2005). Indeed, the compressive stress required to achieve high strength is often very high and localized and delamination between layers(i.e. edge cracking) may occur(Moon et al 2002). Recently, Sglavo et al. (2001) and Sglavo and Bertoldi(2006) have proposed the design and manufacturing of alumina mullite/zirconia multilayer laminates with an engineered residual stress profile with a maximum compression at a certain depth from surface, obtaining a low-scattered strength and a higher reliability in comparison to monolithic ceramics. In the present work both energy-based and local approaches are applied to evaluate the behavior of the aforementioned ceramic laminate, applying the finite element method. First of all, the residual stress field and the effective toughness are put on the role of the residual stress field in the Weibull type statistical andlysis, developed. Particular emphasis will be evaluated. Subsequently a statistical analysis based on the Weibull approach is 2. Finite element model to determine effective toughness and strength of ceramic laminates The residual stress field and the effective toughness of the laminate have been calculated for bars having nominal length of 60 mm and a rectangular cross section(7.5 mm height and 1.45 mm thickness). a nine layer symmetric composite has been considered in this paper. Composite materials with alumina, mullite and zirconia constituents were considered. Thickness and material composi- tions of each layer are reported in Fig. la. Following notation reported in Sglavo and Bertoldi(2006) each layer is identified as AZw/ly or AMwly in which A, Z and M stand for alumina, zirconia and mullite, respectively; w is the volumetric content of zirconia or mullite and y is the layer thickness in um. The elastic constitutive parameters, i.e. the Young modulus and the poisson ratio, as well as the coefficient of thermal expansion of the composites at different volumetric compositions used in Sglavo et al. (2001)have been assumed(see Table 1). The Young modulus and poisson ratio were estimated through the voigt and reuss bounding values. The thermal expan sion coefficient and fracture toughness for AM and az composites were measured on monolithic samples( bertoldi et al 2003) The residual stress field was obtained by solving the finite element equations in which a thermal expansion, proportional to a temperature variation from stress-free temperature to standard conditions AT=-(1200-25)C, was applied. The coef- ficients of thermal expansion of the materials are reported in Table 1 a plane stress model was used to simulate the cooling process after sintering and the standard four-point bending tests: therefore, the finite element mesh of half length of the material sample has been built accounting for symmetry conditions. Second order plane strain displacement based finite elements were used The smallest element size, in the area close to the crack tip along the crack propagation path, is 2.5 um. The size of the largest element is consistent with the smallest layer thickness, so that in the far region, at least one element in the layer thickness is used The finite element analyses are carried out by setting a fictitious thermal expansion coefficient for each layer eft as in which Azo is the reference value, i.e. the coefficient of thermal expansion of pure alumina(azo), with the purpose to avoid unrealistic out-of-plane residual stress values. The residual stresses far from the crack tip are characterized by a homogenous field in each layer As shown in Fig. 1b, the external Az30 layer is subject to a slight tensile stress, whereas the azo (second layer). AM40 and azo ( third layer) are sub- jected to compressive stress field. The middle layer is subjected to tensile residual stress. 400 6002005). Indeed, the compressive stress required to achieve high strength is often very high and localized and delamination between layers (i.e. edge cracking) may occur (Moon et al., 2002). Recently, Sglavo et al. (2001) and Sglavo and Bertoldi (2006) have proposed the design and manufacturing of alumina/ mullite/zirconia multilayer laminates with an engineered residual stress profile with a maximum compression at a certain depth from surface, obtaining a low-scattered strength and a higher reliability in comparison to monolithic ceramics. In the present work both energy-based and local approaches are applied to evaluate the behavior of the aforementioned ceramic laminate, applying the finite element method. First of all, the residual stress field and the effective toughness are evaluated. Subsequently a statistical analysis based on the Weibull approach is developed. Particular emphasis will be put on the role of the residual stress field in the Weibull type statistical analysis. 2. Finite element model to determine effective toughness and strength of ceramic laminates The residual stress field and the effective toughness of the laminate have been calculated for bars having nominal length of 60 mm and a rectangular cross section (7.5 mm height and 1.45 mm thickness). A nine layer symmetric composite has been considered in this paper. Composite materials with alumina, mullite and zirconia constituents were considered. Thickness and material composi￾tions of each layer are reported in Fig. 1a. Following notation reported in Sglavo and Bertoldi (2006) each layer is identified as AZw/y or AMw/y in which A, Z and M stand for alumina, zirconia and mullite, respectively; w is the volumetric content of zirconia or mullite and y is the layer thickness in lm. The elastic constitutive parameters, i.e. the Young modulus and the Poisson ratio, as well as the coefficient of thermal expansion of the composites at different volumetric compositions used in Sglavo et al. (2001) have been assumed (see Table 1). The Young modulus and Poisson ratio were estimated through the Voigt and Reuss bounding values. The thermal expan￾sion coefficient and fracture toughness for AM and AZ composites were measured on monolithic samples (Bertoldi et al., 2003). The residual stress field was obtained by solving the finite element equations in which a thermal expansion, proportional to a temperature variation from stress-free temperature to standard conditions DT ¼ ð1200 25Þ C, was applied. The coef- ficients of thermal expansion of the materials are reported in Table 1. A plane stress model was used to simulate the cooling process after sintering and the standard four-point bending tests; therefore, the finite element mesh of half length of the material sample has been built accounting for symmetry conditions. Second order plane strain displacement based finite elements were used. The smallest element size, in the area close to the crack tip along the crack propagation path, is 2.5 lm. The size of the largest element is consistent with the smallest layer thickness, so that in the far region, at least one element in the layer thickness is used. The finite element analyses are carried out by setting a fictitious thermal expansion coefficient for each layer an eff as an eff ¼ an aAZ0 ð1Þ in which aAZ0 is the reference value, i.e. the coefficient of thermal expansion of pure alumina (AZ0), with the purpose to avoid unrealistic out-of-plane residual stress values. The residual stresses far from the crack tip are characterized by a homogenous field in each layer. As shown in Fig. 1b, the external AZ30 layer is subject to a slight tensile stress, whereas the AZ0 (second layer), AM40 and AZ0 (third layer) are sub￾jected to compressive stress field. The middle layer is subjected to tensile residual stress. Fig. 1. Architecture of the AMZ laminate (not to scale): symmetry of the laminate structure and symmetry used in FEA are indicated (a); residual stress profile (b). P. Vena et al. / Mechanics Research Communications 35 (2008) 576–582 577