On azimuthal shear of a circular cylindrical tube of compressible elastic material

In this paper the plane strain character of the finite azimuthal shear of a circular cylindrical annulus of compressible isotropic elastic material is used to express the strain energy as a function of two independent deformation invariants. The resulting formulation of the equilibrium equations is applied to the special case of pure azimuthal shear (an isochoric deformation), and explicit necessary and sufficient conditions on the strain-energy function for the material to admit this deformation are obtained. These conditions are examined for two classes of strain-energy functions and in each case complete solutions of the equilibrium equations are obtained. Some known results are recovered as special cases and some new results for particular strain-energy functions are determined and discussed. Numerical results showing the dependence of the applied shear stress on the resulting relative rotation of the boundaries are given for different strain-energy functions for comparison. The theory is then used to show how corresponding solutions can be deduced for incompressible materials. It is pointed out that the method of considering isochoric deformations in compressible elastic materials provides a means of generating classes of strain-energy functions for which closed-form solutions can be found for incompressible materials.