Regularity for shearable nonlinearly elastic rods in obstacle problems

Based on the Cosserat theory describing planar deformations of shearable no nlinearly elastic rods we study the regularity of equilibrium states for pr oblems where the deformations are restricted by rigid obstacles. We start w ith the discussion of general conditions modeling frictionless contact. In particular we motivate a contact condition that, roughly speaking, requires the contact forces to be directed normally, in a generalized sense, both t o the obstacle and to the deformed shape of the rod. We show that there is a jump in the strains in the case of a concentrated contact force, i.e., th e deformed shape of the rod has a corner. Then we assume some smoothness fo r the boundary of the obstacle and derive corresponding regularity for the contact forces. Finally we compare the results with the case of unshearable rods and obtain interesting qualitative differences.