Towards a complete characterization of the effective elasticity tensors of mixtures of an elastic phase and an almost rigid phase

TL;DR

This paper characterizes the set of possible effective elastic tensors for composites with a nearly rigid phase, using microgeometries of wall structures and hierarchical laminates to understand their elastic behavior in the limit of infinite stiffness contrast.

Contribution

It provides a detailed description of microgeometries that achieve minimum energy states for effective elasticity tensors in composites with an almost rigid phase, extending previous algebraic and geometric analyses.

Findings

Microgeometries with wall structures can achieve minimum energy configurations.

The effective elasticity tensors are characterized through finite-dimensional minimization problems.

Hierarchical laminates called 'Avellaneda material' optimize elastic energy in the composite.

Abstract

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with positive definite elasticity tensors $\BC_1$ and $\BC_2=\Gd\BC_0$ comprising the set $U=\{\BC_1,\Gd\BC_0\}$ and mixed in proportions $f$ and $1-f$ is partly characterized in the limit $\Gd\to\infty$. The material with tensor $\BC_2$ corresponds to a material which (for technical reasons) is almost rigid in the limit $\Gd\to \infty$. The paper, and the underlying microgeometries, have many aspects in common with the companion paper "On the possible effective elasticity tensors of 2-dimensional printed materials". The chief difference is that one has a different algebraic problem to solve: determining the subspaces of stress fields for which the thin walled structures can be rigid, rather than determining, as in the companion paper, the subspaces of strain fields for which the thin walled structure is compliant. Recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is almost rigid to $6-p\leq 5$ independent applied stresses, yet is compliant to any strain in the orthogonal space. Thus the walls, by themselves, can support stress with almost no deformation. The region outside the walls contains "Avellaneda material" that is a hierarchical laminate which minimizes an appropriate sum of elastic energies.