The Mathematical Institute, University of Oxford, Eprints Archive

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

Destrade, M. and Goriely, A. and Saccomandi, G. (2010) Scalar evolution equations for shear waves in incompressible solids: A simple derivation of the Z, ZK, KZK, and KP equations. Proceedings of the Royal Society A .

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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.

Item Type:Article
Subjects:D - G > General
Research Groups:Oxford Centre for Collaborative Applied Mathematics
ID Code:1411
Deposited By:Peter Hudston
Deposited On:08 Nov 2011 09:50
Last Modified:09 Feb 2012 15:51

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