On the Mathematical and Geometrical Structure of the Determining Equations for Shear Waves in Nonlinear Isotropic Incompressible Elastodynamics

TL;DR

This paper explores the mathematical structure of shear wave equations in nonlinear isotropic elastodynamics, generalizing known solutions and identifying a broad class of hyperbolic PDEs with similar properties, using symmetry methods.

Contribution

It introduces a natural generalization of shear wave solutions and fully integrates the associated PDE system via generalized symmetries.

Findings

Infinite family of linear solutions for shear waves

Identification of a large class of hyperbolic PDEs with shear wave properties

Complete integration of the transverse wave equations using symmetry techniques

Abstract

Using the theory of $1+1$ hyperbolic systems we put in perspective the mathematical and geometrical structure of the celebrated circularly polarized waves solutions for isotropic hyperelastic materials determined by Carroll in Acta Mechanica 3 (1967) 167--181. We show that a natural generalization of this class of solutions yields an infinite family of \emph{linear} solutions for the equations of isotropic elastodynamics. Moreover, we determine a huge class of hyperbolic partial differential equations having the same property of the shear wave system. Restricting the attention to the usual first order asymptotic approximation of the equations determining transverse waves we provide the complete integration of this system using generalized symmetries.