The Treloar–Kearsley bifurcation problem using a new class of constitutive equations

It has been observed in experiments that, for some materials, when a sheet is subjected to increasing equal biaxial tensile forces on its edges, its deformation can change from homogeneous equal biaxial extension to homogeneous unequal biaxial extension. Such response has been analyzed in the literature as a bifurcation from the base deformation to a new deformation. The analyses have used a constitutive equation that expresses the stress tensor as a function of a deformation tensor. The present work is concerned with analyzing this bifurcation using a new class of constitutive equations in which the deformation tensor is expressed as a function of the stress tensor. Using such a model for an incompressible isotropic nonlinear elastic material, a condition is derived for determining when during an equal biaxial extension of a sheet, there is a bifurcation into an unequal biaxial extension. An example is provided using a constitutive equation that has been fit to experimental data.