Nonlinear Incompressible Shear Wave Models in Hyperelasticity and Viscoelasticity Frameworks, with Applications to Love Waves

TL;DR

This paper develops nonlinear shear wave models for incompressible hyperelastic and viscoelastic materials, applies them to Love wave propagation at interfaces, and performs numerical simulations revealing wave speed behaviors.

Contribution

It introduces a general framework for nonlinear shear wave equations in hyperelasticity and viscoelasticity, including specific models and numerical analysis of wave propagation.

Findings

Wave speed generally satisfies linear Love wave conditions.

Nonlinear effects influence wave speed, tending towards material wave speeds over time.

Numerical simulations confirm theoretical wave behavior in layered materials.

Abstract

General equations describing shear displacements in incompressible hyperelastic materials, holding for an arbitrary form of strain energy density function, are presented and applied to the description of nonlinear Love-type waves propagating on an interface between materials with different mechanical properties. The model is valid for a broad class of hyper-viscoelastic materials. For a cubic Yeoh model, shear wave equations contain cubic and quintic differential polynomial terms, including viscoelasticity contributions in terms of dispersion terms that include mixed derivatives $u_{xxt}$ of the material displacement. Full (2+1)-dimensional numerical simulations of waves propagating in the bulk of a two-layered solid are undertaken and analyzed with respect to the source position and mechanical properties of the layers. Interfacial nonlinear Love waves and free upper surface shear waves are tracked; it is demonstrated that in the fully nonlinear case, the variable wave speed of interface and surface waves generally satisfies the linear Love wave existence condition $c_1 < \abs{v} < c_2$, while tending to the larger material wave speed $c_1$ or $c_2$ for large times.