w. Lee et al. I Composites Science and Technology 66(2006)435-443 Interface Penetrating Crack Fig. 3.(a) Finite element mesh and load-and-boundary conditions used for the numerical analysis (deformed after loading).(b) Meshing details around the crack tip region (o: uniaxial end displacement mary crack. It was assumed that ap =ad as in the case of smaller than the size of the K-dominant field. The sensi- other analyses in existing works [9, 1l-13]and that r=0 tivity of ga/p ratio(crack deflection criteria) on the for comparison purpose. In Fig 4(a), sa/sp ratios(also relative size of the crack extension with respect to the critical R /Rm ratios for crack deflection)calculated by length of the parent crack described herein is qualita- FEM for various ap/h(=aJ/h=a/h)are plotted as func- tively consistent with what explained in Ahn et al. [28 tions of a and corresponding phase angles of the infini- and Leguillon et al. [13] tesimal deflection is also compared in Fig. 4(b). It can be Having shown that convergence of the numerical seen in Fig 4 that as a/h decreases, both the sa/sp ratio olution was achieved when a/h=1x 10-4. effect of and the phase angle tend to converge toward those for a/ the thermal misfit strain was investigated by repeating h=l x 10. The curves for this length scale of the infin- the estimation of sa/sp(and therefore critical R/Rmra tesimal putative crack extension are identical to the tio) for various values of I with this normalised putative crack tip field solutions reported by Martinez and Gup- crack extension length. Fig. 5 shows the numerical re ta [11] and He et al. [ 12] for nt=0 indicating that the sults(Sa/s ratios) plotted as functions of Dundurs numerical solution converged to the crack tip field solu- parameter x. It can be seen in the figure that the tion when a/h was chosen to be 1 x 10 Sals ratio is insensitive to the thermal residual stress One interesting feature found in Fig. 4(a) is that when a <o whilst the effect of the residual stress be. Sa/ p is not sensitive to the choice of length scale for comes more significant as a increases from about zero ad(and also ap) when a<-0.3 whilst its dependence to higher values. It can also be seen that the effect of on a becomes more significant as a increase toward residual stress is more noticeable for negative values of 1.0. This trend can be understood on referring to the di Is. It is predicted that ratio for negative Is are cussion on the size of the K-dominant field of a crack lower than that for I=0 and in particular, when terminating at a bi-material interface presented by Ro- T=-1.0, a/p ratio falls toward 0 as a approaches meo and Ballarini [29], according to which it was pre- about 0.9 meaning that it is almost impossible to achieve dicted that the region of K-dominance becomes larger crack deflection in such a case. Changes in a/p ratio with decreasing o and almost vanishes when o is higher for Is lower than -1. 0 are not shown since the finite ele than about 0.5. Decreasing size of the K-dominant field ment analysis predicted interpenetration of the surfaces means that strain energy release rate of the infinitesimal of the primary crack. In practice, this would result the global loading since K-dominance holds only when due to the negative component of the stress intensity a crack extension would be more likely to be influenced by crack closure and thus modify the crack tip stress fiel the fracture process is confined to the scale substantially tor of the primary crack arising from the crack closuremary crack. It was assumed that ap = ad as in the case of other analyses in existing works [9,11–13] and that C = 0 for comparison purpose. In Fig. 4(a), Gd=Gp ratios (also critical Ri/Rm ratios for crack deflection) calculated by FEM for various ap/h (= ad/h = a/h) are plotted as func￾tions of a and corresponding phase angles of the infini￾tesimal deflection is also compared in Fig. 4(b). It can be seen in Fig. 4 that as a/h decreases, both the Gd=Gp ratio and the phase angle tend to converge toward those for a/ h = 1 · 10
4 . The curves for this length scale of the infin￾itesimal putative crack extension are identical to the crack tip field solutions reported by Martinez and Gup￾ta [11] and He et al. [12] for gt = 0 indicating that the numerical solution converged to the crack tip field solu￾tion when a/h was chosen to be 1 · 10
4 . One interesting feature found in Fig. 4(a) is that Gd=Gp is not sensitive to the choice of length scale for ad (and also ap) when a <
0.3 whilst its dependence on a becomes more significant as a increase toward 1.0. This trend can be understood on referring to the dis￾cussion on the size of the K-dominant field of a crack terminating at a bi-material interface presented by Ro￾meo and Ballarini [29], according to which it was pre￾dicted that the region of K-dominance becomes larger with decreasing a and almost vanishes when a is higher than about 0.5. Decreasing size of the K-dominant field means that strain energy release rate of the infinitesimal crack extension would be more likely to be influenced by the global loading since K-dominance holds only when the fracture process is confined to the scale substantially smaller than the size of the K-dominant field. The sensi￾tivity of Gd=Gp ratio (crack deflection criteria) on the relative size of the crack extension with respect to the length of the parent crack described herein is qualita￾tively consistent with what explained in Ahn et al. [28] and Leguillon et al. [13]. Having shown that convergence of the numerical solution was achieved when a/h = 1 · 10
4 , effect of the thermal misfit strain was investigated by repeating the estimation of Gd=Gp (and therefore critical Ri/Rm ra￾tio) for various values of C with this normalised putative crack extension length. Fig. 5 shows the numerical re￾sults (Gd=Gp ratios) plotted as functions of Dundurs
parameter a. It can be seen in the figure that the Gd=Gp ratio is insensitive to the thermal residual stress when a < 0 whilst the effect of the residual stress be￾comes more significant as a increases from about zero to higher values. It can also be seen that the effect of residual stress is more noticeable for negative values of Cs. It is predicted that Gd=Gp ratio for negative Cs are lower than that for C = 0 and in particular, when C =
1.0, Gd=Gp ratio falls toward 0 as a approaches about 0.9 meaning that it is almost impossible to achieve crack deflection in such a case. Changes in Gd=Gp ratio for Cs lower than
1.0 are not shown since the finite ele￾ment analysis predicted interpenetration of the surfaces of the primary crack. In practice, this would result in crack closure and thus modify the crack tip stress field due to the negative component of the stress intensity fac￾tor of the primary crack arising from the crack closure x y Interface Interface δ Penetrating Crack Deflected Crack a b Fig. 3. (a) Finite element mesh and load-and-boundary conditions used for the numerical analysis (deformed after loading). (b) Meshing details around the crack tip region (d: uniaxial end displacement). 438 W. Lee et al. / Composites Science and Technology 66 (2006) 435–443