Torsional measurement of a complex shear modulus ($K^{\ast }(\omega )=\pi r^4{G}^{\ast }(\omega )/(2\ell )$)

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 $G^{\ast }(\omega )$ gives shear-stress amplitude $G^{\ast }\rho \Theta /\ell$. Integrating the restoring torque givesThe 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 $[-I{\omega }^2+K^{\ast }(\omega )]\Theta =0$ 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.