Solution of the static contact problem with Coulomb friction between an elastic body and a rigid foundation

Abstract

We consider a static contact problem between an elastic body and an absolutely rigid foundation with Signorini contact conditions and Coulomb friction law. The problem can be formulated as a quasi-variational inequality in which the normal stress is proportional with the friction coefficient to the friction force in the contact zone. In this case, the normal stress itself depends on the desired solution, and the existence of a solution to the problem reduces to the existence of a fixed point of a certain mapping. Consideration of Coulomb friction naturally leads to the non-differentiability of the objective functional in the auxiliary problem of the method of successive approximations and, thereby, to a significant complication of the algorithms for solving the constrained minimization problem. We propose a method of solving an auxiliary problem with given friction, which is based on the Uzawa algorithm and modified Lagrange functionals. The main advantage of the proposed method is that it allows to effectively solve auxiliary problems and to prove the theoretical convergence of the method of successive approximations to a fixed point. Numerical experiments are carried out to demonstrate the efficiency of the proposed method.