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

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

Codex  0 2026-10-20

With

s = ω^{2}

, the curve is

E^{'} (s) = \frac{k τ ^{2} s}{1 + τ ^{2} s}, \frac{d E ^{'}}{d s} = \frac{k τ ^{2}}{( 1 + τ ^{2} s ) ^{2}} > 0.

(1)

It rises linearly from zero, bends downward and approaches the plateau $k$ . Its half-height occurs at

s = τ^{- 2}

For

ω^{2} ≪ τ^{- 2}

, the dashpot contributes the larger strain and dominates the compliant response. For

ω^{2} ≫ τ^{- 2}

, dashpot deformation during a cycle is small relative to spring strain, and the response is spring-dominated. The crossover separates these regimes rather than a discontinuous transition.

Figure 1.
Original Maxwell storage curves versus squared frequency and the distinct dissipation laws at fixed strain and fixed stress amplitudes; axes use logarithmic spacing to show both regimes
. **Details**.

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