Torsion constant of a rod. Dimensional derivation

The 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}