinant of the coefficient matrix must vanish. This yields the frequency equation to be in the form 2 4 2 2 11 11 11 11 A D B m m N P L L A π π ω − ¸ _ ¸ _ + · ÷ ÷ ¸ , ¸ , (7.101) Note that the critical buckling load, N cr , corresponds to the minimum value of compressive force N for a specific mode shape m, when the frequency is zero. It is to be mentioned that the approximate analysis methods such as the Ritz method and Galerkin method can be used to obtain solutions for laminated composite beams with various other support conditions for which closed form solutions may not be easily obtainable. 7.9 PLY STRAIN, PLY STRESS AND FIRST PLY FAILURE Once the mid-plane displacement 0 0 1 2 , and u u w are determined, as discussed in the previous sections, the mid-plane strains 0 0 0 1 2 6 , and ∈ ∈ ∈ and curvatures k 1 , k 2 and k 6 are determined using Eqs. 6.47 and 6.48. Next the strains 1 2 6 , and ∈ ∈ ∈ for any ply located at a distance z from the mid-plane (see Fig. 6.16) are computed utilizing Eqs. 6.49. Equations 6.50 are then employed to determine the ply stresses 1 2 6 , and σ σ σ at the same location. In some cases, it is required to determine the stresses in each ply, that correspond to the material axes x 1 ' and x 2 ' (Fig. 6.12). These are obtained using the following relations (see Eqs. A.11and A.19): ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ 1 1 1 ] 1 ¸ − − − · ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ ′ ′ ′ 6 2 1 2 2 2 2 2 2 6 2 1 2 2 σ σ σ σ σ σ n m mn mn mn m n mn n m (7.102) where m = cos ؠand n = sin ؠ In many practical design problems, the first ply failure is usually the design criterion. Once the stresses are determined in each ply of a laminate, one of the failure theories presented in section 6.14 is employed to determine the load at which any one of the laminae in the laminated structure fails first ('first ply failure'). The laminate failure, however, corresponds to the load at which the progressive failure of all plies takes place. The estimation of the laminate strength is more complex. 7.10 BIBLIOGRAPHY 1. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw Hill, NY, 1959. 2. S. G. Lekhnitskii, Anisotropic Plates, Gordon and Breach, N.Y., 1968 3. L.R.Calcote, Analysis of Laminated Composite Structure, Van Nostrand Rainfold, NY, 1969. 4. J.E. Ashton and J.M Whitney, Theory of Laminated Plates, Technomic Publishing Co., Inc., Lancaster, 1984. 5. J.C. Halpin, Primer on Composite Materials: Analysis, Technomic Publishing Co., Inc., Lancaster, 1987. 6. J.M. Whitney, Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing Co., Inc., 1987. 7. R.M. Jones, Mechanics of Composite Materials, McGraw Hill, NY 1975. 8. J.R. Vinson and T. ?W, Chou, Composite Materials and their Use in Structures, Applied Science Publishers, London, 1975. 9. K.T. Kedward and J.M. Whitney, Design Studies, Delware Composites Design Encyclopedia, Vol.5, Technomic Publishing Co., Inc., Lancaster, 1990. 10. J.E. Ashton, Approximate Solutions for Unsymmetrically Laminated Plates, J. Composite Materials, 3, 1969, p. 189. 7.11 EXERCISES 1. Derive the governing differential equations as defined in Eqs.7.7, 7.8 and 7.9. 2. Determine the deflection equation for a simply supported square (axa) symmetric laminated plate subjected to a transverse load q=q 0 x 1 /a. 3. Determine the deflection equation for a square (axa) symmetric laminated plate subjected to a transverse load q=q 0 x 1 /a when the edges at x 1 = 0, a are simply supported and those at x 2 = 0, b are clamped. 4. A simply supported antisymmetric cross-ply laminated (0 0 /90 0 /0 0 /90 0 ) kelvar/epoxy composite square plate (0.5m x 0.5m x 5mm) is subjected to a uniformly distributed load of 500N/m 2 . Determine the deflection an dply stresses at the centre of the plate. Use properties listed in Table 6.1. 5. A simply supported antisymmetric angle-ply laminated (45 0 /-45 0 /45 0 /-45 0 ) carbon/epoxy composite plate (0.75m x 0.5m x 5mm) is subjected to a uniformly distributed transverse load q 0 . Determine the load at which the first ply failure occurs. Use the Tsai-Hill or Tsai-Wu strength criterion. See Table 6.1 and also assume X ' 11 t =1450 MPa, X ' 11 c =1080 MPa, X ' 22 t =60 MPa, X' 22 c =200 MPa and X ' 12 = 80 MPa. 6. Determine the transverse natural frequencies for the plates defined in Problems 4 and 5 above. Neglect the transverse load. 7. Determine the uniaxial compressive buckling loads for the plates defined in problem 4 and 5 above. Neglect the transverse load. 8. Make a comparative assessment between the Ritz method and the Galerkin method. CHAPTER - 8 SANDWICH STRUCTURES 8.1 BASIC CONCEPT 8.2 FACE, CORE AND ADHESIVE MATERIALS 8.3 SANDWICH LAMINATED PLATE THEORY 8.4 SYMMETRIC ORTHOTROPIC SANDWICH LAMINATED PLATES 8.4.1 Bending under Transverse Load q 8.4.2 Transverse Vibration and Buckling 8.5 UNSYMMETRIC ORTHOTROPIC SANDWICH LAMINATED PLATES 8.6 SECONDARY FAILURE MODES 8.7 BIBLIOGRAPHY 8.8 EXERCISES 8.1 BASIC CONCEPT A sandwich structural element is essentially a composite construction, where a relatively thick core layer of low strength, stiffness and density is sandwiched between two thin, face layers of strong and dense materials (Fig. 8.1). In this three-layered Fig. 8.1 construction, the faces and the core along with the adhesive layer have got some distinct roles to play. The main function of the core is to keep the faces apart and stablise them. To this effect, however, the core must possess a certain shear rigidity in planes perpendicular to the faces, being thin but made of much stronger and stiffer materials, resist the main part of the stresses developed in their own planes under action of external forces. In fact, the faces are finally responsible for bearing the loads of structural sandwiches, while the core enables the faces to act accordingly. In this repect, the role played by the adhesive is most vital, as it ensures the composite action of the sandwich system as a whole. Once the adhesive fails, the sandwich action is lost and the faces merely behave as two independent thin members. The main advantage of a sandwich construction is primarily derived from the fact that, as face sheets are separated further and further apart the moment of inertia of faces (I f ) about the mid-plane of the sandwich increases considerably, although there may be a marginal increase in weight (normally 5-10%) due to the higher thickness of the core. This is illustrated in Fig. 8.2. The behaviour resembles as that of a I section. With the increase of c, I f increases at a much higher rate, although the face thickness t remains unchanged. A higher moment of inertia I f in conjunction with a higher face modulus E f may provide very high flexural rigidity (E f I f ) against lateral bending, buckling and vibration. Fig. 8.2 Further in a sandwich construction, various materials can most efficiently be combined together not only to derive the desired structural rigidity but also to achieve improved heat resistance acoustic insulation, vibration isolation, shock resistance and several other properties. 8.2 FACE, CORE AND ADHESIVE MATERIALS The aluminium alloys, magnesium alloys, titanium alloys, stainless steel and structural composities are commonly used face materials for primary structural components. The ply wood, gypsum and paper boards, plastic laminates and similar other materials are also used as face layers in sandwich construction. The most common cores are of honeycomb (cellular) type (Fig. 8.3a). They may be either formable or nonformable (Fig. 8.4). The formable honeycomb cores permit some amount of out-of-plane bending so that they can be employed to fabricate curved structural elements. A honeycomb core of the desired cell size (normally varies between 4-12 mm) is fabricated with very thin layers (foils) of sheet materials of aluminium alloys, titanium alloys, resin impregnated draft paper, plastic film and polymer composites. The honeycomb foil thickness, t h usually ranges between 20-150?m. Fig. 8.3 Fig. 8.4 The cores are fabricated either using expansion method or corrugation method. The expansion method, is however, very common. It is schematically illustrated in Fig. 8.5. A typical expanded core is shown in Fig. 8.6. Figure 8.7 exhibits the node to node bond test of a honeycomb core. The honeycomb core exhibits higher transverse properties (G 13 > G 23 ) along the ribbon directions. A corrugated core (Figs. 8.3b and 8.8) can also be made out of similar materials as those used for making honeycomb cores. The thickness t s , in this case, is normally higher than 200?m. In the case of a corrugated core (G 23 > G 13 ), the Fig. 8.8 corrugation also provides an additional amount of bending rigidity about the x 2 -axis. The other core materials are foams in the form of expanded plastics (e.g., PVC, Phenolic Polyurethene, Polystyrene, etc.), foamed glass, foamed aluminium and so on. As the air bubbles are uniformly dispersed within a foam material, it exhibits an isotropic behaviour (G 13 = G 23 ). The balsa wood has also been used as a core. The airframe of the Mosquito Bomber of World War II frame was of sandwich construction with birch wood faces and balsa wood core. The density of conventional core materials may vary between 0.025-0.35 gm/cc. The transverse shear rigidity may be as low as 4000 Pa in the case of a foamed plastic or as high as 1 MPa in the case of a honeycomb core. A plate shear test (Fig. 8.9) is normally used to determine the transverse shear properties of the core. Fig. 8.9 The adhesive are either in the form of liquids and pastes or films. Modified epoxies, polyimides, nitrile phenolics and modified urethanes are commonly used in a sandwich construction. Sandwich adhesives should have a unique combination of surface-melting, bond line control and controlled flow during curing as well as excellent adhesion and better peel strength properties. The adhesion and peel strength properties of a core material are determined using lap shear and drum peel tests (Figs. 8.10 and 8.11). Fig. 8.10 Fig. 8.11 8.3 SANDWICH LAMINATED PLATE THEORY Here we develop the governing differential equations for bending of a three layered sandwich plate (Fig. 8.12). The plate is subjected to transverse load q per unit area of the plate as well as edge loads 6 2 1 , Ν Ν Ν and per unit length. Note that 0 2 0 1 , u u and w are mid-plane displacement components, and 2 1 ψ ψ and are bending rotations. The following assumptions are made: 1. The displacements are small. 2. The material behaviour is linear and elastic. 3. The faces are thin and may be of different materials (including general laminated composites) and thicknesses. 4. The core is thicker, and its behaviour resembles that of an antiplane core. An antiplane core contributes marginally towards the flexural rigidity of the sandwich plate. This is true in many practical sandwich structures, especially with honeycomb cores. The transverse shear stiffnesses of the face materials are assumed to be infinite,while the core is shear flexible (i.e., for the core (G 1z and G 2z ) are finite). With above assumptions, we may assume the Mindlin type of first order shear deformation behaviour of the core, although one may generalize using higher-order shear deformation theories. Thus, straight lines originally normal to the midsurface, before deformation, remain straight but not normal to the deformed midsurface. The actual rotation w ,1 of a section perpendicular to the x 1 -axis is resulted due to the rotation due to bending. 1 ψ (in which the normal remains perpendicular to the midsurface) and a rotation 4 ∈ ( z 1 ∈ ) due to transverse shear. This is illustrated in Fig. 8.13. Hence 1 1 , 4 ψ + · ∈ w 2 2 , 5 ψ + · ∈ w and also 1 , 1 0 1 , 1 1 ψ z u + · ∈ 2 , 2 0 2 , 12 2 ψ z u + · ∈ (8.1) ) ( 1 , 2 2 , 1 0 1 , 2 0 1 , 1 6 ψ ψ + + + · ∈ z u u Fig. 8.13 The force and moment resultants acting on the positive faces of an infinitesimally small plate element dx 1 dx 2 are shown in Fig. 8.14. These are expressed in terms of midplane strains and curvatures as given by ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ 1 ] 1 ¸ Β Β Α · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ Μ Μ Μ Ν Ν Ν 6 2 1 0 6 0 2 0 1 6 2 1 6 2 1 k k k D ij ij ij ij (8.2) where ∫ − · Β Α 2 / 2 / 2 , ) , , 1 ( ) , , ( h h ij ij ij ij dz z z Q D i, j =1,2,6 (8.3) considering both the faces and the core together as multi-layered laminate, and for the core ¹ ; ¹ ¹ ' ¹ ∈ ∈ 1 ] 1 ¸ · ¹ ; ¹ ¹ ' ¹ 5 4 55 55 45 45 45 45 44 44 5 4 S k S k S k S k Q Q (8.4) where, , 2 / 2 / dz Q S c c ij ij ∫ − · i, j = 4,5 (8.5) and k 44 , k 55 are shear correction factors. For an antiplane core (e.g., honeycomb core) , 0 , 45 2 55 1 44 · ≈ ≈ andS cG S cG S z z if the ribbon direction lies parallel to the x 1 - axis. Also note that 0 1 , 2 0 2 , 1 0 6 0 2 , 2 0 2 0 1 , 1 0 1 ; ; u u u u + · ∈ · ∈ · ∈ 1 , 2 2 , 1 6 2 , 2 2 1 , 1 1 ; ; ψ ψ ψ ψ + · · · k k k (8.6) and 5 4 ∈ ∈ and are defined in Eq. 8.1. Fig. 8.14 From the equilibrium of forces acting on the sandwich plate element (Fig. 8.14), the equations of equilibrium are derived as 0 2 , 6 1 , 1 · Ν + Ν 0 2 , 2 1 , 6 · Ν + Ν 0 4 2 , 6 1 , 1 · − Μ + Μ Q (8.7) 0 5 1 , 6 2 , 2 · − Μ + Μ Q 0 2 12 , 6 22 , 2 11 , 1 2 , 5 1 , 4 · Ν + Ν + Ν + + + w w w q Q Q Substitution of Eqs. 8.1 through 8.6 in Eqs. 8.7 yields five differential equations of equilibrium, in terms of five unknown displacements 0 2 0 1 , u u , w, 2 1 ψ ψ and . These are derived as follows: 0 ) ( 2 ) ( 2 22 , 2 26 12 , 2 66 12 11 , 2 16 22 , 1 66 12 , 1 16 11 , 1 11 0 22 , 2 26 0 12 , 2 66 12 0 11 , 2 16 0 22 , 1 66 0 12 , 1 16 0 11 , 1 11 · Β + Β + Β + Β + Β + Β + Β + Α + Α + Α + Α + Α + Α + + Α ψ ψ ψ ψ ψ ψ u u u u u u 0 2 ) ( 2 ) ( 22 , 2 22 12 , 2 26 11 , 2 66 22 , 1 26 12 , 1 66 12 11 , 1 16 0 22 , 2 22 0 11 , 2 26 0 11 , 2 66 0 22 , 1 26 0 12 , 1 66 12 0 11 , 1 16 · Β + Β + Β + Β + Β + Β + Β + Α + Α + Α + Α + Α + Α + Α ψ ψ ψ ψ ψ ψ u u u u u u 0 ) ( ) ( ) ( ) 2 ( ) ( ) 2 ( 2 2 , 45 44 1 1 , 44 44 22 , 2 26 12 , 2 66 12 12 , 1 11 , 2 16 22 , 1 66 11 , 1 11 0 22 , 2 26 0 12 , 2 66 12 0 12 , 1 0 11 , 2 16 0 22 , 1 66 0 11 , 1 11 · + − + + + + + + + + + Β + Β + Β + + Β + Β + Β ψ ψ ψ ψ ψ ψ ψ ψ w S k w S k D D D D D D u u u u u u (8.8) 0 ) ( ) ( ) ( ) 2 ( ) ( ) 2 ( 2 2 , 55 55 1 1 , 45 45 12 , 1 66 12 12 , 2 22 , 1 26 11 , 1 16 11 , 2 66 22 , 2 22 0 12 , 1 66 12 0 12 , 2 0 22 , 1 26 0 11 , 1 16 0 11 , 2 66 0 11 , 2 22 · + − + + + + + + + + + Β + Β + + Β + Β + Β + Β ϕ ψ ψ ψ ψ ψ ψ ψ w S k w S k D D D D D D u u u u u u 0 2 ) ( ) ( ) ( ) ( 12 , 6 22 , 2 11 , 1 2 , 2 22 , 55 55 2 , 1 12 , 45 45 1 , 2 12 , 45 45 1 , 1 11 , 44 44 · Ν + Ν + Ν + + + + + + + + + w w w q w S k w S k w S k w S k ψ ψ ψ ψ 8.4 SYMMETRIC ORTHOTROPIC SANDWICH LAMINATED PLATES Assume the case (e.g., similar cross-ply laminated faces and orthotropic honeycomb core) when A 16 = A 26 = D 16 = D 26 =S 45 = 0 and due to symmetry B ij =0. The differential equations defined in Eqs. 8.8 reduce to 0 ) ( 0 ) ( 0 22 , 2 22 0 11 , 2 66 12 , 1 0 66 12 0 12 , 2 66 12 0 22 , 1 66 0 11 , 1 11 · Α + Α + Α + Α · Α + Α + Α + Α u u u u u u 0 2 ) ( ) ( 0 ) ( ) ( 0 ) ( ) ( 12 , 6 11 , 2 11 , 1 2 , 2 22 , 55 55 1 , 1 11 , 44 44 2 2 , 55 55 12 , 1 66 12 11 , 2 66 22 , 2 22 1 1 , 44 44 12 , 2 66 12 22 , 1 66 11 , 1 11 · Ν + Ν + Ν + + + + + · + − + + + · + − + + + w w w q w S k w S k w S k D D D D w S k D D D D ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ (8.9) The first two equations of Eqs. 8.9 are expressed in terms of 0 2 0 1 , u u only and are uncoupled with respect to w, 2 1 ψ ψ and . The last three equations of Eqs. 8.9, which are expressed in terms of w, 2 1 ψ ψ and , form the set of governing equations for the bending of symmetric orthotropic sandwich laminated plates. 8.4.1 Bending under Transverse Load q The plate (Fig. 8.14)is assumed to be simply supported along all the edges. The simply supported plate boundary conditions are as follows: 0 , 0 : , 0 2 , 2 12 1 , 1 11 1 1 · + · Μ · · ψ ψ D D w a x 0 , 0 : , 0 2 , 2 22 1 , 1 12 2 2 · + · Μ · · ψ ψ D D w b x (8.10) The solution given by b x m a x m m n mn 2 1 1 1 1 1 sin cos π π ψ ψ ∑∑ ∞ · ∞ · · b x m a x m m n mn 2 1 1 1 2 2 cos sin π π ψ ψ ∑∑ ∞ · ∞ · · (8.11) b x m a x m W w m n mn 2 1 1 1 sin sin π π ∑∑ ∞ · ∞ · · satisfies boundary conditions defined in Eqs. 8.10. b x m a x m q q m n mn 2 1 1 1 sin sin π π ∑∑ ∞ · ∞ · · (8.12) Substitution of Eqs. 8.11 and 8.12 in the last three equations of Eqs. 8.9 ( ) 0 ( 6 2 1 · Ν · Ν · Ν results in three algebraic equations in terms of mn mn mnn andw 2 1 ,ψ ψ given as follows: ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ − · ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ 1 1 1 ] 1 ¸ 2 / 2 1 33 23 13 23 22 12 13 12 11 ) / ( 0 0 h b q W mn h mn mn mn ψ ψ β β β β β β β β β (8.13) where, 2 55 55 2 44 44 33 55 55 23 2 55 55 2 3 66 2 3 22 22 44 44 13 2 3 66 12 12 2 44 44 3 66 2 3 11 11 ) )( ( ) )( ( ) )( ( ) )( ( ) ( ) ( ) )( )( ( ) )( ( ) )( ( ) ( ) ( π π β π β π π β π β π β π π β n h S k a b m h S k n h b h S k h b h S k a b m h D n h D a b m h b h S k a b mn h D D h b h S k h h D a b m h D − − · − · − − − · − · + − · − − − · (8.14) Solving these equations, we obtain mn mn mnn andW 2 1 ,ψ ψ in terms of q mn where q mn for a specific transverse load q(x 1 x 2 ) is determined from Eq. 7.21. Substitution of mn mn mnn andW 2 1 ,ψ ψ in Eqs. 8.11 yields the final solution of displacements 2 1 ,ψ ψ and w. 8.4.2 Transverse Vibration and Buckling For transverse vibration, the last equation of Eqs. 8.9 is modified replacing 'q' by ' tt w , ρ − ' and assuming 0 , 6 2 1 · Ν Ν − · Ν Ν − · Ν and p . Note that ρ is the mass per unit area of the plate. The solution t i m n mn mn e b x m a x m ω π π ψ ψ 2 1 1 1 1 1 sin cos ∑∑ ∞ · ∞ · · t i m n mn mn e b x m a x m ω π π ψ ψ 2 1 1 1 2 2 cos sin ∑∑ ∞ · ∞ · · (8.15) t i m n mn mn e b x m a x m W w ω π π 2 1 1 1 sin sin ∑∑ ∞ · ∞ · · satisfies the simply supported boundary conditions (Eq. 8.10). Substitution of Eqs. (8.15) in the last three equations (replacing 'q' by ' tt w , ρ − ' and 0 , 6 2 1 · Ν Ν − · Ν Ν − · Ν and p ) of Eqs. 8.9 yields a set of three homogeneous, algebraic equations in terms of mn mn mnn andW 2 1 ,ψ ψ , as given by ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ · ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ' ¹ 1 1 1 ] 1 ¸ 0 0 0 / 2 1 33 23 13 23 22 12 13 12 11 h mn mn mn W ψ ψ β β β β β β β β β (8.16) where 23 22 13 12 11 , , , , β β β β β are defined in Eqs. 8.14 and 33 β is expressed as ] ) / ( )} /( ) {( ] ) / ( } ) ( ) [{( )} /( ) {( ) )( ( ) )( ( 2 22 3 22 4 2 2 22 2 2 2 3 22 2 2 2 55 55 2 44 44 33 h b h b h b n p a b m h b n h S k a b m h S k Ε′ Ε′ + Ε′ + Ε′ Ν + − − · ρω π π π π π π β (8.17) The frequency equation is obtained from the condition that the determinant of the coefficients of mn mn mnn andW 2 1 ,ψ ψ is equal to zero. The buckling equation corresponds to the case, when the frequency term vanishes. 8.5 UNSYMMETRIC ORTHOTROPIC SANDWICH LAMINATED PLATES Consider a simply supported unsymmetric orthotropic sandwich laminated platem e.g., a sandwich plate with dissimilar cross-ply faces and a honeycomb core. Then, 0 45 26 16 26 16 26 16 · · · · Β · Β · Α · Α S D D (8.18) Substitution of Eqs. (8.18) into Eqs. 8.8 yields five governing differential equation which are coupled with respect to 2 1 0 2 0 1 , , , ψ ψ and w u u . Let the solution be b x m a x m U u m n mn mn 2 1 1 1 1 1 1 0 1 sin cos ) , ( ) , ( π π ψ ψ ∑∑ ∞ · ∞ · · b x m a x m U u m n mn mn 2 1 1 1 2 2 2 0 2 cos sin ) , ( ) , ( π π ψ ψ ∑∑ ∞ · ∞ · · b x m a x m W w m n mn 2 1 1 1 sin sin π π ∑∑ ∞ · ∞ · · (8.19) that satisfy the simply supported boundary conditions as follows: 0 : , 0 0 2 1 1 1 · · · Μ · · u N w a x 0 : , 0 0 1 2 2 2 · · · Μ · · u N w b x (8.20) Following the procedure described in section 8.4 the transverse bending, vibration and buckling problems can be solved. 8.6 SECONDARY FAILURE MODES The sandwich structures exhibit various secondary failure modes. Some of these failure modes may precede the primary failure due to bending and overall buckling. A few important secondary failure modes commonly observed in sandwich construction, are illustrated in Fig. 8.15. Intracellular Buckling (Fig. 8.15a) This is a localized mode of buckling failure, when the core is not supported by faces continuously as in the case of a honeycomb core. The supported face within a honeycomb cell may be regarded as a thin plate with the cell walls acting as edge supports. The intracellular buckling strength corresponds to the buckling stress, σ cr of the face plate of cell sizes (Fig. 8.3a). In the case of a thin isotropic face, the critical intracellular buckling stress σ cr is given by 2 2 ) ( 1 0 . 2 s t f f f cr ν η σ − Ε · (8.21) Here η is a plasticity correction factor for metallic faces and the suffix 'f ' relates to faces. Similar expressions for σ cr for general laminated composite faces do not exist. However, one can compute the critical intracellular buckling stress for such cases using the analysis procedures described in chapter 7 or by employing the finite element analysis method (chapter 8). Face wrinkiling (Fig. 8.15b) This is also a localized mode of instability. It is not confined to individual cells of cellular type cores. Further it involves the transverse (normal to facings) straining of the core material. The faces wrinkle in the form of short wavelengths and finally lead to either crushing of the core or tensile rupture of the core and the core-facing bond. The antisymmetric form of wrinkling is normally encountered in a continuous or foam core. The cellular core usually causes symmetric wrinkling. For an isotropic face the critical wrinkling stress σ wr is given as follows: Antisymmetric wrinkling : 3 / 1 2 1 1 1 ] 1 ¸ Ε Ε · f c cz r wr G Q ν η σ (8.22) Where Q = 0.5 provides the lower bound of test results. The suffix 'c ' refers to the core. Symmetric wrinkling: 2 / 1 82 . 0 1 1 ] 1 ¸ Ε Ε Ε · c t f f c f wr σ (8.23) Shear Crimping (Fig. 8.15c) It is a special form of general (overall) instability in which the buckle wavelength is very short. This happens when the core possesses very low shear modulus. This phenomenon occurs quite suddenly and usually causes the core to fail in shear. It may also lead to the face-to-core debond. Unaxial compression: cz crimp G c t t h ) ( 2 1 2 + · σ (8.24) Pure inplane shear : z c z c crimp G G c t t h 2 1 2 1 2 ) ( + · σ (8.25) Fig. 8.15 8.7 BIBLIOGRAPHY 1. Structural Sandwich Composites, MIL-HDBK-23A, 1974. 2. H.G. Allen, Analysis and Design of Structural Sandwich Panels, Pergamon Press, Lodon, 1969. 3. P.K. Sinha, An Investigation on the Bending and Buckling Characteristics of Orthotropic Sandwich Plates, Ph.D Dissertation, I.I.T., Kharagpur, India, 1972. 4. R.D. Mindlin, Influence of Rotary Inertia and Shear Deformation on the Bending of Elastic Plates, J.Applied Mechanics, Trans. ASME, 18, 1951, p.31. 5. E. Reissner and Y. Stavsky, Bending and Stretching of Certain Types of Heterogeneous Aelotropic Plates, , J.Applied Mechanics, Trans. ASME, 83, 1961, p.402. 6. J.M. Whitney and N.J. Pagano, Shear Deformation in Heterogeneous Anisotropic Plates, , J.Applied Mechanics, Trans. ASME, 37, 1970, p.1031. 7. N.J. Pagano, Exact Solutions for Bidirectional Composites and Sandwich Plates, J. Composite Materials, 4, 1970, p. 20. 8. S. Srinivas and A.K. Rao, Bending, Vibration and Buckling of Simply Supported Thick Orthotropic Rectangular Plates and Laminates, Int. J.Solids and Structures, 6, 1970, p. 1463. 9. S. Srinivas, C.V. Joga Rao and A.K. Rao, An Exact Analysis for Vibration of Simply Supported Homogeneous and Laminated Thick Plates, J. Sound and Vibration, 12, 1970, p. 187. 10. P.K. Sinha and A.K. Rath, Transverse Bending of Cross-Ply Laminated Circular Cylindrical plates, J. Mechanical Engg. Science, 18, 1976, p. 53. 11. P.K. Sinha and A.K. Rath, Vibration and Buckling of Cross-ply Laminated Circular Cylindrical Panels, Aeronautical Quarterly, 26, 1975, p.211. 12. C.W. Bert and T. L. C. Chen, Effect of Shear Deformation on Vibration of Antisymmetric Angle-ply Laminated Rectangular Plates, Int. J. Solids and Structures, 14, 1978, p. 455. 13. J. N. Reddy, A simple Higher Order Theory for Laminated Composite Plates, , J.Applied Mechanics, Trans. ASME, 45, 1984, p.745. 14. A.K. Noor and W.S. Burton, Assessment of Shear Deformation Theories for Multilayered Composite Plates, Applied Mechanics Reviews, 42, 1989, 1-13. 15. P.K. Sinha and D. P. Ray, On the Flexural Behaviour of Orthotropic Sandwich Plates, Building Science, 8, 1973, p. 127. 16. P.K. Sinha and A.K. Rath, Frequencies of Free Vibration of Axially Compressed Orthotropic Sandwich plates, J.Sound and Vibration, 33, 1974, p.541. 17. Y. V. K.S. Rao and P.K. Sinha, Vibration of Sandwich Plates under Axial Compression, AIAA J, 12, 1974, p. 1282. 18. A. M. Kulkarni, J.R. Banerjee and P.K. Sinha, Response of Randomly Excited Orthotropic Sandwich Plates, J. Sound and Vibration, 41, 1975, p. 197. 19. B.R. Bhat and P.K. Sinha, Forced Vibration of Simply Supported Orthotropic Sandwich plates J. Acoustic Society of America, 61, 1977, p. 428. 20. S. K. Goyal and P.K. Sinha, A Note on the Free Vibration of Sandwich Beams with Central Masses, J. Sound and Vibration, 49, 1976, p. 437. 21. S.K. Goyal and P.K. Sinha, Transverse Vibration of Sandwich Plates with Concentrated Mass, Spring and Dashpot, J. Sound and Vibration, 51, 1977, p. 570. 8.8 EXERCISES 1. Derive the set of governing equations listed in Eqs. 8.8. 2. Determine the displacement relations for a simply supported symmetric orthotropic sandwich beam under action of transverse load q. 3. Determine the fundamental frequency for the transverse vibration of a simply supported symmetric orthotropic sandwich beam. 4. Determine the compressive buckling load of a simply supported symmetric orthotropic sandwich column. 5. Consider a simply supported unsymmetric orthotropic sandwich beam and solve the cases as defined in Problem 2,3 and 4 above. 6. How will you use the Ritz method and the Galerkin method to analyse sandwich beams and plates. CHAPTER - 9 FINITE ELEMENT ANALYSIS 9.1 INTRODUCTION 9.2 FINITE ELEMENT DISPLACEMENT ANALYSIS 9.3 TWO-DIMENSIONAL HEAT CONDUCTION IN COMPOSITE LAMINATES 9.4 TWO DIMENSIONAL HYGROTHERMAL DIFFUSION IN COMPOSITES 9.5 Bending and Vibration of laminated Composite Plates and Shells 9.6 BIBLIOGRAPHY 9.7 EXERCISES 9.1 INTRODUCTION The solution of a real life problem involving an arbitrary plate geometry and complicated loading and boundary conditions cannot be easily realized using the analytical methods discussed in chapters 7 and 8. A numerical analysis technique, especially the finite element analysis method, is suited most to tackle such problems. Further, unlike the metallic structure, the design of a composite structure is, in most cases, preceded by the design of the composite material with which the structure is made. The hygrothermal environment, the temperature dependence of thermo-mechanical properties, the anisotropy, the stacking sequence and several other parameters involving material, geometry, loading and service conditions endorse the need for the application of the finite element analysis techniques in composite materials and structures. The finite element method is essentially a piecewise application of a variational method. The finite element formulation is, therefore commonly, based on the conventional Rayleigh-Ritz method (Ritz method) and the Galerkin method (weighted residual method). In the Rayleigh-Ritz method we construct a functional that express the total potential energy ∏ of the system in terms of nodal variables d 1 . The problem is solved using the stationary-functional conditions 0 / 1 · ∂ ∏ ∂ d . In the case of the Galerkin method, the functional for the residual R is formed using differential equations of the physical problem. The problem is solved by setting the weighted averages of the residual R to zero, i.e., ∫ · 0 1 RdV W , where W 1 are the weight functions. The Galerkin method find applications in several non-structural physical problems. In structural mechanics, both the Rayleigh-Ritz method and the Galerkin method yield identical results, when both use the same field variables. 9.2 FINITE ELEMENT DISPLACEMENT ANALYSIS The displacements {u} within an element are usually expressed as {u} = [N] {d e } (9.1) where [N] is the shape function and {d e } is the element nodal displacements with respect to the local axis. The strains { ∈ } in an element are defined in terms of the displacements as } ]{ [ } { u ∆ · ∈ (9.2) where [Δ] is an appropriate differential operator. Now, combining Eqs. 9.1 and 9.2, the relations between the strains and nodal displacements are obtained as } ]{ [ } { e d B · ∈ (9.3) where ] ][ [ ] [ N B ∆ · (9.4) The stresses and strains in an element are defined as } ]{ [ } { ∈ · Q σ (9.5) where [Q] are the elastic stiffnesses. Hence the stress-nodal displacement relations are derived as } ]{ ][ [ } { e d B Q · σ (9.6) The total potential of an element is first computed to apply the Rayleigh-Ritz variational approach. This is equal to the sum of the strain energy developed in the element and the work done by the applied forces on the surface of the element. The work done by the body forces within the element is neglected in the present case. Thus, the total potential of the element is given by ∫ ∫ − ∈ · ∏ Ve Se T T e dS q u dV } { } { } { } { 2 1 σ (9.7) where {q} are the surface tractions. Combining Eqs. 9.1, 9.3, 9.6 and 9.7, we obtain ∫ ∫ − · ∏ Ve Se T T e e T T e e dS q N d dV d B Q B d } { ] [ } { } ]{ ][ [ ] [ } { 2 1 (9.8) Applying the principle of Minimum Potential Energy i.e., 0 / · ∂ ∏ ∂ e e d yields [K e ] {d e } ={P e } (9.9) where [K e ] is the element stiffness matrix and {P e } are the element nodal forces. These are defined as ∫ · Ve T e dV B Q B K ] ][ [ ] [ ] [ and ∫ · Se T e dS q N P } { ] [ } { (9.10) Transformation of Eq. (9.9) from the local axes to the global axes and proper assembly of terms over all elements will lead to a set of equilibrium equations for the complete structure, as given by } { } ]{ [ P d K · (9.11) where [K], {d}and {P} correspond to the global axes. When the Galerkin weighed residual approach is employed, the residual equation is expressed in the form ∫ · − ∈ Ve j kl ijkl T dV q Q N 0 ] ) [( ] [ 1 , (9.12) where 0 ) ( 1 , · − ∈ q Q j kl ijkl are the equilibrium equations, and the shape functions [N] are the weight functions. Applying the Green-Gauss theorem to Eq. 9.12 and expanding and then employing Eqs. 9.1 through 9.4, one obtains ∫ ∫ · − Se T e Ve ijkl T dS q N d dV B Q B 0 } { ] [ } { ] )[ ( ] [ 1 (9.13) after eliminating the non-essential boundary conditions. Equations. 9.13 can be written in the form as that given by Eqs. 9.9, where [K e ] and {P e } are defined in Eqs. 9.10. 9.3 TWO-DIMENSIONAL HEAT CONDUCTION IN COMPOSITE LAMINATES The axes system of a rectangular laminated composite plate is presented in Fig. 9.1. It is assumed that the temperature varies along the x 1 x 2 plane, but remains constant through the thickness of the plate. The two-dimensional heat conduction equation for the composite plate then assumes the form t cT T K T , 22 , 22 11 , 11 ρ · + Κ (9.14) where K 11 and K 22 are the laminate conductivities, T is the temperature field, ρ is the mass per unit area of the plate, c is the heat capacity, t is the time and the associated boundary conditions are 2 1 2 , 2 22 1 , 1 11 1 1 S S T S S S q T K T andK T T · · · + · (9.15) with 1 1 and 1 2 are direction cosines and T q is the boundary heat flux. The Galerkin finite element method is now employed. The residual equation for the composite plate takes the form ∫ ∫ · − + 0 ] [ ] [ 2 1 22 , 22 11 , 11 dx dx cT T K T K N T ρ (9.16) where [N] is the shape function matrix and a dot denotes differentiation with respect to time. Fig. 9.1 Integration of Eq. 9.16 yields 0 ] [ ) 1 , 1 ( ] [ ) ] [ ] ([ 2 1 2 , 2 22 1 , 1 11 2 1 2 , 22 2 , 1 , 11 1 , · − + + + − ∫ ∫ ∫ ∫ dx dx T N c dS T K T K N dx dx T K N T K N T T T T  ρ (9.17) Expressing the temperature variables as {T} = [N] {T e } and substituting in Eq. 9.17 one obtains ∫ ∫ ∫ ∫ ∫ · + + dS q N T dx dx N N c T dx dx N K N N K N T T e T e T T ] [ } ]{ ] [ ] [ [ } ]{ ]) [ ] [ ] [ ] ([ [ 2 1 2 1 2 , 22 2 , 1 , 11 1 ,  ρ (9.18) Equation 9.18 finally reduces to ∫∫ · + · } { } ]{ ) [( } { ] ) [( Te e e cp e e cn Q T K T K  (9.19) where 2 1 ] [ ] [ ] [ ] ) [( dx dx B K B K T e cn ∫∫ · (9.20) 2 1 ] [ ] [ ] ) [( dx dx N N c K T e cp ∫∫ · ρ (9.21) and ∫ · 2 ] [ } { S T T Te dS q N Q (9.22) Note that the conductivity matrix for the laminate is given as 1 ] 1 ¸ · 22 11 0 0 ] [ K K K (9.23) and ∑ · 1 ] 1 ¸ Ν Ν · p i i i B 1 2 , 1 , ] [ (9.24) where N i is the shape function of an element with p nodes. Consider a laminated composite plate (Fig. 9.1), where each side is of length a i.e., a = b. The temperature specified on the boundaries are x 1 =0,a and x 2 = 0 : T = 273k x 2 = a : T = 773k The laminate conductivities are K 11 = 10.03 W/cm K and K 22 = 1.71 W/cm K The analytical solution to the problem is found to be ∑ ∞ · − ∏ · 1 2 1 sin sin ) / sinh( cosh 1 2 N g ar x n a x n r n n T T π π π π (9.25) where 11 22 / K K r · The problem is also analysed using the Galerkin finite element approach. Four elements (Fig. 9.2) are employed. Fig. 9.2 The shape functions are given as follows: 3-noded triangular element in area coordinates (LP3): N i = A i /A, i = 1,2,3 (9.26) 4-noded bilinear isoparametric element (LP4) : , 4 / ) 1 )( 1 ( η η ξ ξ i i i + + · Ν i = 1,2,3,4 (9.27) 8-noded quadratic isoparametric element (QP8): , 4 / ) 1 )( 1 )( 1 ( − + + + · Ν i i i i i ηη ξξ η η ξ ξ 4 ≤ i , 2 / ) 1 )( 1 ( 2 η η ξ i i + − · Ν i = 5,7 (9.28) , 2 / ) 1 )( 1 ( 2 ξ ξ η i i + − · Ν i = 6,8 9-noded quadratic isoparametric element (QP9) , 4 / ) 1 )( 1 ( ξη η η ξ ξ i i i + + · Ν 4 ≤ i , 2 / ) 1 )( 1 ( 2 η η η η ξ i i i + − · Ν i = 5,7 (9.29) , 2 / ) 1 )( 1 ( 2 ξ ξ ξ ξ η i i i + − · Ν i = 6, 8 ), 1 )( 1 ( 2 2 η ξ − − Ν i i = 9 The transformation of coordinates from the Cartesian system x 1 x 2 to the isoparametric system ξη is necessary for the use of isoparametric planar elements (Fig. 9.2). For example, in the isoparametric coordinates, Eq. 9.20 assumes the form ∫ ∫ − − · 1 1 1 1 ] ][ [ ] [ ] ) [( η ξd Jd B K B K T e cn (9.30) in which the Jacobian J arises due to the change of coordinates. The matrix [B] is also expressed as 1 ] 1 ¸ · − η ξ , , 1 ] [ ] [ i i N N J B (9.31) where the Jacobian matrix is given by 1 ] 1 ¸ Ν Ν Ν · ∑ ∑ ∑ ∑ i i i i i i i i y x y x N J η η ξ ξ , , , , ] [ (9.32) The comparision between the analytical solution using Eq. 9.25 and the finite element solution employing four planar finite elements with meshes corresponding to nearly identical degrees of freedom, are presented in Table 9.1. The results depict the steady state temperature at four locations along the line x 1 = a/2. The Lagrangian QP9 element exhibits close proximity to analytical results. The steady state temperature distribution in two antisymmetrically laminated (0 0 /30 0 /0 0 /30 0 ) GT75/Nickel and S i C/6061 Al metal matrix composite plates are plotted in Figs. 9.3 and 9.4. The laminate conductivities are derived using Eq. 4.45 (replace 'd' with ' k' ) and Eq. 11c of Table 4.1, where K f = 10.03 W/cmK, K m = 0.62 W/cmK and V f = 0.5 for the GT75/Nickel composite and K f = 0.16 W/cmK, K m = 1.71 W/cmK and V f =0.5 for the S i C/6061 Al composite. Fig. 9.3 Fig. 9.4 Table 9.1 : Comparison of results Locations Element FEM Analytical %error X 1 = a/2 LP3 289.27 +1.12 X 2 = a/2 LP4 284.64 -0.49 QP8 286.59 286.06 +0.18 QP8 286.64 +0.20 x 1 = a/2 LP3 313.39 +1.20 x 2 = 5a/8 LP4 304.26 -1.74 QP8 312.64 309.66 +0.96 QP9 309.67 +0.00 x 1 = a/2 LP3 372.19 +1.30 x 2 = 3a/4 LP4 359.30 -2.18 QP8 361.53 367.32 -1.58 QP9 364.53 -0.76 x 1 = a/2 LP3 508.31 +0.12 x 2 = 7a/8 LP4 506.64 -0.20 QP8 507.85 507.68 -0.03 QP9 507.67 -0.00 Mesh size: LP3: 16x16, LP4: 8x8, QP8: 4x4, QP9: 4x4 9.4 TWO DIMENSIONAL HYGROTHERMAL DIFFUSION IN COMPOSITES Here we discuss the finite element analysis procedure where the spectral distribution of temperature and moisture are determined simultaneously. This is essentially an uncoupled problem. The temperature field is first evaluated using the procedure discussed in section 9.3. The diffusion of moisture is then analysed with the updated material diffusivity data, which are dependent on temperature. The finite element formultation of moisture diffusion is identical to that of heat conduction. The temperature dependent two-dimensional moisture diffusion equation is given by d 11 C ,11 + d 22 C ,22 = C ,t (9.33) where d 11 and d 22 are the moisture diffusivities and are dependent on temperature and C is the moisture concentration at a time t. The boundary conditions are 2 1 2 , 2 22 1 , 1 11 1 1 S S C S S S q C d C andd C C · · · + · (9.34) where C q is the boundary moisture flux. Applying the Galerkin finite element approach, the residual equation assumes the form 0 ] [ ] [ 2 1 , 22 , 22 1 , 11 · − + ∫∫ dx dx C C d C d N t T (9.35) The moisture field within the element is assumed as {C} =[N] {C e } (9.36) where [N] is the shape function matrix and {C e } are the nodal moisture concentration vector for the element. Expanding Eq. 9.35 and substituting Eq. 9.36 in Eq. 9.35 yield, for the element } ) {( } { ) [( } ]{ ) [( e C e e dc e e df Q C K C K · +  (9.37) where ∫∫ · 2 1 ] ][ [ ] [ ] ) [( dx dx B d B K T e df (9.38) ∫∫ · 2 1 ] [ ] [ ] ) [( dx dx N N K T e dc (9.39) dS q N Q S C T e c ∫ · 2 ] [ } ) {( (9.40) The diffusivity matrix [d] is given by 1 ] 1 ¸ · 22 11 0 0 ] [ d d d (9.41) Consider the diffusion of moisture through a single fibre polymer composite model (V f = 0.385) as shown in Fig. 9.5. The composite model consists of a carbon fibre embedded in an epoxy matrix. The fibre is assumed to be impermaeable to moisture. The matrix is initially saturated with 2% moisture concentration i.e., C i = 0.002. The initial temperature of the composite is specified as T i = 300K. The outside opposite faces of the composite model are now exposed to zero moisture concentration level (C s = 0) and a sudden temperature rise of 573K (T s = 573K). Only a quarter of the composite model is considered for the finite element analysis. The boundary conditions considered are given by for the fibre: 0 : 0 ; 0 : 0 2 2 1 1 · ∂ ∂ · · ∂ ∂ · x T x x T x for the matrix: 0 : 0 1 1 1 · ∂ ∂ · ∂ ∂ · x T x C x x 1 = 0 : C S = 0, T S = 573K x 2 = 0, a: 0 2 2 · ∂ ∂ · ∂ ∂ x T x C and for the fibre-matrix interface : 0 · ∂ ∂ n C The relevant hygrothermal material parameters are presented in Table 9.2. The diffusivity d m of the matrix is assumed to be Table 9.2: Hygrothermal parameters for fibre and matrix Material Conductivity, k Coefficient of thermal d 0 A W/cm K expansion α x 10 -6 /K cm 2 /sec K Fibre 10.03 -1.1 0 - Matrix 0.0072 22.5 1.637 5116 Temperature dependent as given by d m = d 0 e -A/T (9.42) where d 0 is the diffusivity at 300 K. The distribution of transient moisture and temperature along the x 1 axis are plotted in Fig. 9.5. Eight noded isoparametric elements (QP8) are employed for the analysis. An implicit time integration scheme, based on the Galerkin method (? = 2/3), is employed to determine the transient temperature and moisture field. The temperature is observed to be almost saturated at 573 K through the fibre and the matrix at t = 1 sec., but the moisture diffuses through the matrix at a much slower rate. Fig. 9.5 9.5 Bending and Vibration of laminated Composite Plates and Shells Consider a doubly curved laminated composite shallow shell of thickness h and principal radii of curvature R 11 , R 22 and R 12 approach infinity. The shell laminate consists of n number of arbitrarily oriented laminae. The lamina behaviour accounts for the first order transverse shear deformation of the Reissner-Mindlin type. This permits the use of the present analysis to solve composite sandwich plates and shells as well, when one or more plies assume the elastic properties of core materials. Fig. 9.6 The stress resultants on a composite shell element are shown in Fig. 9.7 and are expressed in terms of mid-plane strains and curvature as ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ ∈ ∈ 1 1 1 ] 1 ¸ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ Μ Μ Μ Ν Ν Ν 0 5 0 4 6 2 1 0 6 0 2 0 1 5 4 6 2 1 6 2 1 0 0 0 0 K K K S K D B B A Q Q ij ij ij ij ij ij (9.43) where A ij , B ij , D ij , S ij ,K ij are defined in Eqs. 8.3 through 8.5. Also, for each lamina φ φ 2 2 2 1 44 sin cos z z G G Q ′ + ′ · φ φ 2 2 2 1 45 sin cos ) ( z z G G Q ′ − ′ · (9.44) φ φ 2 2 2 1 55 cos sin z z G G Q ′ + ′ · where ' refers to the principal material axes x 1 'x 2 '. Fig. 9.7 Assume an eight-noded quadratic isoparametric doubly curved shell element (Fig.9.8), where the mid-plane displacements i.e., degrees of freedom per node are defined as three translational displacements andw u u 0 2 0 1 , and two bending rotations 2 1 ϕ ϕ and (see section 8.3). These are expressed in terms of their nodal values by the element shape functions and are given by ) , , , , ( ) , , , ( 8 1 2 1 2 1 2 1 0 2 0 1 ∑ · Ν · i i i i i i i w u u w u u ϕ ϕ ϕ ϕ (9.45) where the shape functions are defined in Eq. 9.28. Fig. 9.8 The strain-displacement relations based on an improved shallow shell theory using the modified Donnell's approximations, are expressed as ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ + ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ ∈ ∈ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ∈ ∈ ∈ ∈ ∈ 0 0 6 2 1 0 5 0 4 0 6 0 2 0 1 5 4 6 2 1 K K K Z (9.46) where } { 0 ∈ correspond to the mid-surface strains an d{K} are the curvatures, and are given by ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ + + − + − − · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ; ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ' ¹ ·∈ ∈ ·∈ ∈ ∈ ∈ ∈ 2 2 , 1 1 , 12 0 1 , 2 0 2 , 1 22 0 2 , 2 11 0 1 , 1 5 0 5 4 0 4 0 6 0 2 0 1 / 2 / / ) ( ) ( ϕ ϕ w w R w u u R w u R w u (9.47) and 1 , 2 2 , 1 6 2 , 2 2 1 , 1 1 , ϕ ϕ ϕ ϕ + · · · andK K K (9.48) The strain nodal displacement relations for the element is given by } ]{ [ } { e d B · ∈ (9.49) where T e T w u u w u u d K K K ] ...... [ } { , ] [ } { 28 18 8 28 18 21 11 1 21 11 5 4 6 2 1 6 2 1 ϕ ϕ ϕ ϕ · ∈ ∈ ∈ ∈ ∈ · ∈ and the matrix [B] is obtained substituting Eq. 9.45 in Eqs. 9.46 through 9.49 and is expressed as ∑ · 1 1 1 1 1 1 1 1 1 1 1 ] 1 ¸ Ν Ν Ν Ν Ν Ν Ν Ν Ν − Ν Ν Ν − Ν Ν − Ν · 8 1 2 , 1 , 1 , 2 , 2 , 1 , 12 2 , 1 , 22 2 , 11 1 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 / 2 0 0 / 0 0 0 / 0 ] [ i i i i i i i i i i i i i i i i R R R B (9.50) The element stiffness matrix is given by ∫∫ · 2 1 ] ][ [ ] [ ] [ dx dx B D B K T e (9.51) Note that the stiffness matrix [D] in the above relation is expressed as 1 1 1 ] 1 ¸ · ij ij ij ij ij ij S K D B B A D 0 0 0 0 ] [ (9.52) as given in Eq. 9.43. The element mass matrix is expressed as ∫∫ Ν Ρ Ν · Μ 2 1 ] ][ [ ] [ ] [ dx dx T e (9.53) where ∑ · 1 1 1 1 1 1 ] 1 ¸ Ν Ν Ν Ν Ν · Ν 8 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] [ i i i i i i (9.54) and ∑ ∞ · 1 1 1 1 1 1 ] 1 ¸ · 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] [ i I I P P P P (9.55) with ∑ ∫ · − · Ρ n k z z k k dz 1 1 ρ and ∑ ∫ · − · n k z z k k dz z I 1 2 1 ρ (9.56) Here ρ is the mass density. The element level load vector due to transverse load {q} per unit area is obtained ∫∫ Ν · 2 1 } { ] [ } { dx dx q P T e (9.57) The stiffness matrix [K e ] and [M e ] are evaluated first by expressing the integrals in the local isoparametric coordinates ξ and η of the element and then performing numerical integration employing the 2x2 Gauss quadrature. The element matrices are assembled after performing appropriate transformations due to the curved shell surface to obtain the global matrices [K] and [M]. Thus, one obtains for the static case [K] {d} = {P} (9.58) and for the free vibration case 0 ] [ ] [ 2 · − M K mn ω (9.59) Table 9.2 shows the results in non-dimensional form a q a q h W w 0 1 1 4 0 4 3 22 / , / 10 ) ( Ν · Ν Ε′ · and 2 0 2 1 1 / 10 ) ( a q M · Μ at the centre (x 1 = a/2, x 2 = a/2) of a simply supported laminated [(0 0 /90 0 ) 4 /0 0 ] paraboloid of revolution shell (Fig. 9.9) with , 100 / 1 / , 10 / , 25 . 0 6 . 0 , 10 12 22 12 22 11 · · · · ′ Ε′ · ′ Ε′ · Ε′ h anda b a a R G ν when it is acted upon by a uniformly distributed transverse load q 0 . The shear of deformation of the laminae is, however, neglected here. The convergence of results seems to be reasonably good. Fig. 9.9 Table 9.2 : Non-Dimensional displacement and force and moment resultants Mesh size w 1 N 1 M 2 x 2 2.839 3.063 5.313 3 x 3 2.771 3.032 5.277 4 x 4 2.746 3.021 5.259 6 x 6 2.729 3.013 5.245 8 x 8 2.722 3.009 5.240 The non-dimensional fundamental frequencies 2 / 1 2 22 2 } / { h a mn Ε′ · ρ ω ω for simply supported laminated composite spherical shells (R 11 = R 22 = R, R 12 = 0) with 11 Ε′ = 25 22 Ε′ , 12 G′ = 13 G′ = 0.5 25 . 0 , 2 . 0 , 12 22 23 22 · ′ Ε′ · ′ Ε′ ν G , a/b = 1, a/h =100 are presented in Table 9.3. A 6 x 6 mesh is used to discretize the shell. Note that R 11 = R 22 = ∞ corresponds to a flat plate. Table 9.3 : Non-dimensional fundamental frequencies h / R 11 0 0 /90 0 0 0 /90 0 /0 0 1/300 45.801 47.035 1/400 35.126 36.890 1/500 28.778 30.963 1/1000 16.706 20.356 Plate 9.689 15.192 The frequencies are found to be comparatively lower for the (0 0 /90 0 ) laminate due to the bending-stretching coupling effect. Further, as the curvature reduces, the frequency comes down. The dynamic stiffness is the lowest for the flat plate. The coupling effect is more pronounced in the case of a plate. 9.6 BIBLIOGRAPHY 1. R. D. Cook, D.S. Malkus and M.E. Plesha, Concepts and Applications of Finite Element Analysis, Wiley, NY, 1989. 2. K.J. Bathe, Finite Element Procedures in Engineering Analysis Preintice- Hall of India, New Delhi, 1990. 3. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, Vol.2, McGraw-Hill Bood Co., NY, 1991. 4. S.S. Rao, The Finite Element Method in Engineering, Peergamon, NY, 1989. 5. J.N. Reddy, An Introduction to Finite Element Method, McGraw Hill, NY.,1992. 6. W.Jost, Diffusion in Solids, Liquids and Gases, Academic Press, 1960. 7. J. Crank, Mathematical Theory of Diffusion, Oxford Press, London, 1975. 8. J.S. Carslaw and J.C. Jaegar, Conduction of Heat in Solids, Clarendon, Oxford, 1959. 9. K.S. Sai Ram and P.K. Sinha, Hygrothermal Effects on the Bending Characteristics of Laminated Composite Plates, Computers and Structures, 40, 1991, p. 1009. 10. K.S Sai Ram and P.K. Sinha, Hygrothermal Effects on the Free Vibration of Laminated Composite Plates, J. Sound and Vibration, 158, 1992, p. 133. 11. K.S Sai Ram and P.K. Sinha, Hygrothermal Effects on the Buckling of Laminated Composite Plates, Int. J. Composite Structures, 21, 1992, p.233. 12. A. Dey, J.N. Bandyopadhyay and P.K. Sinha, Finite Element Analysis of Laminated Composite Paraboloid of Revolution Shells, Computers and Structures, 44, 192, p. 675. 13. A. Dey, J.N. Bandyopadhyay and P.K. Sinha, Finite Element Analysis of Laminated Composite Conoidal Shell Structures, Computers and Structures, 43, 1992, p. 469. 14. K.S Sai Ram and P.K. Sinha, Hygrothermal Bending of Laminaated Composite Plates with a Cutout, Computers and Structures, 43, 1992, p. 1105. 15. K.S Sai Ram and P.K. Sinha, Hygrothermal Effects on Vibration and Buckling of Laminated Plates with a Cutout, AIAAJ, 30, 1992, p. 2353. 16. N. Mukherfee and P.K. Sinha, A Finite Element Analysis of Inplane Thermostructural Behaviour of Composite Plates, J. Reinforced Plastics and Composites, 12, 1993, p. 1026. 17. N. Mukherfee and P.K. Sinha, A Finite Element Analysis of Thermostructural Bending Behaviour of Composite Plates, J. Reinforced Plastics and Composites, 12, 1993, p. 1221. 18. N. Mukherfee and P.K. Sinha, A Comparative Finite Element Heat Conduction Analysis of Laminated Composite Plates, Computers and Structures, 52, 1994, p. 505. 19. N. Mukherfee and P.K. Sinha, Three Dimensional Thermostructural Analysis of Multidirectiional Fibrous Composite Plates, Int. J. Composite Structures, 28, 1994, p. 333. 20. A. Dey, J.N. Bandyopadhyay and P.K. Sinha, Behaviour of Paraboloid of Revolution Shell Using Cross-ply and Antisymmetric Angle-ply Laminates, Computers and Structures, 52, 1994, p. 1301. 21. D. Chakravorty, J.N. Bandyopadhyay and P.K. Sinha, Finite Element Free Vibration Analysis of Point Supported Laminated Composite Cylindrical Shells, J. Sound and Vibrations, 18, 1995, p. 43. 22. D.K. Maiti and P.K. Sinha, Bending and Free Vibration of Shear Deformable Laminated Composite Beams by Finite Element Method, Int. J. Composite Structures, 29, 1994, p. 421. 23. T. V. R. Choudary, S. Parthan and P.K. Sinha, Finite Element Flutter Analysis of Laminated Composited Panels, Computers and Structures, 53, 1994, p. 245. 24. D. Chakravorty, J.N. Bandyopadhyay and P.K. Sinha, Free Vibration Analysis of Point Supported Laminated Composite Doubly Curved Shells ? A Finite Element Approach, Computers and Structures, 54, 1995, p. 191. 25. P.K. Sinha and S. Parthan (Eds.), Computational Structural Mechanics, Applied Publishers Ltd., New Delhi, 1994. 9.7 EXERCISES 1. What are shape functions? Distinguish between the Rayleigh-Ritz approach and the Galerkin weighted residual method in finite element analysis. 2. Develop the finite element governing equations for a one-dimensioal heat conduction problem using a two-noded isoparametric element. 3. Develop the finite element governing equations for a one-dimensional moisture diffusion problem using a two-noded isoparametric element. 4. Develop the finite element governing equations for the transverse bending of an unsymmetrically laminated composite sandwich beam using a three-noded isoparametric element. CHAPTER - 10 ENVIRONMENTAL EFFECTS 10.1 INTRODUCTION 10.2 GROUND ENVIRONMENT 10.2.1 Corrosion 10.2.2 Moisture Diffusion 10.2.3 Foreign Object Impact 10.3 SPACE ENVIRONMENT 10.3.1 Radiation and Thermo-Vacuum Environments 10.3.2 Meteororoid Environment 10.4 BIBLIOGRAPHY 10.5 EXERCISES 10.1 INTRODUCTION Composite materials and structures experience two distinct external environmental conditions (i) ground (and near ground) environment and (ii) space environment. The former is primarily dominated by the temperature and the humidity. The presence of oxygen, sodium chloride, sulphuric acid and certain other aggressive chemicals in the lower atmosphere is also of major concern. All ground based materials and structures including aircraft and other flight vehicle structural components arelikely to be affected by ground environments during their service, manufacturing and storage. The damage due to foreign body impact is also to be viewed with all seriousness, as it may lead to abort a mission or call for a major repair. The impact damage may occur owing to accidental dropping of a hard object (tool, nut, bolt, etc.) from a height. Aircraft are also quite often hit by birds (soft objects) while flying through hailstone forming clouds. The engine fans, compressor blades or the aircraft body is normally affected by such impact. The major environmental problems that are encountered by space vehicles during their flight and orbit in space are due to radiation, thermo-vacuum environment and meteoroid impact. In the present chapter we discuss some of these problems involving both ground and space environments and their effects on composite materials and structures. 