C R Chen et al. Acta Materialia 55(2007)409-421 curves of the real composite are generally shifted to F= Ffr can be taken as the apparent fracture toughness higher Jtip values. In addition, the shapes of the curves Jfr of the specimen which would be measured in the differ when the crack approaches the interfaces: near experiment. For an elastically inhomogeneous material interface I, Jtip is enhanced; near interface 2, Jtip is without residual stresses, the far-field J-integral of the specimen with zero crack length Far(0)=0 and, therefore, As discussed in Section 5.2, it can be assumed that the Far is path-independent if the path I crosses all the inter- specimen fractures when Jtip reaches the value of the intrin- faces and if Ly is not too small. In the case of a residual sic fracture toughness Jo of the material where the crack tip stress discontinuity, the size of Far(0) depends on the load is located. With this condition, the fracture loads Frr of F and therefore Jfar becomes strongly path-dependent pecimens with arbitrary crack lengths can be evaluated. This is the reason why the Jfar value which correspond Fr= Fr/(2B)on the crack length a. For comparison, apparent fracture toughness of the specimen. Instead the corresponding curves of the elastically inhomogeneous we use an"apparent far-field J-integral"Jfr which is specimen without residual stresses are also given. Note that determined as follows: first the fracture load Ffr is deter in a homogeneous specimen Ffr should be proportional to mined so that for the real composite Jtip equals Jo. This a-1/. It is seen that the compressive residual stresses in Fr value is then applied to the model without the thermal layer I greatly increase Fir. This is so not only when the residual stresses and, subsequently, the path-independent crack tip is located in layer l, but also for some crack value of Far is evaluated. This Far value represents the length after crossing interface l For the crack tip in layer apparent fracture toughness Jfr of the specimen with the 2, the tensile residual stresses in the layer make the slope of residual stress state the Ffr vS a curve distinctly larger than that of the model In Fig. 9 the apparent Jfr values are plotted as a function without the residual stresses. When the crack tip has passed of the crack length a. For comparison, the intrinsic fracture the middle of layer 2, the fracture load falls below the Frr toughness values Jo of the A- and AZ-material and the Jfr value of the model without the residual stresses. After the values of the elastically inhomogeneous specimen without crack tip has penetrated interface 2, the compressive resid- residual stresses are also included. The latter demonstrate ual stresses in layer 3 quickly enhance Ffr over that of the the effect of the inhomogeneity of the elastic modulus on model without the residual stresses. According to this the apparent fracture toughness. This effect is rather small urve, even a stable crack extension should be possible compared with the effect of the thermal residual stresses for an initial crack with its tip close to interface 2. For however, for a crack approaching interface 2, it still leads all other initial crack lengths, unstable crack growth will to an increase of the fracture toughness of about 35% occur and the specimen will fail catastrophically when the The benefit of the compressive thermal residual stresses crack starts to grow in layer I is largest for a crack approaching interface 1 The experimentally measured fracture loads of the spec- for the crack tip in the layer 1, Jfr increases almost line imens 1, 2, and 3 are also indicated in Fig. 8. The experi- with increasing a, reaching a final value more than four mental data match the numerically predicted values that of the intrinsic fracture toughness. After having pene- s For a specimen without residual stresses, the far-field trated interface 1, the apparent fracture toughness J-integral of the specimen evaluated at the fracture load decreases sharply, reaching a minimum value for the crack homogeneous material with and without residual stress material with and without residual stress 4-with residual stress J, with residual stress 去 000.050.100.150200250.300.350.40045050055060 000050.100.150200250.300.35040045050055060 Fig. 8. Numerically predicted and experimentally measured specific Fig. 9. Numerically predicted fracture toughness Jfr, computed for the fracture loads F/2B which make Juip equal to the intrinsic fracture composite with and without thermal residual stresses, and experimentally toughness Jo of the individual layers. measured JIc valuescurves of the real composite are generally shifted to higher Jtip values. In addition, the shapes of the curves