Effective moduli of an anisotropic material with elliptical holes of arbitrary orientational distribution

Abstract Effective moduli of a two-dimensional anisotropic solid with elliptical holes having an arbitrary (non-random) orientational distribution are given in closed form. The results are derived in the non-interacting approximation. Besides being rigorous at small defect densities, this approximation constitutes the basic building block for various approximate schemes. Proper tensorial parameters of defect density (dependent on ellipses’ eccentricity and their orientations relative to the matrix anisotropy axes) are identified. When derived in terms of such parameters, expressions for the effective moduli cover, in a unified way, mixtures of holes of diverse eccentricities and arbitrary orientational distribution. A number of special cases (circles, cracks of various orientational distributions) are discussed. If the field of defects is “geometrically isotropic” (holes of the circular shapes, or randomly oriented cracks), it reduces the matrix anisotropy.

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