= 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 has
$$
F=\begin{pmatrix}\lambda^{-1/2}&0&0\\0&\lambda^{-1/2}&\alpha R\lambda^{-1/2}\\0&0&\lambda\end{pmatrix}.
$$
The <principal stretches> follow from the eigenvalues of $F^TF$. For a cylinder of reference radius $A$, stored energy per reference length is $\mathcal E=2\pi\int_0^A W(F)R\,dR$. 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.
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