Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM

TL;DR

This paper develops a boundary element method-based approach to solve quasistatic contact problems involving visco-elastic bodies with Coulomb friction, formulating the problem as a recursive minimization with a QP structure in 2D.

Contribution

It introduces a semi-implicit discretization combined with SGBEM for visco-elastic contact problems, transforming the problem into a quadratic programming framework for 2D cases.

Findings

The method effectively models visco-elastic contact with Coulomb friction.

Numerical tests demonstrate stability and convergence.

The approach simplifies the contact problem to a boundary-only minimization.

Abstract

The quasistatic normal-compliance contact problem of isotropic homogeneous linear visco-elastic bodies with Coulomb friction at small strains in Kelvin-Voigt rheology is considered. The discretization is made by a semi-implicit formula in time and the Symmetric Galerkin Boundary Element Method (SGBEM) in space, assuming that the ratio of the viscosity and elasticity moduli is a given relaxation-time coefficient. The obtained recursive minimization problem, formulated only on the contact boundary, has a nonsmooth cost function. If the normal compliance responds linearly and the 2D problems are considered, then the cost function is piecewise-quadratic, which after a certain transformation gets the quadratic programming (QP) structure. However, it would lead to second-order cone programming in 3D problems. Finally, several computational tests are presented and analysed, with additional discussion on numerical stability and convergence of the involved approximated Poincar\'e-Steklov operators.