Effective-medium theory for infinite-contrast, 2D-periodic, linear composites with strongly anisotropic matrix behavior: dilute limit and cross-over behavior

TL;DR

This paper develops an effective-medium theory for 2D anisotropic composites with voids, revealing a crossover between regular and singular dilute regimes influenced by anisotropy and porosity, with implications for elastic behavior.

Contribution

It introduces a Hashin-Shtrikman type model that accounts for elastic interactions in anisotropic composites and analyzes the crossover behavior in dilute limits.

Findings

Identification of a crossover between regular and singular dilute regimes.

The singular regime exhibits a leading correction of order f^{1/2}.

The model aligns well with numerical solutions and exact results in certain limits.

Abstract

The overall behavior of a 2D lattice of voids embedded in an anisotropic matrix is investigated in the limit of vanishing porosity f. An effective-medium model (of the Hashin-Shtrikman type) which accounts for elastic interactions between neighboring voids, is compared to Fast Fourier Transform numerical solutions and, in the limits of infinite anisotropy, to exact results. A cross-over between regular and singular dilute regimes is found, driven by a characteristic length which depends on f and on the anisotropy strength. The singular regime, where the leading dilute correction to the elastic moduli is an O(f^{1/2}), is related to strain localization and to change in character - from elliptic to hyperbolic - of the governing equations.