= Solution
A viscoelastic material stores part of the deformation <energy> and dissipates part of it. In <linear viscoelasticity>, a small imposed shear <strain> produces a <stress> determined by its history, $\tau(t)=\int_{-\infty}^tG(t-t')\dot\gamma(t')\,dt'$. A shear step probes <stress relaxation>; a <stress> step probes <creep compliance>. A purely elastic solid has persistent <stress> after a held <strain>, while a <Newtonian fluid> relaxes that <stress> and creeps continuously under constant <stress>.
For a <Maxwell model>, a <spring> of <shear modulus> $G_0$ and a <dashpot> of <viscosity> $\eta$ are in series:
$$
\dot\gamma=\frac{\dot\tau}{G_0}+\frac\tau\eta,\qquad
G(t)=G_0e^{-t/\tau_M},\qquad \tau_M=\eta/G_0.
$$
For the convention $\gamma\propto e^{i\omega t}$,
$$
\boxed{G'=G_0\frac{(\omega\tau_M)^2}{1+(\omega\tau_M)^2},\qquad
G''=G_0\frac{\omega\tau_M}{1+(\omega\tau_M)^2}.}
$$
The <storage modulus> dominates at short times or high frequencies; the <loss modulus> dominates at long times or low frequencies. The <energy> dissipated per cycle per <volume> is $\pi G''\gamma_0^2$. A <Kelvin-Voigt model> places <spring> and <dashpot> in parallel and instead gives bounded creep $J(t)=(1-e^{-G_0t/\eta})/G_0$: a simple illustration of a viscoelastic solid. Real materials can require a spectrum of relaxation times rather than one element.
A small-amplitude oscillatory <rheometer> measures <stress> amplitude and phase relative to imposed <strain>, giving $G'=|\tau_0/\gamma_0|\cos\delta$ and $G''=|\tau_0/\gamma_0|\sin\delta$. First sweep <strain> amplitude to identify the linear range; control <temperature>, sample ageing, <wall slip> and instrument inertia. Frequency-dependent phase lag alone is insufficient if these apparatus effects are unresolved.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2013/paper-1-viscoelastic-response.png]
{title=Original normalized Maxwell relaxation and frequency response}
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