Calculation of the Relaxation Modulus in the Andrade Model by Using the Laplace Transform

TL;DR

This paper derives an analytical expression for the relaxation modulus in the Andrade model using Laplace transforms, Mittag-Leffler functions, and asymptotic analysis, with numerical validation of the results.

Contribution

It provides explicit formulas for the relaxation modulus in the Andrade model for rational parameters, linking it to special functions and validating through numerical methods.

Findings

Analytical expressions for G_α(t) in terms of Mittag-Leffler, Rabotnov, and Miller-Ross functions.

Asymptotic behaviors of G_α(t) at t→0+ and t→∞ derived using Tauberian theorem.

Numerical validation of analytical results via Volterra integral equation solutions and Talbot's method.

Abstract

In the framework of the theory of linear viscoelasticity, we derive an analytical expression of the relaxation modulus in the Andrade model $G_{\alpha }\left( t\right) $ for the case of rational parameter \mbox{$\alpha =m/n\in (0,1)$} in terms of Mittag--Leffler functions from its Laplace transform $\tilde{G}_{\alpha }\left( s\right) $. It turns out that the expression obtained can be rewritten in terms of Rabotnov functions. Moreover, for the original parameter $\alpha =1/3$ in the Andrade model, we obtain an expression in terms of Miller-Ross functions. The asymptotic behaviours of $G_{\alpha }\left( t\right) $ for $t\rightarrow 0^{+}$ and $t\rightarrow +\infty $ are also derived applying the Tauberian theorem. The analytical results obtained have been numerically checked by solving the Volterra integral equation satisfied by $G_{\alpha }\left( t\right) $ by using a successive approximation approach, as well as computing the inverse Laplace transform of $\tilde{G}_{\alpha }\left( s\right) $ by using Talbot's method.