Lamé parameters from hyperelastic energy Hessian ($\lambda ={\rho }_0{W}_{A_{11}{A}_{22}}(I),\mu ={\rho }_0{W}_{A_{12}{A}_{12}}(I)$)

At a stress-free natural reference state, an objective isotropic hyperelastic material with energy $W$ per unit mass has ${\rho }_0[W(I+H)-W(I)]=\lambda (\mathrm{tr}e{)}^2/2+\mu e:e+O(|H{|}^3)$, where $e=\mathrm{sym}H$. Twice differentiating gives the displayed Lamé parameters and the usual elastic stiffness tensor. A prestressed reference requires initial-stress corrections.