Availableonlineatwww.sciencedirect.com DIRECTO COMPOSITES SCIENCE AND TECHNOLOGY ELSEVIER Composites Science and Technology 66(2006)435-443 Effect of thermal misfit stress on crack deflection at planar interfaces in layered systems Woong Lee Jae-Min Myoung Yo-Han Yoo, Hyunho Shin Department of Materials Science and Engineering, Yonsei Unirersity, 134 Shinchonl-Dong, Seoul 120-749 First R&D Centre, Agency for Defence Development, P.O. Box 35.1, Yoosung, Taejeon 305-600, Repi Department of Ceramics Engineering, Kangnung National Unirersity, 120 Gangneung Daehangno, Kangnung, Kangwon-do 210-702, Republic of Korea Received 1 April 2005: received in revised form 13 July 2005: accepted 14 July 2005 Available online 26 august 2005 Abstract Deflection of a crack at a planar bi-material interface in a layered system was investigated by considering the effects of the in-plane residual thermal misfit stress. a new parameter based on strains due to mismatch of thermal expansion coefficients was introduced to describe residual stress state independent of length scale. From a numerical analysis, it was predicted that introducing ompressive residual stress in the stiffer intact layers of a composite laminate ahead of a growing primary crack would favour crack deflection by allowing advantageous energetic conditions, which indicates that stronger interfaces can be introduced in layered sys- tems this way to improve overall mechanical properties. It was also predicted that the residual stress effect is negligible if the intact layer is more compliant than the cracked layer supporting a previous analysis and discussion reported elsewher c 2005 Elsevier Ltd. All rights reserved Keywords: Crack deflection; Interface; Residual stress; Layered systems 1. Introduction primary crack terminating at a bi-material interface of layered composites, would also undergo In many materials systems, residual stresses are often the influence of the residual stresses. introduced during fabrication processes due to the mis- Crack deflection at bi-material interfaces in brittle match of coefficient of thermal expansion (CtE)be- matrix composite systems is important as it is a prere- tween constituent materials. The representative quisite to the operation of many toughening mecha examples include layered systems such as ceramic lami- nisms or it is itself practically the only toughening nates [1, 2], thin-film deposited device materials [3, 4] mechanism in some layered systems [7, 8]. It is often as and engineering structures with protective coatings sumed that deflection of a crack terminating at an inter [5, 6]. Presence of residual stress, superposed on the face occurs by a competition between crack penetration stress field due to applied load, will modify the stress dis- across the interface or deflection along the interface(see tributions within the systems thereby causing the Fig. 1). Then the energetic condition for crack deflection changes in the fracture behaviour of these materials. It would be satisfied when strain energy released due to is therefore expected that crack deflection criteria, deter- crack penetration across the interface, p, is lower than mined by the fracture process within the stress field of a the bulk fracture energy, Rm whilst that due to crack deflection along the interface, d, is higher than the Corresponding author. Tel :+82 2 2123 2843/119 813 8192; fax: Interface fracture energy, R i.e. when [9] +8223652680/23652680 E-lmail addresses: woong lee(@yonsei ac kr, cwll0001 Ri ga (x,B) -3538/S- see front matter 2005 Elsevier Ltd. All rights reserved. 016/j. compscitech. 2005.07.015
Effect of thermal misfit stress on crack deflection at planar interfaces in layered systems Woong Lee a,*, Jae-Min Myoung a , Yo-Han Yoo b , Hyunho Shin c a Department of Materials Science and Engineering, Yonsei University, 134 Shinchon-Dong, Seoul 120-749, Republic of Korea b First R&D Centre, Agency for Defence Development, P.O. Box 35-1, Yoosung, Taejeon 305-600, Republic of Korea c Department of Ceramics Engineering, Kangnung National University, 120 Gangneung Daehangno, Kangnung, Kangwon-do 210-702, Republic of Korea Received 1 April 2005; received in revised form 13 July 2005; accepted 14 July 2005 Available online 26 August 2005 Abstract Deflection of a crack at a planar bi-material interface in a layered system was investigated by considering the effects of the in-plane residual thermal misfit stress. A new parameter based on strains due to mismatch of thermal expansion coefficients was introduced to describe residual stress state independent of length scale. From a numerical analysis, it was predicted that introducing compressive residual stress in the stiffer intact layers of a composite laminate ahead of a growing primary crack would favour crack deflection by allowing advantageous energetic conditions, which indicates that stronger interfaces can be introduced in layered systems this way to improve overall mechanical properties. It was also predicted that the residual stress effect is negligible if the intact layer is more compliant than the cracked layer supporting a previous analysis and discussion reported elsewhere. 2005 Elsevier Ltd. All rights reserved. Keywords: Crack deflection; Interface; Residual stress; Layered systems 1. Introduction In many materials systems, residual stresses are often introduced during fabrication processes due to the mismatch of coefficient of thermal expansion (CTE) between constituent materials. The representative examples include layered systems such as ceramic laminates [1,2], thin-film deposited device materials [3,4] and engineering structures with protective coatings [5,6]. Presence of residual stress, superposed on the stress field due to applied load, will modify the stress distributions within the systems thereby causing the changes in the fracture behaviour of these materials. It is therefore expected that crack deflection criteria, determined by the fracture process within the stress field of a primary crack terminating at a bi-material interface of layered composites, would also undergo changes under the influence of the residual stresses. Crack deflection at bi-material interfaces in brittle matrix composite systems is important as it is a prerequisite to the operation of many toughening mechanisms or it is itself practically the only toughening mechanism in some layered systems [7,8]. It is often assumed that deflection of a crack terminating at an interface occurs by a competition between crack penetration across the interface or deflection along the interface (see Fig. 1). Then the energetic condition for crack deflection would be satisfied when strain energy released due to crack penetration across the interface, Gp, is lower than the bulk fracture energy, Rm whilst that due to crack deflection along the interface, Gd, is higher than the interface fracture energy, Ri, i.e. when [9] Ri Rm < Gd Gp ¼ f ða; bÞ; ð1Þ 0266-3538/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.07.015 * Corresponding author. Tel.: +82 2 2123 2843/119 813 8192; fax: +82 2 365 2680/236 526 80. E-mail addresses: woong.lee@yonsei.ac.kr, cwl10001@hanmail.net (W. Lee). Composites Science and Technology 66 (2006) 435–443 COMPOSITES SCIENCE AND TECHNOLOGY www.elsevier.com/locate/compscitech
