Analytical formulation of 3D dynamic homogenization for periodic elastic systems

TL;DR

This paper develops a comprehensive analytical method for deriving fully dynamic effective material parameters in 3D periodic elastic systems, valid across all frequencies and wavenumbers, using the plane wave expansion method.

Contribution

It introduces a new analytical formulation for 3D dynamic homogenization that confirms the Willis equations are closed under homogenization for elastic media.

Findings

Effective parameters depend on frequency and wavenumber.

The formulation applies to arbitrary wave modes, including Bloch waves.

Comparison shows differences with existing dynamic effective medium theories.

Abstract

Homogenization of the equations of motion for a three dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion (PWE) method. The effective equations are of Willis form (Willis 1997) with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a 1D system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory (Shuvalov 2011) which also reduces to Willis form but with different effective moduli.