 Solution

ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2018/paper-2/section-b/21b/v/solution

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 2 / Section B / 21B / v / Solution by

Codex  0 2026-10-20

For

ω τ ≪ 1

G^{'} (ω) ≃ \frac{η _{M}^{2}}{G} ω^{2}, G^{''} (ω) ≃ (η_{M} + η_{s}) ω,

(1)

and

η^{*} (ω) = \frac{G ^{*}}{iω} = η_{s} + \frac{η _{M}}{1 + iω τ} ⟶ η_{0} = η_{M} + η_{s} .

(2)

Thus the material flows at sufficiently low frequency: its static shear modulus vanishes and its terminal response is viscous. The dashpot in the Maxwell branch allows its spring stress to relax, so there is no permanent elastic path through the parallel network.

The fact that both

G^{'}

and

G^{''}

tend to zero is not evidence of a force-free material: stress at fixed shear rate tends to

η_{0} \overset{γ}{˙}

. At higher frequency the spring stores energy, but a finite-frequency elastic response does not make the long-time material a solid.

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