The incompressible limit in nonlinear elasticity
type: Item
instance of: scholarly article
title: The incompressible limit in nonlinear elasticity
zbMATH Open document ID: 0651.73004
DOI: 10.1007/BF01229377
author: Steven Schochet
published in: Communications in Mathematical Physics
publication date: 1985-01-01T00:00:00Z
review text: The incompressible limit in nonlinear elasticity is shown to fall under the theory of singular limits of quasilinear symmetric hyperbolic systems developed by \textit{S. Klainerman} and \textit{A. Majda} [Commun. Pure Appl. Math. 34, 481-524 (1981; Zbl 0476.76068)]. Specifically, initial-value problems for a family of hyperelastic materials with stored energy functions \[ W(\frac{\partial x}{\partial X})=W_{\infty}(\frac{\partial x}{\partial X})+\lambda^ 2w(\det \frac{\partial x}{\partial X}) \] are considered, where X and x are reference and deformed coordinates, respectively. Under the assumption that the elasticity tensor \[ A^{ij}_{kl}\equiv \frac{\partial^ 2W_{\infty}}{\partial (\partial x^ i/\partial X^ k)\partial (\partial x^ j/\partial X^ l)} \] is positive definite near the identity matrix and that \(w''(1)>0\), the following results are proven for appropriate initial data: i) existence of solutions of the corresponding evolution equations on a time interval independent of \(\lambda\) as \(\lambda\) \(\to \infty\), and ii) convergence as \(\lambda\) \(\to \infty\) of the solutions to a solution of the incompressible elastodynamics equations.
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35L15
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 4062671
zbMATH Keywords: singular limits of quasilinear symmetric hyperbolic systems
zbMATH Keywords: existence of solutions
zbMATH Keywords: evolution equations
zbMATH Keywords: convergence
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4062671