The incompressible limit in nonlinear elasticity
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.
singular limits of quasilinear symmetric hyperbolic systems
scientific article; zbMATH DE number 4062671