10.2 GROUND ENVIRONMENT 10.2.1 Corrosion Corrosion of a composite material involves the chemical or physical deterioration of its constituents (including the fibre matrix interface and the protective surface coat) when exposed to a hostile environment. Chemical degradation means weakening and/or breakage of chemical bond due to reaction of constituents with each other or the corrosive medium. The process of chemical degradation is irreversible. Physical deterioration involves only physical changes and the process is reversible. For example, the swelling of a polymer composite due to absorption of moisture is a physical degradation process, and the swelling is removed when the moisture is fully desorbed. Corrosive environments are essentially of two types: gaseous and aqueous. The potential reactants in the gaseous environments are O 2 , C, Cl 2 and S 4 which are abundant in the ground environment. Oxidation is the most commonly observed corrosive phenomenon in a gaseous environment. Almost all metal matrices form oxides with gaseous oxygen. The rise of temperature may accelerate the oxide nucleation process. The nucleation of the oxide normally takes place at favourable sites on the metal oxygen interface. The rate of oxidation primarily depends on the physical state of oxides formed, the nature of transport processes within it and the extent of physical contact with the oxidizing environment. The oxide so formed may be solid, liquid or gaseous. A thin solid oxide layer may grow to form a thick, compact adherent scales owing to diffusion of ions through it. The formation of solid oxide scales may slow down the oxidation rate by preventing direct contact of the metal matrix with the oxygen. Liquid oxides usually flow off and volatile oxides vapourise and therefore offer little resistance to oxidation. Ceramic matrices, on the other hand possess excellent oxidation resistant properties. The oxides such as alumina, beryllia, mullite (3Al 2 O 3 . 2SiO 2 ), silica, titanium oxide, niclel oxide, zirconia, hafnia, ytrria and thorium oxide are more stable in air at high temperatures compared to other cermics such as nitrides, carbides or sulphides. Carbon and glass fibres exhibit corrosion when subjected to CO and CO 2 vapours. Carbon fibres are more susceptible to oxidation in presence of oxygen. Oxidation occurs on the surface and in the cracks and cavities of the fibre. The rate of oxidation is dependent on the fibre texture, inclusions, porosity and surface condition. Glass fibres with higher silica content exhibit more resistance to corrosion. Thermoplastics, in general, possess excellent oxidation resistant properties. Oxidation of thermosets occur due to chain reactions of free radicals with oxygen that diffuses into the composite. The rate of oxidation increases with ingress of more oxygen due to a rise in temperature. It also depends on the chemical bond of different polymers. Antioxidants are normally added to reduce this type of oxidation. Oxidation results in polymer chain scissions and thereby lowering the molecular weight. Mechanical properties reduce due to a decrease in molecular weight. Water is the most corrosive electrolyte that causes all common types of corrosion. The dissolved oxygen in aqueous solution is the major driving force to initiate corrosion at crevices that are formed by settlement of sand, debrises and marine growths on the composite surfaces or that exist in joints and other flaw sites. Pitting corrosion occurs when some localized areas are exposed due to erosion or corrosion of the protective surface layer and come in contact with aqueous solutions containing aggressive anions (strong acids) such as chlorides, bromides, perchlorates, sulphates and nitrates. However, the chloride ion is the most aggressive one. Seawater is more corrosive than natural water due to its Cl - content. Galvanic corrosion is also predominant in seawater owing to its higher electrical conductivity. Other forms of corrosion like corrosion fatigue, stress corrosion and hydrogen embrittlement are induced due to presence of applied or residual stresses. Graphite fibre reinforced aluminium matrix (AA6061) composites exhibit accelerated corrosion when both graphite and aluminium are exposed to saline environment due to pitting, crevice corrosion or erosion of the protective aluminium surface layer. The formation of aluminium carbides at the fibre-matrix interface alters the properties of the aluminium bond layer in the affected regions and accelerates the corrosion process. The wedging of the Al(OH) 3 corrosion products within the composite results in exfoliation which in turn aggravates the corrosion phenomenon. The presence of intermetallic compounds containing iron, copper, manganese and aluminium that are formed during material processing is responsible for exfoliation of the aluminium alloy (AA6061) matrix. Silicon carbide fibre reinforced aluminium composites are also affected when exposed to sea water. Pitting, both localized and distributed, is very common and the corrosion rate is relatively higher compared to graphite/aluminium composites. The discontinuous silicon carbide reinforcements (particulates or short fibres) exhibit crevice corrosion at the reinforcement-matrix interface which is responsible for subsequent excessive pitting. The corrosion of silicon carbide/aluminium composite is more severe in sea water than in marine environment. The formation of an aluminium boride bond layer at the fibre matrix interface of boron/aluminium composite leads to corrosion at the interface. Sites containing disbands and fissures may exhibit an accelerated rate of corrosion. The fibre/matrix interface corrosion is also noticed in alumina/aluminium (AA6061) composites. On the other hand, the Li 2 O. 5Al 2 O 3 bond layer in the alumina/aluminium-lithium composite prevents corrosion, when the composite is immersed in NaCl solutions. However, under such conditions, the Mg 5 Al 8 precipitates in alumin/aluminium-magnesium composites initiates pitting at the fibre-matrix interfaces. A couple of general observations can be made based on similar studies of several metal matrix composites: 1. Most fibres and matrices corrode when exposed to a corrosive environment. 2. Erosion of protective layer, pitting and crevice corrosion accelerate the corrosion process. 3. The properties of the reinforcement/matrix interface may control the corrosion nucleation process at favourable sites. 4. The existence of disbonds, fissures, elemental segregation, inhmogeneity, nonuniform plastic deformation, cold worked regions, residual stresses, etc., may influence the corrosion process. An appropriate protective coating, in most cases, provides a barrier between the composite and the corrosive medium and slows down the corrosion rate. Sulphuric acid anodizing and organic coatings have been found to be effective for corrosion control of graphite filament reinforced aluminium composites, whereas electroplated or vapour deposited nickel and titanium coatings may create highly unfavouable anodic (aluminium) and cathodic (nickel or titanium) area ratio at the coating flaw sites for corrosion acceleration. A surface coat of flame or arc sprayed aluminium (plus an organic top coat) may decelerate corrosion in silicon carbide/aluminium composites. One undesirable characteristic of glass fibres is that they are soluble with either very high or very low pH level. The corrosion rate is much higher in basic solutions compared to acidic solutions. Some basic solutions such as sodium hydroxide, potassium hydroxide and sodium hypochlorite with low pH levels contain much higher hydroxyl and hydrogen ions. The corrosion of a glass fibre takes place when the negatively charged hydroxyl ions attack the positively charged silicon ions and the positively charged hydrogen ions attack the nonbridging negatively charged oxygen ions. Suitable surface treatment (sizing) is generally made to control such corrosion. Reinforced plastic boat hulls exhibit blistering (also known as boat pox, aquatic acne or plastic plague) at the outer surfaces. The blistering is caused due to osmotic effects. An osmotic pressure builds up due to the presence of a solute within the composite, when a solvent (water or dilute sodium chloride solution) diffuses through the thin gel coat (which acts as a semipermeable membrane) and causes formation of blisters. Polyester resins which are normally used in fabrication of boat hulls contain traces of water soluble substances. Expansion of trapped gases or liquids within the voids in the composite, when exposed to hot sun, may also create differential pressure which distorts the gel coat and forms blisters. Blisters, as such, are not harmful, for they are formed on the gel coat. But their formation can activate the process of deterioration of composites due to entrapment of oxygen, water vapour and other corrosive substances within the blisters. One way to control blisters is to do away with the gel coat or use resins which do not contain solutes. The formation of microvoids is another phenomenon which is normally associated with absorption of water by polyesters and epoxies when immersed in water. These microvoids are formed as a result of chemical reactions. The swelling and plasticization of the resins occur with the absorption of moisture. Increased moisture content also results in a decrease in the glass transition temperature. The mechanical properties of composites, especially transverse and shear properties (those which are matrix-dominateed) are greatly reduced with the increase in the moisture content (Fig. 10.1). Moisture and temperature may also introduce hygrothermal stresses and deformation in laminated composite structures. Fig. 10.1 10.2.2 Moisture Diffusion All polymers absorb moisture in humid environment or when immersed in water. The measured critical surface tension of most polymers (18-50 dyn/cm) is lower compared to the surface tension of water (72.8 dyn/cm). All polymes are therefore hydrophobic. Diffusion is the process by which water is absorbed by a polymer. The extent of moisture absorption by a particular resin depends on the affinity of its polar functional groups for water molecules. The process of diffusion is distinctly different from the capillary action by which water is transported through fissures, cracks, voids, etc. In the case of moisture diffusion, water is not absorbed in the liquid form, but in the form of molecule or groups of molecules linked by hydrogen bond to the polymer. The absorption or desorption of moisture is governed primarily by the moisture gradients that arise due to non-uniform distribution of moisture. The transportation of mass takes place from the region of higher concentration to one of lower concentration. There are several other factors such as temperature, relative humidity, area of exposed surface, fibre and matrix diffusivities, resin content and fibre shapes that influence the moisture diffusion process. Moisture absorption or desorption characteristics can be modeled using a simple one-dimensional form of Fick's diffusion equation 2 2 2 22 x C d t C ∂ ∂ · ∂ ∂ (10.1) where C is the moisture concentration (g/m 3 ), d 22 is the diffusivity (m 2 /s) and t is the time (s). The Fick's relation defined in Eq. 10.1 is fundamentally equivalent to Fourier's heat conduction equation. The diffusion coefficient, d 22 is a material constant and gives the measure of the rate at which moisture diffuses through the composite along the x 2 -axis. Consider a semi-infinite composite body of thickness h along the x 2 -axis (Fig. 10.2). The boundary conditions are C = C i for 0 < x 2 < h when t ≤ 0 C = C 0 at x 2 = 0 and h when t > 0 (10.2) Assume C 0 > C i then moisture diffuses into the composite through the faces x 2 = 0 and x 2 = h. The other faces are assumed to be impermeable to moisture. Let each of the faces (x 2 = 0, h) has an exposed area, A, through which moisture diffuses. The solution of Eq. 10.1 is then obtained as 2 / 1 22 2 0 2 ) ( 2 1 ) , ( t d x erf C C C t x C i i − · − − (10.3) The total weight of moisture that diffuses through the two exposed faces is given by dt x C Ad g t x t ∫ · ∂ ∂ − · Μ 0 0 2 22 2 ) ( 2 (10.4) Combining Eqs. 10.3 and 10.4 and noting that ∫ − · ) ( 2 / 0 2 22 2 22 2 ) exp( 2 ) ( 2 t d x d t d x erf η η π (10.5) one obtains M t = 4Ag (C 0 ? C i ) (d 22 t / π ) 1/2 (10.6) The moisture content (or the percent weight gain) at a time t is given as 100 100 x W x W W W d t d d Μ · − · Μ (10.7) where W is the weight of the moist composite after a period of time t and W d is the weight of the dry bone composite. Note that W d = Ahρg with ρ is the density of the dry composites. Hence, from Eqs. 10.6 and 10.7, 2 / 1 22 0 ) / ( 4 π ρ t d h C · Μ (10.8) The maximum moisture concentration within the composite reaches the level of C 0 so as to be equilibrium with the outside moisture concentration (i.e., C 0 ), when it is exposed for a long time. Hence the maximum moisture content is given by M m = C 0 / ρ (10.9) Substitution of Eq. 10.9 in Eq. 10.8 yields 2 / 1 22 ) / ( 4 π t d h M m · Μ (10.10) Eq. 10.10 can be used to determine the diffusion coefficient d 22 for a composite that permits the Fickian moisture diffusion. Figure 10.3 illustrates the moisture absorption behaviour of a typical polymer composite. In the initial phase of moisture absorption the moisture content increases linearly with the square root of the time of exposure following the Fick 's law as defined in Eq. (10.10). The diffusion coefficient, d 22 is then determined from the slope as shown in Fig. 10.3, and expressed as 2 1 2 1 2 2 22 ) ( ) 4 ( t t h d m − Μ − Μ Μ · π (10.11) After a long exposure, the moisture content of the composite reaches asymptotically to the maximum moisture content, M m . Note that the value of M m is a material constant, when the composite is immersed in water. When the composite is exposed to humid environment, M m varies with the relative humidity, φ as given by M m = a φ b (10.12) Where the constants a and b (for a particular composite) are determined from the best fit curve for the M m vs. φ plot. The diffusion coefficient d 11 and d 33 can also be determined in a similar manner. The moisture diffusion coefficient is normally dependent on temperature, T and can be expressed as d = d 0 exp (-A/T) (10.13) The spatial distribution of moisture concentration, C(x 2 , t) at a time t can be obtained for particular boundary conditions solving Eq. 10.1 analutically (for example, see Eq. 10.3) or using the finite element method (see section 9.4) or other numerical analysis techniques. The analytical solutions are however, available for simple one- dimensional and a few two-dimensional problems. The finite element analysis technique, on the other hand, can be extended to tackle three-dimensional moisture diffusion problems and is a convenient means to solve hygrothermal (both moisture and temperature) diffusion in a composite body having complicated geometry with multidirectional fibre orientations and complex boundary conditions. The changes in the moisture concentration and temperature may introduce thermal stresses and strains in a laminate (see section 6.13). Figure 10.4 illustrate the effects of moisture concentration on the bending, free vibration and buckling of a simply supported laminated composite plate. The results are obtained using the finite element method and employing eight nodded isoparametric quadratic elements. Fig. 10.4 Practical composites may exhibit non-Fickian diffusion behaviour. The anomalous Fickian diffusion behaviour is observed, if cracks, voids, delaminations and fibre matrix interface debonds exist in the composite and when the matrix itself exhibits non-Fickian behaviour. The existence of cracks, voids, etc., increases the moisture absorption at a faster rate. The non-Fickian behaviour of the polymer is observed when the relaxation processes inside the polymer progress at a rate comparable to the diffusion processes. The absorbed moisture may decrease the T g thereby affecting the diffusion process. Fick's law is generally applicable to rubbery polymers, but fails to characterize the diffusion process in glassy polymers. It is, in general, observed that there exists an upper limit of humidity or temperature at which moisture diffusion deviates from that governed by Fick's law. However, much more research effort is needed to understand and analyse the effects of non-Fickian diffusion is polymer composites. 10.2.3 Foreign Object Impact The impact considered here, is defined as the phenomenon involving collision of two elastic bodies, in which the striking object (or the impactor) has relatively less mass compared to that of the target. Examples of such impact involving composite structures (i.e., targets) are too many. A few typical examples are a bird striking an aircraft engine blade, a hailstone impacting on the aircraft wing skin, a bullet hitting a composite vest or car body panel, or a ball bounching off a composite hockey stick. The impacting velocity in these cases mostly range from Match 1 down to a few metres per second. Such an impact phenomenon is normally termed as low velocity impact. The material behaviour of the striker and the target even in the highly stressed impacting region is assumed to follow the constitutive relations defined in the realm of solid mechanics. Consider a simple case of an elastic spherical mass impacting at the mid-span of a unidirectional composite beam (Fig. 10.5). The transverse vibrational response of the beam is then governed by the following relation: D 11 W, 1111 + Pw = F c (t) (10.14) Where F c (t) is the time-dependent contact force exerted by the impacting mass at the contacting mid-span of the beam. Fig. 10.5 The impactor motion is governed by M i w i + F i (t) = 0 (10.15) Where the subscript 'i' refers to the impactor. The contact force F e (t) during loading can be determined by the modified Hertzian contact law as given by F c (t) =n α 3/2 (10.16) Where α is the local indentation on the target at the contact point and n is the modified Yang-Sun contact stiffness. Note that the value of α is time dependent and it is the difference between the impactor displacement, w i and the target displacement, w at the contact point at a time t after the initial contact. The parameter n is given by ] / 1 / ) 1 [( 1 3 4 2 1 zz i i r n Ε′ + Ε − · ν (10.17) where, r i is the radius of the impactor, i ν and E i are elastic constants of the impactor, and E' zz is the transverse modulus of the uppermost layer of the composite beam. During the unloading and reloading processes the contact force is modified as 5 . 