Simple Shear and Applied Piola-Kirchhoff Shear Stress

The kinematical consequences of assuming a pure simple applied shear in terms of the Piola-Kirchhoff stress for homogeneous isotropic incompressible hyperelastic materials are explored. It is shown that for materials that depend only on the first invariant of the Cauchy-Green deformation tensors that the corresponding deformation is one of plane strain and that the sum of the in-plane shear terms can be written as a function of the applied Piola-Kirchhoff stress. Ericksen’s concept of maximum orthogonal shear is chosen as the strain measure that quantifies the effect of the applied shear stress. The analytic difficulties faced when both of the invariants of the deformation tensors are included are illustrated by consideration of the Mooney-Rivlin material. For this material a contractive out-of-the-plane stretch is required. There is now a coupled relationship between the stretch, the sum of the in-plane shear terms of the deformation tensor and the applied shear stress.