On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

TL;DR

This paper characterizes the set of possible effective elastic tensors for 2D and 3D printed composites with voids, providing microgeometries that achieve minimum energy states and describing their properties.

Contribution

It offers a partial characterization of effective elastic tensors for composites with voids and describes microgeometries that attain energy minimums in various cases.

Findings

Microgeometries are unions of walls supporting specific strain modes.

Minimum energy configurations can be computed via finite-dimensional numerical problems.

Hierarchical laminate structures minimize complementary energies.

Abstract

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors $\BC_1>0$ and $\BC_2=0$ comprising the set $U=\{\BC_1,\BC_2\}$ and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor $\BC_2=0$ corresponds to a material which is void. (For technical reasons $\BC_2$ is actually taken to be nonzero and we take the limit $\BC_2\to 0$). Specifically, 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 easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.