w. Lee et al. I Composites Science and Technology 66(2006)435-443 t the Interface Crack Penetrating Fig. 1. Schematic illustration of the geometry used for the investigation of crack deflection: (a)crack penetrating the interface and (b) crack residual stress acting parallel to the interface: ap and ad: lengths of putative crack ctively, of the layer number i(i=1,2);o: in-plane or tangential along the interface [E, H; and vi- elastic and shear moduli, and P( etration into the where a and B are Dundurs' parameters [10] represent ing the extent of elastic modulus mismatch across the t interface, which are expre Poissons ratios of each layer (see Fig. 1),Hi and v where ki is a stress intensity-like factor, a, is the in-plane or tangential residual stress acting parallel to the inter- s[l6] 1(1-v2)-2(1-w 1(1-2)+2(1-v +B2 coS人T p1(1-2v2)-2(1-2v) 1+ (4) 2[1(1-2v2)+2(1-2v) Use of n]t to describe the state of residual stress in any sys- Work on the effect of residual stress on crack deflec. tem requires the measurement of the in-plane residual tion criteria is relatively scarce although there have been stress,ot, and the choice of proper length scale for ap, plenty of investigations into the crack deflection problem which is somewhat arbitrary[12]. Consequently, depend itself, notably by He and Hutchinson [9] and Martinez ing on the choice of ap, many nt can be defined for a given and Gupta [ll]. Effect of residual stress was first consid- magnitude of the residual stress, or, since Eq (3)is pro- ered by He et al. [12]. Based on crack tip field solutions, portional to the ith power of the length scale ap [13] they predicted that compressive residual stress acting To avoid these difficulties, in the present study, alterna parallel to the interface would increase critical interface tive parameter is defined in the following way The misfit strain can be found from CTEs of each fracture energy required for crack deflection, i.e. R:/Rm layer, ai and a2, respectively and the difference between ratio in Eq (1), and vice versa for a tensile residual stress he processing temperature(usually elevated tempera for all ranges of a. To the contrary, in a later analytical ture)and the room temperature, AT, as work on the effect of residual stress. Leguillon et al [13] argued that the presence of residual stresses has no Er=(1-22)AT effect on the crack deflection criteria for a 0(intact layer is stiffer). Since no fur- can be shown that the residual stresses developed in each ther work is available, it would be difficult to decide layer, lr and 2r, due to er, are which criterion is to be chosen as a design guideline espe- cially for laminated composite systems in which crack OIr= E1E2 ENEx deflection tends to occur under the condition of negative e.fer and a2r=- E1 a[14, 15]. Therefore, it is now necessary to investigate the where fi and f2 are volume fractions of inner and outer problem further and in the current work, these aspects of layers, respectively, E1 and E2 are elastic moduli of inner crack deflection, related to residual stress, are considered and outer layers, respectively, and EL is the effective using a finite element method(FEM) elastic modulus of the laminate in the direction of the plane of the layers(length-wise direction) given by EL Eifi E2f2 2. Description of the thermal misfit stress From Eq(6), it follows that the strain changes due to In the analysis of the effect of residual stresses on relaxation of the residual stress after debonding of the crack deflection, He et al. [12] introduced a dimension lamination)are less parameter for the geometry shown in Fig. l, which EL 2Er fiEr is expressed as E1
where a and b are Dundurs parameters [10] representing the extent of elastic modulus mismatch across the interface, which are expressed using shear moduli and Poissons ratios of each layer (see Fig. 1), li and mi (i = 1, 2), as a ¼ l1ð1 m2Þ l2 ½ ð1 m1Þ l1ð1 m2Þ þ l2 ½ ð1 m1Þ and b ¼ 1 2 l1ð1 2m2Þ l2 ½ ð1 2m1Þ l1ð1 2m2Þ þ l2 ½ ð1 2m1Þ : ð2Þ Work on the effect of residual stress on crack deflection criteria is relatively scarce although there have been plenty of investigations into the crack deflection problem itself, notably by He and Hutchinson [9] and Martinez and Gupta [11]. Effect of residual stress was first considered by He et al. [12]. Based on crack tip field solutions, they predicted that compressive residual stress acting parallel to the interface would increase critical interface fracture energy required for crack deflection, i.e. Ri/Rm ratio in Eq. (1), and vice versa for a tensile residual stress for all ranges of a. To the contrary, in a later analytical work on the effect of residual stress, Leguillon et al. [13] argued that the presence of residual stresses has no effect on the crack deflection criteria for a 0 (intact layer is stiffer). Since no further work is available, it would be difficult to decide which criterion is to be chosen as a design guideline especially for laminated composite systems in which crack deflection tends to occur under the condition of negative a [14,15]. Therefore, it is now necessary to investigate the problem further and in the current work, these aspects of crack deflection, related to residual stress, are considered using a finite element method (FEM). 2. Description of the thermal misfit stress In the analysis of the effect of residual stresses on crack deflection, He et al. [12] introduced a dimensionless parameter for the geometry shown in Fig. 1, which is expressed as gt ¼ rtak p k1 ; ð3Þ where k1 is a stress intensity-like factor, rt is the in-plane or tangential residual stress acting parallel to the interface and k is stress singularity expressed as [16] cos kp ¼ 2ðb aÞ 1 þ b ð1 kÞ 2 þ a þ b2 1 b2 : ð4Þ Use of gt to describe the state of residual stress in any system requires the measurement of the in-plane residual stress, rt, and the choice of proper length scale for ap, which is somewhat arbitrary [12]. Consequently, depending on the choice of ap, many gt can be defined for a given magnitude of the residual stress, rt, since Eq. (3) is proportional to the kth power of the length scale ap [13]. To avoid these difficulties, in the present study, alternative parameter is defined in the following way. The misfit strain can be found from CTEs of each layer, a1 and a2, respectively and the difference between the processing temperature (usually elevated temperature) and the room temperature, DT, as er ¼ ða1 a2ÞDT : ð5Þ Using the well-known cut and weld technique [17], for the symmetrical laminate geometry shown in Fig. 2, it can be shown that the residual stresses developed in each layer, r1r and r2r, due to er, are r1r ¼ E1E2 EL f2er and r2r ¼ E1E2 EL f1er; ð6Þ where f1 and f2 are volume fractions of inner and outer layers, respectively, E1 and E2 are elastic moduli of inner and outer layers, respectively, and EL is the effective elastic modulus of the laminate in the direction of the plane of the layers (length-wise direction) given by EL ¼ E1f1 þ E2f2: ð7Þ From Eq. (6), it follows that the strain changes due to relaxation of the residual stress after debonding of the layer (delamination) are e1r ¼ E2 EL f2er and e2r ¼ E1 EL f1er: ð8Þ ap #2 (E , µ , ν ) 2 2 2 #1 (E , µ , ν ) 1 1 1 Interface Primary Crack ad Crack Deflectied at the Interface Crack Penetrating the Interface ad Primary Crack #2 (E , µ ν ) 2 2 2 #1 (E , µ , ν ) 1 1 1 σt σt , a b Fig. 1. Schematic illustration of the geometry used for the investigation of crack deflection: (a) crack penetrating the interface and (b) crack deflected along the interface. [Ei, li and mi: elastic and shear moduli, and Poissons ratio, respectively, of the layer number i (i = 1,2); rt: in-plane or tangential residual stress acting parallel to the interface; ap and ad: lengths of putative crack penetration into the layer #1 and deflection along the interface, respectively.] 