KI Evaluation by the Displacement Extrapolation Technique

Engineering Fracture Mechanics 66 (2000) 243±255www.elsevier.com/locate/engfracmech KI evaluation by the displacement extrapolation technique Gustavo V. Guinea*, Jaime Planas, Manuel Elices Departamento de Ciencia de Materiales, Universidad PoliteÂcnica de Madrid, E.T.S.I. Caminos Universitaria s/n, 28040-Madrid, Spain Received 8 October 1999; received in revised form 1 February 2000; accepted 1 February 2000 Abstract This paper shows the in¯uence of element size, element shape, and mesh arrangement on numerical values of KI obtained by the displacement method, and gives some guidelines to obtain KI values as good as the most accurate energy based estimations, typically within a few percent di€erence of the exact value. Three di€erent displacementbased extrapolation techniques are analyzed. The in¯uence of stress state is also shown. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Stress intensity factor; Finite elements; Displacement extrapolation techniques 1. Introduction The evaluation of stress intensity factors in 2D geometries by means of the ®nite element method is a technique widely used for non standard crack con®gurations. Basically, there are two groups of estimation methods, those based on ®eld extrapolation near the crack tip [1,2] and those which make use of the energy release when the crack propagates. This latter group includes the J-contour integration, the elemental crack extension, the sti€ness derivative method [3] and the recent energy domain integral formulation [4,5]. All require special post-processing routines, and it is often very dicult, if not impossible, to separate KI and KII components in mixed mode problems. Their main advantage is that relatively coarse meshes can give accurate estimations for the stress intensity factor. In contrast, methods based on near-tip ®eld ®tting procedures require ®ner meshes to produce a good numerical representation of crack-tip ®elds, the most accurate methods being those based on nodal displacements, which are a primary output of the ®nite element program. To represent adequately the singular strain ®eld near the crack tip, a modi®ed isoparametric element is introduced, as suggested by * Corresponding author. Tel.: +34-91-336-66-79; fax: +34-91-336-66-80. E-mail address: [email protected] (G.V. Guinea). 0013-7944/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 0 ) 0 0 0 1 6 - 3 With the advent of energy-based methods to compute stress intensity factors. and that the new energy-based techniques are more accurate and ecient for a given mesh geometry [1. the in¯uence of stress state and Poisson's coecient e€ect have been systematically ignored in the literature. / Engineering Fracture Mechanics 66 (2000) 243±255 Barsoum [6]. [10]. The 1a r linear-elastic singularity for stresses and strains is obtained by shifting a quarter to the crack tip the midside nodes of all surrounding elements. n the Poisson's ratio. quarter-point isoparametric elements are used as p suggested by Barsoum [6] and Hensell and Shaw [11] (Fig. 1). no conclusive arguments have been presented. Displacement extrapolation procedures The displacement extrapolation method is based on the nodal displacements around the crack tip. The paper also investigates the e€ect of stress state on the numerical analysis. and only old references with a limited analysis have been handled (as Ref. COSMOS1) recommend energy-based methods as the most ecient for KI computing. When some simple requirements Ð described in the paper Ð are met. displacement methods are still widely used because the continuous improvement of computers has reduced memory and computing time requirements. The normal displacement at crack tip. The paper closes in Section 5 with some comments and conclusions. In Section 4 the in¯uence of mesh and other computational parameters on the numerical results of KI are analyzed.8]. stress intensity factors are easily computed directly from nodal displacements. v…r ˆ 0†. Guinea et al. In addition. Although many reference books in fracture mechanics ([9] for example) and commercial ®nite element codes (ABAQUS1. element shape. without resort to specialized post-processing routines. .V. and they have lower precision. and r and y are the polar coordinates. the asymptotic expression for the displacement