= Inflation and extension of an incompressible tube
{title2=$r^2=a^2+(R^2-a_0^2)/z$}
For reference radii $a_0,b_0$, current inner radius $a$ and uniform axial stretch $z>0$, <incompressibility> integrates to $r^2=a^2+(R^2-a_0^2)/z$. The <principal stretches> are $r_{,R}=1/(z\lambda)$, $\lambda=r/R$ and $z$. Write $\widehat W(\lambda,z)=W((z\lambda)^{-1},\lambda,z)$ and $\mathcal E=2\pi\int_{a_0}^{b_0}\widehat W R\,dR$ per reference length. <Virtual work> gives $p=\mathcal E_{,a}/(2\pi az)$ and the total axial wall load $N_{\rm wall}=\mathcal E_{,z}$, both derivatives holding the other displayed parameter fixed. For pressurized closed ends the independently applied axial load is $N_{\rm ext}=N_{\rm wall}-\pi a^2p$. The distinction follows from the axial <pressure> power $\pi a^2p\dot z$.
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