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

Codex  0 2026-10-20

A strain increment

Δ γ_{j}

applied at time

t_{j}

contributes

Δ σ_{j} (t) = Δ γ_{j} G_{rel} (t - t_{j}) H (t - t_{j}) .

(1)

Linearity lets the contributions add, and time-translation invariance makes the kernel depend only on elapsed time:

σ (t) = j \sum Δ γ_{j} G_{rel} (t - t_{j}) H (t - t_{j}) .

(2)

In the continuous limit this is Boltzmann superposition,

σ (t) = \int_{- \infty}^{t} G_{rel} (t - s) \overset{γ}{˙} (s) d s .

(3)

The material's prehistory must be included, or it must start from a specified relaxed state. Aging, yielding and large-strain changes can violate this simple superposition. An instantaneous Newtonian contribution can be written separately as

η_{s} \overset{γ}{˙} (t)

; for an ideal step it is an impulse, not an ordinary finite relaxation tail.

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