P. Vena et al /Mechanics Research Communications 35(2008) 576-582 b10 n 0 10 0 100 283MP 30m=10 -200 5.6 6.4 n(o1) n(o1) Fig. 5. Weibull plots for m= 35 at different values of Go (a): Weibull plots for o= 283 MPa at different values of m(b 525.66646.8 6.4 6.5 6.6 Fig. 6. Numerical Weibull and experimental(squares) results for pure alumina and the AMz laminate(m=20)(a); enlargement of the plot showing numerical Weibull (solid line) and experimental results(circles) for the AMZ laminate(b) In the case of laminates with oo #0 a non-linear plot is obtained and it is singular for In(d)=In(oo)(i. e. it exhibits a ve ical asymptote). In Fig 5a, Weibull plots for different values of the parameter do are reported. The Weibull plots, as obtained by using(8)and(12). for three different values of the parameter m, are reported in Fig 5b. The plots for all values of the parameter m exhibit a bend at oi=330 MPa, which is the stress level beyond which the rela tionship (10) does not hold anymore. This behavior is consistent with the results outlined in the recent work dealing with tatistical numerical simulations for materials with residual stress fields(danzer et al, 2007 The slope(m1)of the Weibull plot calculated for f=0(i.e. a failure probability of approximately 63. 2%), which is a mea- sure of the reliability of the material strength, can be easily related to the Weibull parameter m in(11)and( 12)according to the following relationship For ao=0(i.e no residual stress), mi= m is obtained. This implies that experimental results exhibiting slope mi=35 in the Weibull plot of AMZ laminates can be fit by using, according to(13), m= 20. 1(Fig. 6). In this paper, a numerical method suitable to study and design high-reliability AMZ ceramic laminates characterized by residual stresses is presented. It involves both energy-based(global) and statistical (local)approaches, with the purpose to determine the fracture strength in a four-point bending stress and to discuss the effect of the thermal residual stress. The energy approach in the finite element model allowed to estimate the effective toughness during crack propagation, ointing out the stable propagation of cracks from the surface of the specimen subjected to the four-point bending test along the direction perpendicular to the interfacesIn the case of laminates with r0 6¼ 0 a non-linear plot is obtained and it is singular for lnðrT 1Þ ¼ lnðr0Þ (i.e. it exhibits a ver￾tical asymptote). In Fig. 5a, Weibull plots for different values of the parameter r0 are reported. The Weibull plots, as obtained by using (8) and (12), for three different values of the parameter m, are reported in Fig. 5b. The plots for all values of the parameter m exhibit a bend at rT 1 ¼ 330 MPa, which is the stress level beyond which the rela￾tionship (10) does not hold anymore. This behavior is consistent with the results outlined in the recent work dealing with statistical numerical simulations for materials with residual stress fields (Danzer et al., 2007). The slope ðm1Þ of the Weibull plot calculated for f ¼ 0 (i.e. a failure probability of approximately 63.2%), which is a mea￾sure of the reliability of the material strength, can be easily related to the Weibull parameter m in (11) and (12) according to the following relationship: m1 m ¼ 1 þ a r0W r0 ð13Þ For r0 ¼ 0 (i.e. no residual stress), m1 ¼ m is obtained. This implies that experimental results exhibiting slope m1 ¼ 35 in the Weibull plot of AMZ laminates can be fit by using, according to (13), m ¼ 20:1 (Fig. 6). 4. Conclusions In this paper, a numerical method suitable to study and design high-reliability AMZ ceramic laminates characterized by residual stresses is presented. It involves both energy-based (global) and statistical (local) approaches, with the purpose to determine the fracture strength in a four-point bending stress and to discuss the effect of the thermal residual stress. The energy approach in the finite element model allowed to estimate the effective toughness during crack propagation, pointing out the stable propagation of cracks from the surface of the specimen subjected to the four-point bending test along the direction perpendicular to the interfaces. Fig. 5. Weibull plots for m ¼ 35 at different values of r0 (a); Weibull plots for r0 ¼ 283 MPa at different values of m (b). Fig. 6. Numerical Weibull and experimental (squares) results for pure alumina and the AMZ laminate ðm ¼ 20Þ (a); enlargement of the plot showing numerical Weibull (solid line) and experimental results (circles) for the AMZ laminate (b). P. Vena et al. / Mechanics Research Communications 35 (2008) 576–582 581