Nonlinear dispersive waves in soft elastic laminates under finite magneto-deformations

TL;DR

This paper develops a homogenised nonlinear wave model for soft magneto-active layered media, revealing how magnetic fields and microstructure influence shear wave propagation and solitary wave formation.

Contribution

It extends existing models by incorporating nonlinear dispersive effects in magneto-elastic laminates using homogenisation and a modified KdV approach.

Findings

Magnetic fields can tune shear wave speed and band gaps.

Nonlinear dispersive waves form solitary waves in the medium.

The model aligns with numerical simulations of impact problems.

Abstract

Layered media can be used as acoustic filters, allowing only waves of certain frequencies to propagate. In soft magneto-active laminates, the shear wave band gaps (i.e., the frequency intervals for which shear waves cannot propagate) can be adjusted after fabrication by exploiting the magneto-elastic coupling. In the present study, the control of shear wave propagation in magneto-active stratified media is revisited by means of homogenisation theory, and extended to nonlinear waves of moderate amplitude. Building upon earlier works, the layers are modelled by means of a revised hard-magnetic material theory for which the total Cauchy stress is symmetric, and the incompressible elastic response is of generalised neo-Hookean type (encompassing Yeoh, Fung-Demiray, and Gent materials). Using asymptotic homogenisation, a nonlinear dispersive wave equation with cubic nonlinearity is derived, under certain simplifying assumptions. In passing, an effective strain energy function describing such laminates is obtained. The combined effects of nonlinearity and wave dispersion contribute to the formation of solitary waves, which are analysed using the homogenised wave equation and a modified Korteweg-de Vries (mKdV) approximation of the latter. The mKdV equation is compared to direct numerical simulations of the impact problem, and various consequences of these results are explored. In particular, we show that an upper bound for the speed of solitary waves can be adjusted by varying the applied magnetic field, or by modifying the properties of the microstructure.