arXiv subjects
Publications and source records attributed to Enrico Masina.
In this paper we survey the properties of the Schelkunoff modification of the Exponential integral and we generalize it with the Mittag-Leffler function. So doing we get a new special function (as far as we know) that may be relevant in linear viscoelasticity because of its complete monotonicity properties in the time domain. We also consider the generalized sine and cosine integral functions
Special functions have always played a central role in physics and in mathematics, arising as solutions of particular differential equations, or integrals, during the study of particular important physical models and theories in Quantum Mechanics, Thermodynamics, Fluid Dynamics, Viscoelasticity, et cetera. The aim of this short series of lectures is to provide a good treatise about the Exponential Integral special function, covering also a few selected topics like its generalisation, other related special functions (Sine Integral, Cosine Integral and the Logarithmic Integral) and the theory of Asymptotic Series.
We present a new rheological model depending on a real parameter $ν\in [0,1]$ that reduces to the Maxwell body for $ν=0$ and to the Becker body for $ν=1$. The corresponding creep law is expressed in an integral form in which the exponential function of the Becker model is replaced and generalized by a Mittag-Leffler function of order $ν$. Then, the corresponding non-dimensional creep function and its rate are studied as functions of time for different values of $ν$ in order to visualize the transition from the classical Maxwell body to the Becker body. Based on the hereditary theory of linear viscoelasticity, we also approximate the relaxation function by solving numerically a Volterra integral equation of the second kind. In turn, the relaxation function is shown versus time for different values of $ν$ to visualize again the transition from the classical Maxwell body to the Becker body. Furthermore, we provide a full characterization of the new model by computing, in addition to the creep and relaxation functions, the so-called specific dissipation $Q^{-1}$ as a function of frequency, which is of particularly relevance for geophysical applications