Friedemann Schuricht

Global injectivity and topological constraints for spatial nonlinearly elastic rods

Summary. In this paper we study the local and global injectivity of spatial deformations of shearable nonlinearly elastic rods. We adopt an analytical condition introduced by Ciarlet & Nečas in nonlinear elasticity to ensure global injectivity in that case. In particular we verify the existence of an energy-minimizing equilibrium state without self-penetration which may also be restricted by a rigid obstacle. Furthermore we consider the special situation where the ends of the rod are glued together. In that case we can still impose topological restrictions such as, e.g., that the shape of the rod belongs to a given knot type. Again we show the existence of a globally injective energy minimizer which now in addition respects the topological constraints. Note that the investigation of super-coiled DNA molecules is an important application of the presented results.

Contact between nonlinearly elastic bodies

We study the contact between nonlinearly elastic bodies by variational methods. After the formulation of the mechanical problem we provide existence results based on polyconvexity and on quasiconvexity. Then we derive the Euler-Lagrange equation as a necessary condition for minimizers. Here Clarke’s generalized gradients are the essential tool to treat the nonsmooth obstacle condition.

The Critical Role of the Base Curve for the Qualitative Behavior of Shearable Rods

In this paper, we treat several aspects of the induced geometrically exact theory of shearable rods, of central importance for contact problems, for which the regularity of solutions depends crucially on the presence of shearability. (An induced theory is one derived from three-dimensional theory by the imposition of constraints. Because the role of thickness enters into our theory in an essential way, it is an exact version of what has been called the theory of “moderately thick” rods.) In particular, we study how the theory and constitutive restrictions depend upon the choice of the base curve, and we show how this choice has major qualitative consequences, which are illustrated with several concrete examples.

Paradoxical Bending Behavior of Shearable Nonlinearly Elastic Rods

In nonlinear elasticity the exact geometry of deformation is combined with general constitutive relations. This allows a very sophisticated interaction of deformations in different material directions. Based on the Cosserat theory for planar deformations of nonlinearly elastic rods we demonstrate some paradoxical bending effects caused by a nontrivial interaction of extension, flexure, and shear. The analytical results are illustrated by numerical examples.

Straight configurations of shearable nonlinearly elastic rods

In contact problems for elastic rods sometimes we have to look for a solution with some prescribed shape, in particular where some material curve has to be straight. While this question is a triviality for the Euler rod (or simplifications of it), the problem becomes much more subtle within a theory which describes planar deformations of nonlinearly elastic rods that can bend, stretch, and shear. For some selected material curve of the rod we assume that it is constrained to be straight by suitable external forces orthogonal to that straight axis. It is shown that such configurations satisfy a second-order system of ordinary differential equations. In the case where this system is homogeneous a very rich structure can be observed by phase plane analysis. Finally some applications for rods which are in contact with a straight obstacle are discussed, and interesting new effects can be derived.