Random composite media: homogenization, modeling, simulation, and material symmetry
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DepartmentMathematics and Statistics
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We present a systematic study of homogenization of diffusive and elastic random media with emphasis on tile-based random microstructures. We give detailed examples of several such media starting from their physical descriptions, then construct the associated probability spaces and verify their ergodicity. After a discussion of material symmetries of random media, we derive criteria for the isotropy of their homogenized limits. Furthermore, we study the periodization algorithm for the numerical approximation of the homogenized diffusion tensor and study the algorithm's rate of convergence. For one-dimensional tile-based media, we prove a central limit result, giving a concrete rate of convergence for periodization. We also provide numerical evidence for a similar central limit behavior in the case of two dimensional tile-based structures. Moreover, we study symmetries of elastic random media, and develop a theory for analyzing questions of isotropy; this allows us to derive minimal conditions on a symmetry group of a linearly elastic material that implies its isotropy. A natural setting for the formulation and analysis is provided by the group representation theory where the necessary and sufficient conditions for isotropy are expressed in terms of the irreducibility of certain group representations. Our approach allows for deriving isotropy results in contexts of hyperelastic, non-hyperelastic and polar media. Our theory also allows us to rederive several known results under a unified theory.