Normal uniform deformations of monotropic hyperelastic rods

Abstract

A directed rod theory describes the deformation of a long, thin body called a rod. The rod is modeled as curve (the rod axis) with additional structure provided by a triad of vectors at each point along the rod axis. A hyperelastic rod is one which is associated with a scalar function called the strain energy density. Field equations and constitutive restrictions form the theory describing the deformation of hyperelastic rods. The field equations are derived from a variational principle relating the virtual work (which depends on the strain energy density) to an arbitrary virtual displacement of the rod. Two constitutive restrictions are assumed to apply to the rod undergoing deformation. Material frame indifference states that strain energy density is invariant under a rotation of the rod system following a deformation. Monotropic symmetry states that strain energy density is invariant under special rotations and reflections prior to deformation. Uniform rods are those which have a constant twist about therod axis described by a constant skew-symmetric tensor. Normal rods are those in which the cross-section of the rod is perpendicular to the rod axis at every point. Normal uniform rods have a limited number of possible shapes: straight, circular and helical. Normal uniform deformations are those deformations in which the initial and final configurations of the rod are normal and uniform. Four normal uniform deformations are solved where the initial state is straight and untwisted.