436 W. Lee et al. / Composites Science and Technology 66 (2006) 435–443
w. Lee et aL. / Composites Science and Technology 66(2006)435-443 Outer Layer(Material 2) Edge crack 2h Innner Layer(Material #1 Outer Layer(Material 2 Edge Crack Fig. 2. Geometry of the modelled laminate structure used for the analysis (a c: remotely applied mechanical load) It would be practically more convenient to measure these changes in strains of the layers and estimate the y≡tan (10) misfit strain to characterise the residual stress Thus stress.Thus, where K, and K, are opening and shear stress intensity the relative misfit strain E with respect to the mechanical factors respectively, were calculated by estimating K1 ad-induced strain, eapp, as and K2 using the interaction integral method [26]. Finite element mesh used for the calculations is shown in Fig 3. Typical 8-node isoparametric elements were used and quarter-point node elements [27]were Positive I corresponds to compressive residual stress in employed around the crack tip region to model crack the inner intact layer (layer #1) and tensile residual tip singularity. To model symmetry boundary cond stress in the outer cracked layers(layer#2), and vice ver- tions, normal displacement degrees of freedom(DOF) sa. This term will be used to describe the residual stress of the node points on the symmetry lines were fixed state of the layered system considered in the current whilst their tangential displacement DOFs were re- study(Fig. 2) leased. Dundurs parameter varied from -0.85 to 0.85 whilst keeping B equal to zero for simplicity since it has been demonstrated and suggested that effect of B is 3. Numerical analysis not significant [25, 28]. Applied load was simulated by displacing the node points in the right end face of the Double edged notch specimen geometry shown in model along x-direction by a predetermined distance Fig. 2, loaded under a uniform tensile loading on its Residual stress due to the misfit strain was simulated end faces by a constant displacement 8, was chosen for by applying stresses estimated from Eq.(6) within the the numerical analysis for the convenience of modelling. range of r between -l0 and 1.0. It is believed that r The laminate shown in Fig. 2 consists of two outer lay in this range covers er estimated for some practical lam- ers with the thickness h and an inner layer with the inate systems such as SiC/C, SiC/B,C, SiC/TiB2 and thickness 2h. It was assumed that the primary cracks ZrOz/Al]o3 laminates [l]. A general-purpose finite ele- in the outer layers, terminated at the interface, are dou- ment analysis package ABAQUS was used for the bly deflected as usually observed in many real systems calculations [14, 18-20]. Only the quarter section of the geometry was modelled for the analysis due to the symmetry The residual stress terms(Eq (6) were separately cal- culated for various Is and superposed to the mechanical Before discussing the effect of residual stress on the loading term using principle of superposition to obtain crack deflection criteria, it is first necessary to confirm their combined contributions to the fracture parameters whether the numerical solution converges to the analyt governing crack deflection such as strain energy release ical solutions obtained for infinite geometry since the rates, a and p(and equivalent stress intensity factor numerical model used for the analysis herein is based and phase angle of loading. a and s, were calculated on the geometry having finite dimensions. This was done through the estimation of J-integral [21, 22] based on by estimating the a and p for various lengths of infin the virtual crack extension/domain integral methods tesimal crack penetration ap and crack deflection ad rel 23, 24] assuming linear plane strain elasticity. Phase an- ative to a reference global length scale, the thickness of gle of loading, which is an indicator of the relative con the outer layer h which is equal to the length of the pri tribution of shear and opening terms to the total crack driving force and defined as [25] Product of hibbit. Karlsson sore
It would be practically more convenient to measure these changes in strains of the layers and estimate the misfit strain to characterise the residual stress. Thus, magnitude of the residual stress is defined herein as the relative misfit strain er with respect to the mechanical load–induced strain, eapp, as C ¼ er eapp : ð9Þ Positive C corresponds to compressive residual stress in the inner intact layer (layer #1) and tensile residual stress in the outer cracked layers (layer #2), and vice versa. This term will be used to describe the residual stress state of the layered system considered in the current study (Fig. 2). 3. Numerical analysis Double edged notch specimen geometry shown in Fig. 2, loaded under a uniform tensile loading on its end faces by a constant displacement d, was chosen for the numerical analysis for the convenience of modelling. The laminate shown in Fig. 2 consists of two outer layers with the thickness h and an inner layer with the thickness 2h. It was assumed that the primary cracks in the outer layers, terminated at the interface, are doubly deflected as usually observed in many real systems [14,18–20]. Only the quarter section of the geometry was modelled for the analysis due to the symmetry (Fig. 2). The residual stress terms (Eq. (6)) were separately calculated for various Cs and superposed to the mechanical loading term using principle of superposition to obtain their combined contributions to the fracture parameters governing crack deflection such as strain energy release rates, Gd and Gp (and equivalent stress intensity factors) and phase angle of loading. Gd and Gp were calculated through the estimation of J-integral [21,22] based on the virtual crack extension/domain integral methods [23,24] assuming linear plane strain elasticity. Phase angle of loading, which is an indicator of the relative contribution of shear and opening terms to the total crack driving force and defined as [25] W tan1 K2 K1 ; ð10Þ where K1 and K2 are opening and