Stress-free linearization of an objective energy ($D^{2}W(I)[K]=0(K^T=-K)$)

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 $D^{2}W(I)[H]$ 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 $DW(I)H^T$.