Kearsley-type instabilities in finite deformations of transversely isotropic and incompressible hyperelastic materials

Abstract

We assume that the strain energy density, W, for a transversely isotropic and incompressible hyperelastic solid is a complete quadratic function of components of the right Cauchy-Green strain tensor, C. We first discuss restrictions on the seven material parameters appearing in the expression for W. It is shown that for the reference configuration to be stress-free, both terms linear in (I4 − 1) and (I5 − 1) must simultaneously appear in the expression for W where I4 = A • CA, I5 = A • C2A, and A is a unit vector along the axis of transverse isotropy. Subsequently, we show that for a prismatic body comprised of this material deformed either in dead-load or displacement-controlled uniaxial tension/compression along A, depending upon the sign of the material parameter h14 multiplying the term (I1 − 3)(I4 − 1) in the expression for W, the two lateral stretches may bifurcate from being equal to being unequal for a stable solution defined as the one that has lower free energy. Here I1 = tr(C) where tr is the trace operator. This is similar to Kearsley's instability in an isotropic square Mooney-Rivlin membrane deformed with equal biaxial dead loads on the edges for which the stable deformed configuration shifts from having equal to unequal biaxial stretches.