shear stress intensity factors respectively, were calculated by estimating K1 and K2 using the interaction integral method [26]. Finite element mesh used for the calculations is shown in Fig. 3. Typical 8-node isoparametric elements were used and quarter-point node elements [27] were employed around the crack tip region to model crack tip singularity. To model symmetry boundary conditions, normal displacement degrees of freedom (DOF) of the node points on the symmetry lines were fixed whilst their tangential displacement DOFs were released. Dundurs parameter a was varied from 0.85 to 0.85 whilst keeping b equal to zero for simplicity since it has been demonstrated and suggested that effect of b is not significant [25,28]. Applied load was simulated by displacing the node points in the right end face of the model along x-direction by a predetermined distance. Residual stress due to the misfit strain was simulated by applying stresses estimated from Eq. (6) within the range of C between 1.0 and 1.0. It is believed that C in this range covers er estimated for some practical laminate systems such as SiC/C, SiC/B4C, SiC/TiB2 and ZrO2/Al2O3 laminates [1]. A general-purpose finite element analysis package ABAQUS1 was used for the calculations. 4. Results and discussion Before discussing the effect of residual stress on the crack deflection criteria, it is first necessary to confirm whether the numerical solution converges to the analytical solutions obtained for infinite geometry since the numerical model used for the analysis herein is based on the geometry having finite dimensions. This was done by estimating the Gd and Gp for various lengths of infinitesimal crack penetration ap and crack deflection ad relative to a reference global length scale, the thickness of the outer layer h which is equal to the length of the priOuter Layer (Material # 2) Edge Crack Symmetry Line Edge Crack Innner Layer (Material #1) h 2h h Outer Layer (Material # 2) δ8 δ8 Fig. 2. Geometry of the modelled laminate structure used for the analysis (r1: remotely applied mechanical load). 1 Product of Hibbit, Karlsson & Sorensen, Inc. W. Lee et al. / Composites Science and Technology 66 (2006) 435–443 437
w. Lee et al. I Composites Science and Technology 66(2006)435-443 Interface Penetrating Crack Fig. 3.(a) Finite element mesh and load-and-boundary conditions used for the numerical analysis (deformed after loading).(b) Meshing details around the crack tip region (o: uniaxial end displacement mary crack. It was assumed that ap =ad as in the case of smaller than the size of the K-dominant field. The sensi- other analyses in existing works [9, 1l-13]and that r=0 tivity of ga/p ratio(crack deflection criteria) on the for comparison purpose. In Fig 4(a), sa/sp ratios(also relative size of the crack extension with respect to the critical R /Rm ratios for crack deflection)calculated by length of the parent crack described herein is qualita- FEM for various ap/h(=aJ/h=a/h)are plotted as func- tively consistent with what explained in Ahn et al. [28 tions of a and corresponding phase angles of the infini- and Leguillon et al. [13] tesimal deflection is also compared in Fig. 4(b). It can be Having shown that convergence of the numerical seen in Fig 4 that as a/h decreases, both the sa/sp ratio olution was achieved when a/h=1x 10-4. effect of and the phase angle tend to converge toward those for a/ the thermal misfit strain was investigated by repeating h=l x 10. The curves for this length scale of the infin- the estimation of sa/sp(and therefore critical R/Rmra tesimal putative crack extension are identical to the tio) for various values of I with this normalised putative crack tip field solutions reported by Martinez and Gup- crack extension length. Fig. 5 shows the numerical re ta [11] and He et al. [ 12] for nt=0 indicating that the sults(Sa/s ratios) plotted as functions of Dundurs numerical solution converged to the crack tip field solu- parameter x. It can be seen in the figure that the tion when a/h was chosen to be 1 x 10 Sals ratio is insensitive to the thermal residual stress One interesting feature found in Fig. 4(a) is that when a <o whilst the effect of the residual stress be. Sa/ p is not sensitive to the choice of length scale for comes more significant as a increases from about zero ad(and also ap) when a<-0.3 whilst its dependence to higher values. It can also be seen that the effect of on a becomes more significant as a increase toward residual stress is more noticeable for negative values of 1.0. This trend can be understood on referring to the di Is. It is predicted that ratio for negative Is are cussion on the size of the K-dominant field of a crack lower than that for I=0 and in particular, when terminating at a bi-material interface presented by Ro- T=-1.0, a/p ratio falls toward 0 as a approaches meo and Ballarini [29], according to which it was pre- about 0.9 meaning that it is almost impossible to achieve dicted that the region of K-dominance becomes larger crack deflection in such a case. Changes in a/p ratio with decreasing o and almost vanishes when o is higher for Is lower than -1. 0 are not shown since the finite ele than about 0.5. Decreasing size of the K-dominant field ment analysis predicted interpenetration of the surfaces means that strain energy release rate of the infinitesimal of the primary crack. In practice, this would result the global loading since K-dominance holds only when due to the negative component of the stress intensity a crack extension would be more likely to be influenced by crack closure and thus modify the crack tip stress fiel the fracture process is confined to the scale substantially tor of the primary crack arising from the crack closure
mary crack. It was assumed that ap = ad as in the case of other analyses in existing works [9,11–13] and that C = 0 for comparison purpose. In Fig. 4(a), Gd=Gp ratios (also critical Ri/Rm ratios for crack deflection) calculated by FEM for various ap/h (= ad/h = a/h) are plotted as functions of a and corresponding phase angles of the infinitesimal deflection is also compared in Fig. 4(b). It can be seen in Fig. 4 that as a/h decreases, both the Gd=Gp ratio and the phase angle tend to converge toward those for a/ h = 1 · 104 . The curves for this length scale of the infinitesimal putative crack extension are identical to the crack tip field solutions reported by Martinez and Gupta [11] and He et al. [12] for gt = 0 indicating that the numerical solution converged to the crack tip field solution when a/h was chosen to be 1 · 104 . One interesting feature found in Fig. 4(a) is that Gd=Gp is not sensitive to the choice of length scale for ad (and also ap) when a < 0.3 whilst its dependence on a becomes more significant as a increase toward 1.0. This trend can be understood on referring to the discussion on the size of the K-dominant field of a crack terminating at a bi-material interface presented by Romeo and Ballarini [29], according to which it was predicted that the region of K-dominance becomes larger with decreasing a and almost vanishes