normal to the crack plane. typically within a few percent di€erence of the exact value. Nevertheless. For a bidimensional crack under mode I loading.7. To obtain a good representation of the crack-tip ®eld. This paper analyzes the in¯uence of element size. On the other hand. and mesh arrangement on the numerical values of KI obtained by the displacement method. de®ned in Fig.244 G. the precision of the displacement method is comparable to the most accurate energy-based estimation. which are not always implemented in the commercial programs. in the authors' opinion. for most commercial ®nite element codes. These are applied in Section 3 to two well known test geometries: the center cracked panel and the three-point bending beam. and further speci®c research is needed in this area. The paper is organized as follows: in Section 2 three di€erent methods to evaluate KI are reviewed. k an elastic parameter equal to 3±4n for plane strain and (3Àn)/(1+n) for plane stress. cited in the book by Kaninen and Popelar [9]). v. most authors claimed that displacement-based methods had become obsolete. is given by [9]: r& ' 1 ‡ n 2r y 3y A 1 …1 ‡ n †r …k À 3 † sin y …2k ‡ 1 † sin À sin ‡ v ˆ KI E 4E p 2 2 & ' A2 …1 ‡ n †r3a2 …2k À 1 † 3y y À sin sin ‡ FFF …1† ‡ E 2 2 3 where E is the modulus of elasticity. 1. is seen to be zero as prescribed by the symmetry of mode I. Stress ®eld extrapolation is not used because stresses are computed from the nodal displacement solution. 2. Ai are parameters depending on the geometry and load on the specimen. and shows that its in¯uence on the ®nite element results is far from negligible. (2) and (3) can be solved for KI and A2. r 3/2.V. [1].G. 1). Particularizing Eq. / Engineering Fracture Mechanics 66 (2000) 243±255 245 When the displacement v is evaluated along the crack faces …y ˆ 2p). making the extrapolation more accurate. (2):  r  r 2E 2p E H 2p …6† v ˆ vA KI ˆ …1 ‡ n †…k ‡ 1 † – A 2 – Finally. Guinea et al. r 5/2 etc. we have: r 2 …1 ‡ n †…k ‡ 1 † p A2 …1 ‡ n †…k ‡ 1 † 3a2 – ‡ O…–5a2 † –À …2† vA ˆ K I 4E 12E p r 2 …1 ‡ n †…k ‡ 1 † p 2A2 …1 ‡ n †…k ‡ 1 † 3a2 –À – ‡ O…–5a2 † vB ˆ KI 2E 3E p …3† where – is the length of the element side TB. The value of the stress intensity factor is then: r  E 2p …8vA À vB † …4† KI ˆ 3…1 ‡ n †…k ‡ 1 † – or EH KI ˆ 12 r  2p …8vA À vB † – …5† where E ' is the e€ective elastic modulus de®ned as equal to E for plane stress and Ea…1 À n 2 † for plane strain. Eq. (1) only contains terms in r 1/2. a di€erent estimation of KI can be produced if the term in r1a2 of the displacement expansion along the upper crack face is matched with the corresponding term of the element interpolation function Fig. Eqs. Quarter-point singular elements and coordinates for near crack-tip ®eld description. 1. as ®rst suggested by Chan et al. Ignoring higher order terms. A simpler estimation of KI can be made by means of the quarter node displacement. (1) for nodes A and B on the singular element at the upper face of the crack (see Fig. . vA . if terms in –3a2 and higher are neglected in Eq. Numerical analysis The three procedures for evaluating KI reviewed in Section 2 are now applied to two well known geometries: the center cracked plate and the three-point bending beam. only a quarter of the panel was modeled. Guinea et al.1. / Engineering Fracture Mechanics 66 (2000) 243±255 for v…r†X The displacement ®eld along the crack edge y ˆ p for a singular eight-node or six-node isoparametric element is a function of the nodal displacements vA and vB . The in¯uence of the number and size of the elements. The stress intensity factor for this problem was calculated by Isida [12] as equal to: Fig. of the stress state and Poisson's ratio are analyzed for each case. Center cracked plate geometry. a similar result would be obtained for the lower face element. In the following section we evaluate the performance of these three KI estimates. 3. Center cracked plate The center cracked plate geometry is shown in Fig.V. . 1): r r r …7† v…r† ˆ …4vA À vB † À …4vA À 2vB † – – p By setting y ˆ p in Eq. Due to symmetry.246 G. 2. (5). 3. 2. Given the symmetry. and is given by (see Fig. (1) and identifying terms with r in Eqs. (1) and (7) we obtain: r r 2 …1 ‡ n †…k ‡ 1 † p r r ˆ …4vA À vB † …8† KI ˆ 2E p – and the stress intensity factor is now: r  r  E 2p E H 2p …4vA À vB † ˆ …4vA À vB † KI ˆ …1 ‡ n †…k ‡ 1 † – – 4 …9† Eqs. (6) and (9) are three di€erent estimates of KI based on the nodal displacements of the quarterpoint element located on the upper face of the crack. 