Obstacle problems in elasticity

Obstacle problems in elasticity are usually described by variational inequalities where the set of admissible deformations is convex in some function space. This is, however, too restrictive for applications on the basis of realistic models. Furthermore it is unsatisfactorily that variational inequalities do not have enough structure to provide a detailed description of the forces exerted by the obstacle. An other point is that in obstacle problems concentrated contact forces actually arise while in standard continuum mechanics body forces and surface tractions with integrable densities are considered only. It seems that finer nonsmooth methods than variational inequalities and a more general approach to forces are necessary to handle these difficulties. On the basis of the very general Cosserat theory describing planar deformations of shearable nonlinearly elastic rods we demonstrate how this can be done. Using Clarke's calculus of generalized gradients we derive the Euler‐Lagrange equations for very general obstacle problems and we obtain a very natural representation of the contact reactions, including concentrated forces. This finally serves as basis for further regularity results.