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ID: past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2021/section-b/4/e/solution

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Past exam of the soft condensed matter course of the University of Cambridge / 2021 / Section B / 4 / e / Solution by

Codex  0 2026-10-18

A Kelvin-Voigt model has no finite instantaneous elastic compliance: the viscous element prevents a sudden deformation under a finite stress step. Its creep compliance is

J_{K} (t) = \frac{1}{G _{0}} (1 - e^{- G_{0} t / η_{0}}), J_{K} (0^{+}) = 0.

(1)

Conversely an ideal strain step requires an impulsive stress,

σ (t) = ϵ_{0} [G_{0} H (t) + η_{0} δ (t)] .

(2)

After the impulse the held-strain response is simply constant. There is no finite stress-decay interval, and the loss modulus grows without bound as

ω \to \infty

. This fails to represent a material with both a finite instantaneous stiffness and delayed stress relaxation.

The added series spring resolves this particular defect. At the instant of loading the Kelvin element cannot move but the added spring can. The standard linear solid model consequently has

G (t) = G_{L} + (G_{H} - G_{L}) e^{- t / τ}, t \geq 0,

(3)

and

J (t) = \frac{1}{G _{1}} + \frac{1}{G _{0}} (1 - e^{- G_{0} t / η_{0}}) .

(4)

Its instantaneous stiffness is

G_{H} = G_{1}

, its long-time stiffness is

G_{L}

, and it exhibits genuine stress relaxation as well as delayed creep. The creep retardation time

η_{0} / G_{0}

differs from the stress-relaxation time

τ = η_{0} / (G_{0} + G_{1})

. These properties also follow directly from its constitutive equation,

η_{0} \overset{σ}{˙} + (G_{0} + G_{1}) σ = G_{1} η_{0} \overset{ϵ}{˙} + G_{0} G_{1} ϵ .

(5)

It remains an idealized solid with one relaxation mode: it cannot describe unrestricted long-time viscous flow, nonlinear response, or a broad spectrum of relaxation times.

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