Critical friction for wedged configurations

Abstract

A wedged configuration with Coulomb friction is a nontrivial equilibrium state of a linear elastic body in a frictional unilateral contact with a rigid body under vanishing external loads. We analyze here the relation between the geometry of the elastic body and the friction coefficient for which wedged configurations exist in a 3-D context. The critical friction coefficient, μ w , is defined as the infimum of a supremal functional defined on the set of admissible normal displacement and tangential stresses. For friction coefficients μ with μ > μ w the wedged problem has at least a solution and for μ < μ w there exits no wedged configurations. For the in-plane problem we discuss the link between the critical friction and the smallest real eigenvalue μ s which is related to the loss of uniqueness. The wedged problem is stated in a discrete framework using a mixed finite element approach and the (discrete) critical friction coefficient μ h w is introduced as the solution of a global minimization problem involving a non differentiable and non-convex functional. The existence of Φ h * , the displacement field of a critical wedged state, is proved and a specific numerical method, based on a genetic algorithm, was developed to compute the critical wedged configurations. Some techniques to handle the discontinuities of the normal vector on the contact surface are presented and the analysis is illustrated with three numerical simulations.