Energy dissipation in linear viscoelasticity

RESUMO

We compare the specific dissipation $Q^{-1}\left( \omega \right) $ for the Becker, Lomnitz, and Lambert models in linear viscoelasticity. Graphically, the specific dissipation of these models behaves similarly as $\omega \rightarrow 0^{+}$ and $\omega \rightarrow +\infty$. Furthermore, we obtain analytic formulas for the asymptotic behavior of $Q^{-1}\left( \omega \right) $ in the Becker and Lomnitz models as $\omega \rightarrow 0^{+}$ and $\omega \rightarrow +\infty$, 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_{\nu }^{-1}\left( \omega \right) $, when the parameter $\nu \in \left( 0,1\right] $ is rational. When $\nu \rightarrow 0$, we recover the specific dissipation of the Maxwell model, whereas when $\nu =1$, 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_{\alpha }^{-1}\left( \omega \right) $, the first being valid for $\alpha \in \left( 0,1\right] $, and the second for $\alpha \in \left( -\infty ,1\right] $. When $\alpha \rightarrow 0$, we recover the specific dissipation of the Lomnitz model, whereas when $\alpha =1$, we recover that of the Maxwell model. Finally, we conclude that the asymptotic behavior of $Q_{\alpha }^{-1}\left( \omega \right) $ as $\omega \rightarrow +\infty $ coincides with the specific dissipation of the Maxwell model. Keywords: