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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(cracked layer is stiffer) whilst reporting similar results to those USing the well-known cut and weld technique [17], for the symmetrical laminate geometry shown in Fig. 2, it by he et al. 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 E1where 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 (cracked layer is stiffer) whilst reporting similar results to those by He et al. 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