Past exam of the soft condensed matter course of the University of Cambridge / 2021 / Section B / 4 / d / Solution

For elements in series the stress is common and the strains add. If ${\widehat{ϵ}}_K$ and ${\widehat{ϵ}}_1$ are the Kelvin element and added spring amplitudes, thenThe compliances therefore add, yieldingThis is a standard linear solid model. DefineWriting $x=\omega \tau$, its storage modulus and loss modulus areThus $G^{\prime }$ increases from the low-frequency plateau $G_L$ to the high-frequency plateau $G_H$. The loss modulus has low-frequency slope $+1$ and high-frequency slope $-1$ on a log-log plot, with maximum $\Delta G/2$ at $\omega =1/\tau$. At that frequency $G^{\prime }=(G_L+G_H)/2$. There need not be a crossing of the two curves for arbitrary spring constants.