 Solution

Here the complex viscoelastic modulus is a stress-to-strain response coefficient, rather than a complex number's absolute value. Use strain $\gamma (t)=\mathrm{Re}({\gamma }_0{e}^{i\omega t})$ and stress $\tau (t)=\mathrm{Re}[G^{\ast }(\omega ){\gamma }_0{e}^{i\omega t}]$, withThe storage modulus $G^{\prime }$ measures the elastic component; the loss modulus $G^{\prime \prime }$ measures dissipation. The mean power per volume is $\omega {\gamma }_0^2{G}^{\prime \prime }/2$, and the energy dissipated per cycle is $\pi {\gamma }_0^2{G}^{\prime \prime }$, fixing the sign convention.

A Hookean solid has $G^{\ast }=G_0$, while a Newtonian fluid has $G^{\ast }=i\omega \eta$. The Maxwell model interpolates with $G^{\ast }=G_{0}i\omega {\tau }_R/(1+i\omega {\tau }_R)$, giving $G^{\prime }=G_0(\omega {\tau }_R{)}^2/[1+(\omega {\tau }_R{)}^2]$ and $G^{\prime \prime }=G_0\omega {\tau }_R/[1+(\omega {\tau }_R{)}^2]$. It is fluid-like at low frequency and elastic on times short compared with relaxation time ${\tau }_R$.

A rheometer imposes small-amplitude oscillatory shear and resolves the torque amplitude and phase relative to the imposed motion. The torsional measurement of a complex shear modulus makes the conversion from torque and twist to $G^{\ast }$ explicit. A frequency sweep locates relaxation scales; keeping the strain small tests the linear-response assumption.