| We consider long, thin elastic structures possessing a uniform helical micro-structure in a natural state. Examples motivating our work include manufactured ropes and cables, and biological filaments occurring in nature, e.g., DNA molecules and mammalian tendon. We adopt a nonlinearly elastic, Cosserat rod model for initially straight filaments of such material. The helical symmetry leads to natural restrictions on the stored energy function. Assuming that the period of the helical structure is much smaller than the length of any rod under consideration, we obtain, by simple averaging, a homogeneous rod whose stored energy function is invariant under all proper rotations about the centerline. We call such rods (transversely) hemitropic. We also propose a new definition of (transverse) isotropy for straight Cosserat rods, which distinguishes the latter from hemitropy. We show that hemitropic rods have built-in chirality or ^Óhandedness^Ô. That is, in contrast to isotropic rods, the hemitropic model captures the essential difference between right-handed and left-handed helical micro-structures. In particular, hemitropic rods are characterized by mechanical coupling between extension and twist. We consider the analysis of two classes of problems for straight, hemitropic rods under end thrust, demonstrating that the post-buckling behavior depends crucially upon the end conditions. In particular, we show that a ^Ófixed-free^Ô hemitropic rod responds much like an isotropic one, while a ^Ófixed-fixed^Ô rod exhibits non-planar behavior, which is markedly different from the isotropic case. We also consider the numerical determination of global equilibria of elastic Cosserat rods in general. Like the pure analysis of such problems, as above, the main difficulty here stems from the effective treatment of the rotation field. Like many others before, we parametrize the rotations via unit quaternions (Euler parameters), which induces a single explicit constraint. Not surprisingly, this leads to inconsistencies in the prescription of boundary conditions within in a general formulation. We present an effective formulation, which eliminates the constraint and enables consistent specification of boundary conditions. The heart of our approach is inspired by the well known Liapunov Center Theorem, which has been used in the numerical treatment of periodic solutions of systems of ODE's by S. Doedel. We present several concrete examples demonstrating the effectiveness of our formulation. |