C R Chen et al. Acta Materialia 55(2007)409-421 Homogeneous material with and without residual stress fold effect of the residual stresses the first effect is that 6=20 Nmm compared with the fully homogeneous material, the term Jfar faro) is generally reduced. This is due to the com- pressive residual stresses in the outer layer which restrain the opening of the crack. The second effect arises from with residual stress the shielding(Fig. 5a)or anti-shielding(Fig. 5b) of the crack tip due to the inhomogeneity of the residual stress distribution in the specimen. This makes the effective crack driving force Jtip differ from the term Far -Far(0). Eq. (3) can be extended to the relation J tip-Ufar-Jfar())=Cinh-Cinh(0) where Cinh(0) denotes the material inhomogeneity term for a component with zero crack length. Note that the terms amy o-15020025 Far(0)and Cinh(0) are used in the figures only to make the Far and Cinh values path-independent, and that the relation Far(0)=-Cinh(O) holds, see Appendix A2.2 and Homogeneous materi and without residual stress Eq(A2.5). The material inhomogeneity term Cinh reaches F2B= 10 N/mm a local extremum if the crack just penetrates an interface The material inhomogeneity term Cinh is negative and J and J crack tip shielding occurs for a crack located at interface l; a positive Cinh and strong anti-shielding occurs for a crack located at interfac 5.3. Multilayer composite with inhomogeneity in elastic modulus and CTE In this section the numerical results of the actual multi layer composite are presented and compared with the experimental results. Besides the spatially var stresses due to the spatial CTE-variation, also the different 025 elastic moduli of the a-and Az-laminae influence the frac- ture behavior. Fig. 6 shows the Jtip vs. F curves for speci- Fig. 5. n and the path-independent far-field J-integral term J -Jao) mens with a=0. 18 mm and a=0.20 mm. The curves of a function of the crack length a for the homogeneous specimen with and the elastically inhomogeneous specimen without residual without residual stresses. (a) For F/B=20N/mm;(b) for F/2B= stresses are also given. The comparison with Fig 4 delivers the following findings: The inhomogeneity of the elastic modulus does not influence the origin of the curves however, it generally increases the slopes of the curves so with a crack, Jfar(a) depends also on Ly, but the term Jfar -Far(O) is path-independent; see Eq(A2.6) For comparison, the Jtip and Far VS a curves of the com pletely homogeneous specimen without residual stresses are Iso given. The two curves coincide and Far(0)=0. Fig 5a lows the curves for a crack close to interface 1 and F=20 N/mm; Fig 5b shows the curves for a crack close to interface 2 and F=10 N/mm. (The loads were chosen so that j in has a realistic size. not far from the size of the intrinsic fracture toughness values. )As deduced in Eq (A2.7), the effective crack driving force tip shows an approximately linear dependence on the crack length a The small deviation from non-linearity appears since the 亠a= parameter K(see Appendix A2.3, Eqs.(A2.7) and(A2. 12) is slightly dependent on the crack length a. Note that 1618202224 Jtip Far t Cinh does not depend on the length of the inte- gration path Ly, and that, for a component with zero crack Fig. 6. Effective crack driving force Juip as a function of F/2B for the length, JtiplO)=0; see Eq(A2.5). The curves reveal a two- elastically inhomogeneous composite with and without residual stresseswith a crack, Jfar(a) depends also on Ly, but the term Jfar Jfar(0) is path-independent; see Eq. (A2.6). For comparison, the Jtip and Jfar vs. a curves of the com￾pletely homogeneous specimen without residual stresses are also given. The two curves coincide and Jfar(0) = 0. Fig. 5a shows the curves for a crack close to interface 1 and F b ¼ 20 N=mm; Fig. 5b shows the curves for a crack close to interface 2 and F b ¼ 10 N=mm. (The loads were chosen so that Jtip has a realistic size, not far from the size of the intrinsic fracture toughness values.) As deduced in Eq. (A2.7), the effective crack driving force Jtip shows an approximately linear dependence on the crack length a. The small deviation from non-linearity appears since the parameter j (see Appendix A2.3, Eqs. (A2.7) and (A2.12) is slightly dependent on the crack length a. Note that Jtip = Jfar + Cinh does not depend on the length of the inte￾gration path Ly, and that, for a component with zero crack length, Jtip(0) = 0; see Eq. (A2.5). The curves reveal a two￾fold effect of the residual stresses: the first effect is that, compared with the fully homogeneous material, the term Jfar Jfar(0) is generally reduced. This is due to the com￾pressive residual stresses in the outer layer which restrain the opening of the crack. The second effect arises from the shielding (Fig. 5a) or anti-shielding (Fig. 5b) of the crack tip due to the inhomogeneity of the residual stress distribution in the specimen. This makes the effective crack driving force Jtip differ from the term Jfar Jfar(0). Eq. (3) can be extended to the relation Jtip ð Þ¼ Jfar Jfarð0Þ Cinh Cinhð0Þ; ð7Þ where Cinh(0) denotes the material inhomogeneity term for a component with zero crack length. Note that the terms Jfar(0) and Cinh(0) are used in the figures only to make the Jfar and Cinh values path-independent, and that the relation Jfar(0) = Cinh(0) holds, see Appendix A2.2 and Eq. (A2.5). The material inhomogeneity term Cinh reaches a local extremum if the crack just penetrates an interface. The material inhomogeneity term Cinh is negative and crack tip shielding occurs for a crack located at interface 1; a positive Cinh and strong anti-shielding occurs for a crack located at interface 2. 5.3. Multilayer composite with inhomogeneity in elastic modulus and CTE In this section, the numerical results of the actual multi￾layer composite are presented and compared with the experimental results. Besides the spatially varying residual stresses due to the spatial CTE-variation, also the different elastic moduli of the A- and AZ-laminae influence the frac￾ture behavior. Fig. 6 shows the Jtip vs. F b curves for speci￾mens with a = 0.18 mm and a = 0.20 mm. The curves of the elastically inhomogeneous specimen without residual stresses are also given. The comparison with Fig. 4 delivers the following findings: The inhomogeneity of the elastic modulus does not influence the origin of the curves; however, it generally increases the slopes of the curves so 0.00 0.05 0.10 0.15 0.20 0.25 0 20 40 60 80 100 120 140 160 180 interface 1 no residual stress Jtip and Jfar Homogeneous material with and without residual stress F/2B = 20 N/mm with residual stress Jtip Jfar - Jfar(0) Jtip and Jfar - Jfar(0) [J/m2 ] a [mm] 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0 20 40 60 80 100 120 140 160 180 interface 2 interface 3 no residual stress Jtip and Jfar Homogeneous material with and without residual stress F/2B = 10 N/mm with residual stress Jtip Jfar - Jfar(0) Jtip and Jfar - Jfar(0) [J/m2 ] a [mm] Fig. 5. Jtip and the path-independent far-field J-integral term Jfar Jfar(0) as a function of the crack length a for the homogeneous specimen with and without residual stresses. (a) For F/2B = 20 N/mm; (b) for F/2B = 10 N/mm. 0 2 4 6 8 10 12 14 16 18 20 22 24 0 5 10 15 20 25 30 35 40 45 50 55 60 with residual stress a = 0.18 mm a = 0.20 mm J0 (A) J0 (AZ) Inhomogeneous material with and without residual stress no residual stress a = 0.18 mm a = 0.20 mm Jtip [J/m2 ] F/2B [N/mm] Fig. 6. Effective crack driving force Jtip as a function of F/2B for the elastically inhomogeneous composite with and without residual stresses. C.R. Chen et al. / Acta Materialia 55 (2007) 409–421 415