Analysis of a time discretization for an implicit variational inequality modelling dynamic contact problems with friction (Q2726641)

The authors solve the dynamic problems for a frictional contact of a viscoelastic body with a rigid foundation. Two types of problems are considered. It is a normal compliance model with Coulomb friction and a bilateral contact problem with a Tresca friction law. Strong formulations of the problems are as follows. Find \(u=u(t,x)\) such that \(u(0,x)=u_0(x)\), \(\dot u(0,x)=\dot u_0(x)\) and div\(\mathbf \sigma + F_1=\rho \ddot u\;\text{in} \Omega\), \(\mathbf \sigma=\mathbf A\mathbf \varepsilon(u)+\mathbf B\mathbf \varepsilon(\dot u)\), \(u=0\) on \(\Gamma_u\), \(\mathbf \sigma\cdot n=F_2\) on \(\Gamma_f\) with the elliptic and symmetric tensors \(\mathbf A=(a_{ijkl})\), \(\mathbf B=(b_{ijkl})\). The friction conditions are given at the part \(\Gamma_c\) of the boundary \(\partial \Omega=\Gamma\). The normal compliance law with a Coulomb friction law is expressed by \(\quad \sigma_N=-C_N(u_N-q)_+\), \(|\sigma_T|< C_T(u_N-q)_+\Rightarrow \dot u_T=0\), \(|\sigma_T|= C_T(u_N-q)_+>0\Rightarrow \exists \lambda\leq 0\), \(\dot u_T=-\lambda \sigma_T,\) where \(q\) represents the initial gap. In the case of bilateral contact with a time-dependent Tresca friction law the conditions on the part \(\Gamma_c\) are of the form \(\;u_N=0\;\text{on} \Gamma_c\), \(|\sigma_T|< g\Rightarrow \dot u_T=0\), \(|\sigma_T|= g \Rightarrow \exists \lambda \leq 0\), \(\dot u_T=-\lambda \sigma_T,\) where \(g\) is a real values mapping defined on \(]0,T[\times \Gamma_c\). The abstract variational formulation involving both problems is formulated. The discrete formulation is obtained by a time discretization and a sequence of implicit elliptic variational inequalities follows. The convergence of the incremental solution is established and the limit is shown to be a unique solution of the pseudoparabolic variational inequality.