= Twist and inflation of an incompressible tube
{title2=$r^2=\rho^2+c,\quad\theta=\Theta+\alpha Z,\quad z=Z$}
In cylindrical orthonormal bases the <deformation gradient> is $F=\begin{pmatrix}\rho/r&0&0\\0&r/\rho&\alpha r\\0&0&1\end{pmatrix}$ and has unit <determinant>. One <principal stretch> is $\rho/r$; the squares of the other two are the roots of $z^2-[(r/\rho)^2+\alpha^2r^2+1]z+(r/\rho)^2=0$. The reference-volume energy per height is $E=2\pi\int\rho W\,d\rho$. <Virtual work> gives end <torque> $M=\partial_\alpha E$ and inner pressure $p=(2\pi a)^{-1}\partial_aE$ at fixed reference radii and height.
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