w. Lee et al./ Composites Science and Technology 66(2006)435-443 =5.0 r=10.0 I=0.5 12 L⊥,⊥A⊥,⊥f,⊥⊥,⊥A .8-0.6 0 Fig. 5. Change in sa/sp ratio plotted as a function of Dundurs'z for various residual stress magnitude parameter I [r: non-dimensionalised residual stress magnitude parameter defined as the relative misfit strain e with respect to the mechanical load-induced strain, eapp, i.e., I=-(e/eapp)] gle of loading of the deflected crack, regardless of the va- interface between layer A and B. Since the crack is ter lue of T, as shown in Fig. 6. The in-plane residual stress minating at stiff to compliant interface, a x-0. 75(Eq considered herein is uniaxial in two-dimensional plane (2). From Fig. 5, it is expected that the crack propa strain geometry and therefore its presence changes only gates across the interface into the interlayer(layer B) the magnitude of the crack driving force of the primary regardless of the magnitude of the residual stress and crack, not its fracture mode, which is assumed to be in then reaches the interface between layer B and A.As pure opening(mode I). Hence, it can be understood that the crack is terminated at compliant to stiff interface, the phase angle of the deflected crack, which is deter- a.75. Under this condition, if r=0, as R; / Rm mined by the fracture mode of the primary crack pro-(=1.05)is higher than a/p(=0.96)in Fig. 5, crack vided that all the fracture process is confined well deflection is still not possible; however, if residual stress within the K-dominant field of the primary crack, will is introduced such that r=0.5 for instance, R/Rm undergo practically no noticeable change whatever the (=1.05)is lower than ga/p(=1.07)allowing crack magnitude of the residual stress is deflection at this interface. Alternatively, it is conceiv The result shown in Fig. 5 is qualitatively consistent able that similar effect can be obtained by choosing sti with the work of Leguillon et al. [13]. in which it was sta- fer material as interlayer and then introducing ted that crack deflection criterion is almost insensitive to compressive residual stress in the interlayer to allow he magnitude of residual stress for a <0 whilst notice- the condition of positive a and but such laminate will able effect is expected for a>0. Further, the trend that have limited use due to inferior overall stifiness-related presence of compressive residual stress in the intact layer would be beneficial for crack deflection in the positive ox From the results and discussions so far, it is expected regime is also qualitatively consistent with the predic- that higher degree of modulus mismatch as well as larger tions by Leguillion et al. [13]. Applying these results to misfit strain will result in improved crack deflection cri- the design of ceramic laminate systems, which are pro- teria. However, there will be certain limits to the design duced by stacking relatively thick stiff matrix lavers w1 of materials systems this way. First, increasing the resid thin compliant interlayers alternately to introduce ual stress through the selection of the materials with lar crack-deflecting interfaces, it is possible to increase the ger difference in CTE may result in the spontaneous critical fracture energy of the crack-deflecting interface delamination during fabrication process as observed in by introducing the compressive residual stress in the stiff ZrO2/Al2O3 system since excessive CTE mismatch may matrix layers. This concept of laminate design is sche- provide enough driving force for the growth of delani matically illustrated in Fig. 7. nation crack initiating from the laminate edges [1]. An For the example shown in Fig. 7, suppose that the other limiting factor is the magnitude of misfit strain, layers denoted as a are 7 times stiffer than those denoted the maximum value of which would be determined by as B and that R /Rm=1.05. Now, further suppose that a the fracture strain of the constituent materials. For in- primary crack in the outer layer is reaching the first stance, if fi <f2 I as in the case of typical ceramicgle of loading of the deflected crack, regardless of the va￾lue of C, as shown in Fig. 6. The in-plane residual stress considered herein is uniaxial in two-dimensional plane strain geometry and therefore its presence changes only the magnitude of the crack driving force of the primary crack, not its fracture mode, which is assumed to be in pure opening (mode I). Hence, it can be understood that the phase angle of the deflected crack, which is deter￾mined by the fracture mode of the primary crack pro￾vided that all the fracture process is confined well within the K-dominant field of the primary crack, will undergo practically no noticeable change whatever the magnitude of the residual stress is. The result shown in Fig. 5 is qualitatively consistent with the work of Leguillon et al. [13], in which it was sta￾ted that crack deflection criterion is almost insensitive to the magnitude of residual stress for a < 0 whilst notice￾able effect is expected for a > 0. Further, the trend that presence of compressive residual stress in the intact layer would be beneficial for crack deflection in the positive a regime is also qualitatively consistent with the predic￾tions by Leguillion et al. [13]. Applying these results to the design of ceramic laminate systems, which are pro￾duced by stacking relatively thick stiff matrix layers with thin compliant interlayers alternately to introduce crack-deflecting interfaces, it is possible to increase the critical fracture energy of the crack-deflecting interface by introducing the compressive residual stress in the stiff matrix layers. This concept of laminate design is sche￾matically illustrated in Fig. 7. For the example shown in Fig. 7, suppose that the layers denoted as A are 7 times stiffer than those denoted as B and that Ri/Rm = 1.05. Now, further suppose that a primary crack in the outer layer is reaching the first interface between layer A and B. Since the crack is ter￾minating at stiff to compliant interface, a 
0.75 (Eq. (2)). From Fig. 5, it is expected that the crack propa￾gates across the interface into the interlayer (layer B) regardless of the magnitude of the residual stress and then reaches the interface between layer B and A. As the crack is terminated at compliant to stiff interface, a  0.75. Under this condition, if C = 0, as Ri/Rm (=1.05) is higher than Gd=Gp (=0.96) in Fig. 5, crack deflection is still not possible; however, if residual stress is introduced such that C = 0.5 for instance, Ri/Rm (=1.05) is lower than Gd=Gp (=1.07) allowing crack deflection at this interface. Alternatively, it is conceiv￾able that similar effect can be obtained by choosing stif￾fer material as interlayer and then introducing compressive residual stress in the interlayer to allow the condition of positive a and C, but such laminate will have limited use due to inferior overall stiffness-related property. From the results and discussions so far, it is expected that higher degree of modulus mismatch as well as larger misfit strain will result in improved crack deflection cri￾teria. However, there will be certain limits to the design of materials systems this way. First, increasing the resid￾ual stress through the selection of the materials with lar￾ger difference in CTE may result in the spontaneous delamination during fabrication process as observed in ZrO2/Al2O3 system since excessive CTE mismatch may provide enough driving force for the growth of delami￾nation crack initiating from the laminate edges [1]. An￾other limiting factor is the magnitude of misfit strain, the maximum value of which would be determined by the fracture strain of the constituent materials. For in￾stance, if f1  f2  1 as in the case of typical ceramic -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 α p d Γ = -1.0 Γ = -0.5 Γ = 0 Γ = 0.5 Γ = 1.0 Γ = 2.0 Γ = 5.0 Γ = 10.0 Fig. 5. Change in Gd=Gp ratio plotted as a function of Dundurs
a for various residual stress magnitude parameter C [C: non-dimensionalised residual stress magnitude parameter defined as the relative misfit strain er with respect to the mechanical load-induced strain, eapp, i.e., C =
(er/eapp)]. 440 W. Lee et al. / Composites Science and Technology 66 (2006) 435–443