Overall Dynamic Constitutive Relations of Micro-structured Elastic Composites

TL;DR

This paper introduces a homogenization method for elastic composites that accurately derives frequency-dependent dynamic constitutive relations, applicable to both periodic and non-periodic microstructures, enabling precise modeling of wave dispersion and field solutions.

Contribution

It presents a novel homogenization approach that directly computes overall dynamic relations for micro-structured elastic composites, extending applicability beyond periodic structures.

Findings

Accurately evaluates frequency-dependent constitutive relations.

Provides point-wise solutions to elasto-dynamic equations.

Applicable to non-periodic microstructures.

Abstract

A method for homogenization of a heterogeneous (finite or periodic) elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations. It is shown that when the spatial variation of the field variables is restricted by a Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress modeshapes and, necessarily, the dispersion relations. It also gives as a matter of course point-wise solution of the elasto-dynamic field equations, to any desired degree of accuracy. The resulting overall dynamic constitutive relations however, are general and need not be restricted by the Bloch-form periodicity. The formulation is based on micro-mechanical modeling of a representative unit cell of the composite proposed by Nemat-Nasser and coworkers; see, e.g., [1] and [2].