Azimuthal shear of a transversely isotropic elastic solid

In this paper we study the problem of (plane strain) azimuthal shear of a circular cylindrical tube of incompressible transversely isotropic elastic material subject to finite deformation. The preferred direction associated with the transverse isotropy lies in the planes normal to the tube axis and is at an angle with the radial direction that depends only on the radius. For a general form of strain-energy function the considered deformation yields simple expressions for the azimuthal shear stress and the associated strong ellipticity condition in terms of the azimuthal shear strain. These apply for a sense of shear that is either "with" or "against" the preferred direction (anticlockwise and clockwise, respectively), so that material line elements locally in the preferred direction either extend or (at least initially) contract, respectively. For some specific strain-energy functions we then examine local loss of uniqueness of the shear stress—strain relationship and failure of ellipticity for the case of contraction and the dependence on the geometry of the preferred direction. In particular, for a reinforced neo-Hookean material, we obtain closed-form solutions that determine the domain of strong ellipticity in terms of the relationship between the shear strain and the angle (in general, a function of the radius) between the tangent to the preferred direction and the undeformed radial direction. It is shown, in particular, that as the magnitude of the applied shear stress increases then, after loss of ellipticity, there are two admissible values for the shear strain at certain radial locations. Absolutely stable deformations involve the lower magnitude value outside a certain radius and the higher magnitude value within this radius. The radius that separates the two values increases with increasing magnitude of the shear stress. The results are illustrated graphically for two specific forms of energy function