S Guicciardi et al. /Journal of the European Ceramic Sociery 27(2007)351-356 structure. By fracture mechanics analysis, the effects of the stress field superposition can be estimated by considering that the stress-intensity factor, KI, of a crack of length a due to an E arbitrary distribution of crack-line stress o(x)is KI= h(-,a o(r)dx where h(x/a, a)is a weight function given by 3 {-+4(-)-0 and a is the crack length normalized for the bar thickness(alW). Fig. 5. Apparent fracture toughness profile and superimposition of experimental ness W and width B subjected to 3-pt bending over a span of s is given by2I 4. Conclusions a(x)=Flex(x)+or(x) (11) A layered ceramic composite in the AIN-SiC-MoSiz system where o flex(x)is given by Eq (5)and or (x)is the residual stress was prepared by tape-casting. The composite was prepared by at point x given by alternatively stacking electrical insulating and conductive lami- o,(x)=o for x corresponding to the I layers nae. The stacking was such that the outer layers were underresid ual compressive stress. The magnitude of the residual stresses (123 MPa, in the present case) was estimated through the lamination theory. The mechanical 02 for x corresponding to the c layers properties of the constituent materials ite were measured. with respect to the stress-free outer material, (+146 MPa, in the present case) the fracture strength of the laminated composite increased by an amount comparable to the compressive residual stress calcu For the calculations of the apparent fracture toughness pI lated by the lamination theory. The apparent fracture toughness file, the coefficients A in Eq (6)were taken from Ref.,which of the laminated composite was estimated by an analytical frac- considered the case of a flexural bar made of a homogeneous ture mechanics model and a good agreement was found betweer material without variation of Youngs modulus across its sec- experimental values and the predictions of fracture mechanics tion. However, as shown in Ref, even with a strong gradient the section of the bar the dif- References ference in the calculated stress intensity factor remains <10% The apparent fracture toughness profile of the laminated com- 1. Marshall, D B. Ratto, J.J. and Lange, F E, Enhanced fracture toughness posite was calculated as follows. For any crack length a, a in layered microcomposites of Ce-ZrO and Al2O3. J. Am. Ceram. So corresponding critical bending load, Pc, was calculated so that, 2. Sathy amoorthy, R, Virkar, A. V. and Cutler, R. A, Damage-resistant integrating Eq (9)with the stress distribution given by Eq (11), SiC-AIN layered composites with surface compressive stresses. J. Am. the intrinsic (i.e. stress-free) Klc for the material containing Ceran.Soe,1992,75,1136-1141. the tip of the crack a was obtained. This couple(a, Pc) was 3. Chartier, T, Merle, D and Besson, J.L. Laminar ceramic composites. J. then inserted in Eqs. (1)and (2)to find out the corresponding 4. saijgalik, P, Lences, Z and Dusza, J, Layered Si NA composites with apparent fracture toughness. By varying the crack length and the corresponding Pc, the apparent fracture toughness profile as 5.Rao, M P, Sanchez-Herencia, A.I. Beltz, G.E., McMeeking, R.Mand a function of crack length a can be drawn. In order to obtain a Lange, F. F, Laminar ceramics that exhibit a threshold strength. Science, unique Kle profile, the thickness of the layers were in this case averaged. This apparent KIc profile is shown in Fig. 5 along with 6. Kovar, D, Thouless, M D. and Halloran, J. w, Crack deflection and propa- the experimental data points. As can be seen, the apparent KI 1998,81,1004-1012. is an increasing function of the crack length in the compressive 7. Cai, PZ, Green, D.J. and Messing, GL, Mechanical layer and a decreasing function of the crack length in the tensile Al2O3/ZrO2 hybrid laminates. J. Eur Ceram Soc., 1998, 5, 2025-2034 layer with values ranging from 0.9 to 7.2 MPam.Recognis- 8.Rao, M. P and Lange, E F, Factors affecting threshold strength in laminar ing all the approximations involved in the calculation of the 1222-1228 apparent fracture toughness, the agreement between the theoret 9. Blattner. A. J. Lakshmina ical curve and the experimental points can be considered quite layered ceramic composites with residual surface compression: effects of yer thickness. Eng. fract. Mech, 2001, 68, 1-7S. Guicciardi et al. / Journal of the European Ceramic Society 27 (2007) 351–356 355 structure.23 By fracture mechanics analysis, the effects of the stress field superposition can be estimated by considering that the stress-intensity factor, KI, of a crack of length a due to an arbitrary distribution of crack-line stress σ(x) is KI = a 0 h x a , α σ(x) dx (9) where h(x/a, α) is a weight function given by23 h x a , α =  2 πa 1 (1 − (x/a))1/2(1 − α) 3/2 ×  (1 − α) 3/2 +Aij 1 − x a i+1 αj  (10) and α is the crack length normalized for the bar thickness (a/W). The in-depth stress distribution σ(x) in a laminated bar of thick￾ness W and width B subjected to 3-pt bending over a span of S is given by21 σ(x) = σflex(x) + σr(x) (11) where σflex(x) is given by Eq. (5) and σr(x) is the residual stress at point x given by σr(x) = σ1 for x corresponding to the I layers (−123 MPa, in the present case) = σ2 for x corresponding to the C layers (+146 MPa, in the present case) For the calculations of the apparent fracture toughness pro- file, the coefficients Aij in Eq. (6) were taken from Ref.24, which considered the case of a flexural bar made of a homogeneous material without variation of Young’s modulus across its sec￾tion. However, as shown in Ref.,25 even with a strong gradient profile of Young’s modulus across the section of the bar, the dif￾ference in the calculated stress intensity factor remains <10%. The apparent fracture toughness profile of the laminated com￾posite was calculated as follows.23 For any crack length a, a corresponding critical bending load, Pc, was calculated so that, integrating Eq. (9) with the stress distribution given by Eq. (11), the intrinsic (i.e. stress-free) KIc for the material containing the tip of the crack a was obtained. This couple (a, Pc) was then inserted in Eqs. (1) and (2) to find out the corresponding apparent fracture toughness. By varying the crack length and the corresponding Pc, the apparent fracture toughness profile as a function of crack length a can be drawn. In order to obtain a unique KIc profile, the thickness of the layers were in this case averaged. This apparent KIc profile is shown in Fig. 5 along with the experimental data points. As can be seen, the apparent KIc is an increasing function of the crack length in the compressive layer and a decreasing function of the crack length in the tensile layer with values ranging from 0.9 to 7.2 MPa m0.5. Recognis￾ing all the approximations involved in the calculation of the apparent fracture toughness, the agreement between the theoret￾ical curve and the experimental points can be considered quite good. Fig. 5. Apparent fracture toughness profile and superimposition of experimental data points. 4. Conclusions A layered ceramic composite in the AlN–SiC–MoSi2 system was prepared by tape-casting. The composite was prepared by alternatively stacking electrical insulating and conductive lami￾nae. The stacking was such that the outer layers were under resid￾ual compressive stress. The magnitude of the residual stresses was estimated through the lamination theory. The mechanical properties of the constituent materials and the layered compos￾ite were measured. With respect to the stress-free outer material, the fracture strength of the laminated composite increased by an amount comparable to the compressive residual stress calcu￾lated by the lamination theory. The apparent fracture toughness of the laminated composite was estimated by an analytical frac￾ture mechanics model and a good agreement was found between experimental values and the predictions of fracture mechanics. References 1. Marshall, D. B., Ratto, J. J. and Lange, F. F., Enhanced fracture toughness in layered microcomposites of Ce–ZrO2 and Al2O3. J. Am. Ceram. Soc., 1991, 74, 2979–2987. 2. Sathyamoorthy, R., Virkar, A. V. and Cutler, R. A., Damage-resistant SiC–AlN layered composites with surface compressive stresses. J. Am. Ceram. Soc., 1992, 75, 1136–1141. 3. Chartier, T., Merle, D. and Besson, J. L., Laminar ceramic composites. J. Eur. Ceram. Soc., 1995, 15, 101–107. 4. Sajgal ˇ ´ık, P., Lenceˇ s, Z. and Dusza, J., Layered Si ˇ 3N4 composites with enhanced room temperature properties. J. Mater. Sci., 1996, 31, 4837–4842. 5. Rao, M. P., Sanchez-Herencia, A. J., Beltz, G. E., McMeeking, R. M. and ´ Lange, F. F., Laminar ceramics that exhibit a threshold strength. Science, 1998, 286, 102–105. 6. Kovar, D., Thouless, M. D. and Halloran, J. W., Crack deflection and propa￾gation in layered silicon nitride/boron nitride ceramics. J. Am. Ceram. Soc., 1998, 81, 1004–1012. 7. Cai, P. Z., Green, D. J. and Messing, G. L., Mechanical characterization of Al2O3/ZrO2 hybrid laminates. J. Eur. Ceram. Soc., 1998, 5, 2025–2034. 8. Rao, M. P. and Lange, F. F., Factors affecting threshold strength in laminar ceramic containing thin compressive layers. J. Am. Ceram. Soc., 2002, 85, 1222–1228. 9. Blattner, A. J., Lakshminarayanan, R. and Shetty, D. K., Toughening of layered ceramic composites with residual surface compression: effects of layer thcickness. Eng. Fract. Mech., 2001, 68, 1–7.