Two Different Modes of Shear Stress Response in Simple Shear of Isotropic Incompressible Hyperelastic Materials

The classic deformation known as simple shear has been investigated within the framework of nonlinear elasticity for isotropic incompressible hyperelastic materials in a large variety of contexts. For example, this deformation has been widely used in recent years in the analysis of the mechanical behaviour of biological soft matter. In this paper, we compare two different representations for the shear stress response that arise in the context of simple shearing: the first is the response arising in the classical strain or displacement based formulation while the second is the response arising in the stress or traction based formulation where the only nonzero stress is the shear stress. The former is also called a state of pure shear stretch where one stretch ratio is fixed at unity so that the other two principal stretches are a reciprocal of one another while the latter is a state of pure shear stress. Two different representations for the shear stress versus maximum stretch are obtained. The results are illustrated on using specific strain-energy densities for homogeneous isotropic incompressible hyperelastic materials.