2 0 0 ) ( 1 ] 1 ¸ − − · α α α α m m c F t F (unloading) (10.18) 5 . 1 0 0 ) ( 1 ] 1 ¸ − − · α α α α m m c F t F (reloading) (10.19) where F m is the maximum contact force just before unloading, α m is the maximum local permanent indentation and α 0 is defined as α 0 = 0, when α m < α cr α 0 = α m [1- (α cr / α m ) 0.4 ] , when α m ≥ α cr (10.20) Note that α cr is the critical indentation beyond which permanent indentation will occur. The typical value of α cr for a carbon/epoxy composite is 8.0264 x 10 -5 m. The transient vibrational response problem defined in Eqs. 10.4 through 10.20 can be solved using numerical integration techniques. Figure 10.6 shows the impact response characteristics of a simply supported unidirectional ( φ =0) compositebeam (L = 0.1905m, W= 0.0127 m and h=3.175x10 -5 m), when a spherical steel (E i =210 GPa, ν i = 0.3, ρ i = 7800 kg/m 3 ) ball of 1.27 cm dia strikes the beam at the mid-span with an initial velocity (v 0 ) of 35 m/s. The composite properties are assumed as follows: E' 11 = 129.207 GPa, E' 22 = 9.425 GPa, G' 12 = 5.157 GPa ν' 12 = 0.3, ρ = 1550 Kg/m 3 . Fig. 10.6 10.3 SPACE ENVIRONMENT 10.3.1 Radiation and Thermo-Vacuum Environments Figure 10.7 illustrates various regions of space. Upto an altitude of 200km from the earth's surface, the atmosphere, at an altitude of 200 km to 1000 km, is composed of atomic nitrogen and oxygen with some traces of helium between 700 km to 1000 km. High energy protons and electrons, some charged particles as well as neutral and atomic hydrogen are found at an altitude of 1000km and above. Protons and electrons may cause severe radiation damage to space vehicles because of their penetrability and higher intensity. One of the important radiation sources for low earth orbit (LEO) satellite missions is the existence of magnetically trapped radiation consisting of electrons and protons. This radiation extends from about an altitude of 200 km to beyond synchronous orbit altitude (35, 900 km). High energy electrons are present in both the inner and outer belts tetween 5000 km to about 35,000 km. Electrons of lower energy are found at higher altitude between 40,000 km to 50,000 km. High energy protons are contained in the inner belt upto an altitude of 15000 km and lower energy protons extend into the outer belt. Fig. 10.7 Other major prevailing radiation sources are solar cosmic rays and galactic cosmic rays. They consist of mostly protons. Solar cosmic rays are ejected sporadically from the Sun during solar flare events which at present cannot be precisely predicted. However, solar cosmic ray and galactic cosmic ray environments are hazardous for lunar and interplanetary space vehicles and for near polar or high altitude orbital missions. Figure 10.8 describes the typical distribution of particle concentration surrounding the earth. Figure 10.9 illustrates the pressure variation upto an altitude of 800 km. it may be mentioned that as the particle concentration goes down at higher altitude, the pressure falls and vacuum is created. The pressure between 10 -2 to 10 -7 torr is termed as high vacuum and that from 10 -8 and below is usually noted as ultra high vacuum. Besides high or ultrahigh vacuum environment, space vehicles are also subjected to extreme temperature fluctuations. The body temperature in space is the equilibrium temperature due to direct radiation from the sun, radiation (albedo effect) from the earth and radiation by the body into space (which is a heat sink of about 4K). The body temperature is usually controlled by the materials, location of parts and thermal balance system chosen for the vehicle, and may vary between -120 0 C and 150 0 C, depending on whether the body is on the shadoe region, or it faces the sun. Fig. 10.8 Fig. 10.9 The principal effect of radiation in polymers (both thermosetting plastics and thermoplastics) is the formation of new and irreversible chemical bonds due to cross linking between two adjacent polymer molecules. Cross linking may lead to appreciable variations in mechanical, thermal and electrical properties, as well as changes in chemical and physical states. Characteristics of these changes vary from polymer to polymer. Chain scission or fracture of polymer molecules is another phenomenon associated with radiation. This often results in decrease in Young's modulus, strength and hardness, and increase in elongation, thermal conductivity and so on. Studies carried out to characterize effects of radiation in glasses, graphites, etc., in their bulk form, indicate that the density of most of the silica system increases with the increase of radiation and approaches a limiting value of 2200 kg/m 3 . The thermal conductivity of invalidated fused silica reaches a limiting value of about twice the initial value. Changes in mechanical properties such as Young's modulus, shear modulus, etc., are however less than 5 percent. The neutron irradiated graphites exhibit increase in strength, hardness and chemical reactivity, and substantial loss in thermal conductivity. The outgassing in materials under thermo-vacuum environment is a common phenomenon. Outgassing is significant in some polymeric materials. Besides polymeric constituents, polymers contain additional substances such as solvents, catalysts, etc. Loss of gases as well as sublimation or evaporation of volatile substances occur in a high/ultrahigh vacuum environment. The process is accelerated due to elevated temperature and prolonged time of operation in such an environment. This may affect some of the important material properties such as elastic moduli, mechanical and fracture strength, coefficients of thermal expansion etc. 10.3.2 Meteroroid Environment Meteroroid impact can cause considerable damage to space vehicles. The type and extent of damage depends on the vehicle size, structural configuration and materials, position and exposure time in space, as well as meteoroid characteristics such as velocity, density, mass flux (i.e., number of particles per unit area per unit time), and angle of impact. Meteoroids are solid particles that are of cometory and asteroidal origin. Lunar efecta that are created by the impact of cometary particles on the lunar surface also fall under the broad category of meteoroids. Meteoroids are classified as (i) sporadics when their orbits are random, and (ii) streams or showers when most of them have nearly identical orbits. The meteoroid environment of cometary origin is found at one astronomical unit (1AU) from the sun near the ecliptic plane. The lunar ejecta environment exists from the lunar surface to an altitude of 30 km. The meteoroid environment of asteroidal origin is commonly found in the interplanetary space, particularly in the asteroidal belt between Mars and Jupiter. In the near-earth region (1AU) asteroidal particles are assumed to be non-existent and hence particles of cometary origin are of major concern. The cometory meteoroid is porous, highly frangible and often described as dust ball or a conglomerate of dust particles bound together by frozen gases/ices. The particle mass ranges from 10 -12 to 1 gm for sporadic meteoroids and 10 -6 to 1 gm for stream meteoroids. The mass density is assessed to be about 0.5 gm/cm 3 . The geometric velocity varies between 11 and 72 km/sec. An average atmospheric entry velocity of 20 km/sec is assumed as the average velocity of sporadic meteoroids. The velocity of stream meteoroids varies from stream to stream. For example Leonid has a stream velocity of 72 km/s, while the velocity of Bielids is 16 km/s, although both appear during the month of November. An average total meteoroid (average sporadic plus a derived average stream) environment can be assumed in the initial design phase. This average cumulative meteoroid flux-mass model is expressed as 10 -6 ≤ m ≤ 10 0 : log 10 N = -14.37 ? 1.213 log 10 m 10 -12 ≤ m ≤ 10 -6 : log 10 N = -14.339 ? 1.584 log 10 m ? 0.063 (log 10 m) 2 (10.21) where N = number of particle of mass m or greater per square meter per second m = particle mass in grams. The following Poisson distribution equation describes the probability of impact by meteoroids. ∑ · − ≤ · Ρ n r r NAt n x r NAt e 0 ! ) ( (10.22) where n x≤ Ρ = probability of impact by n meteoroids or less N = flux, particles/m 2 as defined in Eq. 10.21 (accounting for the effects of gravitational focusing and of shielding by planetary bodies or by the parts of the vehicle, if applicable). A = exposed area, m 2 t =exposure time, second The flux-mass relations given in Eq. 10.14 and Eq. 10.15 should be used to establish the probability of impact by n or less meteoroids of a particular mass or greater. When a meteoroid hits a target in the hypervelocity range 1 , the following phenomenon occurs. An enormous shock pressure is built up at the interface between the target and projectile. This causes the material at the point of contact to be compressed almost instantaneously to an extremely high pressure. Fracture occurs when the stress exceeds a critical value. As the shock passes through the target, the pressure decreases rapidly and the material expands adiabatically. This process creates irreversible shock heating which may even lead to fusion, liquefaction and vapourisation. As the shock pressure propagates away, its intensity reduces. Then the pressure fails below the strength of the material and the damage is arrested. However, for a brittle target, the initial damage may lead to catastrophic failure. The response of a structure under meteoroid impact depends primarily on the structure material, thickness, type of construction (laminated, sandwich, etc.) and temperature as well as the meteoroid characteristics. Foot Note 1: Subordinate range : 25 to 500 m/s Nominal ordance range : 500 to 1300 m/s Ultraordnance range : 1300 to 3000 m/s Hypervelocity range : >3000 m/s Both experimental studies and analytical studies have been carried out to assess the damage due to the simulated meteoroid impact on metallic targets. The experimental study involves use of particle accelerators. The pellet materials are aluminium and glass. Syntactic foam (hollow glass spheres embedded in plastic matrix) has also been used as a pellet with the density as low as 0.7 gm/cm 3 . Two major limitations of experimental studies are : realization of higher velocity and simulation of the desired range of meteoroid density. Analytical method employ incompressible and compressible hydrodynamic theories that provide excellent understanding of the penetration of hypervelocity particles. In the former case, large stresses developing due to the hypervelocity impact permit neglect of rigidity and compressibility of striking bodies and the impact is viewed as fluid flow. This simplifies the description of material properties to a great extent. The refined analyses consider materials to be compressive fluids and also take into account effects of material strength. The results of impact damage phenomenon in composites, that are available in the open literature, mostly relate to FCD (Foreign Object Damage). Both analytical and experimental investigations have been carried out to study the transient and steady state responses and damage mechanics. However, these results are not applicable to meteoroid impact as the foreign object impact velocity considered in these studies fall well below the hypervelocity range. Hence there is considerable need to investigate and identify the impact damage mechanics in composites in the hypervelocity range and to establish appropriate methodology for design of composite structural elements against hypervelocity impact. 10.4 BIBLIOGRAPHY 1. W. K. Bond and F.W.Fink, Corrosion of Metals in Marine Environments, Metals and Cermamics Information Centre,OH, 1978. 2. C.R. Crowe, Localised Corrosion Currents from Gr/Al and Welded SiC/Al Metal Matrix Composites, NRL Report 5415, 1985. 3. P.P. Trzasloma, The Corrosion Behaviour of a Graphite Fibre/Magnesium Metal Matrix in Aqueous Chloride Solution, NRL Report 5640, 1985. 4. D.M. Aylor, Metal Matrix Composites: Corrosion, in International Encyclopedia of Composites (S.M. Lee, Ed.), Vol. 3, VCH, NY, 1990, p.228. 5. P. Schweitzer (Ed.), Corrosion and Corrosion Protection Handbook, Marcel Dekker, NY,1983. 6. W. Schnabel, Polymer Degrdation : Principles and Practical Applications, Macmillan, NY, 1988. 7. J.H.Mallinson, Corrosion-Resistant Plastic Composites in Chemical Plant Design, Marcel Dekker, NY, 1988. 8. G.S. Springer (Ed.), Environmental Effects in Composite Materials, Vols. 1-3, Technomic Publ. Co. Inc., Lancaster, 1981, 1984 and 1988. 9. B.C. Ray, A. Biswas and P.K. Sinha, Freezing and Thermal Spikes Effects on ILSS values of Hygrothermally Conditioned Fibre/Epoxy Composites, J. Materials Science Letters, 11, 1992, p. 508. 10. P.K. Aditya and P.K. Sinha, Diffusion Coefficient of Polymeric Composites Subjected to Periodic Hygrothermal Exposures, J. Reinforced Plastics and Composites, 11, 9, 1992, p. 1035. 11. P.K. Aditya and P.K. Sinha, Effects of Fibre Shapes on Moisture Diffusion Coefficients, J. Reinforced Plastics and Composites, 12, 9, 1993, p. 973. 12. P.K. Aditya and P.K. Sinha, Effects of Arbitrarily Shaped Fibres on Moisture Diffusion Coefficient of Polymeric Composites, J. Reinforced Plastics and Composites, 13, 2, 1994, p. 142. 13. P.K. Aditya and P.K. Sinha, Effects of Fibre Permeability on Moisture Diffusion Coefficients of Polymeric Composites, Polymers and Polymer Composites, 1, 5, 1993, p. 341. 14. P.K. Sinha, Failure of Composites, Report No. AEM-H1-20, Department of Aerospace Engineering and Mechanics, Univ. of Minnesota, MN, 1978. 15. P. K. Sinha, A Review of Moisture Diffusion in Composites, Report No. AEM- H1-19, Department of Aerospace Engineering and Mechanics, Univ. of Minnesota, MN, 1978. 16. J.S. Carslaw and J.C. Jaegar, Conduction of Heat in Solids, Clarendon, Oxford, 1959. 17. W. Jost, Diffusion in Solids, Liquids and Gases, Academic Press, 1960. 18. J. Crank, Mathematical Theory of Diffusion, Oxford Press, London, 1975. 19. K.S. Sai Ram and P.K. Sinha, Hygrothermal Effects on the Bending Characteristics of Laminated Composite Plates, Computers and Structures, 40, 1991, p. 1009. 20. K.S. Sai Ram and P.K. Sinha, Hygrothermal Effects on the Free Vibration of Laminated Composite Plates, J. Sound and Vibration, 158, 1992, p. 133. 21. K.S. Sai Ram and P.K. Sinha, Hygrothermal Effects on the Buckling of Laminated Composites Plates, Int. J. Composite Structures, 21, 1992, p. 233. 22. R.B. Hetnarski (Ed.), Thermal Stresses, I and II, Elsevier, Amsterdam, 1991. 23. Space Radiation Protection, NASA SP 8054, 1970. 24. D. J. Santeler et al., Vacuum Technology and Space Simulation, NASA SP 105, 1967, p. 197. 25. Nuclear and Space Radiation Effects on Materials, NASA SP 8053, 1970. 26. J. Bourriean and A. Paillous, Effect of Radiations on Polymers and Thermal Control Coatings, ESA SP-145, 1979, p. 227. 27. R. C. Tennyson, Composite Materials in a Simulated Space Environment, AIAA Paper No. 80-0678, 1980. 28. R. C. Tennyson et al., Space Environmental Effects on Polymer Matrix Composites, ESA SP-145, 1979, p. 263. 29. Meteoroid Environment Model ? 1969 (Near Earth to Lunar Surface), NASA SP- 8013, 1969. 30. W. Goldsmith, Impact, Edward Arnold (Publishers)Ltd., London, 1960. 31. A.C. Eringer, Transverse Impact on Beams and Plates, J. Applied Mechanics, Trans. ASME, 72, 1950, p. 461. 32. E. H. Lee, The Impact of a Mass Striking a Beam, J. Applied Mechanics, Trans. ASME, 62, 1940, p. A67. 33. M.E. Backman and W. Goldsmith, The Mechanics of penetration of Projectiles into Targets, Int. J. Engng. Sci., 16, 1978, p.1. 34. A.J. Cable, High Velocity Impact Phenomena, Academic Press, NY, 1970. 35. Meteoroid Damage Assessment, NASA SP-8042, 1970. 36. Foreign Object Impact Damage to Composites, ASTM STP 568, 1975. 37. F.C. Moon, Wave Propagation and Impact in Composite Materials, Composite Materials, (Eds. L.J. Broutman and R.H. Krock), Vol. 7, Part I, Academic Press, NY, 1975, p. 260. 38. W.S. de Rosset, Fracture of Boron Epoxy Composite due to Impact Loading, J. Composite Materials, 9, 1975, p. 114. 39. J. Awerbuch and H. T. Hahn, Hard Objet Impact Damage of Metal Matrix Composites, J. Composite Materials, 10, 1976, p. 231. 40. C. T. Sun, Propagation of Shock Waves in Anisotropic Composite Plates, J. Composite Materials, 7, 1973, p. 366. 41. Y. Oved and G. E. Luttwak, Shock Wave Propagation in Layered Composites, J. Composites, J. Composite Materials, 12, 1978, p. 84. 42. S. H. Yang and C. T. Sun, Indentation Law for Composite Laminates, NASA CR-165460, 1981. 43. C.T. Sun and J.K. Chen, On the Impact of Initially Stressed Composite Laminates, J. Composite Materials, 19, 1985, p. 490. 44. D.S. Cairrons and P.A. Lagace, A Consistant Engineering Methodology for the Treatment of Impact in Composite Materials, J. Reinforced Plastics and Composites, 11, 1992, p. 395. 45. N. Y. Choi and F. K. Chang, A Model for Predicting Graphite /Epoxy Laminated Composites Resulting from Low-Velocity point Impact, J. Composite Materials, 62, 1992, p. 2134. 