Evaluating the Effect of the Second Invariant of Deformation Tensor in The Axial and Azimuthal Shear Deformations

The purpose of the present paper is to investigate the effect of the second invariant of the deformation tensor on the axial and azimuthal shear deformation of an incompressible hyperelastic solid with various strain energy functions. To this end, the axial shear deformation of an incompressible cylinder with the modified Gent-Thomas, Gent-Thomas, Gent-Gent, and Carroll strain energies subjected to an axial shear traction is considered, where the displacement field is determined analytically for the first three models and numerically for the fourth model. The phenomenon of strain hardening at large elastic deformations, predicted either by the limiting chain extensibility condition for the modified Gent-Thomas and Gent-Gent models or phenomenologically by the Carroll model, is observed and it is shown that the second invariant of deformation increases the strain hardening experienced by such materials. Next, the azimuthal shear deformation of an incompressible annular wedge with the modified Gent-Thomas, Gent-Thomas, Gent-Gent, and Carroll models is considered, where the annular wedge is subjected to a controllable azimuthal shear deformation and the angular displacement is determined analytically for all the above models. Again, the second invariant of the deformation tensor is shown to have a significant effect on the azimuthal shear deformation as reflected in the increase of the strain hardening of the material in such deformation. In addition, the annular wedge with the modified Gent-Thomas and Carroll models is shown to have a higher resistance in azimuthal shear deformation than the other models mentioned above.