= Pure torsion of an incompressible cylinder
{title2=$r=R,\ \theta=\Theta+\tau Z,\ z=Z$}
A <deformation map> twisting every cross-section of a <cylinder> through an <angle> proportional to its reference axial position, without radial or axial extension. In the local current and reference <cylindrical coordinate systems>, the <deformation gradient> is $F=\begin{pmatrix}1&0&0\\0&1&\tau r\\0&0&1\end{pmatrix}$. Its <determinant> is one. The twist per unit reference length is $\tau$; an end separation $L$ gives end rotation $\varphi=\tau L$. The physical <torque> is $N=2\pi\int_0^A T_{\theta z}r^2\,dr$.
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