= Solution
The requested model is the <standard linear solid model>: a <spring> of modulus $E_0>0$ in parallel with a Maxwell branch consisting of a <spring> $E_1>0$ in series with a <dashpot> of <viscosity> $\eta>0$. Both parallel branches have the same <strain>; their stresses add. In the Maxwell branch the <spring> and <dashpot> have the same <stress> and their strains add. The equilibrium <spring> prevents indefinite flow at long times.
For harmonic <strain> $\epsilon(t)=\operatorname{Re}[\epsilon_0e^{i\omega t}]$, define the <dynamic modulus> by $\widetilde\sigma=E^*(\omega)\widetilde\epsilon$. The Maxwell branch has compliance $1/E_1+1/(i\omega\eta)$. Defining $\tau=\eta/E_1$ therefore gives
$$
\boxed{E^*(\omega)=E_0+\frac{i\omega\eta E_1}{E_1+i\omega\eta}
=E_0+E_1\frac{i\omega\tau}{1+i\omega\tau}.}
$$
The same mechanical model can represent a <shear modulus> if the appropriate shear stresses and strains are used consistently.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2017-q2c-zener-response.png]
{title=A spring parallel to a Maxwell element gives finite low- and high-frequency storage moduli and one loss-modulus peak}
Writing $E^*=E'+iE''$, its <storage modulus> and <loss modulus> are
$$
\boxed{E'=E_0+E_1\frac{(\omega\tau)^2}{1+(\omega\tau)^2},
\qquad E''=E_1\frac{\omega\tau}{1+(\omega\tau)^2}.}
$$
With this time convention the physical <stress> is $\epsilon_0[E'\cos\omega t-E''\sin\omega t]$ and leads the <strain> by the loss angle. The dissipated energy per unit volume per cycle is $\pi E''\epsilon_0^2\ge0$. Thus $E'$ describes elastic storage and $E''$ describes dissipation.
At low <frequency>, $\omega\tau\ll1$, the <dashpot> flows during a cycle and relaxes the Maxwell <spring>, leaving $E'\to E_0$, $E''\sim E_1\omega\tau$. At high <frequency>, $\omega\tau\gg1$, the <dashpot> hardly deforms within a cycle, so both springs carry elastic load: $E'\to E_0+E_1$, while $E''\sim E_1/(\omega\tau)$. The <loss modulus> is maximal at $\omega\tau=1$, with peak value $E_1/2$.
The equivalent constitutive equation is
$$
\sigma+\tau\dot\sigma=E_0\epsilon+\tau(E_0+E_1)\dot\epsilon.
$$
After a sudden <strain> $\epsilon_s$ is imposed and held, the <stress> relaxes as $\sigma(t)=\epsilon_s[E_0+E_1e^{-t/\tau}]$. It falls from the instantaneous modulus $E_0+E_1$ to the nonzero relaxed modulus $E_0$. This single-time relaxation is the simplest improvement over a purely elastic solid or a Maxwell fluid with zero long-time modulus.
The <loss tangent> is $\tan\delta=E''/E'$. Its maximum need not coincide with the loss-modulus maximum: the <peak frequencies in a standard linear solid> give $\omega\tau=\sqrt{E_0/(E_0+E_1)}$ for the <loss tangent>. \b[The model captures one viscoelastic crossover between two finite elastic plateaux, with dissipation concentrated near the relaxation <frequency>.]
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