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

M. Destrade, A. Goriely, G. Saccomandi

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20 Scopus citations


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. © 2010 The Royal Society.
Original languageEnglish (US)
Pages (from-to)1823-1834
Number of pages12
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2131
StatePublished - Dec 8 2010
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
Acknowledgements: This 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.).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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