Stretch, Flexure and Twist in Finite Elasticity

Abstract

The classical stretching, torsion and flexure solutions of linear elasticity are important ingredients of asymptotic treatments of slender structures. They each describe solutions in a cylindrical elastic region, with zero traction applied over the lateral boundary. For rods, bars and tubes of arbitrary cross-section, anisotropy and transverse non-uniformity, they provide the relevant flexural, torsional and extensional rigidities, which relate resultant forces and moments to appropriate curvatures, twist and stretch in the deformed configuration. To leading order, these then characterize the constitutive behaviour.In finite elasticity, it is less clear how to choose the canonical states of deformation which should yield equivalent nonlinear constitutive laws in a one-dimensional theory. The helical solutions of Ericksen[1] provide a two-parameter set of such solutions, but this family must be enlarged by considering other solutions with no body forces or lateral tractions. Static solutions having strain a periodic function of the axial material coordinate provide such a four-parameter set of canonical deformations of a typical cross-section. These solutions are shown to play a central rôle in the asymptotic description of slender elastic bodies, within which both the rotation and strain may become large.KeywordsFinite elasticityasymptoticsanisotropyrodshelicalperiodic strain