been correlated with engineered gradients created by co-sintering a depth-dependent admixture of constituent particles with different elastic moduli. In recent years, a lot of work has been carried out to study the behavior of FGM. Specifically, the finite element method has gained increasing use to determine the overall mechanical response of the materials to given solicitations. Usually FEM approaches are applied on the scale of the entire structure, the macro scale, using commercial codes such as ABAQUS (see for example [2] and [4]). On the other hand, unit cell models based on the FE analysis have been considered. However, these models cannot account for spatial variability of the constituents, due to the assumption that a structure is composed of the same microstructural representative volume element (RVE) in every part. Thus, the standard micromechanics approaches based on the concept of RVE are not suitable in the analysis of FGM, since the RVE cannot be univocally defined because of continuously changing properties through thickness. In fact, such engineered composites have average gradients that match the calculated optimal gradient on the macro scale, but the composite microstructure will have variations in its lateral properties and variations about the optimal gradient. The discrete nature of particulate composites that have been examined experimentally results in "stochastic gradients" in both microstructural directions. Neither the effect of lateral variations or perturbations about a prescribed gradient have been studied analytically. The aim of this work is to investigate the microstructural randomness and discreteness for fixed macroscopic material gradients. Thus, a discrete computational micromechanical model is adopted. An image based finite element code, OOF, is adopted. The peculiarity of this tool is that it is able to analyze arbitrary microstructures, by mapping digitized images of microstructures and their local properties to a two-dimensional finite element mesh. In this way microstructural features can be readily modeled. To the authors' knowledge, the influence of randomness of microstructure on the macroscopic global response of FGM has been studied in literature only for the problem of thermal residual stresses [6], using a physically based micromechanics model. Here we analyze the effect of microstructural discreteness on the fracture and damage behavior of FGM, coupling OOF that calculates the local stress state with a statistical approach for brittle fracture, as described in the following section. METHOD AND MATERIAL As cited in the previous section, the image-based computational tool OOF is used in conjunction with a statistical representation of failure. In order to study the crack propagation, a new finite element has been implemented [7], based on a probabilistic approach for brittle fracture: the two-parameters Weibull law [8]. The microstructural mesh generation is performed using OOF and the microstructural stresses are calculated for the given loading conditions. Depending on the local probabilities of failures, this element loses stiffness as it undergoes damage and microstructural stresses are redistributed among the next elements. Thus, the effect of damage accumulation due to failure of the material can be calculated iteratively and damage can be accumulated. The FGM considered in this study is the one experimentally characterized by Jitcharoen et al. [2]. This material is a graded alumina-glass composite whose Young modulus increases with depth beneath the surface. The thermo-mechanical properties of the constituents are summarized in Table 1. TABLE 1 Young modulus (GPa) CTE (10-6 °C-1) Poisson's ratio Alumina 386 8.8 0.22 Glass 72 8.8 0.22 The coefficients of thermal expansion (CTE) of the two phases are approximately the same, so that thermal residual stresses do not arise upon cooling from the processing temperature.
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