Numerical analysis and simulation of a quasistatic frictional bilateral contact problem with damage, long-term memory and wear

We present a mathematical model describing the equilibrium of a viscoelastic body with long-term memory in frictional contact with a sliding foundation. The process is quasistatic, and material damage resulting from excessive stress or strain is captured by a damage function. We assume the material is inhomogeneous, leading to multiple contact boundary conditions. The contact interface is partitioned into two segments: One part takes into account the wear of the contact surface, utilizing Archard's law. Here, contact is modeled with a normal compliance condition with unilateral constraints, coupled with a sliding version of Coulomb's law of dry friction. In the other part, contact is modeled with a nonmonotone condition involving normal compliance and a subdifferential frictional boundary condition. Variational formulation of the model is governed by a coupled system consisting of a variational–hemivariational inequality for the displacement field, a parabolic variational inequality for the damage field and an integral equation for the wear function. We study a fully discrete scheme for numerical approximation with an error estimation of the solution to this problem. Optimal error estimates for the linear finite element method are derived, followed by numerical simulations illustrating the behavior of the model.