= Stress-free linearization of an objective energy
{title2=$D^2W(I)[K]=0\quad(K^T=-K)$}
For an objective smooth <strain energy density> $W(QF)=W(F)$, differentiation along $Q=e^{tK}$ shows $DW(F)F^T$ is symmetric. Hence the associated <Cauchy stress tensor> is symmetric. At a stress-free reference $DW(I)=0$, differentiation of this identity shows $D^2W(I)[H]$ is symmetric for every $H$. In components this gives output minor symmetry. The <Hessian> has pair-interchange symmetry, which then gives the input minor symmetry and the displayed annihilation of skew increments. The stress-free condition matters: at a prestressed reference the derivative includes the extra term $DW(I)H^T$.
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