Coexistent phase mixtures in the anti-plane shear of an elastic tube (Q1328419)

The structure of the deformed configuration of an elastic tube, whose cross-section is a ring subjected to a prescribed axial displacement of its lateral boundaries, is considered. The material is assumed to have a non-convex stored energy function. Special case when there is no minimizer in the associated anti-plane shear minimalization problem, but the energy functional has an infimum, is treated. The main variational problem is reduced to a free boundary problem, in which the subdomain where the strict convexity of the stored energy function breaks down must be determined as a part of the solution. The characterization of the intermediate phase mixture region is one of the purposes of this work.