Blow-up of the solution to the initial-value problem in nonlinear three-dimensional hyperelasticity
Volume 35 / 2008
Abstract
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.