differ when the crack approaches the interfaces: near interface 1, Jtip is enhanced; near interface 2, Jtip is reduced. As discussed in Section 5.2, it can be assumed that the specimen fractures when Jtip reaches the value of the intrin￾sic fracture toughness J0 of the material where the crack tip is located. With this condition, the fracture loads Ffr of specimens with arbitrary crack lengths can be evaluated. Fig. 8 shows the dependency of the specific fracture load F bfr ¼ F fr=ð2BÞ on the crack length a. For comparison, the corresponding curves of the elastically inhomogeneous specimen without residual stresses are also given. Note that in a homogeneous specimen Ffr should be proportional to a1/2. It is seen that the compressive residual stresses in layer 1 greatly increase Ffr. This is so not only when the crack tip is located in layer 1, but also for some crack length after crossing interface 1. For the crack tip in layer 2, the tensile residual stresses in the layer make the slope of the Ffr vs. a curve distinctly larger than that of the model without the residual stresses. When the crack tip has passed the middle of layer 2, the fracture load falls below the Ffr value of the model without the residual stresses. After the crack tip has penetrated interface 2, the compressive resid￾ual stresses in layer 3 quickly enhance Ffr over that of the model without the residual stresses. According to this curve, even a stable crack extension should be possible for an initial crack with its tip close to interface 2. For all other initial crack lengths, unstable crack growth will occur and the specimen will fail catastrophically when the crack starts to grow. The experimentally measured fracture loads of the spec￾imens 1, 2, and 3 are also indicated in Fig. 8. The experi￾mental data match the numerically predicted values. For a specimen without residual stresses, the far-field J-integral of the specimen evaluated at the fracture load F = Ffr can be taken as the apparent fracture toughness Jfr of the specimen which would be measured in the experiment. For an elastically inhomogeneous material without residual stresses, the far-field J-integral of the specimen with zero crack length Jfar(0) = 0 and, therefore, Jfar is path-independent if the path C crosses all the inter￾faces and if Ly is not too small. In the case of a residual stress discontinuity, the size of Jfar(0) depends on the load F and therefore Jfar becomes strongly path-dependent. This is the reason why the Jfar value which corresponds to the critical force Ffr cannot be directly taken as the apparent fracture toughness of the specimen. Instead, we use an ‘‘apparent far-field J-integral’’ Jfr which is determined as follows: first the fracture load Ffr is deter￾mined so that for the real composite Jtip equals J0. This Ffr value is then applied to the model without the thermal residual stresses and, subsequently, the path-independent value of Jfar is evaluated. This Jfar value represents the apparent fracture toughness Jfr of the specimen with the residual stress state. In Fig. 9 the apparent Jfr values are plotted as a function of the crack length a. For comparison, the intrinsic fracture toughness values J0 of the A- and AZ-material and the Jfr values of the elastically inhomogeneous specimen without residual stresses are also included. The latter demonstrate the effect of the inhomogeneity of the elastic modulus on the apparent fracture toughness. This effect is rather small compared with the effect of the thermal residual stresses; however, for a crack approaching interface 2, it still leads to an increase of the fracture toughness of about 35%. The benefit of the compressive thermal residual stresses in layer 1 is largest for a crack approaching interface 1: for the crack tip in the layer 1, Jfr increases almost linearly with increasing a, reaching a final value more than fourfold that of the intrinsic fracture toughness. After having pene￾trated interface 1, the apparent fracture toughness decreases sharply, reaching a minimum value for the crack Fig. 8. Numerically predicted and experimentally measured specific fracture loads Ffr/2B which make Jtip equal to the intrinsic fracture toughness J0 of the individual layers. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0 20 40 60 80 100 120 140 160 180 200 220 Experimental JIC-data Inhomogeneous material with and without residual stress interface 1 interface 2 interface 3 Jfr with residual stress Jfr without residual stress J0 J-integral [J/m2 ] a [mm] Fig. 9. Numerically predicted fracture toughness Jfr, computed for the composite with and without thermal residual stresses, and experimentally measured JIC values. C.R. Chen et al. / Acta Materialia 55 (2007) 409–421 417