5. 4. 4.G. For the rest of the plate. four (eight) in B-2. as shown schematically in Fig. each di€ering in the number of elements surrounding the crack tip. two elements (four for the whole problem if we take the symmetry into account) were placed around the crack tip. Some of these meshes are shown in Fig. or in triangular elements Ð B-2 con®guration Ð for the Q-2 meshes. ending in quadrilaterals around the tip Ð A con®guration Ð for the Q-1 mesh type. some quadrilateral meshes were re®ned in the crack tip zone by continued subdivision of the crack tip elements into two. 6. Meshes with 3. Fig. six (12) in B-3 and eight (16) in con®guration B-4. 5. 7 and 9 elements around the crack tip and sizes from D/5 to D/20 were analyzed. / Engineering Fracture Mechanics 66 (2000) 243±255 247 p KI ˆ 1X334s pa …10† which is claimed to be accurate up to four signi®cant ®gures. Three-point bending beam The ordinary three-point bending beam with a span±depth ratio of four was the second geometry selected for the analysis. The beam dimensions are shown in Fig. 3. 3. only half of the beam was modeled. one formed by six-node isoparametric triangles (T) of average size H/10. Several mesh con®gurations around the crack tip were analyzed.V. and another made of 10  10 eight-node quadrilaterals (Q). two kinds of mesh were used. varying the size and the number of elements around the crack tip. Due to the symmetry. . 3. Two rectangular eight-node isoparametric elements were used for con®guration A. Guinea et al. In con®guration B-1. Both meshes are plotted in Fig. The stress intensity factor for this geometry was computed by Srawley up to four signi®cant ®gures as [13] KI ˆ 1X775 6F BD1a2 …11† Triangular meshes with six-node isoparametric elements were used for the numerical analysis. Crack-tip modeling for the center cracked panel.2. To study the e€ect of element size. as shown in Fig. whereas six-node elements were used for con®gurations B-1±B-4. 4. the two programs gave the same numerical outputs for all the signi®cant ®gures. 4.1. Meshes for the center cracked panel. 5.V. Comparing the results. The in¯uence of these parameters is speci®cally investigated in Section 4. Default values of E=1. Full gaussian numerical integration (three-points) was used for the triangular elements whereas reduced integration (2  2 points) was applied to the quadrilateral elements. To discard the possibility of spurious results not imputable to the discretization scheme. a parallel analysis with the ®nite element code ABAQUS1 [15] was carried out for a set of selected cases. Analysis and discussion All the cases were solved with the commercial ®nite element code ANSYS1 [14]. Guinea et al. / Engineering Fracture Mechanics 66 (2000) 243±255 Fig. 4. n ˆ 0 and a plane stress state were assumed in all the computations.248 G. In¯uence of the shape of the ®nite element The numerical results show that the shape of the elements itself has no noticeable in¯uence on the Fig. . Three-point bending geometry.4. 1 +1.3 +5.9 +0. The size – and number of the elements n around the crack tip are shown.1 KI ÀKI Isida KI Isida . (6) Eq. based only on the displacement of the quarter point. / Engineering Fracture Mechanics 66 (2000) 243±255 249 Fig. 4. Crack-tip con®gurations and mesh types are shown in Figs. (6). Some of the meshes used for the three-point bending beam geometry.V. Nevertheless.3 T/B-2 +1. The most accurate results are obtained in this problem with the simplest estimation method.5 +9. It appears that the accuracy of KI depends more on other mesh features such as the number of the elements surrounding the crack tip. It should be noted that the accuracy on KI is good for all the methods. 7 for the three-point bending Table 1 E€ect of element shape for the center cracked panel. as shown in the following section. well under 5% for almost all the con®gurations. Eq.1 +4. 