Fractional Calculus And Its Applications To Viscoelastic Materials And Nuclear Magnetic Resonance

This paper focuses on fractional calculus and its applications to viscoelastic ma-terials and nuclear magnetic resonance. It is composed of four chapters, which are independent and correlative to one another. The first chapter is an introduction, a brief introduction of the fractional calculus theory, special functions and fractional opera-tors in a variety of complex systems. The history of the development of fractional calculus and the applications of fractional calculus to various fields are introduced in§1.1. Riemann-Liouville fractional integral operator t0Dt-β、Riemann-Liouville fractional derivative operator t0Dtα, and Caputo fractional operator Ct0Dtα are defined and their main properties are discussed. The Laplace transforms of Riemann-Liouville and Caputo fractional derivative are introduced. Mittag-Leffler function and Fox H-function are briefly discussed in§1.2. There are four kinds of Mittag-Leffler func-tions:Mittag-Leffer function in one parameter Eα.(z)、Mittag-Leffler function in two parameters Eα,β(z)、generalized Mittag-Leffler function Eα,βγ(z), and Mittag-Leffler function in four parameters Eα,βγ,q(z). The importance of Fox H-function lies in the fact that nearly all the special functions occurring in applied mathematics and statistics are its special cases. Besides, Mittag-Leffler function, Meijer G function, the general-ization of the hypergeometric functions and Wright generalized Bessel function are all special cases of the H-function. The properties, the series expression and some special cases of the H-function are also mentioned in§1.2. The H-function plays an important role in fractional calculus. The applications of fractional calculus to non-Newtonian fluid mechenics, biophysics and biomechanics, and the theory of anomalous diffu-sion and random walk are briefly discussed in§1.3. This chapter is the basis of the following chapters of this thesis.A five-parameter generalized Zener model is discussed in Chapter2. The physi-cal background of Zener model and a four-parameter fractional Zener model are intro- duced in§2.1. A five-parameter generalized Zener model σ+a.σ(σ)=Eε+Ebε(β)(0.0.15) is introduced in§2.2. The stress relaxation and the strain creep of the model are dis-cussed in the next two sections. By using Laplace transform techniques, the ana-lytical solutions of the relaxation and creep are obtained: a(t)=Eε0-Eε0Eα(-a-1t(?)) and The fractional4-parameter Zener model is a special case of the5-parameter general-ized Zener model. The experimental data fit with the relaxation and creep expressions respectively in the two sections The fitting results show that the5-parameter gener-alized Zener model fits the experimental data better than The fractional4-paramcter Zener model. Now, usually in literature the tests are performed considering creep test only or relaxation test only and the direct connection between the two tests is not evidenced. The results obtained in this chapter show that β,a,b, a take the same val-ue in the two fittings mentioned above. This indicates the validity of the5-parameter generalized Zener model. Frequency response is also discussed in§2.5, and the limit of the loss factor as the frequency ω approaches to infinity is governed by the difference between the order of time derivatives of strain and stress and this result is consistent with the experiments.A model for solid materials with memory σ+aσ(α)+bσ(2α)=E(ε+cε(β)+dε(2β))(0.0.19) is discussed in Chapter3. By using Laplace transform techniques, the analytical so- lutions of the relaxation and creep are obtained: andFitting results show that the experimental data fit the relaxation and the creep si-multaneously with α=0.2685, β=0.2710, a=0.130, b=0.900, c=0.187, d=1.000. Usually in literature the tests are performed considering creep test only or re-laxation test only and the direct connection between the two tests is not evidenced. The results obtained in this chapter show that α,β, a, b, c and d take the same value in the two fittings mentioned above. This indicates the validity of the model (0.0.19). The limit of the loss factor as the frequency ω approaches to infinity is governed by the difference between the order of time derivatives of strain and stress and this result is consistent with the experiments.fractional derivative Bloch equations with three fractional derivative parameters are discussed in Chapter4. By using Laplace transform techniques, the analytical solution of the equations are obtained (β≥γ. The case of β≤γ see Chapter4): The analytical solution is helpful to the theory of nuclear magnetic resonance. The fig-ures in this paper show that the solution of the classical Bloch equations is the special case of the fractional Bloch equations discussed in this paper and that the smaller the values of γ,β, the faster the decay of the magnetization component in the xy plane, i.e., the cross-section.