Reciprocal-Jacobian quadratic hyperelastic energy ($W(F)=\mu (\frac{1}{2}|F{|}^2+(\det F{)}^{-1})$)

On positive-determinant deformation gradients, the first Piola-Kirchhoff stress is $\mu (F-J^{-1}{F}^{-T})$. Its zero set is $SO(3)$: multiplication by $F^T$ gives $F{F}^T=J^{-1}I$, and determinants give $J^5=1$. These are also the global minimizers, since $|F{|}^2\ge 3J^{2/3}$ and $3J^{2/3}/2+J^{-1}$ has its unique minimum $5/2$ at $J=1$. The minimum energy is $5\mu /2$ rather than zero; an additive constant changes no stress.