Frontiers in homogenization methods towards generalized continua for architected materials

Abstract

Architected materials have witnessed tremendous research activity in the last two decades, due to their ability to achieve unprecedented effective properties. The up-scaling of the mechanical response towards an effective continuum is challenging, due to the possible emergence of a homogenized macroscopic behavior that may substantially differ from that of the base material at the micro level. The need for enriched continuum models beyond Cauchy’s theory arises due to the existence of a new class of artificial materials exhibiting static and dynamic attributes typically not encountered in natural materials, deserving the name meta-materials. In such a situation with a failure of the strict scale separation hypothesis, the enrichment of the Cauchy continuum proceeds by adding new intrinsic parameters and internal length scales representative of the micro-structural impact at the level of the effective medium. Generalized continua including either new kinematic variables or higher gradients of the displacement have proven their ability to provide a faithful description of the response of materials with micro-structural effects; their derivation from a micro-mechanical approach starting directly from the scale of the microstructure raises specific difficulties, in contrast to the phenomenological standpoint, which however lacks a predictive capacity. A survey of the scientific issues raised by higher-order homogenization methods towards generalized continua is provided in this contribution. The main scientific issues and challenges raised by higher-order homogenization methods are reviewed in a critical manner, and solutions to these scientific issues are proposed in light of the more recent contributions. Specific attention is devoted to the elaboration of enriched models for architected materials in both static and dynamic regimes.