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

Handle URI:
http://hdl.handle.net/10754/599559
Title:
Scalar evolution equations for shear waves in incompressible solids: a simple derivation of the Z, ZK, KZK and KP equations
Authors:
Destrade, M.; Goriely, A.; Saccomandi, G.
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. © 2010 The Royal Society.
Citation:
Destrade 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.
Publisher:
The Royal Society
Journal:
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
8-Dec-2010
DOI:
10.1098/rspa.2010.0508
Type:
Article
ISSN:
1364-5021; 1471-2946
Sponsors:
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.).
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Full metadata record

DC FieldValue Language
dc.contributor.authorDestrade, M.en
dc.contributor.authorGoriely, A.en
dc.contributor.authorSaccomandi, G.en
dc.date.accessioned2016-02-28T05:53:21Zen
dc.date.available2016-02-28T05:53:21Zen
dc.date.issued2010-12-08en
dc.identifier.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.en
dc.identifier.issn1364-5021en
dc.identifier.issn1471-2946en
dc.identifier.doi10.1098/rspa.2010.0508en
dc.identifier.urihttp://hdl.handle.net/10754/599559en
dc.description.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.en
dc.description.sponsorshipThis 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.).en
dc.publisherThe Royal Societyen
dc.subjectIncompressible materialsen
dc.subjectShear wavesen
dc.subjectSoft solidsen
dc.titleScalar evolution equations for shear waves in incompressible solids: a simple derivation of the Z, ZK, KZK and KP equationsen
dc.typeArticleen
dc.identifier.journalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciencesen
dc.contributor.institutionNational University of Ireland Galway, Galway, Irelanden
dc.contributor.institutionUniversity College Dublin, Dublin, Irelanden
dc.contributor.institutionUniversity of Oxford, Oxford, United Kingdomen
dc.contributor.institutionUniversita degli Studi di Perugia, Perugia, Italyen
kaust.grant.numberKUK-C1-013-04en
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