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ID: past-exam-of-the-materials-science-course-of-the-university-of-cambridge/2018/paper-1/6/b/solution

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Past exam of the materials science course of the University of Cambridge / 2018 / Paper 1 / 6 / b / Solution by

Codex  0 2026-10-20

Interpret the supplied real quantity as the storage modulus

E^{'}

, with

k

the spring modulus in this continuum representation and

η

the corresponding viscosity. Let

τ = t_{R} = η / k

. Then

E^{'} = \frac{k ω ^{2} τ ^{2}}{1 + ω ^{2} τ ^{2}} .

At high frequency, $ω τ ≫ 1$ , the response approaches the spring stiffness; at low frequency it becomes much softer:

E^{'} ≃ k (ω τ ≫ 1), E^{'} ≃ k ω^{2} τ^{2} = \frac{η ^{2} ω ^{2}}{k} (ω τ ≪ 1) .

In the series Maxwell model, the spring and dashpot carry the same stress while their strains add. A rapidly changing load allows little viscous flow, so the spring dominates the strain response. Over a long cycle the dashpot has time to deform, giving a small storage modulus and a fluid-like low-frequency response. This is generic viscoelastic mechanics, independent of the biological parts deferred above.

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