Abstract
We consider a mathematical model which describes the dynamic evolution of a viscoelastic body in frictional contact with an obstacle. The contact is modelled with a combination of a normal compliance and a normal damped response law associated with a slip rate-dependent version of Coulomb’s law of dry friction. We derive a variational formulation and an existence and uniqueness result of the weak solution of the problem is presented. Next, we introduce a fully discrete approximation of the variational problem based on a finite element method and on an implicit time integration scheme. We study this fully discrete approximation schemes and bound the errors of the approximate solutions. Under regularity assumptions imposed on the exact solution, optimal order error estimates are derived for the fully discrete solution. Finally, after recalling the solution of the frictional contact problem, some numerical simulations are provided in order to illustrate both the behavior of the solution related to the frictional contact conditions and the theoretical error estimate result.
Highlights
Dynamic contact problems abound in industry and everyday life
The normal compliance condition introduced in [7] remains one of the most popular contact models used in the literature; see [4, 8,9,10,11]
It represents a regularization of the so-called Signorini contact condition, expressed in terms of unilateral constraints for the displacement field, and described the contact with a deformable foundation
Summary
Dynamic contact problems abound in industry and everyday life. For this very reason, special care has been given to the modelling, mathematical analysis, and numerical solution of such problems for several decades, be it in the engineering or the mathematical literature. We insist on the fact that, in the problem considered in this paper, the behavior of the foundation is described by a Kelvin-Voigt-like foundation modelled by a combination of normal compliance and normal damped response Such considerations lead to nonstandard Coulomb’s law of dry friction where the threshold depends on the tangential velocity, through the coefficient of friction, and both the normal displacement and normal velocity, because of the normal compliance and normal damped response, respectively. Another trait of novelty arises from the mathematical analysis with in particular the numerical analysis of such a problem by considering a dynamic process.
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