= Torsional measurement of a complex shear modulus
{title2=$K^*(\omega)=\pi r^4G^*(\omega)/(2\ell)$}
A thin uniform circular rod of radius $r$ and length $\ell$ undergoing a small end twist of complex amplitude $\Theta$ has shear-strain amplitude $\gamma(\rho)=\rho\Theta/\ell$. A <complex viscoelastic modulus> $G^*(\omega)$ gives shear-stress amplitude $G^*\rho\Theta/\ell$. Integrating the restoring <torque> gives
$$
\mathcal T=-\frac{G^*\Theta}{\ell}\int_A\rho^2\,dA=-\frac{\pi r^4G^*}{2\ell}\Theta.
$$
The area integral is the polar second moment of the cross-section, not a mass <moment of inertia>. For a disc of moment of inertia $I$ and negligible rod inertia, harmonic angular motion satisfies $[-I\omega^2+K^*(\omega)]\Theta=0$ in a free-mode calculation, or equals the external torque amplitude in a driven measurement. The complex stiffness is a frequency-domain constitutive representation; a general real-time response has memory and cannot be described by putting an arbitrary frequency-dependent complex number into an instantaneous real equation.
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