C R Chen et al. Acta Materialia 55(2007)409-421 crack tip comes close to an interface. All these points show Homogeneous materal with and without residual stress the advantages of the configurational force concept over the conventional J-integral approach 5. Results and discussion In the following, the results of the numerical analyses for our multilayer composite are presented. Corresponding of the results are shown in Appendix In Appendix ion strain energy density and the thermal residual stresses are ●a=0.18mm derived for the composite under plane strain conditions. a=0. In Appendix 2 useful analytical estimates of the crack driv ing force are given 51. Thermal residual stresses crack driving force materia A )t wis nd In Figs. 2 and 3 the results of the 3D and 2D computa tions are compared. As already experienced in a study by Shan et al. [30] nearly 80% of the cross-section measured specific crack lengths, a=0.18 mm (crack tip 0.01 mm in the z-direction shows residual stresses o,Ar according hind the first interface) and a=0.20 mm(crack tip to the plane strain model. In the side-surface region, say 0.01 mm beyond of the first interface). For comparison, within 1.0< 2< 1.25 mm, the residual stresses can be the curves of the completely homogeneous specimen with- roughly approximated by the plane stress distribution. out residual stresses are also given, and these exhibit the Directly at the side-surface the absolute values of o,Ar common quadratic dependency on the load. The residual can, however, become significantly smaller than the plane stresses shift the origin of the curves. Since the residual stress values. It should be mentioned that at the positions stress state in the first layer is a compressive one and the where the interfaces 1-6 impinge the surface, weak stress bending stresses o,b are tensile, the crack will open at a singularities for o,Ar occur which can be ignored load Fo when the bending stresses balance the residual Since 80% of the specimen is controlled by plane strain stresses O, bma=CAar; compare Eq.(A2. 6). From this conditions, we keep this simple eigenstress distribution for equation, the quadratic dependency of Jtip on (F- Fo) Gy. Ar as the relevant one for our further fracture mechanics can be also deduced. In Fig. 4 the intrinsic fracture tough- studies. The thermal residual stresses in the A- and AZ-lay- ness values of the A- and AZ-material are indicated (com- ers are denoted dAar <0 and oMar >0. An analytical esti- pare Table 1). The fracture load can be estimated as the mate of the strain energy density and the thermal residual intersection point of these horizontal lines with the Jtip ver- stresses for the actual configuration under plane strain con- sus F-curve. Note that the material is rather brittle; it thus ditions is given in Appendix 1. For the data at hand the exhibits only a small process zone in front of the crack tip evaluation yields GAar=-144 MPa and oMAr=166 MPa. where the microscopic processes of micro-crack formation and growth take place, which lead to brittle fracture 5.2. Elastically homogeneous material with inhomogeneity in Therefore, it can be assumed that the fracture resistance CTE of the composite is primarily determined by the material in which the process zone is located. Due to the residual There are two sources of shielding and anti-shielding stre esses specific fracture load of the specimen with effects in our multilayer composite: the spatially varying a=0.20 mm increases from Fr 2.5N/mm to residual stresses and the different elastic moduli of the lam- Ffr A 20 N/mm. For the specimen with a=0.18 mm the inae. To separate the two effects, we will first present the shift of the fracture load is even larger, from results for an elastically homogeneous material with spa- Ffr R 10 N/mm to Ffr R 22 N/mm. tially varying thermal residual stresses. It is assumed that The influence of the crack length a on the crack driving the whole specimen has the elastic properties of material force at a constant loading is shown in Fig 5a and b Plot A. The CtE shows a spatial variation with values of a ted are the effective crack driving force Jtip and the term as defined in Section 4.2. To get in the elastically homoge- Far -Jar(0). Far denotes the far-field J-integral for the leous composite exactly the same residual stresses as composite with crack length a, and the expression Faro) ppear in the elastically inhomogeneous composite, describes the far-field J-integral for the composite with zero Mar =-144 MPa and d,ar=166 MPa, the effective tem- crack length(see Appendix A2. 2). Note that for a loaded perature difference was set to△T=-1007.3°C component which contains residual stresses Faro) is non- In Fig 4 the crack driving force Jtip is plotted against the zero and depends on the length of the integration path specific load F=F/(2B) Presented are the curves for two L,, see Appendix A2. 2 and Eq(A2.3). For a componentcrack tip comes close to an interface. All these points show the advantages of the configurational force concept over the conventional J-integral approach. 