C R Chen et al. Acta Materialia 55(2007)409-421 sider the line integral along a contour Ifar around the crack Generally, both Far and Cinh depend on the crack length ip following Eq (4)(see Fig. 1). The following specifica- a. They also depend on Ly, but Eq (3)delivers Jtip values tions can be made which are independent of Ly. In the further context the notation Jtip(a) and Far(a) is sometimes used Due to the symmetry condition, the path along 0≤x≤h,y=0 provides no contribution to Far A2. 2. Crack driving force for zero crack length The x-component of the unit normal vector to the inte- gration path nx is zero along<x< h, y=L, note that In the case of zero crack length. the stress state at x=0 we refer here to the(undeformed) reference configura- is tion when calculating material forces. The x-component of the traction vector t is zero if Ly is so large that both = Oy.b the stress-relieved zone in the wake of the crack and the and at x=h is rack tip field hav ve neglig train conditions: Ly> Ko ca, where k is a positive factor a,l=h=-oy, bar + yar in the order of magnitude I. The y-component of the where o, bma denotes the maximum stresses in the y-direc traction vector ly is equal to ay. Therefore, only the term tion due to the bending and dAar the thermal residual stres- The y-component of the displacement vector ly, is dis- A follow (-layers. From Eq (A2. 1)witha=0in material ly au/ax is of relevand tributed linearly in the x-direction; au/ax is constant The integral of the stresses due to bending far(0)=-(4Ly/EA)oy, bmar OAT (A23) 5o Oy, b(au, /ax)dr is 0, since o, b has an antimetric distri- Thus, if one of the two stress components is zero, the far bution with respect to x= h/2. The integral of the ther- field J-integral for the component without a crack becomes mal stresses Jo o, Ar(Ouy /ax)dr is 0, since equilibrium zero. This is the case for an elastically inhomogeneous enforces Jo Oy, Ar dx=0. Therefore, the integral Eq (4) specimen without residual stresses at arbitrary loading, as becomes zero for0≤x≤h,y=L well as for an unloaded specimen with residual stresses . The traction vector t must be zero at the free surfaces at Next we consider the material inhomogeneity term. We x=0 and x=h introduce a dimensionless coordinate,0≤ξ≤1,with E=0 at x=h/2 and with x=l at x=h. Then xi corre- The consequence is that the far-field J-integral Eq (4) sponds to -S3, x2 to -52, x] to -51, x4 to 51, xs to 52, depends only on the distribution of the strain energy den- and x6 to s3. Following this notation, the term sity along the two surfaces and can be rewritten as (x=xir, y)-(x=xil, D)=-l(xi, y)I Jfr=2/((r=h, y)-(r=0,y)dy (A2. 1) withi=1, 2,..., 6 transforms to-Ip(S, y)lor-[P(-s, y)I ith j=1, 2, 3. Inserting the stresses due to bending and For a further analysis the specific elastic strain energy den- thermal misfit, we get for the jump of the strain energy den- sity can be inserted from Eq(A1.3) sity at the interface for j= l and 3 中=(/E+E(x△T)/2 with E and a belonging to either the A- or AZ-material (m=-[+哈 Note again that or depends on both the external loading (O, bma .EAZ/EA. 5+oAZ-)2.EA/EAZ/2EA and the thermal expansion/contraction Next we consider the material inhomogeneity term Cinh -EAz(x△)2/2, (see Eq (5)and Fig. 1). The second term in the integrand of (-m=-[+kmE/E+%) Cinh. i becomes zero, since: the stress components in the x-direction are small and nore the stress components in the z-direction do not play a The sum of both yields after some algebra role as the corresponding total strain components are -[d(5,y)+d(-/,y)]=20 zero the strain components [EyaJl=[[Eylllex have no jump. For an even number of interfaces 1, ensuring compressive residual stresses in the surface A-layers, the sum of the The consequence is that Cinh. i can be written as jumps above yields with Eqs.(5)and(A2. 2) 2/(p(=xir, y)-(x=xu, y)dy (A22)Cm(O)/2y Mr)/EA·∑(-1)y5 with xir and xi denoting the position at the right and left side of the ith interface, respectively (A24)sider the line integral along a contour Cfar around the crack tip following Eq. (4) (see Fig. 1). The following specifica￾tions can be made:  Due to the symmetry condition, the path along 0 6 x 6 h, y = 0 provides no contribution to Jfar.  