Hypoelastic stress analysis of a pure shear deformation

In view of that a representative pure shear deformation was studied widely on its kinematic necessary condition for providing a plane pure shear Cauchy stress and seldom on its stress response under different constitutive characterizations, the present work renders a systematic hypoelastic stress analysis of grade zero of the deformation. On the basis of a concise description of the pure shear deformation, a novel trigonometric formulation of the deformation is presented. Then, the necessary deformation and rate tensors are derived, and a common representation equation of the material spins is formulated for the pure shear deformation. With these results in hand, the hypoelastic differential equations based on eight representative objective stress rates are derived and solved systematically, and multiple sets of analytical solutions are formulated. The obtained solutions can establish an analytical foundation for further numerical and experimental studies on the pure shear deformation. In order to show and compare the solutions directly and visually, all the solutions are coded in a MATLAB program and calculated numerically. For each hypoelastic equation, the four stress components included in the solutions are plotted vs. a given range of shear. On the basis of the numerical results and graphs, all the stress solutions are compared and analyzed. It can be found that all the hypoelastic equations related to the corotational stress rates can provide in-plane pure shear stresses, but only the one based on the logarithmic corotational description provides the pure shear stress components observed within the standard reference frame. The equations based on the non-corotational descriptions cannot present pure shear stress.