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

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 (I-4 - 1) and (I-5 - 1) must simultaneously appear in the expression for W where I-4 = A center dot CA, I-5 = A center dot C(2)A, 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 h(14) multiplying the term (I-1 - 3)(I-4 - 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 I-1 = 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. (C) 2020 Elsevier Ltd. All rights reserved.