Scalar evolution equations for shear waves in incompressible solids: a simple derivation of the Z, ZK, KZK and KP equations
KAUST Grant NumberKUK-C1-013-04
Online Publication Date2010-12-08
Print Publication Date2011-07-08
Permanent link to this recordhttp://hdl.handle.net/10754/599559
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AbstractWe 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. © 2010 The Royal Society.
CitationDestrade M, Goriely A, 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: Mathematical, Physical and Engineering Sciences 467: 1823–1834. Available: http://dx.doi.org/10.1098/rspa.2010.0508.
SponsorsThis work is supported by a Senior Marie Curie Fellowship awarded by the Seventh Framework Programme of the European Commission to the first author. This publication is based on work supported in part by award no. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST) (A.G.), and also based in part upon work supported by the National Science Foundation under grant DMS-0907773 (A.G.).
PublisherThe Royal Society