| This dissertation addresses a classical and largely unsolved problem: given a structural component constructed of a heterogeneous elastic material that is in equilibrium under the action of applied loads, determine local micromechanical features of its response (e.g. local stresses and displacements in or around phase boundaries or in inclusions) to an arbitrary preset level of accuracy, it being understood that the microstructure is a priori unknown, may be randomly distributed, may exist at multiple spatial scales, and may contain millions, even billions, of microscale components.; The approach described in this work begins with a mathematical abstraction of this problem in which the material body is modeled as an elastic solid with highly-variable, possibly randomly distributed, elastic properties. Information on the actual character of the microstructure of given material bodies is determined by X-ray Computerized Tomography (CT) imaging. A procedure is given for determining the effective material properties from imaging data.; A theory of a posteriors estimation of modeling errors---errors that result from the use of homogenized models---in quantities of interest is then presented. These quantities of interest may, for example, represent averaged stresses on the surface of inclusions or mollifications of pointwise stresses or displacements, or, in general, local features of the fine-scale solution characterized by continuous linear functionals. Based on this theory, an algorithm is developed for determining local quantities of interest to arbitrary accuracy relative to the fine-scale model.; A new computational environment for implementing such analyses is presented which employs parallel, adaptive, hp finite element methods, CT interfaces, automatic meshing procedures, and, effectively, adaptive modeling schemes. Within the basic premises on which the approach is based, results of arbitrary accuracy can be obtained, independently of the number of microscale components and constituents. The results of several numerical experiments are presented. |
