Energy dissipation in linear viscoelasticity
We compare the specific dissipation $Q^{-1}\left( ω\right) $ for the Becker, Lomnitz, and Lambert models in linear viscoelasticity. Graphically, the specific dissipation of these models behaves similarly as $ω\rightarrow 0^{+}$ and $ω\rightarrow +\infty$. Furthermore, we obtain analytic formulas for the asymptotic behavior of $Q^{-1}\left( ω\right) $ in the Becker and Lomnitz models as $ω\rightarrow 0^{+}$ and $ω\rightarrow +\infty$, where the latter limit coincides with the specific dissipation of the Maxwell model. We also derive a novel closed-form expression for the specific dissipation in the generalized Becker model, $Q_{ν}^{-1}\left( ω\right) $, when the parameter $ν\in \left( 0,1\right] $ is rational. When $ν\rightarrow 0$, we recover the specific dissipation of the Maxwell model, whereas when $ν=1$, we recover that of the original Becker model. In addition, we derive two new closed-form expressions for the specific dissipation for the extended Jeffreys--Lomnitz model $Q_{α}^{-1}\left( ω\right) $, the first being valid for $α\in \left( 0,1\right] $, and the second for $α\in \left( -\infty ,1\right] $. When $α\rightarrow 0$, we recover the specific dissipation of the Lomnitz model, whereas when $α=1$, we recover that of the Maxwell model. Finally, we conclude that the asymptotic behavior of $Q_{α}^{-1}\left( ω\right) $ as $ω\rightarrow +\infty $ coincides with the specific dissipation of the Maxwell model. Keywords: