Dynamic frictional contact of a viscoelastic beam
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.
A dynamic contact problem for a thermoelastic diffusion beam with the rotational inertia
Abstract We study the dynamic behaviour of a thermoelastic diffusion beam with the rotational inertia, clamped at one end and free to move between two stops at the other. The contact with the stops is modelled with the normal compliance condition. The system, recently derived by Aouadi (2015) [2] , describes the behaviour of thermoelastic diffusion thin plates. This problem poses new mathematical difficulties due to the nonlinear boundary conditions. We prove the existence and uniqueness of weak solution using the Faedo–Galerkin method as well as the exponential stability at a rate proportional to the rotational inertia parameter. We propose a finite element approximation and we prove that the associated discrete energy decays to zero. Finally, we give an error estimate assuming extra regularity on the solution and we present some results of our numerical experiments.
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