when a is higher than about 0.5. Decreasing size of the K-dominant field means that strain energy release rate of the infinitesimal crack extension would be more likely to be influenced by the global loading since K-dominance holds only when the fracture process is confined to the scale substantially smaller than the size of the K-dominant field. The sensitivity of Gd=Gp ratio (crack deflection criteria) on the relative size of the crack extension with respect to the length of the parent crack described herein is qualitatively consistent with what explained in Ahn et al. [28] and Leguillon et al. [13]. Having shown that convergence of the numerical solution was achieved when a/h = 1 · 104 , effect of the thermal misfit strain was investigated by repeating the estimation of Gd=Gp (and therefore critical Ri/Rm ratio) for various values of C with this normalised putative crack extension length. Fig. 5 shows the numerical results (Gd=Gp ratios) plotted as functions of Dundurs parameter a. It can be seen in the figure that the Gd=Gp ratio is insensitive to the thermal residual stress when a < 0 whilst the effect of the residual stress becomes more significant as a increases from about zero to higher values. It can also be seen that the effect of residual stress is more noticeable for negative values of Cs. It is predicted that Gd=Gp ratio for negative Cs are lower than that for C = 0 and in particular, when C = 1.0, Gd=Gp ratio falls toward 0 as a approaches about 0.9 meaning that it is almost impossible to achieve crack deflection in such a case. Changes in Gd=Gp ratio for Cs lower than 1.0 are not shown since the finite element analysis predicted interpenetration of the surfaces of the primary crack. In practice, this would result in crack closure and thus modify the crack tip stress field due to the negative component of the stress intensity factor of the primary crack arising from the crack closure x y Interface Interface δ Penetrating Crack Deflected Crack a b Fig. 3. (a) Finite element mesh and load-and-boundary conditions used for the numerical analysis (deformed after loading). (b) Meshing details around the crack tip region (d: uniaxial end displacement). 438 W. Lee et al. / Composites Science and Technology 66 (2006) 435–443
w. Lee et aL. / Composites Science and Technology 66(2006)435-443 He et a.(1994)&a/h=1x10 12 a/h= 2x10 0.8 a/h=1x103 6 h= 5x 6-04-020.00.2 0.6 60 50 He et al. (1994) &a/h=1x10 a/h=1x102 -0.4 -0.20.0 0.2 0.6 Fig 4. (a)Change in sa/sp ratio plotted as a function of a for various putative extension length a normalised with respect to the thickness he outer layer, h(which is also equal to the length of the primary crack) and(b)corresponding change in phase angle of the deflected crack, p [sd and sp: strain energy released due to crack deflection along the interface(d)and that due to crack penetration across the interface(ap), respectively; a: putative crack propagation length(=ad=ap)] force, the effect of which is not the scope of the current fore, it is expected that compressive residual stress in stud. the inner intact layer would be beneficial to crack deflec- From the above result, it is suggested that introduc- tion due to the improved deflection criteria whilst tensile ing residual stress such that r0, i.e. intact layer is stiffer than the cracked layer the contrary, higher sa/s, ratio, i.e. critical R /Rmra Whilst residual stress has some effects on the sa/sp tio, is resulted in the positive a regime if T>0. There- ratio, there was no appreciable change in the phase an
force, the effect of which is not the scope of the current study. From the above result, it is suggested that introducing residual stress such that C 0. Therefore, it is expected that compressive residual stress in the inner intact layer would be beneficial to crack deflection due to the improved deflection criteria whilst tensile one is unfavourable against crack deflection, especially as C approaches 1.0. However, since Gd=Gp ratio is insensitive to C when a 0, i.e. intact layer is stiffer than the cracked layer. Whilst residual stress has some effects on the Gd=Gp ratio, there was no appreciable change in the phase an- -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1.0 1.2 1.4 a / h = 0.01 a/ h = 5x10-3 a / h = 1x10-3 a / h = 2x10-4 He et al. (1994) & a/ h = 1x10-4 α p d -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0 10 20 30 40 50 60 70 80 90 Phase Angle α a / h = 1x10-2 a / h = 1x10-3 He et al. (1994) & a / h = 1x10-4 a b Fig. 4. (a) Change in Gd=Gp ratio plotted as a function of a for various putative extension length ad = ap = a normalised with respect to the thickness of the outer layer, h (which is also equal to the length of the primary crack) and (b) corresponding change in phase angle of the deflected crack, W. [Gd and Gp: strain energy released due to crack deflection along the interface (ad) and that due to crack penetration across the interface (ap), respectively; a: putative crack propagation length (=ad = ap)]. W. Lee et al. / Composites Science and Technology 66 (2006) 435–443 439
w. Lee et al./ Composites Science and Technology 66(2006)435-443 =5.0 r=10.0 I=0.5 12 L⊥,⊥A⊥,⊥f,⊥⊥,⊥A .8-0.6 0 Fig. 5. Change in sa/sp ratio plotted as a function of Dundurs'z for various residual stress magnitude parameter I [r: non-dimensionalised residual stress magnitude parameter defined as the relative misfit strain e with respect to the mechanical load-induced strain, eapp, i.e., I=-(e/eapp)] gle of loading of the deflected crack, regardless of the va- interface between layer A and B. Since the crack is ter lue of T, as shown in Fig. 6. The in-plane residual stress minating at stiff to compliant interface, a x-0. 75(Eq considered herein is uniaxial in two-dimensional plane (2). From Fig. 5, it is expected that the crack propa strain geometry and therefore its presence changes only gates across the interface into the interlayer(layer B) the magnitude of the crack driving force of the primary regardless of the magnitude of the residual stress and crack, not its fracture mode, which is assumed to be in then reaches the interface between layer B and A.As pure opening(mode I). Hence, it can be understood that the crack is terminated at compliant to stiff interface, the phase angle of the deflected crack, which is deter- a.75. Under this condition, if r=0, as R; / Rm mined by the fracture mode of the primary crack pro-(=1.05)is higher than a/p(=0.96)in Fig. 5, crack vided that all the fracture process is confined well deflection is still not possible; however, if residual stress within the K-dominant field of the primary crack, will is introduced such that r=0.5 for instance, R/Rm undergo practically no noticeable change whatever the (=1.05)is lower than ga/p(=1.07)allowing crack magnitude of the residual stress is deflection at this interface. Alternatively, it is conceiv The result shown in Fig. 5 is qualitatively consistent able that similar effect can be obtained by choosing sti with the work of Leguillon et al. [13]. in which it was sta- fer material as interlayer and then