Non-Uniqueness, Eigenvalue Solutions and Wedged Configurations Involving Coulomb Friction

It is well known that the conventional Coulomb friction condition can lead to non-uniqueness of solution in elastostatic solutions if the friction coefficient is sufficiently high. Interest in this field has centered on discrete formulations, particularly with reference to the finite element method. More recently Hild has demonstrated the existence of a multiplicity of non-unique solutions to a simple problem in two-dimensional continuum elasticity and showed how to determine the conditions for such states to exist by formulating an eigenvalue problem. Both the discrete and continuum examples of non-uniqueness seem to be related to the well known physical phenomenon whereby a frictional system can become locked or ‘wedged’ in a state of stress even when no external loads are applied (the homogeneous problem), but the equivalence is not complete because of the influence of unilateral inequalities in the physical problem. We shall discuss the relations between these concepts in the context of simple continuum and discrete problems in two-dimensional linear elasticity.