Design of graded two-phase microstructures for tailored elasticity gradients

Being one of new generation of composites, functionally graded materials (FGMs) possess gradually changed physical properties due to their compositional and/ or microstructural gradients. In literature, exhaustive studies have been carried out in compositional modeling and design, while limited reports are available for microstructural optimization. This article presents an inverse homogenization method for the design of two-phase (solid/ void) FGM microstructures, whose periodic base cells (PBCs) vary in a direction parallel to the property gradient but periodically repeat themselves in the perpendicular direction. The effective elasticity tensor at each PBC is estimated in terms of the homogenization theory. The overall difference between the effective tensor and their target is minimized by seeking for an optimal PBC material topology. To preserve the connectivity between adjacent PBCs, three methods, namely connective constraint, pseudo load, and unified formulation with nonlinear diffusion are proposed herein. A number of two-dimensional examples possessing graded volume fraction and Young’s modulus but constant positive or negative Poisson’s ratios are presented to demonstrate this computational design procedure