46. D.K. Maiti and P.K. Sinha, Impact Behaviour of Thick Laminated Composite Beams, J. Reinforced Plastics and Composites, 14, 1995, p. 255. 47. D.K. Maiti and P.K. Sinha, Bending, Free Vibration and Impact Response of Thick Laminated Composite Plates, Computers and Structures (Communicated). 48. D.K. Maiti and P.K. Sinha, Finite Element Impact Analysis of Doubly Curved Laminated Composite Shells, J. Reinforced Plastics and Composites (communicated). 10.5 EXERCISES 1. Describe the corrosion behaviour of some common metal matrix composites. 2. What is blistering and how it is formed ? 3. Which are the important degrading effects of temperature and moisture on polymer composites? 4. How do you determine the moisture diffusion coefficient d' 22 of a polymer composite ? 5. Describe briefly the effects of space environments on composite materials and structures. APPENDIX A TRANSFORMATION MATRICES A.1 Transformation of Coordinates A.2 Transformation of Displacements A.3 Transformation of Strains A.4 Transfomation Stresses A.5 Transformation due to Rotation of Axis A.6 Transformation of Two-Dimensional Case A.7 Transformation of Elastic Constants and Compliances A.1 Transformation of Coordinates The coordinates are vectors. Hence the rules associated with transformation of vectors can be used for transformation of coordinates. Figure A.1illustrates two mutually perpendicular coordinate system x 1 x 2 x 3 and x 1' x 2' x 3' (not necessarily Cartesian coordinates) oriented with respect to each other such that (A.1) where (A.2) is the rotation matrix and its elements represent the direction cosines of angles between axis systems x 1 x 2 x 3 and x 1' x 2' x 3' . These are listed below: (A.3) Fig. A.1 Note that rotational angles are considered positive when measured from the x 1 x 2 x 3 system to x 1' x 2' x 3' system. The rotational matrix [T r ] is orthogonal i.e., [T r ] -1 . Thus, when a transformation is sought from the x 1' x 2' x 3' coordinates to the x 1 x 2 x 3 coordinated, one can write (A.4) A.2 Transformation of Displacements Consider that u 1 u 2 and u 3 are displacement components with respect to the coordinate system x 1 x 2 x 3 and are those corresponding to the x 1' x 2' x 3' system. As displacements are also vectors, similar to coordinates, one can write (A.5) The rotation matrix [T r ] can also be used for coordinate transformation of other vectors such as rotational displacements, forces and moments. A.3 Transformation of Strains Here it is intended to relate strain components corresponding to the x 1' x 2' x 3' coordinates to strain components corresponding to the x 1 x 2 x 3 coordinate system. Now, Using chain rule of differentiation, (A.6a) with . In a similar way, (A.6b) (A.6c) Noting from Eq. A.2 that and substituting Eqs/ (A.6) in it, we obtain or Proceeding in a similar way, it can be shown that or, (A.7) where, (A.8) with Conversely, (A.9) A.4 Transfomation Stresses For stress transformation we relate stress components in the x 1 x 2 x 3 coordinates to stress components in the x 1' x 2' x 3' coordinates. Let be virtual strain components in two coordinate systems. The work done by stresses due to virtual displacements does not change when computed in two coordinate systems. Equating the work computed in two coordinate systems. i.e ., (A.10) Conbersely, (A.11) Note that (A.12a) It follows that (A.12b) (A.12c) To determine , we know from Eq. (A.8) that Then, (A.13) A.5 Transformation due to Rotation of Axis Consider the case of a simple inplane rotation about the x 3 axis (the axis x' 3 is assumed to coincide with the x 3 axis as shown in Fig. A.2. The rotation matrix [T r ] is then reduced to (A.14) with m = cos and n = sin Fig. A.2 The stress and strain transformation matrices and then take the following forms: (A.15) and (A.16) A.6 Transformation of Two-Dimensional Case If transformation is required from the two-dimensional x 1 x 2 coordinate system to the x 1' x 2' system only (Fig. A.3), the rotation matrix further simplifies to (A.17) with m = cos and n = sin Fig. A.3 Then (A.18) (A.19) A.7 Transformation of Elastic Constants and Compliances The stress-strain relations are expressed as in the x 1 x 2 x 3 coordinates (A.20) and in the x 1' x 2' x 3' coordinates (A.21) Here we express [C] in terms of [C']. From Eq. A.10, we have or [From Eq. A.21] [From Eq. A. 7] or with (A.22) Proceeding in a similar manner, but using stress strain relations in terms of compliances as follows (A.23) and , it can be shown that (A.24) APPENDIX B GENERAL BIBLIOGRAPHY GENERAL BIBLIOGRAPHY 1. K.H.G. Ashbee, Fundamental Principle of Fiber Reinforced Composites (2 nd Edition), Technomic Publishing AG, Switzerland, 1993. 2. N.K. Naik, Woven Fabric Composites, Technomic Publishing AG, Switzerland, 1993. 3. G.S. Springer and S.R. Finn, Composite Plates Impact Damage: An Atlas, Technomic Publishing Co., Lancaster, 1991. 4. R.A. Kline, Nondestructive Characterization of Composite Media, Technomic Publishing Co., Lancaster, 1992. 5. A. Brent Strong, High Performance and Engineering Thermoplastic Composites, Technomic Publishing Co., Lancaster, 1993. 6. S.M. Lee, Dictionary of Composite Materials Technology, Technomic Publishing Co., Lancaster, 1989. 7. G. Cederbaum, B. Gurion, I. Elishakoff, J. Aboudi and L. Librescu, Random Vibration and Reliability of Composite Structures, Technomic Publishing Co., Lancaster, 1992. 8. A.M. Skudra, Structural Analysis of Composite Beam Systems, Technomic Publishing Co., Lancaster, 1991. 9. P. Zinoviev and Y.N. Ermakov, Energy Dissipation in Composite Materials, Technomic Publishing AG, Switzerland, 1994. 10. S.V. Hoa, Analysis for Design of Fibre Reinforced Plastic Vessels and Piping, Technomic Publishing Co., Lancaster, 1991. 11. P.W.R. Beaumont, R.L. Crane and J.T. Ryder, Fracture and Damage Mechanics of Composite Materials, Technomic Publishing Co., Lancaster, 1992. 12. S.C. Tan, Stress Concentrations in Laminated Composites, Technomic Publishing Co., Lancaster, 1994. 13. L. Hollaway (Ed.), Handbook of Polymer Composites for Engineers, Woodhead Publishing Ltd., Cambridge, 1994. 14. G.C. Eckold, Design and Manufacture of Composite Structures, Woodhead Publishing Ltd., Cambridge, 1994. 15. J. Maxwell, Plastics in the Automotive Industry, Woodhead Publishing Ltd., Cambridge, 1994. 16. A. Miravete, Optimisation of Composite Structures Design, Woodhead Publishing Ltd., Cambridge, 1995. 17. G. Cuff, Fibre Reinforced Industrial Thermoplastic Composites, Woodhead Publishing Ltd., Cambridge, 1995. 18. B. Harris, Engineering Composite Materials, Broodfield Publishing, Brookfield, 1986. 19. R. Talreja, Fatigue of Composite Materials, Technomic Publishing Co., Lancaster, 1986. 20. D.H. Kaelble, Computer-Aided Design of Polymers and Composites, Dekker, NY, 1985. 21. T.L. Richardson, Composites: A Design Guide, Industrial Pres, NY, 1987. 22. R. M. Hussein, Composite Panels/Plates: Analysis and Design, Technomic Publishing Co., Lancaster, 1986. 23. P.K. Mallick, Fiber-Reinforced Composite Materials, Manufacturing and Design, Dekker, NY, 1987. 24. J.B. Donnett and R.C. Bansal, Carbon Fibres, Dekker, NY, 1984. 25. A. Watt and B.V. Perov (Eds.), Strong Fibres, Elsevier, NY, 1985. APPENDIX C PUBLICATIONS & PRESENTATIONS Journals 1. Sinha, P.K. and Ray, D.P., On the Flexural Behaviour of Orthotropic Sandwich Plates, Building Science, 8, 127-136, 1973. 2. Sinha, P.K. and Rath, A.K., Frequencies of Free Vibration of Axially Compressed Orthotropic Sandwich Plates, Journal of Sound and Vibration, 33(4), 541-547, 1974. 3. Rath, A.K. and Sinha, P.K., Evaluation of Stiffness Coefficients for Fibre- Reinforced Laminated Composites, Fibre Science and Technology, 7(3), 185- 198, 1974. 4. Rao, Y.V.K.S. and Sinha, P.K., Vibration of Sandwich Plates under Axial Compression, AAIA J., 12, 1282-1284, 1974. 5. Reddy, M.N. and Sinha, P.K., Stresses in Adhesive Bonded Joints, Fibre Science and Technology, 8(1), 33-48, 1975. 6. Kulkarni, A.M., Banerjee, J.R. and Sinha, P.K., Response of Randomly Exicted Orthotropic Sandwich Plates, Journal of Sound and Vibration, 41(2), 197-205, 1975. 7. Sinha, P.K. and Rath, A.K., Vibration and Buckling of Cross-ply Laminated Circular Cylindrical Panels, The Aeronautical Quarterly, 211-218, 1975. 8. Sarkar, K and Sinha, P.K., Stresses in Diametrically Compressed Composite Circular Disks, Transactions of Japan Society of Composite Materials, 1(1), 17-20, 1975. 9. Sinha, P.K. and Reddy, M.N., Thermal Analysis of Composite Bonded Joints, Fibre Science and Technology, 9, 153-159, 1976. 10. Reddy, M.N. and Sinha, P.K., Free Vibration of Laminated Circular Cylindrical Panels, Journal of Structural Engineering, 4, 57-62, 1976. 11. Sinha, P.K. and Rath A.K., Transverse Bending of Cross-ply Laminated Circular Cylindrical Plates, Journal of Mechanical Engineering Science, 18(2), 53-56, 1976. 12. Goyal, S.K. and Sinha, P.K., A Note on the Free Vibration of Sandwich Beams with Central Masses, Journal of Sound and Vibration, 49(3), 437-441, 1976. 13. Bhat, B.R. and Sinha, P.K., Forced Vibration of Simply Supported Orthotropic Sandwich Plates, Journal of Acoustic Society of America, 61(2), 428-435, 1977. 14. Goyal, S.K. and Sinha, P.K., Transverse Vibration of Sandwich Plates with Concentrated Mass, Spring and Dashpot, Journal of Sound and Vibration, 51(4), 570-573, 1977. 15. Sinha, P.K., Acoustic Emission and Stress Wave Propagation in Composites, ISVR report, University of Southampton, 1977. 16. Sarkar, K. and Sinha, P.K., Shear Stresses in an Adhesive Bonded Lap Joint, Journal of Structural Engineering, 5(2), 91-94, 1977. 17. Sinha, P.K., A Review of Moisture Diffusion in Composites, Structural Inelasticity Report AEM-H1-19, Department of Aerospace Engineering and Mechanics, University of Minnesota, 1978. 18. Sinha, P.K., Failure of Composites, Structural Inelasticity Report AEM-H1- 20, Department of Aerospace Engineering and Mechanics, University of Minnesota, 1978. 19. Sinha, P.K. and Rao, B.N., Report on the Design Methodology Sub- Committee of Fracture Mechanics Task Team, Part I,1980. 20. Sinha, P.K. and Rao, B.N., Tables for Fracture Strength Curves, Part II, 1980. 21. Sai Ram, K.S. and Sinha, P.K., Hygrothermal Effects on the Bending Characteristics of Laminated Composite Plates, Computers & Structures, 40 (4), 1009-1015, 1991. 22. Ray, B.C., Biswas, A. and Sinha, P.K., Hygrothermal Effects on the Behaviour of Fibre-reinforced Polymeric Composites, Journal of Metals, Materials and Processes, 3(2), 99-108, 1991. 23. Sai Ram, K.S. and Sinha, P.K., Hygrothermal Effects on the Free Vibration of Laminated Composite Plates, Journal of Sound and Vibration, 158(1), 133- 148, 1992. 24. Sai Ram, K.S. and Sinha, P.K., Hygrothermal Effects on the Buckling of Laminated Composite Plates, Composite Structures, 21(4), 233-247, 1992. 25. Sai Ram, K.S. and Sinha, P.K., Hygrothermal Bending of Laminated Composite Plates with a cutout, Computers & Structures, 43 (6), 1105-1115, 1992. 26. Sai Ram, K.S. and Sinha, P.K., Hygrothermal Effects on Vibration and Bukling of Laminated Plates with a cutout, AIAA Journal, 30 (9), 2353-2355, 1992. 27. Dey, A., Bandyopadhyay, J.N. and Sinha, P.K., Finite Element Analysis of Laminated Composite Conoidal Shell Structures, Computers & Structures, 43(3), 469-476, 1992. 28. Dey, A., Bandyopadhyay, J.N. and Sinha, P.K., Finite Element Analysis of Laminated Composite Paraboloid of Revolution Shells, Computers & Structures, 44(3), 675-682, 1992. 29. Ray, B.C., Biswas, A. and Sinha, P.K., Freezing and Thermal Spikes Effects on ILSS Values of Hygrothermally Conditioned Fibre/epoxy Composites, Journal of Materials Science Letters, 11, 508-509, 1992. 30. Chattopadhyay, B., Sinha, P.K. and Mukhopadhyay, M., Finite Element Free Vibration Analysis of Eccentrically Stiffened Composite Plates, Journal Reinforced Plastics and Composites, 1003-1034, 11, 1992. 31. Aditya, P.K. and Sinha, P.K., Diffusion Coefficients of Ploymeric Composites Subjected to Periodic Hygrothermal Exposures, Journal of Reinforced Plastics and Composites, 11, 1035-1047, 1992. 32. Paul, T.K. and Sinha, P.K., Design of Hat-Stiffened Composite Panels Loaded in Axial Compression, Composite Structures, 21(4), 205-209, 1992. 33. Chattopadhyay, B., Sinha, P.K. and Mukhopadhyay, Finite Element Analysis of Blade-Stiffened Composite Plates under Transverse Loads, Journal of Reinforced Plastics and Composites, 12(1), 76-100, 1993. 34. Aditya, P.K. and Sinha, P.K., Effects of Fibre Shapes on Moisture Diffusion Coefficients, Journal of Reinforced Plastics and Composites, 12(9), 973-986, 1993. 35. Aditya, P.K. and Sinha, P.K., Effects of Fibre Permeability on Moisture Diffusion Coefficients of Polymeric Composites, Polymers and Polymer Composites, 1(5), 341-348, 1993. 36. Mukherjee, N. and Sinha, P.K., A Finite Element Analysis of In-plane Thermo-Structural Behaviour of Composite Plates, Journal of Reinforced Plastics and Composites, 12(10), 1026-1042, 1993. 37. Mukherjee, N. and Sinha, P.K., Finite Element Analysis of Thermo-Structural Bending Behaviour of Composite Plates, Journal of Reinforced Plastics and Composites, 12(11), 1221-1238, 1993. 38. Mukherjee, N. and Sinha, P.K., A Comparative Finite Element Heat Conductor Analysis of Laminated Composite Plates, Computers & Structures, 52(3), 505-510, 1994. 39. Aditya, P.K. and Sinha, P.K., Effects of Arbitrary Shaped Fibres on Moisture Diffusion Coefficients of Polymeric Composites, Journal of Reinforced Plastics and Composites, 13(2), 142-154, 1994. 40. Mukherjee, N. and Sinha, P.K., 3D Thermostructural Response of Thick Laminated Composites: A Finite Element Approach, Journal of Reinforced Plastics and Composites, 13(11), 976-997, 1994. 41. Mukherjee, N. and Sinha, P.K., Three Dimensional Thermostructural Analysis of Multidirectional Fibrous Composite Plates, Composite Structures, 28(3), 333-346, 1994. 42. Dey, A., Bandyopadhyay, J.N. and Sinha, P.K., Behaviour of Paraboloid of Revolution Shell using Cross-ply and Antisymmetric Angle-ply Laminates, Computers & Structures, 52(6), 1301-1308, 1994. 43. Dey, A., Bandyopadhyay, J.N. and Sinha, P.K., Finite Element Analysis of Laminated Composite Hyperbolic Paraboloid Shell Structures, Computers & Structures, (Accepted) 44. Chowdary, T.V.R., Parthan, S. and Sinha, P.K., Finite Element Flutter Analysis of Laminated Composite Panels, Computers & Structures, 53(2), 245-251, 1994. 45. Maiti, D.K. and Sinha, P.K., Bending and Free Vibration of Shear Deformable Laminated Composite Beams by Finite Element Method, Composites Structures, 29(4), 421-431, 1994. 46. Maiti, D.K. and Sinha, P.K., Impact Behaviour of Thick Laminated Composite Beams, Journal of Reinforced Plastics and Composites, 14(3), 255- 279, 1994. 47. Chakravorty, D., Bandyopadhyay, J.N. and Sinha, Finite Element Free Vibration Analysis of Point Supported Laminated Composite Cylindrical Shells, Journal of Sound and Vibration, 181, 43-52, 1995. 48. Chakravorty, D., Bandyopadhyay, J.N. and Sinha, Free Vibration Analysis of Point Supported Laminated Composite Doubly Curved Shells: A Finite Element Approach, Computers & Structures, 54(2), 191-198, 1995. 49. Chattopadhyay, B., Sinha, P.K. and Mukhopadhyay, Geometrically Nonlinear Analysis of Composite Stiffened Plates with Finite Elements, Composite Strucutres, 31(1), 107-118,1995. 50. Chakravorty, D., Bandyopadhyay, J.N. and Sinha, Finite Element Free Vibration Analysis of Conoidal Shells, Computers & Structures, 56(6), 975- 978, 1995. 51. Chowdary, T.V.R., Sinha, P.K. and Parthan, S., Finite Element Flutter Analysis of Composite Skew Panels, Computers & Structures, 58(3), 613-620, 1996. 52. Chakravorty, D., Bandyopadhyay, J.N. and Sinha, Finite Element Free Vibration Analysis of Doubly Curved Laminated Composite Shells, Journal of Sound and Vibration, 191, 491-504, 1996. 53. Mukherjee, N. and Sinha, P.K., Thermoelastic Excitation of Multidirectional Fibrous Composite Cylinders, Journal of Sound and Vibration, 192(4), 807- 820, 1996. 54. Chakravorty, D., Bandyopadhyay, J.N. and Sinha, Free Vibration Analysis of Laminated Composite Hyper Shells Bounded by Straight Lines, Computers & Structures, (Accepted). 55. Aditya, P.K. and Sinha, P.K., Moisture Diffusion in Variously Shaped Fibre Reinforced Composites, Computers & Structures, 59(1), 157-166, 1996. 56. Maiti, D.K. and Sinha, P.K., Finite Element Impact Analysis of Doubly Curved Laminated Composite Shells, Journal of Reinforced Plastics and Composites, 15(3), 322-342, 1996. 57. Maiti, D.K. and Sinha, P.K., Bending, Free Vibration and Impact Response of Thick Laminated Composite Plates, Computers & Structures, 59(1), 115-129, 1996. 58. Maiti, D.K. and Sinha, P.K., Impact Response of Doubly Curved Laminated Composite Shells using Higher Order Shear Deformation Theories, Journal of Reinforced Plastics and Composites, 15(6), 575-601, 1996. 59. Mukherjee, N. and Sinha, P.K., Thermal Shocks in Composite Plates: A Coupled Thermoelastic Finite Element Analysis, Composite Structures, 34, 1- 12, 1996. 60. Chowdary, T.V.R., Sinha, P.K. and Parthan, S., Environmental Effects on Flutter Characteristics of Laminated Composite Rectangular and Skew Panels, Journal of Shock and Vibration, 3, 361-372, 1996. 61. Karmakar, A. and Sinha, P.K., Free Vibration Analysis of Composite Pretwisted Cantilever Plates, Journal of Aero. Soc. of India, 48(1), 1-9,1996. 