introducing ted that crack deflection criterion is almost insensitive to compressive residual stress in the interlayer to allow he magnitude of residual stress for a 0. Further, the trend that have limited use due to inferior overall stifiness-related presence of compressive residual stress in the intact layer would be beneficial for crack deflection in the positive ox From the results and discussions so far, it is expected regime is also qualitatively consistent with the predic- that higher degree of modulus mismatch as well as larger tions by Leguillion et al. [13]. Applying these results to misfit strain will result in improved crack deflection cri- the design of ceramic laminate systems, which are pro- teria. However, there will be certain limits to the design duced by stacking relatively thick stiff matrix lavers w1 of materials systems this way. First, increasing the resid thin compliant interlayers alternately to introduce ual stress through the selection of the materials with lar crack-deflecting interfaces, it is possible to increase the ger difference in CTE may result in the spontaneous critical fracture energy of the crack-deflecting interface delamination during fabrication process as observed in by introducing the compressive residual stress in the stiff ZrO2/Al2O3 system since excessive CTE mismatch may matrix layers. This concept of laminate design is sche- provide enough driving force for the growth of delani matically illustrated in Fig. 7. nation crack initiating from the laminate edges [1]. An For the example shown in Fig. 7, suppose that the other limiting factor is the magnitude of misfit strain, layers denoted as a are 7 times stiffer than those denoted the maximum value of which would be determined by as B and that R /Rm=1.05. Now, further suppose that a the fracture strain of the constituent materials. For in- primary crack in the outer layer is reaching the first stance, if fi <f2 I as in the case of typical ceramic
gle of loading of the deflected crack, regardless of the value of C, as shown in Fig. 6. The in-plane residual stress considered herein is uniaxial in two-dimensional plane strain geometry and therefore its presence changes only the magnitude of the crack driving force of the primary crack, not its fracture mode, which is assumed to be in pure opening (mode I). Hence, it can be understood that the phase angle of the deflected crack, which is determined by the fracture mode of the primary crack provided that all the fracture process is confined well within the K-dominant field of the primary crack, will undergo practically no noticeable change whatever the magnitude of the residual stress is. The result shown in Fig. 5 is qualitatively consistent with the work of Leguillon et al. [13], in which it was stated that crack deflection criterion is almost insensitive to the magnitude of residual stress for a 0. Further, the trend that presence of compressive residual stress in the intact layer would be beneficial for crack deflection in the positive a regime is also qualitatively consistent with the predictions by Leguillion et al. [13]. Applying these results to the design of ceramic laminate systems, which are produced by stacking relatively thick stiff matrix layers with thin compliant interlayers alternately to introduce crack-deflecting interfaces, it is possible to increase the critical fracture energy of the crack-deflecting interface by introducing the compressive residual stress in the stiff matrix layers. This concept of laminate design is schematically illustrated in Fig. 7. For the example shown in Fig. 7, suppose that the layers denoted as A are 7 times stiffer than those denoted as B and that Ri/Rm = 1.05. Now, further suppose that a primary crack in the outer layer is reaching the first interface between layer A and B. Since the crack is terminating at stiff to compliant interface, a 0.75 (Eq. (2)). From Fig. 5, it is expected that the crack propagates across the interface into the interlayer (layer B) regardless of the magnitude of the residual stress and then reaches the interface between layer B and A. As the crack is terminated at compliant to stiff interface, a 0.75. Under this condition, if C = 0, as Ri/Rm (=1.05) is higher than Gd=Gp (=0.96) in Fig. 5, crack deflection is still not possible; however, if residual stress is introduced such that C = 0.5 for instance, Ri/Rm (=1.05) is lower than Gd=Gp (=1.07) allowing crack deflection at this interface. Alternatively, it is conceivable that similar effect can be obtained by choosing stiffer material as interlayer and then introducing compressive residual stress in the interlayer to allow the condition of positive a and C, but such laminate will have limited use due to inferior overall stiffness-related property. From the results and discussions so far, it is expected that higher degree of modulus mismatch as well as larger misfit strain will result in improved crack deflection criteria. However, there will be certain limits to the design of materials systems this way. First, increasing the residual stress through the selection of the materials with larger difference in CTE may result in the spontaneous delamination during fabrication process as observed in ZrO2/Al2O3 system since excessive CTE mismatch may provide enough driving force for the growth of delamination crack initiating from the laminate edges [1]. Another limiting factor is the magnitude of misfit strain, the maximum value of which would be determined by the fracture strain of the constituent materials. For instance, if f1 f2 1 as in the case of typical ceramic -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 α p d Γ = -1.0 Γ = -0.5 Γ = 0 Γ = 0.5 Γ = 1.0 Γ = 2.0 Γ = 5.0 Γ = 10.0 Fig. 5. Change in Gd=Gp ratio plotted as a function of Dundurs a for various residual stress magnitude parameter C [C: non-dimensionalised residual stress magnitude parameter defined as the relative misfit strain er with respect to the mechanical load-induced strain, eapp, i.e., C = (er/eapp)]. 440 W. Lee et al. / Composites Science and Technology 66 (2006) 435–443
w. Lee et aL. / Composites Science and Technology 66(2006)435-443 0004000 all the rs considered L,,,1,,,1,,,⊥,,,王,,,,,,,A,,,L -0.8-0.6-040. 0.0 04 0.8 Fig. 6. Change in phase angle of the deflected crack, Y, plotted as a function of a for various residual stress magnitude parameter I. imary Crack Penetrating the Compressive Residual Stress (a≤G;rs) Tensile Residual Stress Layer A(stiff) Layer B (compliant) Primary Crack Deflected at the Compressive Residual Stress (a>0: T>0; and R/Rm< ga/gp) Tensile Residual stress Layer A(stiff) Layer B(compliant) Fig. 7. Conceptual illustration of the laminate design utilising beneficial role of residual stress in achieving crack deflection laminates, then EL s E2(Eq (7)and from Eq( 8), it is Further, from the narrowing gap between Sa/sp ra expected that EIr A Er. In this case, unless Er is lower than tio curves with increasing I in Fig. 5, it is inferred that the failure strain of layer #l, the laminate will fail dur- the Sa/p ratio converges to a certain value with ing the fabrication process increasing T for a given a. To visualise this, Sa/ ratios