6.3 +1. (5) Eq. 3 and 4 ea …7† K Mesh type/crack-tip con®guration Q/A Eq.2. Guinea et al.G. . eK …7† ˆ 100  element size/panel width = 1/20. (9) a T/B-1 +4. computed values of KI. In¯uence of the size of the ®nite elements The e€ect of the size of the ®nite elements on KI is illustrated in Fig.4 À0. this method shows a remarkable dependence on the size of the singular element. Table 1 shows the percent di€erence between the displacement-based estimates of KI and Isida's [12] solution for the center cracked panel for quadrilateral and triangular meshes of the same element size. such as Q-1. The method least sensitive to the element size is that based on Eq. (5).3. and the element size – relative to the beam depth D. Beyond this point a ®ner mesh may give worse results. 8. In this case. These ®gures agree with the data in Table 1. Guinea et al. the accuracy of practical mesh con®gurations is typically between 5 and 10%. Percent di€erence is de®ned as eK …7† ˆ 100  IKI Srawley X . (6) is of the ®rst order (r 1/2 term). 5. (5) to give more accurate results. with element sizes ranging from 1/20 to 1/5 of beam depth. as is indeed the case shown in Fig.2. 7.V. The largest di€erences in KI are produced by Eq. (5) gives surprisingly good results with very coarse meshes and simple angular discretizations. depicted in Fig. where the stress intensity factor is obtained forcing the r1a2 term of the interpolation function for the singular element to coincide with the theoretical asymptotic expansion. In¯uence of mesh discretization around the crack tip The main in¯uence on KI estimation is due to the angular discretization around the crack tip.250 G. Percent di€erence from Srawley's stress intensity factor for a half-notched three-point bending beam as a function of the number n of elements around the crack tip. The beam geometry is de®ned in K ÀKI Srawley Fig. / Engineering Fracture Mechanics 66 (2000) 243±255 beam geometry. The ®gure shows the results obtained with 75 triangular meshes as described in Section 3. It should be noted that a mesh re®nement with a poor angular discretization. As depicted in Figs. 7. (5) is of the second order (r 1/2 and r3/2 terms). gives results that apparently converge to an erroneous estimate of KI. one might expect Eq. (5) and (6) are based on the series expansion of displacements along the crack faces. This is the main conclusion one can draw from the analysis of the meshes for the center cracked panel. a poor discretization with two/three singular elements (four/six if the Fig. A local re®nement of the mesh in the crack tip zone can improve the estimation of KI up to a maximum for a certain element size. (9). 4. as stated in Section 2 in this paper. although the procedure based on Eq. while Eq. Since Eq. 7 and 8. KI estimations by Eqs. This agrees with the data shown in Fig. 7 for the three-point bending beam geometry. The three methods analyzed in this paper seem to converge properly to the value computed by Srawley [13]. Fig. 5-.4) gives deviations in KI within 0.1) yields a 0. where. This is particularly evident when singular quadrilaterals are used. 4. Percent di€erence is calculated ÀKI Isida as eK …7† ˆ 100  KIKI Isida X . and Irwin's equation G=K2/E '. Percent di€erence from Isida's stress intensity factor for a center cracked panel as a function of the tip-element size – relative to the panel width 2H. 7-. This behaviour of Eq. (5) gives a value very close to the convergence limit even with very rough meshes. Fig. Two mesh con®gurations. the use of Eq.V.2% from Srawley's reference value. From this ®gure it can be con®rmed that the stress intensity factors computed by the two methods Ð energy and displacement Ð are close to Srawley's estimation. Guinea et al. 9 shows the results. It is worth noting that whereas KI estimations by Eqs.4. where a discretization with n = 4 and –a2H ˆ 0X05 (element size/crack length = 0. and that both procedures converge adequately. in appearance. 7 where a coarse mesh with n = 5 and –aD ˆ 0X2 (element size/ crack length = 0. as in Q-1 con®guration for the center cracked panel in Fig. To ensure that the right convergence limit is approached when a proper number of singular elements are placed around the tip. and compares KI values with the output of Eqs. are considered. (5). 4. All meshes were made of triangular six-node elements. all the procedures converge. (6) or (9) converge slowly with mesh re®nement.3% di€erence from Isida's value. / Engineering Fracture Mechanics 66 (2000) 243±255 251 symmetry is considered) can lead to a false value of KI. 8). This is clearly shown in Fig. The formal mathematical equivalence between the stress ®eld in plane stress and plane strain situations is thought to consider the stress state a secondary variable. 