= Isotropic hyperelastic stress invariant representation
{title2=$S=\frac2J[h_1B+h_2(I_1B-B^2)+I_3h_3I]$}
For a <material isotropy> and <material frame indifference> <strain energy density> $W(F)=h(I_1,I_2,I_3)$, take $B=FF^T$, $I_1=\operatorname{tr}B$, $I_2=((\operatorname{tr}B)^2-\operatorname{tr}B^2)/2$, $I_3=\det B=J^2$, $J=\det F>0$. The chain rule and $d(\det F)=J F^{-T}:dF$ give $DW=2h_1F+2h_2(I_1I-B)F+2I_3h_3F^{-T}$. 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 $B$.
Back to article page
