= Equibiaxial nominal tension of an incompressible sheet
{title2=$N=\tfrac12\frac d{d\lambda}W(\lambda,\lambda,\lambda^{-2})$}
A thin isotropic <hyperelastic material> subject to <incompressibility>, with equal in-plane <principal stretches> $\lambda$ has transverse stretch $\lambda^{-2}$. With unloaded transverse faces, $N_{33}=W_3+q\lambda^2=0$. Hence $N_{11}=N_{22}=W_1-\lambda^{-3}W_3$, using isotropy to set $W_1=W_2$. The derivative along the constrained path is $W_1+W_2-2\lambda^{-3}W_3$, proving the displayed formula. The two equal nominal tensions do work $2N\,d\lambda$ per reference volume, exactly the change in <strain energy density>.
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