A STRUCTURED PHASE-TRANSITION FOR THE ANTIPLANE SHEAR OF AN ELASTIC CIRCULAR TUBE

We consider the radially symmetric axial shear of an elastic circular tube. The strain-energy function for the material is non-convex, and the potential energy contains a penalty contribution for large strain gradients. We study the corresponding problem of minimum potential energy, and give a detailed analysis of its solution for the particular case of a trilinear material. In this case, the relevant Euler-Lagrange equation can be completely analysed in terms of special functions. We study the limiting behaviour of the solution as the penalty parameter is made infinitesimally small. Finally, we present numerical results for this solution and, for reasons of comparison, also for the solution in the case of a cubic material.