A hyperbolic framework for shear sound beams in nonlinear solids

TL;DR

This paper develops a hyperbolic framework for modeling shear sound beams in nonlinear solids, transforming complex equations into a form suitable for shock wave computation and validating it through analytical solutions and harmonic generation analysis.

Contribution

It introduces a novel hyperbolic transformation of coupled nonlinear equations for shear waves, enabling efficient shock wave simulation in nonlinear elastic solids.

Findings

Validation against analytical solutions confirms accuracy.

The method captures harmonic generation in shear-wave beams.

Results demonstrate the framework's effectiveness for nonlinear wave analysis.

Abstract

In soft elastic solids, directional shear waves are in general governed by coupled nonlinear KZK-type equations for the two transverse velocity components, when both quadratic nonlinearity and cubic nonlinearity are taken into account. Here we consider spatially two-dimensional wave fields. We propose a change of variables to transform the equations into a quasi-linear first-order system of partial differential equations. Its numerical resolution is then tackled by using a path-conservative MUSCL-Osher finite volume scheme, which is well-suited to the computation of shock waves. We validate the method against analytical solutions (Green's function, plane waves). The results highlight the generation of odd harmonics and of second-order harmonics in a Gaussian shear-wave beam.