The x-component of the unit normal vector to the inte￾gration path nx is zero along 0 6 x 6 h, y = Ly; note that we refer here to the (undeformed) reference configura￾tion when calculating material forces. The x-component of the traction vector tx is zero if Ly is so large that both the stress-relieved zone in the wake of the crack and the crack tip field have negligible effect on the stress and strain conditions: Ly P ja, where j is a positive factor in the order of magnitude 1. The y-component of the traction vector ty is equal to ry. Therefore, only the term tyouy/ox is of relevance.  The y-component of the displacement vector uy is dis￾tributed linearly in the x-direction; ouy/ox is constant. The integral of the stresses due to bending R h 0 ry;bðouy=oxÞdx is 0, since ry,b has an antimetric distri￾bution with respect to x = h/2. The integral of the ther￾mal stresses R h 0 ry;DT ðouy=oxÞdx is 0, since equilibrium enforces R h 0 ry;DT dx ¼ 0. Therefore, the integral Eq. (4) becomes zero for 0 6 x 6 h, y = Ly.  The traction vector t must be zero at the free surfaces at x = 0 and x = h. The consequence is that the far-field J-integral Eq. (4) depends only on the distribution of the strain energy den￾sity along the two surfaces and can be rewritten as Jfar ¼ 2 Z Ly 0 ð/ðx ¼ h; yÞ /ðx ¼ 0; yÞÞdy ðA2:1Þ For a further analysis the specific elastic strain energy den￾sity can be inserted from Eq. (A1.3), / ¼ ðr2 y=E þ Eða DT Þ 2 Þ=2 with E and a* belonging to either the A- or AZ-material. Note again that ry depends on both the external loading and the thermal expansion/contraction. Next we consider the material inhomogeneity term Cinh (see Eq. (5) and Fig. 1). The second term in the integrand of Cinh,i becomes zero, since:  the stress components in the x-direction are small and can be ignored;  the stress components in the z-direction do not play a role as the corresponding total strain components are zero.  the strain components ½½ey;i ¼ ½½ey jx¼xi have no jump. The consequence is that Cinh,i can be written as Cinh;i ¼ 2 Z Ly 0 ð/ðx ¼ xi;r; yÞ /ðx ¼ xi;l; yÞÞdy ðA2:2Þ with xi,r and xi,l denoting the position at the right and left side of the ith interface, respectively. Generally, both Jfar and Cinh depend on the crack length a. They also depend on Ly, but Eq. (3) delivers Jtip values which are independent of Ly. In the further context the notation Jtip(a) and Jfar(a) is sometimes used. A2.2. Crack driving force for zero crack length In the case of zero crack length, the stress state at x = 0 is ry j x¼0 ¼ ry;bmax þ rA y;DT and at x = h is ry jx¼h ¼ ry;bmax þ rA y;DT where ry;bmax denotes the maximum stresses in the y-direc￾tion due to the bending and rA y;DT the thermal residual stres￾ses in the A-layers. From Eq. (A2.1) with a* = 0 in material A follows Jfarð0Þ ¼ ð4Ly=E AÞry;bmax rA y;DT ðA2:3Þ Thus, if one of the two stress components is zero, the far- field J-integral for the component without a crack becomes zero. This is the case for an elastically inhomogeneous specimen without residual stresses at arbitrary loading, as well as for an unloaded specimen with residual stresses. Next we consider the material inhomogeneity term. We introduce a dimensionless coordinate n, 0 6 n 6 1, with n = 0 at x = h/2 and with x = 1 at x = h. Then x1 corre￾sponds to n3, x2 to n2, x3 to n1, x4 to n1, x5 to n2, and x6 to n3. Following this notation, the term ½/ðx ¼ xir; yÞ /ðx ¼ xi;l; yÞ ¼ ½½/ðxi; yÞ with i = 1, 2,..., 6 transforms to [[/(nj,y)]] or [[/(nj,y)]] with j = 1, 2, 3. Inserting the stresses due to bending and thermal misfit, we get for the jump of the strain energy den￾sity at the interface for j = 1 and 3 ½½/ðnj; yÞ ¼ ðry;bmax nj þ rA yDT Þ 2 h ðry;bmax E AZ=E A nj þ rAZ y;DT Þ 2 E A=E AZi =2E A EAZða AZDT Þ 2 =2; ½½/ðnj; yÞ ¼ ðþry;bmax E AZ=E A nj þ rAZ y;DT Þ 2 h ðþry;bmax nj þ rA y;DT Þ 2 E A=E AZi =2E A EAZða AZDT Þ 2 =2: The sum of both yields after some algebra ½½/ðnj; yÞ þ /ðnj; yÞ ¼ 2ry;bmax nj ðrA y;DT rAZ y;DT Þ=E A For an even number of interfaces I, ensuring compressive residual stresses in the surface A-layers, the sum of the jumps above yields with Eqs. (5) and (A2.2) Cinhð0Þ=2Ly ¼ 2ry;bmax ðrA y;DT rAZ y;DT Þ=E A XI=2 j¼1 ð1Þ j nj ðA2:4Þ C.R. Chen et al. / Acta Materialia 55 (2007) 409–421 419