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

In the framework of the theory of linear viscoelasticity, we derive an analytical expres- sion of the relaxation modulus in the Andrade model Gα(t) for the case of rational parameter α = m/n ∈ (0, 1) in terms of Mittag–Leffler functions from its Laplace transform G ̃ α (s). It turns out that the expression obtained can be rewritten in terms of Rabotnov functions. Moreover, for the original parameter α = 1/3 in the Andrade model, we obtain an expression in terms of Miller-Ross functions. The asymptotic behaviours of Gα(t) for t → 0+ and t → +∞ are also derived applying the Tauberian theorem. The analytical results obtained have been numerically checked by solving the Volterra integral equation satisfied by Gα(t) by using a successive approximation approach, as well as computing the inverse Laplace transform of G ̃α(s) by using Talbot’s method.