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

Volume 35 / 2008

J. A. Gawinecki, P. Kacprzyk Applicationes Mathematicae 35 (2008), 193-208 MSC: 35L15, 35L35, 35L45, 35L55, 35L60, 73C50. DOI: 10.4064/am35-2-5

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.

Authors

  • J. A. GawineckiInstitute of Mathematics and Cryptology
    Faculty of Cybernetics
    Military University of Technology
    Kaliskiego 2
    00-908 Warszawa 49, Poland
    e-mail
  • P. KacprzykInstitute of Mathematics and Cryptology
    Faculty of Cybernetics
    Military University of Technology
    Kaliskiego 2
    00-908 Warszawa 49, Poland

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