62. Maiti, D.K. and Sinha, P.K., Low Velocity Impact Analysis of Composite Sandwich Shells using Higher Order Shear Deformation Theories, SADHANA, Academy Proceedings in Engineering Sciences of the Indian Academy of Sciences - Special Issue in Computational Structural Mechanics, 21(5), 597- 622, 1996. 63. Maiti, D.K. and Sinha, P.K., Finite Element Impact Response Analysis of Doubly Curved Composite Sandwich Shells Part-I: Theoretical Formulation, International Journal of Crashworthiness, 1(2), 191-202, 1996. 64. Maiti, D.K. and Sinha, P.K., Finite Element Impact Response Analysis of Doubly Curved Composite Sandwich Shells Part-II: Numerical Results, International Journal of Crashworthiness, 1(3), 233-249, 1996. 65. Sinha, P.K., Environmental Effects on Polymeric Composites, Popular Plastics and Packaging, XL1, 6, 53-58, 1996. 66. Mukherjee, N. and Sinha, P.K., Thermostructural Analysis of Rotationally Symmetric Multidirectional Fibrous Composite Structures, Computers & Structures, 65(6), 809-817, 1997. 67. Rao, D.M. and Sinha, P.K., Finite Element Coupled Thermostructural Analysis of Composite Beams, Computers & Structures, 63(6), 539-549, 1997. 68. Rao, D.M. and Sinha, P.K., Thermostructural Finite Element Analysis of Laminated Doubly Curved Composite Shells, Journal of Reinforced Plastics and Composites, 16(9), 848-868, 1997. 69. Mukherjee, N. and Sinha, P.K., Thermostructural Analysis of Rotationally Symmetric Multidirectional Fibrous Composite Structures, Computers & Structures, 65 (6), 809-817, 1997. 70. Karmakar, A. and Sinha, P.K., Free Vibration Analysis of Rotating Laminated Composite Pretwisted Cantilever Plates, Journal of Reinforced Plastics and Composites, 16, 1461-1491, 1997. 71. Sinha, P.K., Manufacture of Polymeric Composites, Popular Plastics and Packaging, XLII, 7, 69-74, 1997. 72. Karmakar, A. and Sinha, P.K., Finite Element Transient Dynamic Analysis of Laminated Composite Pretwisted Rotating Plates Subjected to Impact, International Journal of Crashworthiness, 3(4), 379-391, 1997. 73. Pal, N. C., Bhattacharya, S.K. and Sinha, P.K., FE Sloshing Analysis of Liquid Filled Containers, Journal of Institution of Engineers, AS, 78, 18-23, 1997. 74. Chakravorty, D., Sinha, P.K. and Bandyopadhyay, J.N., Application of FEM on Free and Forced Vibration of Laminated Shells with Cutouts, Journal of Engg. Mech. Div., ASCE, 124(1), 1-8, 1998. 75. Rao, D.M. and Sinha, P.K., Finite Element Thermostructural Analysis of Laminated Composite Shells of Revolution under Asymmetric Thermal Loading, Journal of Reinforced Plastics and Composites, 17(7), 580-605, 1998. 76. Bhattacharya, P., Suhail, H. and Sinha, P.K., Finite Element Free Vibration Analysis of Smart Laminated Composite Beams and Plates, Journal of Intelligent Material Systems and Structures, 9(1), 20-29, 1998. 77. Karmakar, A. and Sinha, P.K., Finite Element Transient Dynamic Analysis of Laminated Composite Pretwisted Rotating Plates Subjected to Impact, International Journal of Crashworthiness, 3(4), 379-391, 1998. 78. Niyogi, A.G., Laha, M.K. and Sinha, P.K., Finite Element Vibration Analysis of Laminated Composite Folded Plate Structures, Journal of Shock and Vibration, 6, 273-283, 1999. 79. Pal, N. C., Bhattacharya, S.K. and Sinha, P.K., Coupled Slosh Dynamics of Liquid Filled Composite Cylindrical Tanks, Journal of Engineering Mech. ASCE, 125(4), 491-495, 1999. 80. Pal, N. C., Bhattacharya, S.K. and Sinha, P.K., Finite Element Coupled Slosh Analysis of Rectangular Liquid Filled Composite Tanks, Journal of Reinforced Plastics and Composites, 18(15), 1375-1406, 1999. 81. Parhi, P. K., Bhattacharya, S. K. and Sinha, P. K., Dynamic Analysis of Multiple Delaminated Composite Twisted Plates, An International Journal -Aircraft Engineering and Aerospace Technology, 71, 5, 451-461, 1999. 82. Parhi, P. K., Bhattacharya, S. K. and Sinha, P. K., Failure Analysis of Multiple Delaminated Composite Plates due to Low Velocity Impact, International Journal of Crashworthiness, 5, 1,63-77,2000. 83. Neyogi, A. G., Laha, M.K. and Sinha, P.K., Finite Element Vibration Analysis of Laminated Composite Plate Structures, Shock and Vibration Journal, 6, 5/6, 273-253, 2000. 84. Karmakar, A. and Sinha, P.K., Impact Induced Dynamic Failure of Laminated Composite Pretwisted Rotating Plates, Aircraft Engineering and Aerospace Technology, 72, 2, 142-155, 2000. 85. Bhattacharya, I. P., Suhail, H. and Sinha, P.K., Analysis of Smart Laminated Shells and Deflection Control Strategy with Optimal Voltage, J. Reinforced Plastic and Composites, 19, 1293-1316, 2000. 86. Parhi, P. K., Bhattacharya, S. K. and Sinha, P. K., Finite Element Dynamic Analysis of Laminated Composite Plates with Multiple Delaminations, J. Reinforced Plastics and Composites, 19, 3, 863-882, 2000. 87. Niyogi, A. G., Laha, M.K. and Sinha, P.K., A Coupled FE-BE Analysis of Acoustic Cavities Confined inside Laminated Composite Enclosures, Aircraft Engineering and Aerospace Technology -An Int. J., 72, 345-357, 2000. 88. Pal, N. C., Sinha, P.K. and Bhattacharya, S.K., Finite Element Dynamic Analysis of Submerged Laminated Composite Plates, J. Reinforced Plastics and Composites, 20, 547-563. 2001. 89. Pal, N. C., Bhattacharya, S.K. and Sinha, P.K., Experimental Investigation on Slosh Dynamics of Liquid Filled Containers, J. Experimental Mechanics, 45, 63-69,2001. 90. Sinha, P. K., Failure Analysis of Multiple Delaminated Composite Plates due to Bending and Impact, Indian Academy of Sciences: Bulletin of Materials Sciences, 24, 143-149, 2001. 91. Parhi, P. K., Bhattacharya, S. K. and Sinha, P. K., Hygrothermal Effects on the Dynamic Behaviour of Multiple Delaminated Composite Plates and Shells, Journal of Sound and Vibration, 248(2), 195-214, 2001. 92. Pal, N. C., Sinha, P.K. and Bhattacharya, S.K., Finite Element Dynamic Analysis of Submerged Laminated Composite Plates, Journal of Reinforced Plastics and Composites, 20, 547 ?563, 2001. 93. Parhi, P. K., Sinha, P.K. and Bhattacharya, S.K., Dynamic Behaviour and Impact Induced First Ply Failure of Multiple Delaminated Composite Shells, Journal of Reinforced Plastics and Composites, 20, 15, 1276 ?1300, 2001. 94. Karmakar, A. and Sinha, P.K., Failure Analysis of Laminated Composite Pretwisted Rotating Plates, Journal of Reinforced Plastics and Composites, 20, 15, 1326-1357, 2001. 95. Kumari, Shyama and Sinha, P.K., Hygrothermal Analysis of Composite T -Joints, Aircraft Engineering and Aerospace Technology-An Int. J., 74, 1,23 ? 37, 2002. 96. Kumari, Shyama and Sinha, P.K., Finite Element Analysis of Wing T-Joints, Journal of Reinforced Plastics and Composites, 21(17), 1561-1585, 2002. 97. Kumari, Shyama and Sinha, P.K., Hygrothermal Bending of Moderately Thick Laminated Composite Shells, IE(I) Journal-AS, 83, 49-54, 2002. 98. Raja, S., Prathap, G. and Sinha, P.K., Active Vibration Control of Composite Sandwich Beams with Distributed Piezoelectric Extension -Bending and Shear Actuators, Smart Materials and Structures, 11, 1, 63-71, 2002. 99. Raja, S., Sinha, P.K., Prathap, G. and Bhattacharya, P., Influence of One and Two Dimensional Piezoelectric Actuation on Active Vibration Control of Smart Panels, J. Aerospace Science and Technology, 6, 3, 209-216, 2002. 100. Bhattacharya, P., Suhail, H. and Sinha, P.K., Finite Element Analysis and Analysis and Distributed Control of Laminated Composite Shells using LQR/IMSC Approach, J. Aerospace Science and Technology, 6, 273-281, 2002. 101. Biswal, K.C., Pal, N.C., Bhattacharya, S.K. and Sinha, P.K., Sloshing Response of Liquid Stored in a Tank with Baffles in Finite Element Analysis, Aerospace Engineering Division, I.E., 83, 6 ?9, 2002. 102. Ghosh, A. and Sinha, P.K., Stress and Displacement Behaviour of Damaged Laminated Composite Plate under Bending, Aerospace Engineering Division, I.E., 83, 55-63, 2002. 103. Niyogi, A. G., Laha, M.K. and Sinha, P.K., Coupled Dynamic Response of Composite and Sandwich Enclosures Containing on Acoustic Cavity, Advances in Vibration Engineering, 2(1), 86-96, 2003. 104. Raja, S., Sinha, P.K. and Prathap, G., Active Stiffening and Active Damping Effects on Closed Loop Vibration Control of Composite Beams and Plates, J. of Reinforced Plastics and Composites, 22, 1101-1121, 2003. 105. Pal, N.C., Bhattacharyya, S.K. and Sinha, P.K., Non-linear Coupled Slosh Dynamics of Liquid-filled Laminated Composite Containers: A Two Dimensional Finite Element Approach, J. Sound and Vibration, 261, 729-749, 2003. 106. Kumari, Shyma and Sinha, P.K., Effects of Transverse Stitching and Hygrothermal Environment on Composite Wing T-Joints, J. Reinforced Plastics and Composites, 22, 1705-1728, 2003. 107. Biswal, K.C., Bhattacharyya, S.K. and Sinha, P.K., Free-Vibration Analysis of Liquid-Filled Tank with Baffles, Journal of Sound and Vibration, 259(1), 177- 192, 2003. 108. Rao, V.V.S. and Sinha, P.K., Three Dimensional Analysis of Multidirectional Composites Subjected to Low Velocity Impact, International Journal of Crashworthiness, 8(4), 393-400, 2003. 109. Dutta, S., Choudhary, R. N. P. and Sinha, P.K., Ferroelectric Phase Transition in Bi-doped PLZT Ceramics, Mat. Sci. and Engg. B 98, 74, 2003. 110. Dutta, S., Choudhary, R. N. P. and Sinha, P.K., Synthesis and Characterisation of Fe3+ Modified PLZT Ferroelectrics, J. Mat. Sc., Materials in Electronics, 14, 463, 2003. 111. Raja, S., Dwarakanathan, D., Sinha, P.K. and Prathap, G., Bending Behaviour of Piezo-Hygrothermo-Elastic Smart Laminated Composite Flat and Curved Plates with Active Control, J. of Reinforced Plastics and Composites, 23, 265- 290, 2004. 112. Raja, S., Sinha, P.K. and Prathap, G., Thermally Induced Vibration Control of Composite Plates and Shells with Piezoelectric Active Damping, Smart Materials and Structures, 3, 939-950, 2004. 113. Raja, S., Sinha, P.K., Prathap, G. and Dwarakanathan, D., Influence of Active Stiffening on Dynamic Behaviour of Pizeo-Hygrothermo-Elastic Composite Plates and Shells, Journal of Sound and Vibration, 278, 257-283, 2004. 114. Ghosh, A. and Sinha, P.K., Dynamic and Impact Response of Damaged Laminated Composite Plates, Aircraft Engineering and Aerospace Technology- An International Journal, 76, 29-37, 2004. 115. Rao, V. V. S. and Sinha, P.K., Dynamic Response of Multidirectional Composites in Hygrothermal Environments, Composite Structures, 54, 329- 338, 2004. 116. Biswal, K. C., Bhattacharyya, S.K. and Sinha, P.K., Dynamic Response Analysis of a Liquid-filled Cylindrical Tank with Annular Baffle, J. Sound and Vibration, 274,13-37, 2004. 117. Rao, V. V. S., Krishna Veni, K. and Sinha, P.K., Behaviour of Composite Wing T -joints in Hygrothermal Environments, Aircraft Engineering and Aerospace Technology -An International Journal, 76, 404-413, 2004. 118. Rao, V. V. S. and Sinha, P.K., Bending Characteristics of Thick Multidirectional Composite Plates under Hygrothermal Environment, J. Reinforced Plastics and Composites, 23, 14, 1481-1495, 2004. 119. Hossain, S. J., Sinha, P.K. and Sheikh, A.H., A Finite Element Formulation for the Laminated Composites Shells, Computers & Structures, 82, 1623-1638, 2004. 120. Trivedi, S. and Sinha, P.K., Failure Analysis of Arbitrary Shaped Human Skull due to Impact, International Journal of Crashworthiness, 9, 4, 381-388, 2004. 121. Dutta, S., Choudhary, R. N. P. and Sinha, P.K., Ferroelectric Phase Transition in Sol-gel Derived Bi-doped PLZT Ceramics, J. Mats. Sc., 39 (9), 3129, 2004. 122. Dutta, S., Choudhary, R. N. P., Sinha, P.K. and Thakur, A.K., Microstructural Studies of (PbLa)(ZrTi)O3 Ceramics using Complex Impedance Spectroscopy, J. Appl. Phys., 96 (3), 1607, 2004. 123. Dutta, S., Choudhary, R. N. P. and Sinha, P.K., Structural, Electrical and Electromechanical Sensing Properties of Bi- modified PLZT Ceramics, J. Mat. Sc., Materials in Electronics, 15, 685, 2004. 124. Dutta, S., Choudhary, R. N. P., Sinha, P.K. and Thakur, A.K., Investigation of Electrical, Dielectric and Electromechanical Properties of Lanthanum Modified Lead Zirconate Titanate using Impedance Spectroscopy Technique, Ferroelectrics, 306, 55, 2004. 125. Dutta, S., Choudhary, R. N. P. and Sinha, P.K., Structural, Dielectric and Electrical Properties of Al +3 -modified PLZT Ceramics, Mat. Letts., 58, 2735, 2004. 126. Dutta, S., Choudhary, R. N. P. and Sinha, P.K., Studies on Structural, Electrical and Electromechanical properties of Sb +3 -modified PLZT, Mat. Sc. and Engg., B 113, 215, 2004. 127. Dutta, S., Choudhary, R. N. P. and Sinha, P.K., Impedance Sectroscopy Sudies on Ga-ion Modified PLZT Cramics, Physica Status Solidi., 202(6), 172, 2005. 128. Latifa, SK. and Sinha, P.K., Improved Finite Element Analysis of Multilayered, Doubly Curved Composites Shells, J. Reinforced Plastic and Composites, 24, 4, 385-404, 2005. 129. Naidu, N.V.S. and Sinha, P.K., ?Nonlinear Finite Element Analysis of Laminated Composite Shells in Hygrothermal Environments?, Composite Structures, 69 (4): 387-395, 2005. 130. Naidu, N.V.S. and Sinha, P.K., ?Nonlinear Transient Analysis of Laminated Composite Shells in Hygrothermal Environments?, Composite Structures, (In Press). 131. Naidu, N.V.S. and Sinha, P.K., ?Nonlinear Impact behaviour of Laminated Composite Shells in Hygrothermal Environments?, International Journal of Crashworthiness, Vol-10 (4), 389-402, 2005. 132. Naidu, N.V.S. and Sinha, P.K., ?Nonlinear Free Vibration Analysis of Laminated Composite Shells in Hygrothermal Environments?, Composite Structures, (In Press). 133. Sinha, P. K. and Kundu, C. K., ?Nonlinear Finite Element Analysis of Composite Shells, Int. J. of Mechanical Sciences, (Communicated). 134. Sinha, P. K. and Kundu, C. K., ?Nonlinear Transient Analysis of Laminated Composite Shells?, J. of Reinforced Plastics and Composites, (Communicated). International/National Conferences 1. Sinha, P.K. and Rath, A.K., Dynamic Behaviour of Initially Compressed Multilayer Sandwich Plates, 18 th Congress of Indian Society of Theoretical and Applied Mechanics (ISTAM), December 1973, IIT, Madras. 2. Rath, A.K., Avasthi, S.C., Rao, Y.V.K.S., Reddy, M.N. and Sinha, P.K., Analysis of Sandwich Fin, 2 nd Symposium on Space Science and Technology, September 1973, Trivandrum. 3. Rath, A.K. and Sinha, P.K., Dynamic Analysis of Sandwich Rocket Fin, 19 th Congress of Indian Society of Theoretical and Applied Mechanics (ISTAM), December 1974, IIT, Kharagpur. 4. Rath, A.K. and Sinha, P.K., Free Vibration of Stressed Orthotropic Sandwich Plates, Symposium on Structural Mechanics, March 1975, BARC, Trombay. 5. Sinha, P.K. and Rao, Y.V.K.S., Thermoelastic Analysis of Cross-ply Laminated Circular Cylindrical Panels, Symposium on Structural Mechanics, March 1975, BARC, Trombay. 6. Sinha, P.K. and Rath, A.K., Large Deflection Analysis of Heated Composite Circular Plates, Symposium on Structural Mechanics, March 1975, BARC, Trombay. 7. Sinha, P.K. and Rath, A.K., Analysis of Unsymmetrically Laminated Circular Cylindrical Composite Panels, 20 th Congress of Indian Society of Theoretical and Applied Mechanics (ISTAM), December 1975, BHU, Varanasi. 8. Sinha, P.K., Analysis of Composites and Structures, Structural Dynamics Group Meeting, December 1976, ISVR, University of Southampton. 9. Rao, B.N. and Sinha, P.K., Fracture Analysis of Two-Dimensional Crack Problems, Proceeding of the Symposium on Fracture Mechanics, ISTAM, 68- 79, February 1980, Rourkela. 10. Sinha, P.K, Evaluation of Composites in Space environments, ISRO-DFVLR Workshop on Composite Products Technology, VSSC, 1981. 11. Sinha, P.K., Paul, T.K. and Prasand, O.S., Design of Stiffened Composite Panels Loaded in Axial Compression, Proceedings of National Seminar on Aerospace Structures: Advances and Future Trends, November 1988, VSSC, Trivandrum. 12. Sinha, P.K. and Paul, T.K., Moisture Diffusion in Polymeric Sandwich Composites, 33 rd Congress of ISTAM, December 1988, BHU, Varanasi. 13. Ray, B.C, Biswas, A. and Sinha, P.K., Charaterisation of Hygrothermal Diffusion Parameters in Fibre-Reinforced Polymeric Composites, Proceedings of 4 th National Convention of Aerospace Engineers and All India Seminar on Aircraft Propulsion, January 1989, BIT, Ranchi. 14. Paul, T.K. and Sinha, P.K., Computer Code BUSTCOP, National Seminar on Structural Design and Analysis on Personal Computers 89, May 1989, Aeronautical Development Establishment, Bangalore. 15. Paul, T.K. and Sinha, P.K., Computer Code MGAIN, National Seminar on Structural Design and Analysis on Personal Computers 89, May 1989, Aeronautical Development Establishment, Bangalore. 16. Ray, B.C., Biswas, A. and Sinha, P.K., On the Evaluation of Interlaminar Shear-A Critical Design Parameter, Proceedings of Seminar on Science and Technology of Composites, Adhesives, and Sealants, September 1989, Bangalore. 17. Chattopadhyay, B., Sinha, P.K. and Mukhopadhyay, M., Finite Element Free Vibration Analysis of Composite Stiffened Plates, NASAS, 1990, NAL, Bangalore. 18. Ray, B.C., Biswas, A. and Sinha, P.K., Environmental Effects on Mechanical Behaviour of Glass Fibre-Epoxy Composites, Proceedings of 5 th National Convention of Aerospace Engineers and All India Seminar of New Materials in Aerospace, February 1990, Chandigarh. 19. Ray, B.C., Panda, A.K., Ganguly, R.I., Kumar, A. and Sinha, P.K., Effects of Absorbed Moisture on the Loading Rate Sensitivity of Glass-Fibre Composites, Proceedings of 5 th National Convention of Aerospace Engineers and All India Seminar of New Materials in Aerospace, February 1990, Chandigarh. 20. Ray, B.C, Biswas, A. and Sinha, P.K., Hygrothermal Effects on the Degradation of GRP Composites, National Symposium on Advances in Materials and New Materials, May 1990, ATC, Madras. 