Linear constitutive relations in isotropic finite elasticity

For simple shearing and simple extension deformations of a homogeneous and isotropic elastic body, it is shown that a Linear relation between the second Piola-Kirchhoff stress tensor and the Green-St. Venant strain tensor does not predict a physically reasonable response of the body. This constitutive relation implies that the slope of the curve between an appropriate component of the first Piola-Kirchhoff stress tensor and a deformation measure is an increasing function of the deformation measure.