Dynamic Frictional Contact of a Viscoelastic Beam

Abstract. In this paper, we study the dynamic frictional contact of a viscoelastic beam with a deformable obstacle. The beam is assumed to be situated horizontally and to move, in both horizontal and tangential directions, by the effect of applied forces. The left end of the beam is clamped and the right one is free. Its horizontal displacement is constrained because of the presence of a deformable obstacle, the so-called foundation, which is modelled by a normal compliance contact condition. The effect of the friction is included in the vertical motion of the free end, by using Tresca’s law or Coulomb’s law. In both cases, the variational formulation leads to a nonlinear variational equation for the horizontal displacement coupled with a nonlinear variational inequality for the vertical displacement. We recall an existence and uniqueness result. Then, by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives, a numerical scheme is proposed. Error estimates on the approximative solutions are derived. Numerical results demonstrate the application of the proposed algorithm.