C R Chen et al. Acta Materialia 55(2007)409-421 The alternating sign in the sum corresponds to the fact that K,=1. 12(-x bm +rar)via material A on the right and az on the left side of the inter- face switches to AZ on the right and a on the left side. It is utilizing the well-known relation J=K/E. This compar- inhomogeneous specimen without residual stresses at arbi- are weighted averages, tip can be considered as a quadratic trary loading, as well as for a specimen with residual stres- form in the variables F and EAx'AT For a large crack one may assume that the stress state, stresses is interrupted in the region 0<x< a, and the Jp(0)/2L,=-(20yb/EA) rather"distorted"stress distribution is now approximated by a stress distribution r a which shall be linearly distrib A△x+(2△r=M) uted in a≤x≤ h and independent of y for0≤y≤Ka These assumptions yield The expression in the bracket [ l however, is the global E,a=B-/h-a)+B2 equilibrium expression for the thermal stresses ,, AT in E'(B1x/(h-a)+B2-x2△7) the interval 0 < s< l, ensuring[ that must be 0. There fore, the following relations hold with a, E belonging either to the A-or AZ-material and B1, B2 as two constants. The equilibrium conditions requir Jmp(0)≡0,Jar(0)≡-Cmh(0) (A2. 5) that the integration of oy, a over x in the interval sxsa must be zero and the integration of xo, a must yield the moment per unit thickness M/B= F(S1-S2)/2B. This It is interesting to note that a significant thermodynamical task can easily be performed by any finite element pro- provoke an interface motion for interfaces with certain gram. The coefficients P1, B2 are linearly dependent on mobility. However, the total driving force on the crack △x*△T( with E' being△x* weighted averages over tip is zero 0≤x≤a) and the dimension- free moment term M, B1=B1△x△T+B12Mand A2.3. Crack driving force for non-zero crack length B2=B21△x△T+B2M (A20 0≤x≤hand0≤y≤ ka is affected; the original stress h and the, interface coordinates x,,andM state remains undisturbed for y>Ka. According to the M/(E*(h-a)B). The strain energy density a follows proof above, this undisturbed part of the specimen does from Eq(A1.3)with Eq(A2.9)as not influence Jtip. The further conclusion is that Jtip() is path-independent for Ly ka(and provided that the path sa(x)=E(B1/(h-a)+B2-x'AT)/2 does not cross an interface) +E(x*△7)/2 (A2.11 If the crack is very small, its length a being only a small Finally, a (x) becomes a quadratic form in(E'Ax )AT and fraction of the thickness of the first lamina, it can be m. The coefficients of this form are rather complicated ra- assumed that the stress state of the uncracked specimen tional expressions which also include weighted averages of can be used as an approximation for regions not influenced the material data. These are very lengthy expressions and gh estimate not listed in this paper. However, they can be calculated due to the interruption of the stress flow in the y-direction by any mathematics processor e.g. Maple(httP: // influence zone of the crack -ka< ys ka and for x=0. Jfar and Cinh can be evaluated from Eqs. (A2. 1)and Then Far becomes(compare Appendix A2.1) (A2. 2). With Eq (3) then follows the crack driving force Far N Jar(0)+2Ka(-Oy bma,+ GAa)"/(2EA) (A2.6)Jtip, which will appear in the form We can further assume that the material inhomogeneity Jtip 2xa o(h)-((r=xir-pa(r=xi)) term will be not affected by the short crack: Cinh A Cinh(0) From Eq.(3)and the condition Far(0)+ Cinh(0)=0 then 甲p≈2Aa(-aym+2ar)2/(2E) The summation in Eq (A2. 11) starts from i= is, which is the number of the first interface not intersected by the We can compare this solution with the stress intensity fac- crack. It should be noted that Jtip estimated by Eq tor for a small surface crack (A2. 