Effective Willis constitutive equations for periodically stratified anisotropic elastic media

TL;DR

This paper introduces a novel method to derive homogeneous Willis constitutive equations for layered elastic media, capturing frequency-dependent anisotropic inertia and coupling effects, applicable at any frequency and wavenumber.

Contribution

It proposes a new approach linking the effective medium coefficients to the matrix logarithm of the propagator, providing explicit formulas valid across frequencies.

Findings

Effective equations incorporate anisotropic inertia and coupling.

Method valid for any frequency and horizontal wavenumber.

Long wavelength expansions with convergence estimates.

Abstract

A method to derive homogeneous effective constitutive equations for periodically layered elastic media is proposed. The crucial and novel idea underlying the procedure is that the coefficients of the dynamic effective medium can be associated with the matrix logarithm of the propagator over a unit period. The effective homogeneous equations are shown to have the structure of a Willis material, characterized by anisotropic inertia and coupling between momentum and strain, in addition to effective elastic constants. Expressions are presented for the Willis material parameters which are formally valid at any frequency and horizontal wavenumber as long as the matrix logarithm is well defined. The general theory is illustrated using the example of scalar SH motion. Low frequency, long wavelength expansions of the effective material parameters are also developed using a Magnus series and explicit estimates for the rate of convergence are derived.