w. Lee et al./ Composites Science and Technology 66(2006)435-443 C=0.6 34567 Fig. 8. Change in sa/p ratio plotted as a function of I for two selected values of z. were plotted as functions of I for two selected axs, inate composites be designed in such a way that the con namely 0.6 and 0.8, in Fig 8. It is noticed in Fig 8 that dition of positive a and r is maintained to take the slope of the curves, i.e. a(sa/p)/ar, decreases as r advantage of beneficial effect of the misfit stress, i.e. crit increases In particular, when T is higher than about 1.0 ical fracture energy of the crack deflecting interface can for a=0.6 and 3.0 for a=0.9, respectively, be increased through the incorporation of in-plane resid a(sa/p)/ar is close to zero. Similar tendency was pre- ual stress thereby improving overall mechanical proper dicted in the analysis of the behaviour of an interface ties of the laminate crack in a bi-layer system under the presence of in-plane residual stress [30]. Thus, it is deduced that once the magnitude of the residual stress exceeds a certain value Acknowledgements further improvement in the properties of layered system would be marginal This work was supported in part (W. Lee and Pre 5. Conclusions gramme for Young Scientists(Grant No. RO8-2004- 000-10193-0) Effect of thermal misfit stain induced by mismatch of CtE on the crack deflection at planar interfaces in lay red systems was investigated using a finite element References analysis. In a convergence test of the numerical solution, dependence of the deflection criteria on the choice of I] Howard SJ, Stewart RA, Clegg WJ. The delamination of ceramic length scale of the infinitesimal putative penetration laminates due to residual thermal stresses. Key Eng Mater across the interface or deflection along the interface 1996:116-117:331-50 was noticed only in the positive a regime, which is con Blattner AJ, Lakshminarayanan R, Shetty DK. Toughening of sistent with the previous works by Ahn et al. [28] and layered ceramic composites with residual surface compression effects of layer thickness. Eng Fract Mech 2001: 68(1): 1-7 Leguillon et al. [13]. When residual stress effect was con- [3]Hutchinson JW. Delamination of compressed films on curved sidered, it was predicted that the effect of in-plane ther- substrates. J Mech Phys Solids 2001: 49(9): 1847-64 mal misfit stress is appreciable only when the intact layer [4 Pauleau Y. Generation and evolution of residual stresses in is much stiffer than the crack layer(a>0), whilst crack al vapour-deposited thin films. Vacuum 2001; 61(2-4) deflection criteria were insensitive to the magnitude of 5]Tsui Y-C, Clyne Tw. An analytical model for residual the residual stress when a <0, supporting the predic planar tions of Leguillion et al. [13]. It was further predicted netry. Thin Solid Films 1997: 306(1): 23-33 that introducing compressive residual stress in the stiffer 6 Hutchinson JW, Evans AG. On the delamination of thermal intact layer would be beneficial for crack deflection as barrier coatings in a thermal gradient. Surf Coat Technol 2002;14902-3):17984. this provides condition to increase transition toughness [7 Gordon JE. The new science of strong materials. 2nd ed. Lon- ratio for crack deflection. Thus, it is proposed that lamwere plotted as functions of C for two selected as, namely 0.6 and 0.8, in Fig. 8. It is noticed in Fig. 8 that the slope of the curves, i.e. oðGd=GpÞ=oC, decreases as C increases. In particular, when C is higher than about 1.0 for a = 0.6 and 3.0 for a = 0.9, respectively, oðGd=GpÞ=oC is close to zero. Similar tendency was pre￾dicted in the analysis of the behaviour of an interface crack in a bi-layer system under the presence of in-plane residual stress [30]. Thus, it is deduced that once the magnitude of the residual stress exceeds a certain value, further improvement in the properties of layered system would be marginal. 5. Conclusions Effect of thermal misfit stain induced by mismatch of CTE on the crack deflection at planar interfaces in lay￾ered systems was investigated using a finite element analysis. In a convergence test of the numerical solution, dependence of the deflection criteria on the choice of length scale of the infinitesimal putative penetration across the interface or deflection along the interface was noticed only in the positive a regime, which is con￾sistent with the previous works by Ahn et al. [28] and Leguillon et al. [13]. When residual stress effect was con￾sidered, it was predicted that the effect of in-plane ther￾mal misfit stress is appreciable only when the intact layer is much stiffer than the crack layer (a > 0), whilst crack deflection criteria were insensitive to the magnitude of the residual stress when a < 0, supporting the predic￾tions of Leguillion et al. [13]. It was further predicted that introducing compressive residual stress in the stiffer intact layer would be beneficial for crack deflection as this provides condition to increase transition toughness ratio for crack deflection. Thus, it is proposed that lam￾inate composites be designed in such a way that the con￾dition of positive a and C is maintained to take advantage of beneficial effect of the misfit stress, i.e. crit￾ical fracture energy of the crack deflecting interface can be increased through the incorporation of in-plane resid￾ual stress thereby improving overall mechanical proper￾ties of the laminate. Acknowledgements This work was supported in part (W. Lee and J.-M. Myoung) by the National R&D Support Pro￾gramme for Young Scientists (Grant No. R08-2004- 000-10193-0). References [1] Howard SJ, Stewart RA, Clegg WJ. The delamination of ceramic laminates due to residual thermal stresses. Key Eng Mater 1996;116–117:331–50. [2] Blattner AJ, Lakshminarayanan R, Shetty DK. Toughening of layered ceramic composites with residual surface compression: effects of layer thickness. Eng Fract Mech 2001;68(1):1–7. [3] Hutchinson JW. Delamination of compressed films on curved substrates. J Mech Phys Solids 2001;49(9):1847–64. [4] Pauleau Y. Generation and evolution of residual stresses in physical vapour-deposited thin films. Vacuum 2001;61(2–4): 175–181. [5] Tsui Y-C, Clyne TW. An analytical model for predicting residual stresses in progressively deposited coatings. Part 1: planar geometry. Thin Solid Films 1997;306(1):23–33. [6] Hutchinson JW, Evans AG. On the delamination of thermal barrier coatings in a thermal gradient. Surf Coat Technol 2002;149(2–3):179–84. [7] Gordon JE. The new science of strong materials. 2nd ed. Lon￾don: Pitman; 1976. -1 0 1 2 3 4 5 6 7 8 9 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Γ p d α = 0.8 α = 0.6 Fig. 8. Change in Gd=Gp ratio plotted as a function of C for two selected values of a. 442 W. Lee et al. / Composites Science and Technology 66 (2006) 435–443