12)reflects only the global effect of a crack in a layeredThe alternating sign in the sum corresponds to the fact that material A on the right and AZ on the left side of the inter￾face switches to AZ on the right and A on the left side. It is seen from Eq. (A2.4) that the material inhomogeneity term of the component without a crack is zero for an elastically inhomogeneous specimen without residual stresses at arbi￾trary loading, as well as for a specimen with residual stres￾ses if no external loading applies. If we add to Eq. (A2.4) the corresponding term Jfar/2Ly from Eq. (A2.3), we find Jtipð0Þ=2Ly ¼ ð2ry;bmax =E AÞ rA y;DT þ ðrA y;DT rAZ yDT Þ XI=2 j¼1 ð1Þ j nj " #: The expression in the bracket [ Æ ], however, is the global equilibrium expression for the thermal stresses ry,DT in the interval 0 6 n 6 1, ensuring [ Æ ] that must be 0. There￾fore, the following relations hold: Jtipð0Þ 0; Jfarð0Þ Cinhð0Þ XI i¼1 Cinh;i ðA2:5Þ It is interesting to note that a significant thermodynamical force Cinh acts on each interface which would probably provoke an interface motion for interfaces with certain mobility. However, the total driving force on the crack tip is zero. A2.3. Crack driving force for non-zero crack length If we now introduce a crack with length a, only a region 0 6 x 6 h and 0 6 y 6 ja is affected; the original stress state remains undisturbed for y P ja. According to the proof above, this undisturbed part of the specimen does not influence Jtip. The further conclusion is that Jtip(a) is path-independent for Ly P ja (and provided that the path does not cross an interface). If the crack is very small, its length a being only a small fraction of the thickness of the first lamina, it can be assumed that the stress state of the uncracked specimen can be used as an approximation for regions not influenced by the crack. For a rough estimate of Jfar, we assume that, due to the interruption of the stress flow in the y-direction by the crack mouth, the stresses are reduced to zero in the influence zone of the crack ja 6 y 6 ja and for x = 0. Then Jfar becomes (compare Appendix A2.1) Jfar Jfarð0Þ þ 2jaðry;bmax þ rA y;DT Þ 2 =ð2E AÞ ðA2:6Þ We can further assume that the material inhomogeneity term will be not affected by the short crack: Cinh Cinh(0). From Eq. (3) and the condition Jfar(0) + Cinh(0) = 0 then follows Jtip 2jaðry;bmax þ rA y;DT Þ 2 =ð2E AÞ ðA2:7Þ We can compare this solution with the stress intensity fac￾tor for a small surface crack KI ¼ 1:12ðry;bmax þ rA y;DT Þ ffiffiffiffiffi pa p utilizing the well-known relation J ¼ K2 I =E . This compar￾ison yields j 3.94. Since ry,bmax depends linearly on the load F and rA y;DT linearly on EeD gaDT , where Ee and D ga are weighted averages, Jtip can be considered as a quadratic form in the variables F and EeD gaDT . For a large crack one may assume that the stress state, now denominated as ry,a, exists only in the region a 6 x 6 h, 0 6 y 6 ja. In other words, the transfer of the stresses is interrupted in the region 0 6 x 6 a, and the rather ‘‘distorted’’ stress distribution is now approximated by a stress distribution ry,a which shall be linearly distrib￾uted in a 6 x 6 h and independent of y for 0 6 y 6 ja. These assumptions yield ey;a ¼ b1x=ðh aÞ þ b2 ðA2:8Þ ry;a ¼ E ðb1x=ðh aÞ þ b2 a DT Þ ðA2:9Þ with a*, E* belonging either to the A- or AZ-material and b1,b2 as two constants. The equilibrium conditions require that the integration of ry,a over x in the interval 0 6 x 6 a must be zero and the integration of xry,a must yield the moment per unit thickness M/B = F Æ (S1 S2)/2B. This task can easily be performed by any finite element pro￾gram. The coefficients b1, b2 are linearly dependent on D gaDT (with Ee being D ga weighted averages over 0 6 x 6 a) and the dimension-free moment term Me , b1 ¼ b11D gaDT þ b12Me and b2 ¼ b21D gaDT þ b22Me ðA2:10Þ The coefficients b11, b12, b21, b22 are rational functions in a, h and the interface coordinates xi, and Me ¼ M=ðEeðh aÞ 2 BÞ. The strain energy density /a follows from Eq. (A1.3) with Eq. (A2.9) as /aðxÞ ¼ E ðb1x=ðh aÞ þ b2 a DT Þ 2 =2 þ E ða DT Þ 2 =2 ðA2:11Þ Finally, /a(x) becomes a quadratic form in ðEeD ga ÞDT and Me . The coefficients of this form are rather complicated ra￾tional expressions which also include weighted averages of the material data. These are very lengthy expressions and not listed in this paper. However, they can be calculated by any mathematics processor, e.g. MAPLE (http:// www.maple.com). With the strain energy density known, Jfar and Cinh can be evaluated from Eqs. (A2.1) and (A2.2). With Eq. (3) then follows the crack driving force Jtip, which will appear in the form Jtip 2ja /aðhÞ XI i¼is ð/ðx ¼ xi;rÞ /aðx ¼ xi;lÞÞ " # ðA2:12Þ The summation in Eq. (A2.11) starts from i = is, which is the number of the first interface not intersected by the crack. It should be noted that Jtip estimated by Eq. (A2.12) reflects only the global effect of a crack in a layered 420 C.R. Chen et al. / Acta Materialia 55 (2007) 409–421