5. Results and discussion In the following, the results of the numerical analyses for our multilayer composite are presented. Corresponding analytical evaluations which are helpful for the discussion of the results are shown in Appendix. In Appendix 1 the strain energy density and the thermal residual stresses are derived for the composite under plane strain conditions. In Appendix 2 useful analytical estimates of the crack driv￾ing force are given. 5.1. Thermal residual stresses In Figs. 2 and 3 the results of the 3D and 2D computa￾tions are compared. As already experienced in a study by Shan et al. [30], nearly 80% of the cross-section measured in the z-direction shows residual stresses ry,DT according to the plane strain model. In the side-surface region, say within 1.0 6 z 6 1.25 mm, the residual stresses can be roughly approximated by the plane stress distribution. Directly at the side-surface the absolute values of ry,DT can, however, become significantly smaller than the plane stress values. It should be mentioned that at the positions where the interfaces 1–6 impinge the surface, weak stress singularities for ry,DT occur which can be ignored. Since 80% of the specimen is controlled by plane strain conditions, we keep this simple eigenstress distribution for ry,DT as the relevant one for our further fracture mechanics studies. The thermal residual stresses in the A- and AZ-lay￾ers are denoted rA y;DT < 0 and rAZ y;DT > 0. An analytical esti￾mate of the strain energy density and the thermal residual stresses for the actual configuration under plane strain con￾ditions is given in Appendix 1. For the data at hand the evaluation yields rA y;DT ¼ 144 MPa and rAZ y;DT ¼ 166 MPa. 5.2. Elastically homogeneous material with inhomogeneity in CTE There are two sources of shielding and anti-shielding effects in our multilayer composite: the spatially varying residual stresses and the different elastic moduli of the lam￾inae. To separate the two effects, we will first present the results for an elastically homogeneous material with spa￾tially varying thermal residual stresses. It is assumed that the whole specimen has the elastic properties of material A. The CTE shows a spatial variation with values of a* as defined in Section 4.2. To get in the elastically homoge￾neous composite exactly the same residual stresses as appear in the elastically inhomogeneous composite, rA y;DT ¼ 144 MPa and rAZ y;DT ¼ 166 MPa, the effective tem￾perature difference was set to DT = 1007.3 C. In Fig. 4 the crack driving force Jtip is plotted against the specific load F b ¼ F =ð2BÞ. Presented are the curves for two specific crack lengths, a = 0.18 mm (crack tip 0.01 mm behind the first interface) and a = 0.20 mm (crack tip 0.01 mm beyond of the first interface). For comparison, the curves of the completely homogeneous specimen with￾out residual stresses are also given, and these exhibit the common quadratic dependency on the load. The residual stresses shift the origin of the curves. Since the residual stress state in the first layer is a compressive one and the bending stresses ry,b are tensile, the crack will open at a load F0 when the bending stresses balance the residual stresses ry;bmax ¼ rA y;DT ; compare Eq. (A2.6). From this equation, the quadratic dependency of Jtip on (F F0) can be also deduced. In Fig. 4 the intrinsic fracture tough￾ness values of the A- and AZ-material are indicated (com￾pare Table 1). The fracture load can be estimated as the intersection point of these horizontal lines with the Jtip ver￾sus F b-curve. Note that the material is rather brittle; it thus exhibits only a small process zone in front of the crack tip where the microscopic processes of micro-crack formation and growth take place, which lead to brittle fracture. Therefore, it can be assumed that the fracture resistance of the composite is primarily determined by the material in which the process zone is located. Due to the residual stresses, the specific fracture load of the specimen with a = 0.20 mm increases from F bfr ¼ 12:5 N=mm to F bfr 20 N=mm. For the specimen with a = 0.18 mm the shift of the fracture load is even larger, from F bfr 10 N=mm to F bfr 22 N=mm. The influence of the crack length a on the crack driving force at a constant loading is shown in Fig. 5a and b. Plot￾ted are the effective crack driving force Jtip and the term Jfar Jfar(0). Jfar denotes the far-field J-integral for the composite with crack length a, and the expression Jfar(0) describes the far-field J-integral for the composite with zero crack length (see Appendix A2.2). Note that for a loaded component which contains residual stresses Jfar(0) is non￾zero and depends on the length of the integration path Ly; see Appendix A2.2 and Eq. (A2.3). For a component 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 J0 (A) J0 (AZ) with residual stress a = 0.18 mm a = 0.20 mm Homogeneous 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. 4. Effective crack driving force Jtip as a function of the specific loading F/2B for the homogeneous specimen (material A) with and without residual stresses. 414 C.R. Chen et al. / Acta Materialia 55 (2007) 409–421