Variational approach to contact problems in nonlinear elasticity

Abstract

We use variational methods to study problems in nonlinear 3-dimensional elasticity where the deformation of the elastic body is restricted by a rigid obstacle. For an assigned variational problem we first verify the existence of constrained minimizers whereby we extend previous results. Then we rigorously derive the Euler-Lagrange equation as necessary condition for minimizers, which was possible before only under strong smoothness assumptions on the solution. The Lagrange multiplier corresponding to the obstacle constraint provides structural information about the nature of frictionless contact. In the case of contact with, e.g., a corner of the obstacle, we derive a qualitatively new contact condition taking into account the deformed shape of the elastic body. By our analysis it is shown here for the first time rigorously that energy minimizers really solve the mechanical contact problem.