8. G. In¯uence of stress state The stress state is usually neglected in ®nite element computations. (5) is also con®rmed for the center cracked panel (Fig. and this information is frequently neglected when reporting KI results. some computations of KI for the three-point bending beam were performed from Grith's energy release rate. G was computed from the elastic I strain energy for two adjacent crack lengths. The crack tip was shifted 0. 8.G. and 9-element con®gurations were tried.001% of the beam depth …DaaD ˆ 10 À5 † and 3-. (6) and (9) for a dense angular discretization (n = 9). Q-1 and Q-2 shown in Fig. since all standard ®nite element codes solve the elastic problem by ®nding ®rst the displacement ®eld. Poisson's ratio has no in¯uence on the stress ®eld in two-dimensional states in the absence of body load [16]. nX It is well known that from the strict point of view of the theory of elasticity. the displacement ®eld when a crack is present is not strictly represented by T6 or Q8 elements. and its e€ect is greater for a plane strain condition. even using the quarter point technique (to see this. (1) evaluated at y ˆ 2p Ð crack faces Ð and the displacement interpolation function for a singular element. and then computing the stress components for each element.252 G. although this requirement is easily ful®lled with three Gaussian integration points for T6 elements and a 2  2 Gaussian scheme for Q8 quadrilaterals. This phenomenon also applies. and its performance relies exclusively on the proper numerical representation of the elastic displacement ®eld.and displacement-based K ÀKI Srawley methods are compared. Eq. as shown in Fig. a spurious e€ect is introduced by Poisson's ratio if the displacement ®eld is not properly represented. However. Poisson's ratio in¯uences the computation of KI. the data in Fig. The crack-tip con®guration was the one labeled as A in Fig. both plane stress and plane strain computations diverge with increasing n. (7)). The beam geometry was shown in Fig. plane stress and plane strain states are not equivalent except in some very simple loading cases. Energy. to plane stress computations with di€erent values of Poisson's ratio. 9. Percent di€erence is de®ned as eK …7† ˆ 100  IKI Srawley X . the equivalence between plane stress and plane strain is strictly formal. with a more pronounced e€ect for plane strain conditions. With six-node isoparametric triangles (T6) and eight-node quadrilaterals (Q8). Fig. and only in these cases is the equivalence a strict one. As shown in the ®gure. 5. with two singular Q8 elements around the crack. Similar behaviour is observed for the bending beam. Guinea et al. As a result of the numerical approximation. / Engineering Fracture Mechanics 66 (2000) 243±255 Nevertheless. which is even more important. only linear and quadratic displacement ®elds are exactly reproduced. Besides. a numerically di€erent solution is produced for stresses in plane stress and plane strain states. 3. Unfortunately. 10 illustrates this e€ect on the center cracked panel meshed with Q8 elements of size H/10. 11 Fig. Percent di€erence from Srawley's stress intensity factor for a half-notched three-point bending beam as a function of the number of elements around the crack tip n. so. compare Eq. the numerical integration scheme must be accurate enough to integrate exactly the interpolation functions.V. 11. and the element size – relative to the beam depth D. Moreover. 4. 0. The mesh con®guration is given by the number n of elements around the crack tip. ÀKI Isida Percent di€erence is calculated as eK …7† ˆ 100  KIKI Isida X Fig. 3 and 4) with –a2HX ˆ 0X05X Values of n ˆ 0. Percent di€erence from Isida's stress intensity factor for a center cracked panel as a function of Poisson's ratio and stress state. 10. 0. Percent di€erence from Srawley's stress intensity factor for a half-notched three-point bending beam as a function of Poisson's ratio and stress state.V. Q/A mesh con®guration (Figs. Guinea et al. / Engineering Fracture Mechanics 66 (2000) 243±255 253 Fig. 6). and the element size K ÀKI Srawley – relative to the beam depth D (see Fig. 11.45. 0. 