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

TODO: User-deferred biology. No protein or organism-specific solution or research is included.

TODO: User-deferred biology. No protein or organism-specific solution or research is included.

TODO: User-deferred biology. No protein or organism-specific solution or research is included.

The original diagram is a Kelvin-Voigt model: a spring and a dashpot act in parallel. The spring is an ideal elastic element, ${\sigma }_s=E{\epsilon }_s$; under its own sinusoidal stress its strain is in phase with stress and energy is stored reversibly. The dashpot is viscous, ${\sigma }_d=\eta {\dot{\epsilon }}_d$; its strain under an oscillating stress lags by90degrees and mechanical energy is dissipated.

In parallel the strain is common and the branch stresses add:For imposed harmonic stress, the strain amplitude and phase lag areThe short- and long-timescale limits follow under the printed roman labels below.

A sudden constant stress ${\sigma }_0$ gives delayed, bounded creep:The strain is initially zero because an instantaneous change would require an infinite dashpot stress. Initially the dashpot carries the load, with finite strain rate ${\sigma }_0/\eta$; the spring gradually carries more of it as the rate decays. On unloading, the strain recovers exponentially to zero. The network does not produce the instantaneous strain jump or unbounded constant-stress creep of a series Maxwell element.

On a timescale short compared with $\tau =\eta /E$, or for $\omega \tau \gg 1$, the dashpot strongly resists rapid deformation. The imposed oscillating stress produces a small strain, of amplitude approximately ${\sigma }_0/(\eta \omega )$, nearly90degrees behind the stress. The network appears stiff and predominantly viscous in this stress-controlled high-frequency limit. It is not an ideal perfectly rigid body at finite frequency.

For a slowly varying load with $\omega \tau \ll 1$, the dashpot stress is small relative to the spring stress. The strain approaches $\sigma /E$ almost in phase, so the low-frequency response is predominantly elastic. A constant load approaches the finite equilibrium strain ${\sigma }_0/E$; it does not sustain steady viscous flow in this parallel network.

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. If the observed material has both an immediate extension and a further bounded delayed creep, a spring in series with a Kelvin-Voigt element is a simple linear-solid model:The first term is immediate and recoverable; the second is delayed and also recoverable. A single Voigt element misses the immediate compliance, while a free series dashpot would predict unbounded flow that an intact crosslinked rubber need not show.

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.

TODO: Match the model to the unavailable practical-class observations. No particular measured rubber trace, specimen state or fitted constants are supplied.

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.

If experiments show an immediate strain followed by delayed creep, a spring plus a Voigt branch can capture those features. If permanent set remains after unloading, that recoverable model is insufficient and an irreversible deformation branch or damage law is needed. Strength loss, splitting, moisture-dependent response and nonlinear compression cannot be represented by one fixed linear Voigt element.

TODO: Match the network and any irreversible branch to the unavailable practical-class observations. These are conditional materials-mechanics possibilities, not invented laboratory results. No biological structure or organism-specific mechanism is required here.