Overall Dynamic Properties of 3-D periodic elastic composites

TL;DR

This paper introduces a homogenization method for 3-D periodic elastic composites that calculates frequency-dependent dynamic material tensors, revealing hermitian properties and coupling behaviors, validated through numerical examples.

Contribution

It extends 1-D elastodynamic homogenization to 3-D, providing a comprehensive framework for analyzing dynamic properties of complex composites.

Findings

The overall compliance tensor is hermitian.

Mass density tensor is hermitian.

Strain and momentum tensors are coupled and hermitian transpose.

Abstract

A method for the homogenization of 3-D periodic elastic composites is presented. It allows for the evaluation of the averaged overall frequency dependent dynamic material constitutive tensors relating the averaged dynamic field variable tensors of velocity, strain, stress, and linear momentum. The formulation is based on micromechanical modeling of a representative unit cell of a composite proposed by Nemat-Nasser & Hori (1993), Nemat-Nasser et. al. (1982) and Mura (1987) and is the 3-D generalization of the 1-D elastodynamic homogenization scheme presented by Nemat-Nasser & Srivastava (2011). We show that for 3-D periodic composites the overall compliance (stiffness) tensor is hermitian, irrespective of whether the corresponding unit cell is geometrically or materially symmetric.Overall mass density is shown to be a tensor and, like the overall compliance tensor, always hermitian. The average strain and linear momentum tensors are, however, coupled and the coupling tensors are shown to be each others' hermitian transpose. Finally we present a numerical example of a 3-D periodic composite composed of elastic cubes periodically distributed in an elastic matrix. The presented results corroborate the predictions of the theoretical treatment.