In <small-amplitude oscillatory shear>, expand the measured <shear stress> as
$$
\tau=\epsilon a\sin(\omega t+\delta)
=\epsilon\left(a\cos\delta\,\sin\omega t+a\sin\delta\,\cos\omega t\right).
$$
Comparison with the definition
$$
\tau=\epsilon\left(G'\sin\omega t+G''\cos\omega t\right)
$$
of the <storage modulus> and <loss modulus> gives
$$
\boxed{G'=a\cos\delta,\qquad G''=a\sin\delta},
$$
and hence
$$
\boxed{a=\sqrt{(G')^2+(G'')^2},\qquad \tan\delta=\frac{G''}{G'}}.
$$
Thus the measured phase directly gives the <loss tangent>.

At low <angular frequency>, a standard <viscoelasticity>[viscoelastic fluid] has time to relax and responds mainly as a viscous liquid: stress is approximately in phase with strain rate, so $\delta\to\pi/2$ as $\omega\to0$. At high frequency it cannot relax during one cycle and responds mainly as an elastic solid, so stress is approximately in phase with strain and $\delta\to0$ as $\omega\to\infty$.

For a material dominated by one <viscoelastic relaxation time> $\lambda$, the crossover between these regimes occurs when $\omega\lambda$ is of order one. Measure the frequency $\omega_c$ at which $G'=G''$, equivalently $\delta=\pi/4$, and estimate
$$
\boxed{\lambda\simeq\omega_c^{-1}}.
$$
A broad or multi-peaked crossover instead indicates a spectrum of relaxation times.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 352 / 1 / a / Solution

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Viscoelasticity combines viscous dissipation with elastic storage and relaxation of deformation.


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 Incoming links: Viscoelasticity