21. Ray, B.C., Sarangi, B., Sarangi, A., Biswas, A. and Sinha, P.K., Variation of Shear Values of Glass-Epoxy Composites in H 2 SO 4 and NaCl Solution: The SEM Investigation, National Symposium on Advances in Materials and New Materials, May 1990, ATC, Madras. 22. Sinha, P.K., Environmental Effects on the Behaviour of Polymeric Composites, Proceedings of Indo-US Workshop on Composites for Aerospace Application, July 1990, Bangalore. 23. Sinha, P.K. and Paul, T.K., Moisture Diffusion Through Polymeric Composite Laminated and Sandwich Plates, International Congress on Advances in Structural Testing, Analysis and Design, August 1990, Bangalore. 24. Paul, T.K. and Sinha, P.K., Buckling and Optimization of Stiffened Laminated Composite Panels Loaded in Axial Compression, 1 st International Conference on Vibration Problems of Mathematical Elasticity and Physics, October 1990, Jalpaiguri. 25. Ray, B.C., Biswas, A. and Sinha, P.K., Hygrothermal Effects on the Mechanical Behaviour of Fibre-Reinforced Polymeric Composites, ATM (IIM) Conference, November 1990, 26. Sinha, P.K., Investigations on Hygrothermal Effects in Polymer Composites (Invited Lecture), 3 rd Annual Meeting of Materials Research Society of India, February 1992, IISc, Bangalore. 27. Maiti, D.K. and Sinha, P.K., Impact Analysis of Composite Beams, Proceedings 8 th National Convention of Aerospace Engineers, March 1993, IIT, Kharagpur. 28. Aditya, P.K., Mukherjee, N. and Sinha, P.K., On the Hygrothermal Characteristics of Thick Laminated Composites, Proceedings 8 th National Convention of Aerospace Engineers, March 1993, IIT, Kharagpur. 29. Mukherjee, N. and Sinha, P.K., Performance of Advanced Composite Materials for High Temperature Engine Component Design Applications, Proceedings of Conference on Fibre Reinforced Plastics, Composites and their Applications, ISAMPE, August 1993, Bangalore. 30. Aditya, P.K. and Sinha, P.K., Dynamic Behaviour of Hygrothermally Conditioned Composite, Mechanically Fastened Joints, National Seminar on Aero Structures NASAS, December 1993, IIT, Kanpur. 31. Maiti, D.K. and Sinha, P.K, Finite Element Analysis of Thick Laminated Composite Beams, 38 th ISTAM Congress, December 1993, IIT, Kharagpur. 32. Chowdary, T.V.R., Sinha, P.K. and Parthan, S., Free Vibration and Flutter Analysis of Laminated Plates, 38 th ISTAM Congress, December 1993, IIT, Kharagpur. 33. Chakravorty, D., Bandyopadhyay, J.N. and Sinha, P.K., Finite Element Free Vibration Analysis of Conoidal Shells, 38 th ISTAM Congress, December 1993, IIT, Kharagpur. 34. Sinha, P.K. and Mukherjee, N., Thermoelastic Behaviour of Multidirectional Fibre Reinforced Composites in a Highly Heated Envelope, Symposium on Composites for High Temperature Applications-Recent Developments, February 1994, DRDL, Hyderabad. 35. Aditya, P.K. and Sinha, P.K., Dependence of Moisture Diffusion Coefficients on Fibre Shapes on Polymer Composites, 5 th AGM of MRSI, February 1994, DMRL, Hyderabad. 36. Maiti, D.K. and Sinha, P.K., Bending, Free Vibration and Impact Behaviour of Thick Laminated Composite Structures, Proceedings of National Symposium on Developments in Advanced Composites and Structures, September 1994, DRDL, Hyderabad. 37. Chowdary, T.V.R, Sinha, P.K. and Parthan, S., Finite Element Flutter Analysis of Laminated Composite Flat, Skew and Curved Panels, Proceedings of 46 th AGM of the Aeronautical Society of India, December 1994. 38. Chakravorty, D., Bandhopadhyay, J.N. and Sinha, P.K., Dynamic Analysis of Laminated Composite Shells, Computational Structural Mechanics (eds. Sinha, P.K. and Parthan, S.), Allied Publishers Ltd, New Delhi, 1994, 523-533. 39. Aditya, P.K., Mukherjee, N. and Sinha, P.K., Effects of Thermal Shock on Moisture Diffusion at the Fibre/Matrix Interface, Computational Structural Mechanics (eds. Sinha, P.K. and Parthan, S.), Allied Publishers Ltd, New Delhi, 1994, 534-546. 40. Maiti, D.K. and Sinha, P.K., Assessment of Displacement Based Shear Deformation Theories for Composite Plates, Computational Structural Mechanics (eds. Sinha, P.K. and Parthan, S.), Allied Publishers Ltd, New Delhi, 1994, 547-559. 41. Sinha, P.K., Thermo-Mechanical Response of Advanced Composites (Invited Lecture), Symposium on Advanced Plastics and Rubber Composites, MRSI AGM-6, February 1995, IIT, Kharagpur. 42. Aditya, P.K. and Sinha, P.K., Behaviour of Moisture Saturated Composite Mechanically Fastened Joints Subjected to Cyclic Axial Loading, 5 th NASAS, January 1996, IIT, Bombay, (Advances in Testing, Design and Development of Aerospace Structures, Allied Publishers Ltd., New Delhi, 103-113, 1996). 43. Bhattacharya, P., Suhail, H. and Sinha, P.K., Finite Element Free Vibration Response of Smart Laminated Composite Plates, Proceedings of 48 th AGM of the Aeronautical Society of India, Trivandrum, 503-517, 1997. 44. Rao, D.M. and Sinha, P.K., Finite Element Nonlinear Coupled Thermostructural Analysis of Composite Plates, Proceedings of 48 th AGM of the Aeronautical Society of India, Trivandrum, 541-553, 1997. 45. Pal, N.C, Bhattacharyya, S.K. and Sinha, P.K., Finite Element Sloshing Analysis of Liquid Filled Containers, Proceedings of AEROSPACE-97: XII National Conference of Aerospace Engineers, 111-119, 1997. 46. Rao, D.M. and Sinha, P.K, FE Thermostructural Response Analysis of Doubly Curved Laminated Shells, Proceedings of 2 nd Conference on Engineering Application of Solid Mechanics (CEASM ?97), December 1997, Kalpkkam. 47. Pal, N.C., Bhattacharyay, S.K. and Sinha, P.K., Dynamics of Fluid Structure Systems in the Nuclear Induct, Proceedings of 2 nd Conference on Engineering Application of Solid Mechanics (CEASM ?97), December 1997, Kalpkkam. 48. Karmakar, A. and Sinha, P.K, Dynamic Behaviour of Pretwisted Composite Plates ? A Finite Element Analysis, Proceedings of International Symposium on Vibrations of Continuous Systems, August 1998, Colorado, USA. 49. Karmakar, A. and Sinha, P.K., Finite Element Transient Dynamic Analysis of Laminated Composite Pretwisted Rotating Plates Subjected to Impact, IJCRASH Conference, September 1998, Michigan, USA. 50. Sinha, P.K., Modern Composites ? A Renascent Material System, Keynote Lecture, COMPEAT 1998, March 1998, NML, Jamshedpur. 51. Sinha, P.K., A Glide Through the Past, Present and Future of Aerospace Materials, Keynote Lecture, All India Seminar on Materials for 21 st Century, April 1998, REC, Rourkela. 52. Pal, N.C., Bhattacharyay, S.K. and Sinha, P.K., Tube Dynamic of Sloshing Phenomenon: Some Theoretical and Experimental Studies, Proceedings of ICTACEM 98, December 1998, IIT, Kharagpur. 53. Bhattacharya, P., Suhail, H. and Sinha, P.K., Deflection Control of Smart Laminated Plates: A Finite Element Analysis Approach, Proceedings of ICTACEM 98, December 1998, IIT, Kharagpur. 54. Sinha, P.K., Manufacture of Polymer Composites, Plastics Hand Book, Saket Publishers, Ahmedabad, 357-368, 1999. 55. Pal, N.C., Bhattacharyay, S.K. and Sinha, P.K., Numerical Modeling of Nonlinear Coupled Slosh Dynamic Problems, Proceeding of International Conference on Mathematical Modeling of Nonlinear Systems, ICOMMONS 99, 1, 276-290, December 1999, IIT, Kharagpur. 56. Parhi, P.K., Bhattacharyay, S.K. and Sinha, P.K., Failure Analysis of Multiple Delaminated Composite Plates due to Bending and Impact, Proceeding of International Conference of CFFGLACE-99, September 1999, IACS, Calcutta. 57. Parhi, P. K., Bhattacharya, S.K. and Sinha, P.K., Finite Element Impact Analysis of Multiple Delaminated Composite Plates, Proc. Int. Conf. SEC- 2000, 85-93, Jan. 2000. 58. Niyogi, A. G., Laha, M.K. and Sinha, P.K., Finite Element Analysis of Cantilever Laminated Composite and Sandwich Plate Structures, Proc. Int. Conf. SEC-2000, 77-84, Jan. 2000. 59. Kar, T., Kumari, Shyama and Sinha, P.K., Determination of Hygrothermal Parameters for Fibre Reinforced Composite Materials, Proc. 13th Nat. Conf. On Recent Advances in Experimental Mechanics, March 2000, IIT-Kanpur. 60. Bhattacharya, P., Suhail, H. and Sinha, P.K., Finite Element Analysis and Distributed Control of Laminated Composite Shells wing LQR/IMSC Approach, Symposium on Smart Materials and MEMS, SPIE, December 2000, Melbourne, Australia. 61. Raja, S., Prathap, G. and Sinha, P.K., Active Vibration Control of Composite Sandwich Beams with Distributed Piezoelastic Extension-Bending and Shear Actuators, Symposium on Smart Materials and MEMS, SPIE, December 2000, Melbourne, Australia. 62. Niyogi, A. G., Laha, M.K. and Sinha, P.K., Coupled Dynamic Response of Composite and Sandwich Enclosures Containing an Acoustic Cavity, Int. Conf. VETOMAC-I, October 2000, Bangalore, India. 63. Niyogi, A. G., Laha, M.K. and Sinha, P.K., A Finite Element-Boundary Element Coupled Structural Acoustic Formulation to Study the Effects of Structural Damping on Interior Acoustic Response, Int. Conf. On Recent Advances in Mathematical Sciences, December 2000, Kharagpur, India. 64. Kumari, Shyama and Sinha, P.K., Hygrothermal Bending of Moderately Thick Laminated Composite Shells, Paper No.071, ICTACEM 2001 Proceedings, lIT, Kharagpur, Dec 27-30, 2001. 65. Hossain, S. J., Sinha, P.K. and Sheikh, A.H., A New Element Based on Koiter's Shell Theory with Shear Deformation for Laminated Composite Shells, Paper No.052, ICTACEM 2001 Proceedings, lIT, Kharagpur, Dec 27 -30, 2001. 66. Ghosh, A. and Sinha, P.K., Stress and Displacement Behaviour of Damaged Laminated Composite Plate under Bending, Paper No.089, ICTACEM 2001 Proceedings, lIT, Kharagpur, Dec 27 -30, 2001. 67. Biswal, K.C., Pal, N.C., Bhattacharya, S.K. and Sinha, P.K., Sloshing Response of Liquid Stored in a Tank with Baffles: Finite Element Analysis, Paper no.090, ICTACEM 2001 Proceedings, IIT, Kharagpur, Dec 27 -30, 2001. 68. Dutta, S, Choudhary, R. N. P. and Sinha, P.K. Effect of Isovalent Double Doping on the Ferroelectric Phase Transition of PZT Materials, National Conference on Frontiers in Material Science and Technology (FMST 2002), Material Scinece Centre, IIT Kharagpur, February 2002. 69. Dutta, S, Choudhary, R. N. P. and Sinha, P.K. Synthesis and Characterization of Complex PZT Ceramics, International Conference on Inorganic Materials, Konstanz, Germany, September 2002. 70. Dutta, S, Choudhary, R. N. P. and Sinha, P.K., Structural, Dielectric and Electrical Properties of Sol-Gel Derived Bi-Doped PLZT Ceramics, The Nattional Academy of Sciences, India, Seventy Second Annual Session Shillong, October 2002. 71. Biswal, K.C., Bhattacharyya, S.K. and Sinha, P.K., Coupled Dynamic Response of Liquid Filled Composite Cylindrical Tanks with Baffles, Proc. 5th ASME Symposium onFSI, AE & FIV+N, November 2002. 72. Raja, S., Prathap, G. and Sinha, P.K., Modelling, Simulation and Validation for Active Vibration Control of Smart Sandwich Beam with Pizeoelectric Actuation, ISMMACS-2002, A Seminar on Mathematical Modelling and Simulation, C-MMACS, Bangalore, November 14-15, 2002. 73. Raja, S., Sinha, P.K., and Pratap, G., Active Vibration Control of a Laminated Composite Plate with PZT Actuators and Sensors -An Experimental Study, Proc. Int. Conference on Smart Materials, Structures and Systems (ISSS-SPIE 2002), Microart Multimedia Solutions, Bangalore, December 2002, 591-598. 74. Dutta, S, Choudhary, R. N. P. and Sinha, P.K., Synthesis and Characterization of Double Doped PZT Piezo-Ceramics. Proceedings of International Conference on Smart Materials, Structures and Systems, ISSS-SPIE, IISc, Bangalore, SM- 226,139, December 2002. 75. Dutta, S, Choudhary, R. N. P. and Sinha, P.K., Effect of Trivalent ion Substitution on the Structural and Dielectric Properties of PLZT Ceramics, Twelfth National Seminar on Ferroelectrics and Dielectrics, Material Research Centre, IISc, Bangalore, December 2002. 76. Ghosh, A. and Sinha, P.K., Initiation and Propagation of Failure in Damaged Laminated Composites due to Impact Loading, Proc. 8th Int. Conference on Plasticity and Impact Mechanics (IMPLAST 2003), Phoenix Publ. House Ltd., New Delhi, 2003, 696-703. 77. Rao, V. V.S. and Sinha, P .K., Impact Behaviour of Thick Multidirectional Composites, Proc. 8th Int. Conference on Plasticity and Impact Mechanics (IMPLAST 2003), Phoenix Publ. House Ltd., New Delhi, 2003, 704-710. 78. Ghosh, A. and Sinha, P.K., Initiation and Propagation of Damage in Laminated Composite Plate under Dynamic Loading, Proc. Int. Seminar on Indian Aviation-Challenges and Perspectives, Aero. Soc. of India, Paper No.21, Kolkata, December 2003. 79. Biswal, K.C., Bhattacharyya, S.K. and Sinha, P.K., Slosh Response of Liquid Filled Composite Container with Baffle, Proc. Int. Seminar on Indian Aviation ?Challenges and Perspectives, Aero. Soc. of India, Paper No.30, Kolkata, December 2003. 80. Rao, V. V. S. and Sinha, P.K., Evaluation of Critical Indentation for Laminated Composite Plates Subjected to Impact, Proc. Int. Seminar on Indian Aviation -Challenges and Perspectives, Aero. Soc. of India, Paper No. 31, Kolkata, December 2003. 81. Raja, S., Sinha, P.K., Prathap, G. and Dwarakanathan, D., Thermally Induced Vibration Control of Composite Plates and Shells with Piezoelectric Active Damping, Proc. ISMAPE:INCCOM-2 XII NASAS, Bangalore, September 2003, pp 191-205. 82. Kundu, C. K. and Sinha, P.K., A Nonlinear Membrane Element for the Analysis of Thin Inflated Structures, Proc. ISMAPE:INCCOM-2 & XII NASAS, Bangalore, September 2003, pp 290-297. 83. Latifa, S. K. and Sinha, P.K., An Improved Finite Element for the Analysis of Multilayered Composite Shell Dishes, Proc. ISMAPE:INCCOM-2 & XII NASAS, Bangalore, September 2003, pp. 298-306. 84. Hossain, S. J., Latifa, S. K. and Sinha, P.K., A Refined Finite Element Analysis of Laminated Piezothermoelastic Composite Shells, Proc. ISMAPE:INCCOM-2 & XII NASAS, Bangalore, September 2003, pp.307-317. 85. Tribedi, S., Sinha, P.K., Finite Element Analysis of the Human Skull and Composite Protection System, ISMAPE:INCCOM-2 & XII NASAS, Bangalore, September 2003. 86. Hossain, S. J. and Sinha, P.K., Finite Element Analysis of Laminated Composite Shells using Consistent Shear Correction Factors, Proceedings of SEC 2003, IIT Kharagpur, December 2003, 336-348. 87. Rao, V. V. S. and Sinha, P.K., Dynamic and Transient Analysis of Laminated Composites using Three-Dimensional Super Elements, SEC 2003, IIT Kharagpur, December 2003, 349-357. 88. Latifa SK. and Sinha, P.K., Free Vibration of Multilayered Composite Shell Dishes, Proceedings of SEC 2003, IIT Kharagpur, December 2003, 325-335. 89. Dutta, S., Choudhary, R. N. P. and Sinha, P.K., Synthesis and Characterization of Sol-Gel Derived Al-Modified PLZT Ferroelectric for Sensors and Actuators, 4 th Asian Conference on Ferroelectric and Dielectrics, Materials Research Centre, IISc, Bangalore, December 2003. 90. Dutta, S., Choudhary, R. N. P. and Sinha, P.K., Effect of Sb +3 -ion Incorporation on Structural, Dielectric and Electrical Properties of Sol-Gel Derived Smart Materials PLZT, 4 th Asian Conference on Ferroelectric and Dielectrics, Materials Research Centre, IISc, Bangalore, December 2003. 91. Tribedi, S. and Sinha, P.K., Impact Analysis of Human Skull and Femur Bone System Using ANSYS, ANSYS Users' Conference, Bangalore, December 2003. 92. Rao, V. V. S. Rao and Sinha, P.K., Impact Behaviour of Z-Pinned Laminated Composites in Hygrothermal Environments, ICASI 2004 & XIII NASAS, IISc, Bangalore. 93. Naidu, N. V. S. and Sinha, P.K., Nonlinear Finite Element Analysis of Laminated Composite Thick Shallow Shells under Hygrothermal Load, ICASI 2004 & XIII NASAS, IISc, Bangalore. 94. Hossain, S. J. and Sinha, P.K., A Geometrically Exact Shell Model for the Analysis of Flexible Hinges, ICASI 2004 & XIII NASAS, IISc, Bangalore. 95. Tribedi, S. and Sinha, P.K., Finite Element Failure Analysis of Human Skull Due to Impact, ICASI 2004 & XIII NASAS, IISc, Bangalore. 96. Tripathy, S. and Sinha, P.K., Analysis of Large Space Trusses subjected to Thermal Loadings in Space Environments, ETSMC-2003, Rourkela. 97. Naidu, N.V.S. and Sinha, P.K., Hygrothermal Effects On The Geometrically Nonlinear Behaviour of Laminated Composite Shallow Shells, NCDSMMS- 2004, JNTUC, Kakinada. 98. Naidu, N.V.S. and Sinha, P.K., Nonlinear Finite Element Analysis of Laminated Composite Thick Shallow Shells under Hygrothermal Load, ICASI 2004 & XIII NASAS, Bangalore, 2004. 99. Kundu, C. K. and Sinha, P.K., Non-linear Analysis of Thin Composite Shells, Proc. ICCMS-2004, pp 238-245, December 2004. 100. Hossain, S. J., Latifa SK. and Sinha, P. K., A doubly curved element for composite shells undergoing finite rotation, XXI International Congress of Theoretical and Applied Mechanics (ICTAM04), organized by International Union of Theoretical and Applied Mechanics (IUTAM), August, 2004, Warsaw, Poland, Europe. 101. Biswal, K. C., Bhattacharya, S.K. and Sinha, P.K., Finite Element Analysis of Liquid Filled Annular Cylindrical Tanks with Baffles, ICTACEM 2004, December 2004, Paper no-307, IIT, Kharagpur. 102. Kundu, C. K. and Sinha, P.K., Hygrothermal Effects on the Geometrically Nonlinear Bending of Laminated Composite Shells, ICTACEM 2004, December 2004, Paper no-281, IIT, Kharagpur. 103. Naidu, N. V. S and Sinha, P.K., Nonlinear Transient Analysis of Doubly Curved Laminated Composite Shells, ICTACEM 2004, December 2004,Paper no-220, IIT, Kharagpur. 104. Ghosh, A and Sinha, P.K., Initation and Propagation of Failure in Clamped and Simply supported Damaged Laminated Composites due to Impact, ICTACEM 2004, December 2004, Paper no-282, IIT, Kharagpur.