Area-preserving azimuthal shear deformation of an incompressible isotropic hyper-elastic tube

The principal problem of interest in this article is that of the area-preserving azimuthal shear strain of an incompressible isotropic hyper-elastic circular cylindrical tube subjected to homogeneous radial tractions on its both inner and outer boundaries. Pure azimuthal shear strain may be considered as a particular case of the present deformation. However, in the present case, equilibrium requires a change of the inner and the outer tube boundaries which, due to the incompressibility constraint, may take place only in a manner that preserves the area of the tube cross section. Nevertheless, it is assumed that the tube retains its circular cylindrical shape. A considerable part of the solution to this problem is described analytically, but the final part requires numerical treatment; the balance between these two parts depends on the specific form of the strain energy density of the material. An appropriately modified version of the outlined method of solution may be further used for solving the generalized counterpart of this plane strain problem, where the radial tractions applied on the inner and the outer boundaries of the tube are not necessarily homogeneous.