laminates, then EL E2 (Eq. (7)) and from Eq. (8), it is expected that e1r er. In this case, unless er is lower than the failure strain of layer #1, the laminate will fail during the fabrication process. Further, from the narrowing gap between Gd=Gp ratio curves with increasing C in Fig. 5, it is inferred that the Gd=Gp ratio converges to a certain value with increasing C for a given a. To visualise this, Gd=Gp ratios -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 10 20 30 40 50 60 70 80 90 Phase Angle of the Deflected Crack α all the Γs considered Fig. 6. Change in phase angle of the deflected crack, W, plotted as a function of a for various residual stress magnitude parameter C. Layer A (stiff) Compressive Residual Stress Layer B (compliant) Tensile Residual Stress i m d p Layer A (stiff) Compressive Residual Stress Layer B (compliant) Tensile Residual Stress i m d p Fig. 7. Conceptual illustration of the laminate design utilising beneficial role of residual stress in achieving crack deflection. W. Lee et al. / Composites Science and Technology 66 (2006) 435–443 441
w. Lee et al./ Composites Science and Technology 66(2006)435-443 C=0.6 34567 Fig. 8. Change in sa/p ratio plotted as a function of I for two selected values of z. were plotted as functions of I for two selected axs, inate composites be designed in such a way that the con namely 0.6 and 0.8, in Fig 8. It is noticed in Fig 8 that dition of positive a and r is maintained to take the slope of the curves, i.e. a(sa/p)/ar, decreases as r advantage of beneficial effect of the misfit stress, i.e. crit increases In particular, when T is higher than about 1.0 ical fracture energy of the crack deflecting interface can for a=0.6 and 3.0 for a=0.9, respectively, be increased through the incorporation of in-plane resid a(sa/p)/ar is close to zero. Similar tendency was pre- ual stress thereby improving overall mechanical proper dicted in the analysis of the behaviour of an interface ties of the laminate crack in a bi-layer system under the presence of in-plane residual stress [30]. Thus, it is deduced that once the magnitude of the residual stress exceeds a certain value Acknowledgements further improvement in the properties of layered system would be marginal This work was supported in part (W. Lee and Pre 5. Conclusions gramme for Young Scientists(Grant No. RO8-2004- 000-10193-0) Effect of thermal misfit stain induced by mismatch of CtE on the crack deflection at planar interfaces in lay red systems was investigated using a finite element References analysis. In a convergence test of the numerical solution, dependence of the deflection criteria on the choice of I] Howard SJ, Stewart RA, Clegg WJ. The delamination of ceramic length scale of the infinitesimal putative penetration laminates due to residual thermal stresses. Key Eng Mater across the interface or deflection along the interface 1996:116-117:331-50 was noticed only in the positive a regime, which is con Blattner AJ, Lakshminarayanan R, Shetty DK. Toughening of sistent with the previous works by Ahn et al. [28] and layered ceramic composites with residual surface compression effects of layer thickness. Eng Fract Mech 2001: 68(1): 1-7 Leguillon et al. [13]. When residual stress effect was con- [3]Hutchinson JW. Delamination of compressed films on curved sidered, it was predicted that the effect of in-plane ther- substrates. J Mech Phys Solids 2001: 49(9): 1847-64 mal misfit stress is appreciable only when the intact layer [4 Pauleau Y. Generation and evolution of residual stresses in is much stiffer than the crack layer(a>0), whilst crack al vapour-deposited thin films. Vacuum 2001; 61(2-4) deflection criteria were insensitive to the magnitude of 5]Tsui Y-C, Clyne Tw. An analytical model for residual the residual stress when a <0, supporting the predic planar tions of Leguillion et al. [13]. It was further predicted netry. Thin Solid Films 1997: 306(1): 23-33 that introducing compressive residual stress in the stiffer 6 Hutchinson JW, Evans AG. On the delamination of thermal intact layer would be beneficial for crack deflection as barrier coatings in a thermal gradient. Surf Coat Technol 2002;14902-3):17984. this provides condition to increase transition toughness [7 Gordon JE. The new science of strong materials. 2nd ed. Lon- ratio for crack deflection. Thus, it is proposed that lam
were plotted as functions of C for two selected as, namely 0.6 and 0.8, in Fig. 8. It is noticed in Fig. 8 that the slope of the curves, i.e. oðGd=GpÞ=oC, decreases as C increases. In particular, when C is higher than about 1.0 for a = 0.6 and 3.0 for a = 0.9, respectively, oðGd=GpÞ=oC is close to zero. Similar tendency was predicted in the analysis of the behaviour of an interface crack in a bi-layer system under the presence of in-plane residual stress [30]. Thus, it is deduced that once the magnitude of the residual stress exceeds a certain value, further improvement in the properties of layered system would be marginal. 5. Conclusions Effect of thermal misfit stain induced by mismatch of CTE on the crack deflection at planar interfaces in layered systems was investigated using a finite element analysis. In a convergence test of the numerical solution, dependence of the deflection criteria on the choice of length scale of the infinitesimal putative penetration across the interface or deflection along the interface was noticed only in the positive a regime, which is consistent with the previous works by Ahn et al. [28] and Leguillon et al. [13]. When residual stress effect was considered, it was predicted that the effect of in-plane thermal misfit stress is appreciable only when the intact layer is much stiffer than the crack layer (a > 0), whilst crack deflection criteria were insensitive to the magnitude of the residual stress when a < 0, supporting the predictions of Leguillion et al. [13]. It was further predicted that introducing compressive residual stress in the stiffer intact layer would be beneficial for crack deflection as this provides condition to increase transition toughness ratio for crack deflection. Thus, it is proposed that laminate composites be designed in such a way that the condition of positive a and C is maintained to take advantage of beneficial effect of the misfit stress, i.e. critical fracture energy of the crack deflecting interface can be increased through the incorporation of in-plane residual stress thereby improving overall mechanical properties of the laminate. Acknowledgements This work was supported in part (W. Lee and J.-M. Myoung) by the National R&D Support Programme for Young Scientists (Grant No. R08-2004- 000-10193-0). References [1] Howard SJ, Stewart RA, Clegg WJ. The delamination of ceramic laminates due to residual thermal stresses. Key Eng Mater 1996;116–117:331–50. [2] Blattner AJ, Lakshminarayanan R, Shetty DK. Toughening of layered ceramic composites with residual surface compression: effects of layer thickness. Eng Fract Mech 2001;68(1):1–7. [3] Hutchinson JW. Delamination of compressed films on curved substrates. J Mech Phys Solids 2001;49(9):1847–64. [4] Pauleau Y. Generation and evolution of residual stresses in physical vapour-deposited thin films. Vacuum 2001;61(2–4): 175–181. [5] Tsui Y-C, Clyne TW. An analytical model for predicting residual stresses in progressively deposited coatings. Part 1: planar geometry. Thin Solid Films 1997;306(1):23–33. [6] Hutchinson JW, Evans AG. On the delamination of thermal barrier coatings in a thermal gradient. Surf Coat Technol 2002;149(2–3):179–84. [7] Gordon JE. The new science of strong materials. 2nd ed. London: Pitman; 1976. -1 0 1 2 3 4 5 6 7 8 9 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Γ p d α = 0.8 α = 0.6 Fig. 8. Change in Gd=Gp ratio plotted as a function of C for two selected values of a. 442 W. Lee et al. / Composites Science and Technology 66 (2006) 435–443
