Design of isotropic microstructures via a two-scale approach

Architectured materials are promising to reach extreme properties and ultimately address issues related to lightweight or non conventional properties for bulk materials (eg. high specific rigidity, extremal conductivity or auxetism (negative Poisson's ratio)) [1]. A very efficient way to obtain optimal forms is via inverse homogenization, i.e. using shape and topology optimization techniques in order to achieve target material properties [2]. A great number of publications has been devoted to the design of isotropic materials with extreme properties. Isotropy is usually prescribed via a combination of symmetric planes and penalization techniques, which are quite delicate to handle in an optimization framework. In this work, we present an approach for the design of isotropic multi-materials with extremal conductivity via laminate geometries, consisting in anisotropic phases [3]. More specifically, we design composites with extremal conductivity using rank-1 laminates, composed by two orthotropic phases along parallel layers. The second phase is obtained by a 90-degree rotation of the first one, while their volume fractions are explicitly chosen so that the laminate is isotropic. The orthotropic phases are considered to have their own periodic micro-structure, composed by multiple phases. By optimally distributing the different phases in a periodic cell, we can achieve the Hashin-Shtrikman bounds for the isotropic laminate. We present examples in two dimensions using the level-set method for shape and topology optimization [4].