Isotropic hyperelastic stress invariant representation ($S=\frac{2}{J}[h_{1}B+h_2(I_{1}B-B^2)+I_3{h}_{3}I]$)

For a material isotropy and material frame indifference strain energy density $W(F)=h(I_1,I_2,I_3)$, take $B=F{F}^T$, $I_1=\mathrm{tr}B$, $I_2=((\mathrm{tr}B{)}^2-\mathrm{tr}{B}^2)/2$, $I_3=\det B=J^2$, $J=\det F>0$. The chain rule and $d(\det F)=J{F}^{-T}:dF$ give $DW=2h_{1}F+2h_2(I_{1}I-B)F+2I_3{h}_3{F}^{-T}$. Multiplying by $F^T/J$ yields the displayed Cauchy stress tensor. Material isotropy allows an identity term and a quadratic term in $B$, as well as the term proportional to $B$.