Nonlinear transverse waves in deformed dispersive solids

TL;DR

This paper introduces a phenomenological model for nonlinear dispersive solids, enabling the study of various wave types, including exotic solitary waves, and connects these findings to existing asymptotic theories.

Contribution

It proposes a thermomechanically consistent constitutive model for nonlinear dispersive solids that captures a wide range of wave phenomena without microstructural assumptions.

Findings

Solitary waves exist only with linear transverse polarization in certain dispersive solids.

Pre-stretch and hardening influence wave propagation characteristics.

The model links to the vectorial MKdV equation via multiscale expansion.

Abstract

We present a phenomenological approach to dispersion in nonlinear elasticity. A simple, thermomechanically sound, constitutive model is proposed to describe the (non-dissipative) properties of a hyperelastic dispersive solid, without recourse to a microstructure or a special geometry. As a result, nonlinear and dispersive waves can travel in the bulk of such solids, and special waves emerge, some classic (periodic waves or pulse solitary waves of infinite extend), some exotic (kink or pulse waves of compact support). We show that for incompressible dispersive power-law solids and forth-order elasticity solids, solitary waves can however only exist in the case of linear transverse polarization. We also study the influence of pre-stretch and hardening. We provide links with other (quasi-continuum, asymptotic) theories; in particular, an appropriate asymptotic multiscale expansion specializes our exact equations of motion to the vectorial MKdV equation, for any hyperelastic material.