Analytical hypoelastic stress solutions of the right pure shear deformation

The paper attempts to analyze the hypoelastic stress responses of isotropic solids to the right pure shear deformation. After a concise description of the right pure shear deformation, a new expression of the deformation described in forms of trigonometric functions is developed. Thereafter, the necessary deformation and rate-form tensors are formulated. In addition, a detailed derivation of the eigenvectors of the left and right Cauchy–Green deformation tensors and their eigenprojection tensors is presented. On the basis of the representation principle of the material spin tensors, a set of concise representation equations of the spin tensors for the right pure shear deformation is derived. From the derived deformation and rate-form tensors, the hypoelastic equations based on eight representative stress rates are formulated and solved, and multiple sets of analytical solutions are derived. The analytical stress solutions are plotted vs. the shear strain and compared by graphs. A unique and interesting result is that the spins for the Jaumann, Green–Naghdi, logarithmic and \(\varvec{\Omega }^{\text {E}}\) ${\mathbf{\Omega }}^{\text{E}}$ -based stress rates are the same under the right pure shear deformation; hence, the analytical stress responses rendered by the hypoelastic equations based on these spins are the same and consistent with the response shown by the linear isotropic elastic equation. Under the right pure shear deformation, all the corotational hypoelastic equations can render the pure shear stress state, only the pure shear stress components are not observed under the reference Cartesian frame. The hypoelastic equations related to the spatial velocity gradient cannot present the pure shear stress state.