Straight Configurations of Shearable Nonlinearly Elastic Rods

Investigating obstacle problems for elastic rods we are sometimes confronted with the question to look for a solution which has a prescribed shape along some part of it. In the simplest case the rod is enforced to be straight along some contact area (cf., e.g., Gastaldi & Kinderlehrer [3]). Motivated by such applications we study straight configurations of elastic rods in this paper. More precisly, as in the case of frictionless contact, we consider equilibrium configurations of planar rods which are enforced to be straight by special external forces that are orthogonal to the straight axis. The reader might have in mind the mostly used Euler elastica (or simplifications of it), which neglects shear, extension, and thickness, and our subject appears to be very boring, since in that case only the trivial solution is straight, and even a school boy would not spend some attention to such a triviality. However, based on the Cosserat theory which describes planar deformations of nonlinearly elastic rods that can bend, strech, and shear and which takes into acccount an exact twoor three-dimensional geometry, the problem becomes much more subtle. In contrast to the more primitive models, we shall obtain an interesting richness of structure of this appearently simple problem. An important observation for our general rod theory, which does not neglect thickness, is that we must say more precisely what we mean by a straight configuration. A Cosserat rod describes a “slender” twoor three-dimensional elastic body and, by the nontrivial interaction of flexure, shear, and extension, originally parallel material curves of the rod do not remain parallel under deformations in general. Thus some special material curve which we want to be straight in the deformed configuration has to be selected. Usually this will be some curve of centroids or a certain boundary curve. The last case is from particular interest for contact problems with a straight obstacle. After introducing the underlying Cosserat theory for planar rods in Section 2, we show in Section 3 how the problem of straight configurations can be reduced to a second order system of ordinary differential equations. Based on some basic transformation rules for the reference ∗Partially supported by Deutscher Akademischer Austauschdienst