= Solution
For elements in series the <stress> is common and the <strains> add. If $\widehat\epsilon_K$ and $\widehat\epsilon_1$ are the Kelvin element and added spring amplitudes, then
$$
\widehat\epsilon=\widehat\epsilon_K+\widehat\epsilon_1
=\frac{\widehat\sigma}{G_0+i\omega\eta_0}+\frac{\widehat\sigma}{G_1}.
$$
The compliances therefore add, yielding
$$
\boxed{G^*(\omega)=\left[\frac1{G_0+i\omega\eta_0}+\frac1{G_1}\right]^{-1}
=\frac{G_1(G_0+i\omega\eta_0)}{G_0+G_1+i\omega\eta_0}.}
$$
This is a <standard linear solid model>. Define
$$
G_L=\frac{G_0G_1}{G_0+G_1},\qquad G_H=G_1,\qquad
\Delta G=G_H-G_L=\frac{G_1^2}{G_0+G_1},\qquad
\tau=\frac{\eta_0}{G_0+G_1}.
$$
Writing $x=\omega\tau$, its <storage modulus> and <loss modulus> are
$$
\boxed{G'=G_L+\Delta G\frac{x^2}{1+x^2},\qquad
G''=\Delta G\frac{x}{1+x^2}.}
$$
Thus $G'$ increases from the low-frequency plateau $G_L$ to the high-frequency plateau $G_H$. The <loss modulus> has low-frequency slope $+1$ and high-frequency slope $-1$ on a <log-log plot>, with maximum $\Delta G/2$ at $\omega=1/\tau$. At that frequency $G'=(G_L+G_H)/2$. There need not be a crossing of the two curves for arbitrary spring constants.
\Image[/past-exam-of-the-soft-condensed-matter-course-of-the-university-of-cambridge/2021-q4-moduli.png]
{title=<Storage modulus> and <loss modulus> of the <standard linear solid model>}
{description=Original log-log plot of the derived formulas, with illustrative $G_0=1$, $G_1=4$ and frequency expressed as $\omega\tau$. The chosen spring ratio is not specified by the exam. Dashed horizontal lines mark $G_L$ and $G_H$, and the dotted vertical line marks the loss maximum.}
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