A quasistatic contact problem with normal compliance and damage involving viscoelastic materials with long memory.

*(English)* Zbl 1142.74031
Summary: We study a model for quasistatic frictionless contact between a viscoelastic body with long memory and a foundation. The material constitutive relation is assumed to be nonlinear, and the contact is modelled with the normal compliance condition, i.e., the obstacle is assumed deformable. The mechanical damage of the material, caused by excessive stress or strain, is described by the damage function, which is modelled by a nonlinear partial differential equation. We derive a variational formulation for the problem and prove the existence of its unique weak solution. Then, we introduce a fully discrete scheme for the numerical solutions of the problem, based on the finite element method to approximate the spatial variable and on Euler scheme to discretize the time derivatives, and we obtain error estimates for approximate solutions. Finally, some numerical results are presented for two-dimensional test problems.

##### MSC:

74M15 | Contact (solid mechanics) |

74D05 | Linear constitutive equations (materials with memory) |

74H20 | Existence of solutions for dynamical problems in solid mechanics |

74H25 | Uniqueness of solutions for dynamical problems in solid mechanics |

74S05 | Finite element methods in solid mechanics |

74S20 | Finite difference methods in solid mechanics |