= Inflation pressure of a thin incompressible elastic balloon
{title2=$p=\frac{h_0}{a_0\lambda^2}\frac d{d\lambda}W(\lambda,\lambda,\lambda^{-2})$}
At leading thin-shell order, a spherical balloon with reference radius $a_0$, thickness $h_0$ and current radius $a=\lambda a_0$ has reference <elastic energy> $4\pi a_0^2h_0W(\lambda,\lambda,\lambda^{-2})$. Equating its differential to pressure work $p\,4\pi a^2da$ gives the displayed pressure difference. Its thickness becomes $h_0\lambda^{-2}$ by <incompressibility>. The same result follows from the spherical membrane <force> balance $p=2h\sigma_t/a$, with current tangential <Cauchy stress tensor> component $\sigma_t=\lambda N$ and the <equibiaxial nominal tension of an incompressible sheet>. Thin-shell validity requires $h/a\ll1$.
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