This dissertation introduces an efficient numerical technique for modeling micro- and macroscopic behavior of materials containing numerous cracks and randomly distributed circular holes and elastic inclusions. The focus is on the accuracy and efficiency of the numerical method and the possibility of solving problems with large numbers of unknowns. The computer model is based on a numerical solution of a complex hypersingular boundary integral equation in which the boundary parameters are expressed in terms of series expansions of orthogonal functions. All integrations are performed analytically, and the resulting system of algebraic equations is solved iteratively for the unknown coefficients in the series expansions. For a given number of degrees of freedom the method has been shown to be much more efficient, accurate, and numerically stable than standard boundary element techniques based on pointwise collocation.; In addition to the above features, two other important contributions have been made in this dissertation: (1) A combination of an embedding method with a least squares analysis is introduced to adapt the analysis procedure to handle finite bodies with general convex external boundaries. This development has opened the door to a host of improvements on conventional numerical approaches that require domain or boundary discretizations. (2) A fast and accurate algorithm is constructed by combining the boundary integral method with multipole expansions. This algorithm is of linear complexity, which makes it feasible to solve large-scale practical problems (e.g. with one million degrees of freedom) on a personal computer.; One immediate application of this work is determination of the effective properties of nonhomogeneous materials. It is also of interest for the evaluation of stress concentration factors and identification of areas of possible material failure. These capabilities are particularly useful for evaluation, design, and fracture control of brittle materials such as rock, concrete, micro-porous materials, and fiber-reinforced composites, which are widely used in engineering practice.
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