| Due to the spatial variability in heterogenous (random) materials at microscopic level, its effective properties at various length scales have to be determined in a stochastic (probabilistic) fashion. In this dissertation, such a determination is based on a discrete random modeling of microstructure. Three problems of stochastic nature, are presented and methodologies of treatment are proposed. Solutions are performed via computer simulations. In the first problem, the effective elastic moduli of Delaunay networks, modeling two-phase granular media are calculated. Combined with a Delaunay network, two spring models are used to represent interactions between particles. In the first model, central interaction is taken into account, while in the second one both central and angular interactions are considered. Results of numerical simulations are used to identify that self-consistent model which most closely approximates effective elastic properties of two-phase Delaunay networks.; In the second problem, a micromechanics-based stochastic finite element method is developed to account for the variability in material properties at micro level. The method is illustrated through an out-of-plane elasticity problem of a membrane with a microstructure of a spatially random inclusion-matrix under a deterministic load. The key concept introduced here is a random meso scale continuum model. It is found that two bounds on the material properties and in turn on the global response have to be considered in the analysis.; In the last problem, the effective thermal conductivity of functionally graded heterogeneous interphases between fiber and matrix in such composites is determined. The topology of microstructure is taken as a mosaic or a random chessboard where both phases have locally isotropic properties. The resulting meso-continuum model of the interphase is used to calculate the effective macroscopic properties (transverse conductivity or, equivalently, axial shear modulus) of such composite materials. This problem requires the treatment of several length scales; the fine interphase microstructure, its meso-continuum representation, the fiber size and the macroscale level at which the effective properties are defined. |
