Regularity for shearable nonlinearly elastic rods in obstacle problems

In this paper we are interested in regularity results to obstacle problems for shearable nonlinearly elastic rods. We work with the geometrically exact Cosserat theory for planar deformations which describes rods that can suffer not only flexure but also extension and shear and it involves general nonlinear constitutive relations. This is a consistent intrinsically one-dimensional theory which, however, allows a geometrically exact interpretation in a twoor three-dimensional setting. The most obstacle problems studied in the literature are carried out for much simpler models neglecting shear, extension, and thickness and are restricted to small deformations. By these simplifications the set of admissible deformations is usually convex and the problem leads to a variational inequality. This is, meanwhile, a widely investigated subject where the results essentially rest on monotonicity and convexity arguments (cf., e.g., Fichera [6], Frehse [7], Hlaváček, Haslinger, Nečas & Lov́ı̌sek [8], Lewy & Stampacchia [11], Kikuchi & Oden [9], Kinderlehrer & Stampacchia [10], Rodrigues [12] and references therein). For more realistic models, however, a simple observations shows that even “nice” obstacles where the elastic body can move within a convex set do not correspond to a convex set of admissible deformations in a suitable function space. Thus we have to recognize that the theory of variational inequalities is unsuitable for that purpose. We readily see that an obstacle brings a nonsmooth nonlinearity in the problem. But we cannot expect that the classical smooth analysis combined with the roughest nonsmooth tool, namely the variational inequality, is able to describe subtle nonsmooth effects. In particular the structure of the most interesting term describing the contact reactions, which fills the gap between inequality and equality, is not considered in the variational inequality. Therefore it seems to be necessary and natural to study obstacle problems by refined nonsmooth methods. In Schuricht [13] for the very large class of obstacles having Lipschitz boundary a more general nonsmooth ∗Partially supported by Deutscher Akademischer Austauschdienst