Abstract
This paper addresses a model problem of nonlinear homogenization motivated by the study of the shape-memory effect in polycrystalline media. Specifically, it numerically computes the set of recoverable strains in a polycrystal given the set of recoverable strains of a single crystal in the two-dimensional scalar (or antiplane shear) setting. This problem shares a direct analogy with crystal plasticity. The paper considers typical or random polycrystals where the grains are generated by a Voronoi tesselation of a set of random points and are randomly oriented. The numerical results show that for such microstructures, the Taylor bound appears to be the most accurate (though pessimistic) bound when the anisotropy is moderate, and that recent Kohn–Little–Goldsztein outer bounds overestimate the recoverable strains when the anisotropy is large. The results also show that the stress tends to localize on tortuous paths that traverse (poorly oriented) grains as the polycrystal reaches its limit of recoverable strain.
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