Conservation laws in nonlinear elastodynamics
type: Item
instance of: scholarly article
title: Conservation laws in nonlinear elastodynamics
zbMATH Open document ID: 0702.73032
DOI: 10.1016/0020-7225(89)90134-1
author: Elsie Sterbin Gottlieb
published in: International Journal of Engineering Science
publication date: 1989-01-01T00:00:00Z
review text: The paper proposes a derivation of the full set of basic balance laws of nonlinear elastodynamics within the framework of Cartan's exterior calculus of forms. To that purpose a set of appropriate exterior differential forms is defined on the proper manifold. These forms are annulled by the solution surfaces of the field equations. They also generate a closed ideal in the ring of forms. Conservation laws are found as exact forms within this ideal. Fundamental equations that give the conservation law structure of the field equations are found. Their explicit solution is obtained for an arbitrary strain energy function. The paper bases on previous works by \textit{H. D. Wahlquist} and \textit{F. B. Estabrook} [e.g.: J. Math. Phys. 16, 1-7 (1975; Zbl 0298.35012); Classical quantum gravity 6, No.3, 263-274 (1989; Zbl 0672.53035)] and \textit{P. J. Olver} [e.g.: Arch. Ration. Mech. Anal. 85, 111-129 (1984; Zbl 0559.73019), and 131-160 (1984; Zbl 0582.73024); Bull. Am. Math. Soc., New Ser. 18, No.1, 21-26 (1988; Zbl 0643.49030)]. In addition to the classical balance laws of nonlinear elasticity, the conservation of canonical (material) momentum involving Eshelby's energy-momentum tensor is obtained. However, in the present case, this could have been deduced as a consequence of the other balance laws (especially the balance of physical momentum), a fact not noticed by the author.
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 74Axx
Mathematics Subject Classification ID: 58A15
zbMATH DE Number: 4151916
zbMATH Keywords: 4-dimensional sectioning of proper differential forms
zbMATH Keywords: 19-dimensional differentiable manifold
zbMATH Keywords: closed ideal
zbMATH Keywords: ring of forms
zbMATH Keywords: surface elements patched together
zbMATH Keywords: smooth solution submanifold
zbMATH Keywords: full set of basic balance laws
zbMATH Keywords: Cartan's exterior calculus of forms
zbMATH Keywords: exterior differential forms
zbMATH Keywords: arbitrary strain energy function
zbMATH Keywords: conservation of canonical (material) momentum
zbMATH Keywords: Eshelby's energy-momentum tensor
reviewed by: Gérard A. Maugin
MaRDI profile type: Publication
full work available at URL: 0020-7225(89)90134-1
OpenAlex ID: W2147256169
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description: scientific article; zbMATH DE number 4151916