Equibiaxial dead loading of a neo-Hookean prism ($T=2\alpha (v-v^{-5}),(v_1,v_2,v_3)=(v,v,v^{-2})$)

Consider an incompressible neo-Hookean prism with energy $\alpha (|F{|}^2-3)$, zero nominal end traction normal to $e_3$, and equal nominal lateral tractions $TN$ on side faces whose normals enforce both in-plane principal stresses. Within positive diagonal homogeneous deformations the first Piola-Kirchhoff stress tensor has entries $t_i=2\alpha v_i-p/v_i$. Equilibrium makes $p$ constant; end freedom gives $p=2\alpha v_3^2>0$.

The function $2\alpha v-p/v$ is strictly increasing on positive $v$, so equal lateral stresses require $v_1=v_2$. The determinant constraint then gives $v_3=v^{-2}$ and the displayed load relation. Its derivative is $2\alpha (1+5v^{-6})>0$, with limits minus infinity and infinity at the two ends of the positive axis. Thus there is exactly one positive diagonal equilibrium for every signed load. This uniqueness is confined to the homogeneous diagonal class and does not rule out other modes or nonhomogeneous equilibria under a general dead-loading problem.