Stress softening and residual strain in the azimuthal shear of pseudo-elastic circular cylindrical tube

In a recent paper, Ogden and Roxburgh (Proc. R. Soc. London A 455 (1999a) 2861) developed a theory of pseudo-elasticity for the description of damage-induced stress softening in rubberlike solids (the Mullins effect), and the theory was modified to incorporate residual strains by Ogden and Roxburgh (in: A. Dorfmann, A. Muhr (Eds.), Proceedings of the First European Conference on Constitute Models for Rubber, Vienna, 1999, Balkema, Rotterdam, pp. 23–28.). In the present paper this theory is applied to a problem involving non-homogeneous deformation, namely the (plane strain) azimuthal shear of a thick-walled circular cylindrical tube of incompressible material. Loading, effected by application of a specified rotation of the outer surface of the tube relative to the inner one, is described by an isotropic elastic strain-energy function. Unloading, associated with reduction in the applied shearing stress on the outer boundary, is described by a different isotropic elastic strain-energy function, which is inhomogeneous and dependent locally on the extent of the initial loading. It is shown that if the maximum applied shear stress on the outer boundary is below a certain critical value, then there is no residual strain after the shearing stress is removed, while if it is greater than a second critical value then there is residual strain throughout the tube. In the intermediate situation there is residual strain only within a certain radius. Outside this radius the final state of deformation corresponds to a rigid rotation. The results are specialized in respect of a particular material model, which allows the residual strain to be calculated explicitly. Also, the total residual (non-recoverable) energy due to the loading–unloading cycle is calculated. Numerical calculations are used to illustrate the stress softening effect by comparison of the shear stress on unloading with that on loading