Helical shear of an elastic, circular tube with a non-convex stored energy

Abstract

In 1949, in one of his pioneering studies on large elastic deformations, Rivlin [1] applied the general theory of non-linear elasticity for an incompressible, isotropic, homogeneous body to study the helical shearing of a circular tube for a Mooney material. Approximately 25 years later, Ogden, Chadwick & Haddon [2] reconsidered this problem in some detail for more general materials, but in neither of these works was there an emphasis placed upon the structure of the stored energy function and, in particular, on its convexity. In the present work, we wish to emphasize the case of a non-convex stored energy function and the related emergence of equilibrium states with discontinuous deformation gradients. It is most convenient to consider the problem as one of minimization, and we shall seek to characterize the uniqueness, existence, and detailed structural properties of an absolute minimizer.