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

Table of contents
- Solution d

Solution

For elements in series the stress is common and the strains add. If

ϵ_{K}

and

ϵ_{1}

are the Kelvin element and added spring amplitudes, then

ϵ = ϵ_{K} + ϵ_{1} = \frac{σ}{G _{0} + iω η _{0}} + \frac{σ}{G _{1}} .

The compliances therefore add, yielding

G^{*} (ω) = [\frac{1}{G _{0} + iω η _{0}} + \frac{1}{G _{1}}]^{- 1} = \frac{G _{1} ( G _{0} + iω η _{0} )}{G _{0} + G _{1} + iω η _{0}} .

This is a standard linear solid model. Define

G_{L} = \frac{G _{0} G _{1}}{G _{0} + G _{1}}, G_{H} = G_{1}, Δ G = G_{H} - G_{L} = \frac{G _{1}^{2}}{G _{0} + G _{1}}, τ = \frac{η _{0}}{G _{0} + G _{1}} .

x = ω τ

, its storage modulus and loss modulus are

G^{'} = G_{L} + Δ G \frac{x ^{2}}{1 + x ^{2}}, G^{''} = Δ G \frac{x}{1 + x ^{2}} .

Thus

G^{'}

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

Δ G /2

at

ω = 1/ τ

. At that frequency

G^{'} = (G_{L} + G_{H}) /2

. There need not be a crossing of the two curves for arbitrary spring constants.

Figure 1.
Storage modulus and loss modulus of the standard linear solid model
. Original log-log plot of the derived formulas, with illustrative $G_{0} = 1$ , $G_{1} = 4$ and frequency expressed as $ω τ$ . The chosen spring ratio is not specified by the exam. Dashed horizontal lines mark $G_{L}$ and $G_{H}$ , and the dotted vertical line marks the loss maximum. Details.

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