For a rubber-like solid, a spring is a first model of recoverable elasticity, while a viscous element can represent rate-dependent loss and delayed recovery. An immediate extension followed by bounded delayed material creep is represented by series spring and Voigt-branch creep, a standard linear solid model:

ε (t) = \frac{σ _{0}}{E _{0}} + \frac{σ _{0}}{E _{1}} (1 - e^{- E_{1} t / η}) .

The first term is immediate and recoverable; the second is delayed and also recoverable. For a load held until

t = t_{h}

, recovery after a finite-duration stress load predicts an immediate drop of

σ_{0} / E_{0}

on unloading, followed by

ε (t) = \frac{σ _{0}}{E _{1}} (1 - e^{- t_{h} / τ}) e^{- (t - t_{h}) / τ}, τ = \frac{η}{E _{1}}, t > t_{h} .

(2)

This supplies a direct comparison with a loading-and-unloading observation: the model returns completely to its original length, with separate immediate and delayed recovery. A single Kelvin-Voigt model misses the immediate compliance, while a free series dashpot would predict unbounded flow that an intact crosslinked rubber need not show. A spring alone suffices over a timescale on which the measured response is effectively immediate and elastic.

These are local small-strain analogues. Large-strain rubber elasticity is nonlinear, nearly incompressible and sensitive to temperature, loading history and internal damage. A small set of linear springs and dashpots cannot reproduce every amplitude, timescale or hysteresis loop.

No numerical rubber trace or fitted constants are supplied, so these are explicit qualitative model predictions for comparison with the practical observations.

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A physical mechanics model of wood must at least acknowledge anisotropy: stiffness and creep depend on loading direction. Direction-dependent elastic elements can represent different small-strain moduli, and delayed elastic branches can approximate time-dependent strain. A linear-solid network can therefore be a local fit under specified direction, moisture and temperature, rather than a universal scalar constitutive law.

An immediate strain followed by delayed material creep is captured locally by a spring plus a Kelvin-Voigt model branch. If residual deformation remains after unloading, a Burgers material adds a free series dashpot

η_{0}

. Under a constant load,

ε (t) = \frac{σ _{0}}{E _{0}} + \frac{σ _{0}}{E _{1}} (1 - e^{- t / τ}) + \frac{σ _{0} t}{η _{0}} .

(1)

After a hold time

t_{h}

, the two elastic contributions recover and the dashpot retains

ε (\infty) = \frac{σ _{0} t _{h}}{η _{0}} .

(2)

Thus immediate recovery, delayed recovery and a residual set identify different roles for the components; an observed permanent set would rule out a purely recoverable linear-solid fit. A viscous dashpot does not itself model a yield threshold or structural damage. Strength loss, splitting, moisture-dependent response and nonlinear compression cannot be represented by one fixed linear Kelvin-Voigt model. No direction-specific measured trace is supplied, so selecting numerical parameters requires the practical data; the constitutive predictions and their limitations are the model comparison that can be made from the paper.

Past exam of the materials science course of the University of Cambridge / 2015 / Paper 1 / 5 / e

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