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

By Francesco Mainardi

This monograph offers a accomplished evaluate of the author's paintings at the fields of fractional calculus and waves in linear viscoelastic media, inclusive of his pioneering contributions at the functions of certain features of the Mittag-Leffler and Wright kinds. it really is meant to function a normal advent to the above-mentioned parts of mathematical modeling. the reasons within the publication are distinct sufficient to trap the curiosity of the curious reader, and entire adequate to supply the mandatory historical past fabric had to delve additional into the topic and discover the learn literature given within the large normal bibliography. This ebook could be of curiosity to utilized scientists and engineers.

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Extra resources for Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models

Example text

The application of the Fourier transform leads to the so-called dynamic functions (or harmonic responses) related to the storage and dissipation of energy. 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.

Gives new information. 14) with the convention that 0 and +∞ are reciprocal to each other. The remarkable relations allow us to classify the viscoelastic bodies according to their instantaneous and equilibrium responses. In fact, from Eqs. 1. 1 Type I II III IV The four types of viscoelasticity. Jg >0 >0 =0 =0 Je <∞ =∞ <∞ =∞ Gg <∞ <∞ =∞ =∞ Ge >0 =0 >0 =0 We note that the viscoelastic bodies of type I exhibit both instantaneous and equilibrium elasticity, so their behaviour appears close to the purely elastic one for sufficiently short and long times.

E. vanishing for t < 0. Implicitly, we assume that all our causal functions, including J(t) and G(t), are intended from now on to be multiplied by the Heaviside function Θ(t). The limiting values of the material functions for t → 0+ and t → +∞ are related to the instantaneous (or glass) and equilibrium behaviours of the viscoelastic body, respectively. 2a) and Gg := G(0+ ) glass modulus , Ge := G(+∞) equilibrium modulus . 2b) March 1, 2010 19:27 World Scientific Book - 9in x 6in Ch. 2: Essentials of Linear Viscoelasticity fmws 25 From experimental evidence, both the material functions are nonnegative.

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