12 May 2013

Torsion constant of a rod. Dimensional derivation

Diagram of twisted rodThe torsion constant of a circular rod (the torque needed to twist it by a given angle) is easily calculated from the equations of elasticity. Up to a numerical constant, it can also be derived by dimensional analysis, as shown below.
Consider the rod in Figure a), with radius \(r\), length \(L\) and shear modulus \(G\). Its upper end A is clamped. The torque \(T\) needed to turn the free end B is linear1 in the twist angle \(\theta\):
\begin{equation}
T = \kappa \theta
\label{twist}
\end{equation}
Clearly, \(\theta\) depends on \(r\), \(L\) and \(G\). Are these the only relevant parameters? How about the bulk modulus \(B\), for instance? A non-rigorous way of showing its irrelevance is by considering a material with finite \(B\) and \(G = 0\) (e.g. a liquid): in this case the torsion constant is clearly zero2. We can then write:
\begin{equation}
\kappa = K \, G^a r^b L^c
\label{kappa}
\end{equation}

The various units are: \([\kappa ] = \text{Nm}/\text{rad} = \text{Nm} \), \([G] = \text{Pa} = \text{Nm}^{-2}\), \([L] = [r] = \text{m}\) and \( K\) is a dimensionless prefactor. We can deduce from \eqref{kappa} that \( a = 1\) and that \( b + c = 3\), but we need more information to determine \( b \) and \( c \) separately.
[UPDATE: 19/05/2013] In principle, the expression for \(\kappa\)  \eqref{kappa} could also contain an arbitrary function \(f\) of the dimensionless parameter \(r/L\), but the discussion below shows that \(f(r/L)\) is in fact a constant.

To this end, we make a virtual horizontal cut at the midpoint C (Figure b) and write the torque balance. In A, the clamp exerts a torque \(-T\) (to ensure equilibrium of the whole rod). The equilibrium of segments AC and CB requires that they act on each other with torques \(T\) and \(-T\), respectively. They are identical, so each of them twists by \(\theta/2\). In conclusion, for the same torque, a rod half as long twists half as much, so its torsion constant is twice that of the original rod. Hence, \( c = -1\) and then \(b = 4\), yielding:
\begin{equation}
\kappa = K \, G \frac{r^4}{L}
\label{final}
\end{equation}
The complete calculation3 yields \( K = \pi/2 = 1.57\ldots \), not very far from unity. We can see this as a lucky choice of parameters: working with the diameter \(D = 2 r\) instead of the radius would multiply the estimate by 16. On the other hand, \(r\) is the more natural elasticity variable, since it is over this distance that the deformation varies (from 0 to its maximum value).

1. Rigorously speaking, we can only say that \(\theta\) is odd in \(T\) (they change sign together), but for small twists the dependence will be dominated by the linear term \eqref{twist}. We only consider this linear regime.
2. This is an instance of the general principle that liquids cannot transmit static torques.
3. L. D. Landau and E. M. Lifshitz, Theory of Elasticity (volume 7 of the Course of Theoretical Physics), 3rd edition, Elsevier, 1986 (§ 16).

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