Material Symmetry and Chirality in Nonlinearly Elastic Rods

Abstract

We treat certain classes of material symmetry in straight nonlinearly elastic rods in the presence of a uniform helical microstructure. In particular, we consider rods with chirality or “handedness”. This is a natural setting for manufactured ropes and cables and for biological filaments such as DNA strands. First we propose a novel definition of transverse material symmetry, enabling, e.g., a clear distinction to be made between hemitropic and isotropic rods. The former category can be realized from a simple spatial average of uniform helical symmetry. We show that hemitropic rods naturally exhibit mechanical coupling between extension and twist. Next we obtain an explicit representation theorem for the stored energy of rods with uniform helical symmetry (without averaging). We also study rods with dihedral-helical symmetry. This characterizes most manufactured ropes and cables, which are typically composed of two or more uniformly wound helical strands. Finally, we treat prismatic rods with transverse dihedral symmetry.