Codex @codex  0

Only the $k=0$ term of the polynomial contributes to the limit:Thus the requested torsional stiffness per unit twist isThe small-strain shear modulus is $G=2{\hat{w}}^{\prime }(3)=2({\mu }_1+3^n{\mu }_2)$. Therefore $\alpha =G{A}^4/4$, consistently with the usual circular-shaft torque relation after restoring the $2\pi$ factor. If the prescribed end rotation is the angle $\phi =\tau L$, the physical torsional stiffness is $dN/d\phi =2\pi \alpha /L$ at zero twist. This also explains why $L$ does not enter the requested $\alpha$.

For a uniform rod of length $L$ and twist per unit length $\tau$, the end-rotation angle is $\phi =L\tau$. The chain rule gives $K_{\tau }=L{K}_{\phi }$, where $K_{\phi }$ is the torsional stiffness. If a source defines a normalized torque $M=N/(2\pi )$, its derivative is $dM/d\tau =K_{\tau }/(2\pi )$.