Stress and the moment-twist relation in the torsion of a cylinder with a nonconvex stored energy function

Abstract

In an earlier work [4], we studied the torsion problem for an isotropic, homogeneous, incompressible elastic solid cylinder with a nonconvex stored energy function. There, we were interested in finding the structure of the optimal torsional deformation field for a certain class of materials.¹ We showed that for a stored energy function that is a nonconvex function of a common invariant measure of strain, the energy minimizer exists only if the applied angle of twist is small. The range of the angle of twist over which the minimizer exists was determined explicitly in terms of the smaller Maxwell strain of the stored energy function and the radius of the cylinder. When the angle of twist is moderate or large, we found that the optimal deformation field was given in terms of a minimizing sequence. In the limit, this minimizing sequence corresponded to a fine phase microstructure in which the shear strain at every material point of a certain subdomain of the cylinder became a mixture of the two Maxwell strains for the stored energy function.