Pure torsion of an incompressible cylinder ($r=R,\theta =\Theta +\tau Z,z=Z$)

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 $F=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& \tau r\\ 0& 0& 1\end{array}\right)$. Its determinant is one. The twist per unit reference length is $\tau$; an end separation $L$ gives end rotation $\phi =\tau L$. The physical torque is $N=2\pi {\int }_0^A{T}_{\theta z}{r}^{2}dr$.