European Journal of Mechanics A/Solids 26 (2007) 701–711 On the restrictions imposed by non-affine material symmetry groups for elastic rods: application to helical substructures Todd A. Lauderdale, Oliver M. O’Reilly ∗ Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720-1740, USA Received 6 September 2006; accepted 24 October 2006 Available online 5 December 2006 Dedicated to Hans W. Liepmann on the occasion of his ninetieth birthday Abstract Wire ropes, DNA strands and helical springs are among those bodies which can be modeled as an elastic rod with a helical substructure. The resulting form of the strain-energy function is a matter of material symmetry. This symmetry is explored using a novel treatment which combines non-affine transformations and a relabeling of the material coordinates. The restrictions this treatment imposes on the strain-energy function include a periodic dependency on torsional strain. In addition, comparisons are made with results from a recent treatment of helical symmetry by Healey. Finally, conclusions applicable to material symmetry restrictions for other polar elastic continua are presented.  2006 Elsevier Masson SAS. All rights reserved. Keywords: Material symmetry; Elastic rods; Cosserat theory of rods; Mechanics of DNA strands 1. Introduction The use of rod theories to model long slender bodies with helical substructures is ubiquitous. For instance, it is common to model double-stranded DNA as a single elastic rod with the centerline of the rod mimicking the molecular (or duplex) axis of the DNA. Similarly, plied rope is often modeled as a collection of elastic rods, one of which represents the core while the others correspond to the helically wound plies (see, e.g., Costello, 1997). Suppose one models a helical spring, a piece of double-stranded DNA (which is in the form of a double helix) or plied rope as a single elastic rod (see Fig. 1) consisting of a cylindrical core and a helical substructure. Then, a natural question to pose is what restrictions can be placed on the constitutive relations for the rod? Examining the structures shown in Fig. 1, it is possible to envisage two types of material symmetry for these rod- like bodies. In the first, which was discussed by Healey (2002) in the context of Antman’s special Cosserat rod theory, 1 the rod-like body is subject to a uniform rotation and translation. The second approach to the material symmetry * Corresponding author. E-mail address: oreilly@berkeley.edu (O.M. O’Reilly). 1 Healey’s work can be viewed as an extension to earlier treatments which were framed in the context of a so-called worm-like chain (WLC) model for double-stranded DNA (Marko and Siggia, 1994; Marko, 1997; Kamien et al., 1997). 0997-7538/$ – see front matter  2006 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2006.10.003