Energy dissipation in linear viscoelasticity

Abstract

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 $\omega \to 0^{+}$ and $\omega \to +\infty$. Furthermore, we obtain analytic formulas for the asymptotic behavior of $Q^{-1}\left(\omega \right)$ in the Becker and Lomnitz models as $\omega \to 0^{+}$ and $\omega \to +\infty$, 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 $\nu \to 0$, 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 $\alpha \to 0$, 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 $\omega \to +\infty$ coincides with the specific dissipation of the Maxwell model. Keywords:

27 pages, 3 figures

References in corpus (2)

• A note on the Lambert W function: Bernstein and Stieltjes properties for a creep model in Linear Viscoelasticity
• A comparative view of Becker, Lomnitz, and Lambert linear viscoelastic models