Towards a complete characterization of the effective elasticity tensors of mixtures of an elastic phase and an almost rigid phase
Résumé
The set $GU_f$ of possible effective elastic tensors of composites built from two materials with positive definite elasticity tensors ${\bf C}_1$ and ${\bf C}_2=\delta{\bf C}_0$ comprising the set $U=\{{\bf C}_1,\delta{\bf C}_0\}$ and mixed in proportions $f$ and $1-f$ is partly characterized in the limit $\delta\to\infty$. The material with tensor ${\bf C}_2$ corresponds to a material which is almost rigid in the limit $\delta\to \infty$. (For technical reasons the limit $\delta\to \infty$ is taken, rather than considering this phase to be absolutely rigid). 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 almost rigid to $6-p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the walls, by themselves, can support stress with almost no deformation. The region outside the walls contains "Avellaneda material" which is a hierarchical laminate which minimizes the sum of elastic energies.