0.4999 are considered.49 and 0.G.25. Percent di€erence is de®ned as eK …7† ˆ 100  IKI Srawley X . Crack extension modeling with singular quadratic isoparametric elements. given by Eq. International Journal of Fracture 1974. Acknowledgements  The authors gratefully acknowledge ®nancial support by the Spanish Comision Interministerial de Ciencia y TecnologiÂa (CICYT) under grants MAT97-1022 and MAT97-1007-C02-02.g. Then the only general recommendation which can be given is to use a null Poisson's ratio …n ˆ 0† for all the computations. To circumvent this e€ect. with di€erences from the reference value of KI well under 1%. is recommended. Poisson's ratio In the paper it is also shown that the stress state and Poisson's ratio n can negatively in¯uence the results. brings no bene®ts to the accuracy of KI predictions. Besides. . Tuba IS. a null Poisson's coecient is recommended. e. German MD. A sti€ness derivative ®nite element technique for determination of crack tip stress intensity factors.V. International Journal of Fracture 1976. With this null value. and/or coarse angular discretization. and by the stress state situation. / Engineering Fracture Mechanics 66 (2000) 243±255 clearly show that the angular discretization plays a prominent role. (6) and (9) to estimate KI. de Lorenzi HG. Summary and conclusions The numerical work shown in this paper can provide some guidelines to improve KI estimation in bidimensional problems. Even worse. if a good angular discretization is made around the crack tip. 908 or 608 elements. it has been shown that the displacement extrapolation technique can give very accurate predictions. if the angular discretization is too rough. 3 Parks DM. 2 Shih CF. –.254 G. (5). (5). 308 each. and that a ®ner mesh around the crack tip prevents undesired e€ects. As regards the performance of Eqs. the stress state has no in¯uence on the results. Engineering Fracture Mechanics 1970. On the ®nite element method in linear fracture mechanics. large value of n. all of them are in¯uenced to a similar extent by n. a ®ne angular discretization helps to minimize the error. 5. References 1 Chan SK. even for coarse meshes. Mesh re®nement On the other hand. The method based on a two-term extrapolation of the displacement ®eld. Guinea et al.10:487±502. particularly in any of these circumstances: plane strain.2:1±17. Wilson WK. Angular discretization around crack tip First. a wrong KI limit can be reached when – tends to zero.12:647±51. a mesh re®nement attending only to the element size. A minimum angular discretization with six elements around the tip. brings the best results for large element sizes. Goodier JN. Sherman D.35:295±310. . Inc.12:475±6. Popelar CH.9:495±507. Numerical methods in fracture mechanics. Application of quadratic isoparametric ®nite elements in linear fracture mechanics. 6 Barsoum RS. 1985. Southpointe. Swanson Analysis Systems IP.50:653±70.. Engineering Fracture Mechanics 1995. Swansea: Pineridge. RI.1. Computational fracture mechanics: research and application. 1998. Sandhu JS. 11 Henshell RD. 16 Timoshenko SP. New York: Oxford University Press. Shih CF.7:301±16.21:129± 43. Engineering Fracture Mechanics 1985. / Engineering Fracture Mechanics 66 (2000) 243±255 255 4 de Lorenzi HG. 15 ABAQUS1. International Journal for Numerical Methods in Engineering 1975. E€ect of width and length on stress intensity factors of internally cracked plates under various boundary conditions. PA. International Journal of Fracture 1987.32:127±40.5. International Journal of Fracture 1971. USA. A general treatment of crack tip contour integrals. International Journal of Fracture 1976.. Lee JD. Comparison of methods for calculating stress intensity factors with quarter-point elements. p. International Journal of Fracture 1986. New York: McGraw-Hill. 8 Liebowitz H. Pawtucket. Inc. Release 5. 9 Kanninen MF. Guinea et al. Owen DRJ. Menandro FCM. 14 ANSYS1. Release 5. 1970. 12 Isida M. 7 Bank-Sills L. International Journal of Fracture 1974.7. USA. 1997. Theory of elasticity. 13 Srawley JE. Hibbit. Shaw KG. Energy release rate calculations by the ®nite element method. Crack tip ®nite elements are unnecessary. 10 Gallagher RH.V.G. editors. 5 Moran B. 1978. Karlsson and Sorensen. Wide range stress intensity factor expressions for ASTM E399 standard fracture toughness specimens.10:603±5. 1±25. A review of ®nite element techniques in fracture mechanics. In: Luxmore AR. Advanced fracture mechanics.