= Solution
A viscoelastic material stores part of the <work> of deformation and dissipates part. In the linear regime its <stress> depends on the <strain> history, not just the instantaneous <strain> or <shear strain rate>. For simple shear, a causal <relaxation modulus> $G(t)$ gives $\sigma(t)=\int_{-\infty}^tG(t-t')\dot\gamma(t')dt'$. A sudden <strain> then probes <stress relaxation>, while a constant imposed <stress> probes <material creep>; these are different experiments.
For steady sinusoidal <strain> use the convention $\gamma(t)=\gamma_0\cos\omega t$ and $e^{i\omega t}$ complex amplitudes. The <complex viscoelastic modulus> is
$$
\boxed{G^*(\omega)=\frac{\widetilde\sigma}{\widetilde\gamma}
=G'(\omega)+iG''(\omega),}
$$
so $\sigma(t)=\gamma_0[G'\cos\omega t-G''\sin\omega t]$. The <storage modulus> $G'$ is the in-phase elastic component. The <loss modulus> $G''$ multiplies the component in phase with <shear strain rate>; <stress> leads <strain> by $\delta=\operatorname{atan2}(G'',G')$. The <loss tangent> is $\tan\delta=G''/G'$ when $G'>0$.
Multiplying <stress> by <shear strain rate> and averaging gives $\langle P\rangle/V=\tfrac12\omega G''\gamma_0^2$. Hence the <dissipated energy per oscillatory shear cycle> is
$$
\boxed{\frac{W_{\rm cycle}}{V}=\pi G''\gamma_0^2.}
$$
The elastic contribution returns its <work> over the cycle. A passive material has nonnegative <dissipation>, which fixes the sign of the loss term in the chosen Fourier convention. This <complex viscoelastic modulus> is a response function, not the mathematical absolute value of a complex number.
A <Maxwell model> places a <spring> of <shear modulus> $G$ in series with a <dashpot> of <viscosity> $\eta$, so the <strains> add at common <stress>:
$$
\dot\gamma=\frac{\dot\sigma}{G}+\frac{\sigma}{\eta},
\qquad\tau=\frac{\eta}{G}.
$$
The <Maxwell fluid dynamic modulus> is
$$
G^*=\frac{G\,i\omega\tau}{1+i\omega\tau},\qquad
\boxed{G'=\frac{G(\omega\tau)^2}{1+(\omega\tau)^2},\quad
G''=\frac{G\omega\tau}{1+(\omega\tau)^2}.}
$$
At $\omega\tau\ll1$ the response is predominantly viscous, $G''\simeq\eta\omega$ and $G'\simeq G(\omega\tau)^2$. At $\omega\tau\gg1$ it approaches the elastic plateau $G'=G$. The two moduli cross, and $G''$ reaches its maximum, at $\omega\tau=1$. A <Kelvin-Voigt model> instead has the <spring> and <dashpot> in parallel, giving $G^*=G+i\eta\omega$ and no fluid-like zero-frequency terminal flow. Real materials often need a spectrum of <relaxation times> rather than one Maxwell time.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2015/paper-1-viscoelasticity.png]
{title=Original Maxwell storage and loss moduli, with a stress–strain cycle whose enclosed area measures dissipated work}
Oscillatory <rheometry> measures <stress> amplitude and phase while imposing a small <strain>. An <oscillatory-shear amplitude sweep> first identifies a linear regime; an <oscillatory-shear frequency sweep> then measures $G'$ and $G''$. Repeat at controlled <temperature>, suppress evaporation, check <wall slip> and sample aging, and correct instrument <inertia> at high <frequency>. Frequency-dependent structure and a nonlinear response cannot be inferred reliably by fitting one <spring> and <dashpot> to one point.
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