A viscoelastic Timoshenko beam with Coulomb law of friction

Abstract

This work focuses on the analysis and numerical simulations of dynamic frictional contact between a Timoshenko beam and a stationary rigid obstacle. The beam is assumed to be viscoelastic, and is clamped at its left end while it is free at its right end. When the beam moves horizontally and its right end contacts the rigid obstacle, contact forces arise and then the vertical motion of its right end is considered with friction. Thus the right end of the beam satisfies two contact conditions: the Signorini unilateral contact condition and Coulomb law of dry friction. Moreover, the slip rate dependence of the coefficient of friction is taken into account. The existence of weak solutions to the problem is shown by using finite time stepping and the necessary a priori estimates that allow us to vanish the time step in the limiting process. The energy balance in the system is investigated both theoretically and numerically. A fully discrete numerical scheme for the variational formulation of the problem is implemented and numerical results are presented.