C R Chen et al. Acta Materialia 55(2007)409-421 plane strain ri plane stress plane strain z[mm Fig 3. Thermal residual stresses , AT along the z-direction for various values of y for a plane stress, plane strain and three-dimensional (3D) model. Jfar is the classical J-integral of fracture mechanics. For a done by specifying the set of nodes on the interface as crack growing in the x-direction, Jfar is virtual crack tip nodes. Even a contour directly adjacent to the interface yields very accurate results. For the eval- (4) uation of the J-integral, the virtual crack growth direc tion must be specified; this is the(,0)-direction, as for The components t;(t;=fr,fy)of the traction vector t along he evaluation of Je the contour I follow from the stress tensor g with the rel-. Generally, both Far and Cinh depend on the crack length evant components ox, Oy, txy as I=g n. Note that the a. They also depend on Ly, but produce Jtip values stress components are the sum of their contributions due according to Eq .(3)which are independent of L,; for to bending and the residual stress state, e.g. 0,=0x, b+ details see Appendix 2 At. The components u;(u;=ux, ly) of the displacement vector u are differentiated with respect to the crack growth After the finite element stress analysis, the material i direction, i. e. the x-direction. The quantity o is the specific mogeneity term Cinh is calculated from Eq. (5)by a elastic strain energy and nx is the x-component of the unit processing procedure. The integration along the interface normal vector n to the integration path I is performed using the trapezoid formula. For this, the The material inhomogeneity term can be evaluated by node values of the stress and strain components and the strain energy density are taken, which are extrapolated val ues from the Gauss integration points. The far-field J-inte- Cnh=>Cinh i, Cinh =-2/(l,-(g)le)dy (5) gral Jar is calculated using the virtual crack extension method of ABAQUS. Then the near-tip crack driving force The jump [b] and the average (b)of a quantity b at an Jtip is calculated from Eq (3). The numerical results will be interface are defined presented in the following section [b]=b-b,(b)=(b+b) It should be noted that Sun and Wu[29]have calculated the effective crack driving force by replacing the region far where br and b denote the limiting values of the quantity b by subregions, each including only one layer, and applying on the right and left side of the interface, respectively. The the J-integral procedure for each individual layer. The index i refers to the individual interface; the integer I de- strength of the configurational forces concept lies in its gen notes the total number of interfaces in the specimen, in eral applicability. The material inhomogeneity can be our case I =6 either a sharp interface with a discrete jump of the material The following comments may be useful properties or a region where the material properties change continuously. The Cinh-evaluation procedure can be The multiplier 2 in Eqs. (4)and(5)points to the fact that applied to any arbitrary spatial distribution of these mate- only the upper half of a symmetric configuration with rial inhomogeneities in both elastic and elastic-plastic respect to I is considered. materials. In general, the evaluation of Cinh can be per The material inhomogeneity terms Cinh. i can be also formed very accurately. This enables us to evaluate Jtip as found via the J-integral calculation routine provided the sum of Far and Cinh more accurately than it would be by the finite element code by evaluating the J-integral possible from the calculation of Jtip using the conventional around the ith interface Jint. i[26]. In ABAQUS, this is J-evaluation procedures, especially for cases when theJfar is the classical J-integral of fracture mechanics. For a crack growing in the x-direction, Jfar is Jfar ¼ 2 Z C /nx ti oui ox ds: ð4Þ The components ti (ti = tx,ty) of the traction vector t along the contour C follow from the stress tensor r with the rel￾evant components rx, ry, sxy as t = r Æ n. Note that the stress components are the sum of their contributions due to bending and the residual stress state, e.g. ry = ry,b + ry,DT. The components ui (ui = ux,uy) of the displacement vector u are differentiated with respect to the crack growth direction, i.e. the x-direction. The quantity / is the specific elastic strain energy and nx is the x-component of the unit normal vector n to the integration path C. The material inhomogeneity term can be evaluated by [28] Cinh ¼ XI i¼1 Cinh;i; Cinh;i ¼ 2 Z Ly 0 ð½½/i hrii½½eiÞdy ð5Þ The jump [[b]] and the average Æbæ of a quantity b at an interface are defined as ½½b ¼ br bl; hbi¼ðbl þ brÞ=2; ð6Þ where br and bl denote the limiting values of the quantity b on the right and left side of the interface, respectively. The index i refers to the individual interface; the integer I de￾notes the total number of interfaces in the specimen, in our case I = 6. The following comments may be useful:  The multiplier 2 in Eqs. (4) and (5) points to the fact that only the upper half of a symmetric configuration with respect to C is considered.  The material inhomogeneity terms Cinh,i can be also found via the J-integral calculation routine provided by the finite element code by evaluating the J-integral around the ith interface Jint,i [26]. In ABAQUS, this is done by specifying the set of nodes on the interface as virtual crack tip nodes. Even a contour directly adjacent to the interface yields very accurate results. For the eval￾uation of the J-integral, the virtual crack growth direc￾tion must be specified; this is the (1, 0)-direction, as for the evaluation of Jfar.  Generally, both Jfar and Cinh depend on the crack length a. They also depend on Ly, but produce Jtip values according to Eq. (3) which are independent of Ly; for details see Appendix 2. After the finite element stress analysis, the material inho￾mogeneity term Cinh is calculated from Eq. (5) by a post￾processing procedure. The integration along the interface is performed using the trapezoid formula. For this, the node values of the stress and strain components and the strain energy density are taken, which are extrapolated val￾ues from the Gauss integration points. The far-field J-inte￾gral Jfar is calculated using the virtual crack extension method of ABAQUS. Then the near-tip crack driving force Jtip is calculated from Eq. (3). The numerical results will be presented in the following section. It should be noted that Sun and Wu [29] have calculated the effective crack driving force by replacing the region Xfar by subregions, each including only one layer, and applying the J-integral procedure for each individual layer. The strength of the configurational forces concept lies in its gen￾eral applicability. The material inhomogeneity can be either a sharp interface with a discrete jump of the material properties or a region where the material properties change continuously. The Cinh-evaluation procedure can be applied to any arbitrary spatial distribution of these mate￾rial inhomogeneities in both elastic and elastic-plastic materials. In general, the evaluation of Cinh can be per￾formed very accurately. This enables us to evaluate Jtip as the sum of Jfar and Cinh more accurately than it would be possible from the calculation of Jtip using the conventional J-evaluation procedures, especially for cases when the Fig. 3. Thermal residual stresses ry,DT along the z-direction for various values of y for a plane stress, plane strain and three-dimensional (3D) model. C.R. Chen et al. / Acta Materialia 55 (2007) 409–421 413