Localization of elastic deformation in strongly anisotropic, porous, linear materials with periodic microstructures: exact solutions and dilute expansions

TL;DR

This paper derives exact solutions and dilute expansions for the elastic deformation in strongly anisotropic, porous, linear materials with periodic microstructures, revealing localized field patterns and behavior near void close packing.

Contribution

It provides the first exact solutions and dilute expansions for anisotropic porous media with periodic voids, linking localized fields to fractional porosity powers.

Findings

Effective elastic moduli depend on porosity and anisotropy.

Dilute expansions feature half-integer powers of porosity.

Field singularities are characterized statistically.

Abstract

Exact solutions are derived for the problem of a two-dimensional, infinitely anisotropic, linear-elastic medium containing a periodic lattice of voids. The matrix material possesses either one infinitely soft, or one infinitely hard loading direction, which induces localized (singular) field configurations. The effective elastic moduli are computed as functions of the porosity in each case. Their dilute expansions feature half-integer powers of the porosity, which can be correlated to the localized field patterns. Statistical characterizations of the fields, such as their first moments and their histograms are provided, with particular emphasis on the singularities of the latter. The behavior of the system near the void close packing fraction is also investigated. The results of this work shed light on corresponding results for strongly nonlinear porous media, which have been obtained recently by means of the ``second-order'' homogenization method, and where the dilute estimates also exhibit fractional powers of the porosity.