C R Chen et al. Acta Materialia 55(2007)409-421 that all the fracture loads are decreased by approximately values become singular: Cinh oo and Jtip -oo. Near 10%. interface 2, the shielding effect of the compliant-stiff transi- The dependency of the effective crack driving force Jtip tion induces a negative material inhomogeneity term. For and the term Far -Far(0) on the crack length a at a con- the crack ending directly at the interface, we get stant load is presented in Fig. 7a and b. Fig. 7a shows Cinh --Far and Jtip -0[26,31,32]. For a discussion see the curves for a crack near to interface I and also [33]. F=20 N/mm; Fig. 7b shows the curves for a crack near When comparing the curves of the real composite with to interface 2 and F=10 N/mm. The corresponding E- and CTE- inhomogeneity to those with only the E- curves of the elastically inhomogeneous specimen without inhomogenity and, therefore, without residual stresses, it residual stresses are also plotted. These Jfar vs. a curves can be seen that the thermal residual stresses provoke a are continuous curves which are only slightly bent. The general decrease of the apparent crack driving force comparison with Fig. 5a and b shows that these curves The inhomogeneity of the elastic modulus and the Cte ie above the corresponding curves of the completely inhomogeneity have opposite effects on the material inho- homogeneous specimen. The inhomogeneity of the elastic mogeneity term, but obviously the thermal residual stres modulus induces a material inhomogeneity term which is ses have a much stronger influence on the shielding/anti- positive near interface 1, since the stiff-compliant transition shielding behavior than the modulus inhomogeneity. A (EA>E1%) induces an anti-shielding effect and Jtip >Jfar comparison of Fig. 5 with Fig. 7 shows that, compared [26, 28]. For the crack ending directly at the interface, the with the elastically homogeneous specimen, the Jtip vs. a mogeneous material with and without residual stress F/2B=20 N/mm residual stress with residual stress …d(0) 20 a [mm] Inhomogeneous material with and without residual stress F2B= 10 Nmm no residual stress 8 with residual stress 0.3 35 Fig. 7. Jtip and Far -(0)as a function of the crack length a for the elastically inhomogeneous composite with and without residual stresses. (a)For F/2B=20 N/mm;(b) for F/2B=10N/mmthat all the fracture loads are decreased by approximately 10%. The dependency of the effective crack driving force Jtip and the term Jfar Jfar(0) on the crack length a at a con￾stant load is presented in Fig. 7a and b. Fig. 7a shows the curves for a crack near to interface 1 and F b ¼ 20 N=mm; Fig. 7b shows the curves for a crack near to interface 2 and F b ¼ 10 N=mm. The corresponding curves of the elastically inhomogeneous specimen without residual stresses are also plotted. These Jfar vs. a curves are continuous curves which are only slightly bent. The comparison with Fig. 5a and b shows that these curves lie above the corresponding curves of the completely homogeneous specimen. The inhomogeneity of the elastic modulus induces a material inhomogeneity term which is positive near interface 1, since the stiff-compliant transition (EA > EAZ) induces an anti-shielding effect and Jtip > Jfar [26,28]. For the crack ending directly at the interface, the values become singular: Cinh ! 1 and Jtip ! 1. Near interface 2, the shielding effect of the compliant-stiff transi￾tion induces a negative material inhomogeneity term. For the crack ending directly at the interface, we get Cinh ! Jfar and Jtip ! 0 [26,31,32]. For a discussion see also [33]. When comparing the curves of the real composite with E- and CTE-inhomogeneity to those with only the E￾inhomogenity and, therefore, without residual stresses, it can be seen that the thermal residual stresses provoke a general decrease of the apparent crack driving force. The inhomogeneity of the elastic modulus and the CTE inhomogeneity have opposite effects on the material inho￾mogeneity term, but obviously the thermal residual stres￾ses have a much stronger influence on the shielding/anti￾shielding behavior than the modulus inhomogeneity. A comparison of Fig. 5 with Fig. 7 shows that, compared with the elastically homogeneous specimen, the Jtip vs. a 0.00 0.05 0.10 0.15 0.20 0.25 0 20 40 60 80 100 120 140 160 180 200 220 240 interface 1 no residual stress Jtip Jfar Inhomogeneous 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 Jfar Inhomogeneous 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. 7. Jtip and Jfar Jfar(0) as a function of the crack length a for the elastically inhomogeneous composite with and without residual stresses. (a) For F/2B = 20 N/mm; (b) for F/2B = 10 N/mm. 416 C.R. Chen et al. / Acta Materialia 55 (2007) 409–421