Auxetic Properties of the Stiffest Elastic Bodies as a Result of Topology Optimization and Microstructures Recovery Based on Homogenization Method

Abstract

Herein, several semianalytical formulae for the optimal distribution of elastic moduli in 2D or 3D domains occupied by nonhomogeneous, least compliant bodies are presented and three different methods for recovering microstructures predicted by the isotropic material design method are proposed, which is a stress‐based variant of topology optimization for isotropic bodies. In all five presented variants of the topology optimization of elastic bodies, the numerical implementation of these formulae requires only the use of any algorithm of searching for the minimum of functional in multidimensions without any constraints, even box constraints on design parameters. For each variant considered, the formulae defining this functional are defined analytically. All results concern the design of composite structures that maximize their stiffness (equivalently minimize their compliance) while satisfying a certain isoperimetric condition introducing the upper limit of the available material resource. The article pays special attention to the fact that the stiffest, heterogeneous isotropic elastic material has almost always auxetic properties. Based on the asymptotic homogenization method and adopted bending‐free lattice model, an algorithm for the numerical recovering of nonhomogeneous isotropic microstructure in the entire domain occupied by the plate is proposed, with a fairly easy‐to‐implement ability to model an auxetic behavior.