w. Lee et aL. / Composites Science and Technology 66(2006)435-443 B8 Chawla KK. Ceramic matrix composites. L 20] Phillipps AJ, Clegg WJ. Clyne Tw. Fracture behaviour of ceramic laminates in bending. Part Il of model predictions 9 He M-Y, Hutchinson Jw. Crack deflection between dissimilar elastic materials. Int I solids interface data. Acta Metall Mater 1993: 41(3): 819-27 [21] Rice JR A path independent integral and approximation analysi 98925(9):1053-67 f strain concentration by notches and cracks. J Appl Mech [10 Dundurs J. Edge-bonded dissimilar orthogonal elastic wedges. J ppl Mech(Trans ASME)1969: 36(5): 650-2 222 Smelser RE, Gurtin ME. On the J-integral for bi-material bodies. [n Martinez D, Gupta VJ. Energy criterion for crack deflection at an Int j Fracture 1977: 13: 382-4 interface between two orthotropic media. J Mech Phys Solid [23] Parks DM. The virtual crack extension method for nonli 994:42(8:1247-71 material behaviour. Comp Method Appl Mech Eng [2 He M-Y. Evans AG, Hutchinson Jw. Crack deflection at an 1977;12:353-64 interface between dissimilar elastic materials. 224 Shih CF, Moran B, Nakamura T. Energy release rate along resses. Int J Solids Struct 1994; 31(24): 3443-55 hree-dimensional crack front in a thermally stressed body. Int [13] Leguillon D, Lacroix C, Martin E. Crack deflection by an Fracture1986:30:79102. interface asymptotics of the residual thermal stresses. Int I Hutchinson Jw, Suo Z. Mixed mode cracking in layered Solids Structures 2001: 38(42-43): 7423-45 materials. Adv Appl Mech 1991: 29(1): 63-19 14]Clegg WJ. The fabrication and failure of laminar ceramic 6 Shih CF, Asaro RJ. Elastic-plastic analysis of crack on bimaterial composites. Acta Metall Mater 1992: 40(11): 3085-93 interfaces: Part I-small scale yielding. J Appl Mech(Trans [15] Clegg WJ, Howard SJ, Lee W, Phillipps AJ, Stewart RA ASME)1988:55:299316 nterfacial cracking in ceramic laminates. Comp Interface 227 Henshell RD. Shaw KG. Crack tip finite elements are unneces- 1994:2(5):337-49 sary. Int J Num Meth Eng 1975: 9: 495-507. [16] Zak AR, williams ML. Crack point stress singularities at a b- 8 Ahn BK, Curtin WA, Parthasarathy TA, Dutton RE. Criteria for material interface. J Appl Mech(Trans ASME)1963: 27(2): 142-3 crack deflection/penetration for fiber-reinforced ceramic matrix 7 Gere JM, Timoshenko SP. Mechanics of materials. 3rd SI composites. Comp Sci Tech 1998: 58(11): 1775-84 d. London: Chapman and Hall: 1993 229]Romeo A. Ballarini R. A cohesive zone model for cracks 18 Kendall K. Transition between cohesive and interfacial failure in terminating at a bimaterial interface. Int j Solids Structures a laminate. Proc R Soc Lond A 1975: 344: 287-302 1997:34(11):1307-26. 19]Gupta V, Yuan J, Martinez D. Calculation, measurement, and [30] Lee W, Shin H. Effect of in-plane residual stresses on the post- ontrol of interface strength in composites. J Am Ceram Soc deflection behaviour of a T-shaped deflected crack in layered systems. Phil Mag2003:83(8):985-1006
[8] Chawla KK. Ceramic matrix composites. London: Chapman and Hall; 1993. [9] He M-Y, Hutchinson JW. Crack deflection at an interface between dissimilar elastic materials. Int J Solids Struct 1989;25(9):1053–67. [10] Dundurs J. Edge-bonded dissimilar orthogonal elastic wedges. J Appl Mech (Trans ASME) 1969;36(5):650–2. [11] Martinez D, Gupta VJ. Energy criterion for crack deflection at an interface between two orthotropic media. J Mech Phys Solids 1994;42(8):1247–71. [12] He M-Y, Evans AG, Hutchinson JW. Crack deflection at an interface between dissimilar elastic materials: Role of residual stresses. Int J Solids Struct 1994;31(24):3443–55. [13] Leguillon D, Lacroix C, Martin E. Crack deflection by an interface — asymptotics of the residual thermal stresses. Int J Solids Structures 2001;38(42–43):7423–45. [14] Clegg WJ. The fabrication and failure of laminar ceramic composites. Acta Metall Mater 1992;40(11):3085–93. [15] Clegg WJ, Howard SJ, Lee W, Phillipps AJ, Stewart RA. Interfacial cracking in ceramic laminates. Comp Interfaces 1994;2(5):337–49. [16] Zak AR, Williams ML. Crack point stress singularities at a bmaterial interface. J Appl Mech (Trans ASME) 1963;27(2):142–3. [17] Gere JM, Timoshenko SP. Mechanics of materials. 3rd SI ed. London: Chapman and Hall; 1993. [18] Kendall K. Transition between cohesive and interfacial failure in a laminate. Proc R Soc Lond A 1975;344:287–302. [19] Gupta V, Yuan J, Martinez D. Calculation, measurement, and control of interface strength in composites. J Am Ceram Soc 1993;76(2):305–15. [20] Phillipps AJ, Clegg WJ, Clyne TW. Fracture behaviour of ceramic laminates in bending. Part II – comparison of model predictions with experimental data. Acta Metall Mater 1993;41(3):819–27. [21] Rice JR. A path independent integral and approximation analysis of strain concentration by notches and cracks. J Appl Mech (Trans ASME) 1968;35:379–86. [22] Smelser RE, Gurtin ME. On the J-integral for bi-material bodies. Int J Fracture 1977;13:382–4. [23] Parks DM. The virtual crack extension method for nonlinear material behaviour. Comp Method Appl Mech Eng 1977;12:353–64. [24] Shih CF, Moran B, Nakamura T. Energy release rate along a three-dimensional crack front in a thermally stressed body. Int J Fracture 1986;30:79–102. [25] Hutchinson JW, Suo Z. Mixed mode cracking in layered materials. Adv Appl Mech 1991;29(1):63–191. [26] Shih CF, Asaro RJ. Elastic–plastic analysis of crack on bimaterial interfaces: Part I – small scale yielding. J Appl Mech (Trans ASME) 1988;55:299–316. [27] Henshell RD, Shaw KG. Crack tip finite elements are unnecessary. Int J Num Meth Eng 1975;9:495–507. [28] Ahn BK, Curtin WA, Parthasarathy TA, Dutton RE. Criteria for crack deflection/penetration for fiber-reinforced ceramic matrix composites. Comp Sci Tech 1998;58(11):1775–84. [29] Romeo A, Ballarini R. A cohesive zone model for cracks terminating at a bimaterial interface. Int J Solids Structures 1997;34(11):1307–26. [30] Lee W, Shin H. Effect of in-plane residual stresses on the postdeflection behaviour of a T-shaped deflected crack in layered systems. Phil Mag 2003;83(8):985–1006. W. Lee et al. / Composites Science and Technology 66 (2006) 435–443 443