Scalar evolution equations for shear waves in incompressible solids: A simple derivation of the Z, ZK, KZK, and KP equations

TL;DR

This paper derives scalar evolution equations like Z, ZK, KZK, and KP for shear waves in incompressible solids, clarifying their applicability and limitations through a rigorous asymptotic analysis.

Contribution

It provides a simple, consistent derivation of various scalar wave equations in nonlinear incompressible solids and discusses their validity for different material models.

Findings

Z equation is the asymptotic limit for neo-Hookean solids.

Z equation is not generally valid for all elastic materials.

Dispersive and dissipative terms lead to KP, ZK, and KZK equations.

Abstract

We study the propagation of two-dimensional finite-amplitude shear waves in a nonlinear pre-strained incompressible solid, and derive several asymptotic amplitude equations in a simple, consistent, and rigorous manner. The scalar Zabolotskaya (Z) equation is shown to be the asymptotic limit of the equations of motion for all elastic generalized neo-Hookean solids (with strain energy depending only on the first principal invariant of Cauchy-Green strain). However, we show that the Z equation cannot be a scalar equation for the propagation of two-dimensional shear waves in general elastic materials (with strain energy depending on the first and second principal invariants of strain). Then we introduce dispersive and dissipative terms to deduce the scalar Kadomtsev-Petviashvili (KP), Zabolotskaya-Khokhlov (ZK) and Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations of incompressible solid mechanics.