Unstable perturbation of the equilibrium under Coulomb friction: Nonlinear eigenvalue analysis

Equilibrium configurations of elastic structures under Coulomb friction are considered. We use a nonlinear eigenvalue analysis to see when an equilibrium configuration can be (dynamically) unstable or when it admits (quasi-static) bifurcations. We prove that if the nonlinear eigenvalue problem has a smooth solution then there exists a family of (dynamic) perturbations of the equilibrium which have an exponential growth in time. The above results are obtained under some conditions on the boundary between the free, stick and slipping zones of the contact boundary. These conditions are specific to the continuous problem and they are not necessary when we reformulate the same result for the discrete (finite element) problem.We propose a specific numerical algorithm to compute the eigenvalue, and the corresponding eigenfunction for a mixed finite element approach of the nonlinear eigenvalue problem. It makes use of the successive iterates of a nonlinear operator. We proved that the algorithm is consistent (i.e. if it is convergent then its limit is a solution), but we cannot give a mathematical proof of its convergence. We give some numerical experiments to examine the convergence of the proposed algorithm. Other examples illustrate how from the nonlinear eigenvalue analysis we get information on the stability of the equilibrium configurations.