= Stretch-sum degeneracy of an equally dead-loaded incompressible cube
{title2=$v_1+v_2+v_3=\tau/\mu,\quad v_1v_2v_3=1$}
For $\Phi=\mu[\sum_i v_i^2/2+\sum_i v_i^{-1}]$ and positive stretch product one, the energy simplifies to $\mu S^2/2$, where $S=\sum_i v_i$. Equal nominal tensile loading adds potential $-\tau S$. The homogeneous diagonal <equilibrium> equations reduce to $(\mu S-\tau)(v_i-v_j)=0$. Thus the unit triple exists for every load, while the other <equilibria> form the displayed sum-product family. Since $S\geq3$, only the unit triple exists for $\tau\leq3\mu$. Above this load, a continuous family is obtained by choosing $v_1=t$ near one and taking $v_2,v_3$ as the two positive roots with sum $\tau/\mu-t$ and product $1/t$.
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