= Reciprocal-Jacobian quadratic hyperelastic energy
{c}
{title2=$W(F)=\mu(\tfrac12|F|^2+(\det F)^{-1})$}
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 $FF^T=J^{-1}I$, and <determinants> give $J^5=1$. These are also the global <minimizers>, since $|F|^2\geq3J^{2/3}$ 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>.
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