A dynamic contact problem involving a Timoshenko beam model

In this paper, a dynamic contact problem between a viscoelastic beam and a deformable obstacle is considered. The classical Timoshenko beam model is used and the contact is modeled by using the well-known normal compliance contact condition. The variational problem is written as a coupled system composed of a nonlinear variational equation for the displacement field and a linear variational equation for the angular rotation function. The existence of a unique solution is proved by using the Faedo–Galerkin method and employing some a priori estimates and Gronwallʼs inequality. Then, fully discrete approximations are introduced, based on the finite element method to approximate the spatial variable and the implicit Euler scheme to discretize the time derivatives. A priori error estimates are proved, obtaining the linear convergence of the algorithm under an additional regularity condition. Finally, some numerical simulations are presented to show the accuracy of the algorithm and the behavior of the solution.