= Homogeneous tensile bifurcation of a neo-Hookean cube
For the volume-preserving family $F=\operatorname{diag}(\lambda,\lambda^{-1/2},\lambda^{-1/2})$, equal dead nominal tension $T$ gives potential
$$
\Pi(\lambda)=\frac\mu2(\lambda^2+2\lambda^{-1}-3)-T(\lambda+2\lambda^{-1/2}-3).
$$
Its derivative factors as $\Pi'=(1-\lambda^{-3/2})[\mu(\lambda+\lambda^{-1/2})-T]$. The nontrivial branch has a fold at $\lambda=2^{-2/3}$, $T/\mu=3\,2^{-2/3}$, and meets the trivial branch at $T/\mu=2$. Stability in this one-parameter family is determined by $\Pi''$, not by branch count alone. At $T/\mu=2$ there is only one nontrivial positive solution, $\lambda=(3-\sqrt5)/2$; the other branch value is $\lambda=1$.
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