= Peak frequencies in a standard linear solid
{title2=$\omega_{G''}\tau=1,\quad\omega_{\tan\delta}\tau=\sqrt{G_R/G_U}$}
For a <standard linear solid model> with $0<G_R<G_U$, put $x=\omega\tau$. Its <storage modulus> is $G'=G_R+(G_U-G_R)x^2/(1+x^2)$ and its <loss modulus> is $G''=(G_U-G_R)x/(1+x^2)$. The latter peaks at $x=1$, whereas its <loss tangent> is
$$
\tan\delta=\frac{(G_U-G_R)x}{G_R+G_Ux^2}.
$$
Differentiation puts the loss-tangent peak at $x=\sqrt{G_R/G_U}$, with height $(G_U-G_R)/(2\sqrt{G_RG_U})$. A loss-modulus peak and a loss-tangent peak therefore need not occur at the same frequency. For weak relaxation, $G_U/G_R$ is close to one and their frequencies are approximately equal.
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