Torque and axial force of a twisted incompressible cylinder

For axial stretch $\lambda$ and twist per reference length $\alpha$, the deformation in local cylindrical orthonormal bases hasThe principal stretches follow from the eigenvalues of $F^{T}F$. For a cylinder of reference radius $A$, stored energy per reference length is $\mathcal{E}=2\pi {\int }_0^{A}W(F)RdR$. Reversible power $M\dot{\alpha }+N\dot{\lambda }=\dot{\mathcal{E}}$ gives torque $M={\mathcal{E}}_{,\alpha }$ and axial force $N={\mathcal{E}}_{,\lambda }$. The relevant derivative holds the reference twist fixed, not the twist per current length.

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 $N=2\pi {\int }_0^A{T}_{\theta z}{r}^{2}dr$.

For axial stretch $\lambda$ and twist $\alpha$ per reference length, an incompressible neo-Hookean solid has current radius $a=R/\sqrt{\lambda }$. In local cylindrical orthonormal bases the deformation gradient is $A=\left(\begin{array}{ccc}{\lambda }^{-1/2}& 0& 0\\ 0& {\lambda }^{-1/2}& \alpha r\\ 0& 0& \lambda \end{array}\right)$. The Cauchy stress tensor is $\mu A{A}^T+qI$. Radial equilibrium gives $q^{\prime }=\mu {\alpha }^{2}r$, and a traction-free curved surface gives $q=-\mu /\lambda +\mu {\alpha }^2(r^2-a^2)/2$. The nominal end traction is the displayed vector, measured per reference area. Its moment about the axis is $\int (x_1{t}_2-x_2{t}_1)d{S}_0=\mu \alpha \int r^{2}d{S}_0=\mu \pi R^4\alpha /(2\lambda )$. This calculation uses current moment arms with reference-area tractions.