Peak frequencies in a standard linear solid (${\omega }_{G^{\prime \prime }}\tau =1,{\omega }_{\tan \delta }\tau =\sqrt{G_R/G_U}$)

For a standard linear solid model with $0<G_R<G_U$, put $x=\omega \tau$. Its storage modulus is $G^{\prime }=G_R+(G_U-G_R)x^2/(1+x^2)$ and its loss modulus is $G^{\prime \prime }=(G_U-G_R)x/(1+x^2)$. The latter peaks at $x=1$, whereas its loss tangent isDifferentiation puts the loss-tangent peak at $x=\sqrt{G_R/G_U}$, with height $(G_U-G_R)/(2\sqrt{G_R{G}_U})$. 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.