Fractional Calculus and Waves in Linear Viscoelasticity: An by Francesco Mainardi

By Francesco Mainardi

This monograph offers a accomplished evaluation of the author's paintings at the fields of fractional calculus and waves in linear viscoelastic media, along with his pioneering contributions at the purposes of distinctive services of the Mittag-Leffler and Wright kinds. it truly is meant to function a basic creation to the above-mentioned parts of mathematical modeling. the reasons within the e-book are special adequate to seize the curiosity of the curious reader, and whole adequate to supply the mandatory heritage fabric had to delve extra into the topic and discover the examine literature given within the large common bibliography. This ebook may be of curiosity to utilized scientists and engineers.

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1 Introduction We denote the stress by σ = σ(x, t) and the strain by = (x, t) where x and t are the space and time variables, respectively. e. scaled with respect to a suitable reference state {σ∗ , ∗ } . At sufficiently small (theoretically infinitesimal) strains, the behaviour of a viscoelastic body is well described by the linear theory of viscoelasticity. According to this theory, the body may be considered as a linear system with the stress (or strain) as the excitation function (input) and the strain (or stress) as the response function (output).

The Maxwell model exhibits an exponential (reversible) stress relaxation and a linear (non reversible) strain creep; it is also referred to as the relaxation element. Based on the combination rule introduced above, we can continue the previous procedure in order to construct the simplest models of type I and IV that require three parameters. March 1, 2010 19:27 World Scientific Book - 9in x 6in Ch. 2: Essentials of Linear Viscoelasticity fmws 33 The Zener model. The simplest viscoelastic body of type I is obtained by adding a spring either in series to a Voigt model or in parallel to a Maxwell model, respectively.

Littlewood (1917– 1928), H. Weyl (1917), P. L´evy (1923), A. T. L. Post (1930), A. R. Love (1938–1996), A. Erd´elyi (1939–1965), H. V. Widder (1941), M. Riesz (1949), W. Feller (1952). However, it may be considered a novel topic as well. Only since the Seventies has it been the object of specialized conferences and treatises. For the first conference the merit is due to B. D. dissertation on fractional calculus, organized the First Conference on Fractional Calculus and its Applications at the University of New Haven in June 1974, and edited the proceedings, see [Ross (1975a)] .

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