Torsion of a hollow cylinder

The physical problem consists of a hollow cylinder fixed at one end in the translational degrees of freedom. Opposing twists, equal in magnitude, are applied to the inner and outer diameters of the hollow cylinder. Figure 1 shows the model geometry used in this analysis. Most of the meshes used for this problem are uniform; however, since the stresses are a quadratic function of the radius, mesh refinement in regions of high strain gradients is suggested. Two input files have refined meshes near the inner diameter of the hollow cylinder with appropriate mesh refinement multi-point constraints. Some input files use a kinematic coupling to couple the boundary surfaces to reference nodes; the twists are applied to the reference nodes. All axisymmetric solid elements with twist are tested in this problem; coupled temperature-displacement elements are tested in a steady-state coupled temperature-displacement step with arbitrarily applied boundary temperatures and a coefficient of thermal expansion, $\alpha =$ 0, to prevent coupling with the mechanical solution. Mesh convergence studies have not been performed. To test the use of substructures in problems involving axisymmetric elements with twist, the problem is also solved using substructuring with the entire model treated as a single substructure.