Neo-Hookean solid

For an incompressible neo-Hookean solid, a convenient strain energy density is $W=\mu (\mathrm{tr}C-3)/2$, where $C$ is the right Cauchy-Green deformation tensor. An additive constant in $W$ does not affect the stress. The shear modulus $\mu$ fixes the small-strain stiffness.

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.

For the volume-preserving family $F=\mathrm{diag}(\lambda ,{\lambda }^{-1/2},{\lambda }^{-1/2})$, equal dead nominal tension $T$ gives potentialIts derivative factors as ${\Pi }^{\prime }=(1-{\lambda }^{-3/2})[\mu (\lambda +{\lambda }^{-1/2})-T]$. The nontrivial branch has a fold at $\lambda =2^{-2/3}$, $T/\mu =32^{-2/3}$, and meets the trivial branch at $T/\mu =2$. Stability in this one-parameter family is determined by ${\Pi }^{\prime \prime }$, not by branch count alone. At $T/\mu =2$ there is only one nontrivial positive solution, $\lambda =(3-\sqrt{5})/2$; the other branch value is $\lambda =1$.