Blow-up of the solution to the initial-value problem in nonlinear three-dimensional hyperelasticity

J. A. Gawinecki¹ ; P. Kacprzyk¹

¹ Institute of Mathematics and Cryptology Faculty of Cybernetics Military University of Technology Kaliskiego 2 00-908 Warszawa 49, Poland

Applicationes Mathematicae, Tome 35 (2008) no. 2, pp. 193-208

Résumé

We consider the initial value problem for the nonlinear partial differential equations describing the motion of an inhomogeneous and anisotropic hyperelastic medium. We assume that the stored energy function of the hyperelastic material is a function of the point $x$ and the nonlinear Green–St. Venant strain tensor $e_{jk}$. Moreover, we assume that the stored energy function is $C^\infty $ with respect to $x$ and $e_{jk}$. In our description we assume that Piola–Kirchhoff's stress tensor $p_{jk}$ depends on the tensor $e_{jk}$. This means that we consider the so-called physically nonlinear hyperelasticity theory. We prove (local in time) existence and uniqueness of a smooth solution to this initial value problem. Under the assumption about the stored energy function of the hyperelastic material, we prove the blow-up of the solution in finite time.

Détail

DOI : 10.4064/am35-2-5

Keywords: consider initial value problem nonlinear partial differential equations describing motion inhomogeneous anisotropic hyperelastic medium assume stored energy function hyperelastic material function point nonlinear green venant strain tensor moreover assume stored energy function infty respect description assume piola kirchhoffs stress tensor depends tensor means consider so called physically nonlinear hyperelasticity theory prove local time existence uniqueness smooth solution initial value problem under assumption about stored energy function hyperelastic material prove blow up solution finite time