Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 2 / Section B / 21B

In a material creep experiment, apply a small step shear stress ${\sigma }_{0}H(t)$ and measure the resulting strain. The creep compliance is $J(t)=\gamma (t)/{\sigma }_0$. An elastic solid reaches a bounded strain; a fluid can continue deforming.

In a stress-relaxation experiment, impose a small strain step ${\gamma }_{0}H(t)$ and measure stress while holding the deformation. The relaxation modulus is $G_{\mathrm{rel}}(t)=\sigma (t)/{\gamma }_0$ after the loading transient. A fluid's stored shear stress can decay to zero, while an elastic solid retains a nonzero long-time contribution.

The original curves distinguish the controlled input from measured output. The strain/stress must be small enough for linear response, and real finite loading rise time, instrument compliance, wall slip and temperature must be characterized. Otherwise an instrumental transient can be mistaken for a material relaxation mode.

A strain increment $\Delta {\gamma }_j$ applied at time $t_j$ contributesLinearity lets the contributions add, and time-translation invariance makes the kernel depend only on elapsed time:In the continuous limit this is Boltzmann superposition,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 ${\eta }_s\dot{\gamma }(t)$; for an ideal step it is an impulse, not an ordinary finite relaxation tail.

A Kelvin-Voigt model element places a spring and dashpot in parallel, givingUnder a stress step from a relaxed state,It captures delayed but bounded material creep and eventual strain recovery after unloading, rather than steady terminal flow.

For an ideal strain step,After the loading impulse, the stress remains $G{\gamma }_0$. The model does not produce a gradual stress-relaxation tail at held strain. A real finite-rate ramp replaces the impulse by a finite dashpot stress during loading, but the subsequent held-strain stress is still constant. Thus this model describes material creep retardation but not a solid's genuine time-dependent relaxation toward another finite modulus.

Let the Maxwell branch contain spring modulus $G$ and series viscosity ${\eta }_M$, in parallel with a dashpot of viscosity ${\eta }_s$. The branch strains are equal and stresses add. Writing ${\sigma }_M$ for the Maxwell stress,For $\gamma =\mathrm{Re}({\gamma }_0{e}^{i\omega t})$ and $\tau ={\eta }_M/G$,Multiplying the Maxwell fraction by $1-i\omega \tau$ givesThe sign of the imaginary coefficient depends on the phasor convention; these expressions use $e^{i\omega t}$, so $G^{\prime \prime }>0$ represents dissipation. In the time domain, the equivalent constitutive relation isThe original mechanical schematic shows the series spring/dashpot branch separately from the parallel dashpot.

For $\omega \tau \ll 1$,andThus the material flows at sufficiently low frequency: its static shear modulus vanishes and its terminal response is viscous. The dashpot in the Maxwell branch allows its spring stress to relax, so there is no permanent elastic path through the parallel network.

The fact that both $G^{\prime }$ and $G^{\prime \prime }$ tend to zero is not evidence of a force-free material: stress at fixed shear rate tends to ${\eta }_0\dot{\gamma }$. At higher frequency the spring stores energy, but a finite-frequency elastic response does not make the long-time material a solid.