分数阶微积分及其在黏弹性材料与核磁共振中的某些应用
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摘要
本文主要研究分数阶微积分及其在黏弹性材料与核磁共振中的某些应用,由彼此相关又相互独立的四章构成.第一章为引言,简要介绍了分数阶微积分理论、某些特殊函数以及分数阶算子在各种复杂系统中的应用.§1.1节简要介绍了分数阶微积分的发展历史及其在当前各领域的广泛应用,给出了Riemann-Liouville分数阶积分算子t0Dt-β、Riemann-Liouville分数阶微分算子t0Dtα与Caputo分数阶算子Ct0Dtα的定义及其主要性质以及Riemann-Liouville与Caputo分数阶导数的Laplace变换.§1.2节简要介绍了分数阶微积分及其应用中常用的两类特殊函数：Mittag-Leffler函数与Fox H函数.Mittag-Leffler函数包括以下四种类型：单参数Mittag-Leffler函数Ea(z)、双参数Mittag-Leffler函数Eα,β(z)、广义Mittag-Leffler函数Eα,βγ(z)与四参数Mittag-Leffler函数Eα,βγ,q(z)；而Fox H函数为非常重要的一类函数,在应用数学与统计学中出现的几乎所有的函数,都可作为H函数的特例,即使象Mittag-Leffler函数、Meijer G函数、广义超几何函数与Wright广义Bessel函数这样复杂的函数,也都包括在H函数中.该节还介绍了Fox H函数的一些性质、级数表达式及其特例.Fox H函数是研究分数阶微积分的有力工具.§1.3节简要介绍了分数阶算子在非ewton流体力学、生物物理和生物力学与反常扩散及随机游走理论等复杂系统中的应用.本章是以后各章的基础.
第二章讨论应力与应变的分数阶导数阶不相同的分数阶5参数广义Zener模型.§2.1节为引言,介绍了Zener模型的物理背景及4参数分数阶Zener模型.§2.2节引入分数阶5参数广义Zener模型σraσ(a)=Eε+Ebε(β),(0.0.1)在随后的两节里讨论了该模型的应力松弛及应变蠕变.运用离散求Laplace逆变换技术,得到了应力松弛及应变蠕变的解析表达式及分数阶4参数Zener模型为分数阶5参数广义Zener模型的特例.在这两节中,还分别将实验数据与得到的松弛与蠕变表达式进行拟合.与参数广义Zener模型相比,该解与实验数据更好地吻合.值得注意的是,通常松弛与蠕变试验只能单独独进行,尚无证据表明二者存在必然联系.本文拟合的最佳结果农明,β,α,b,a在上述二拟合中取相同值.这印证了本文模型的有效性.本文还讨论了该模型频率域上的性态,得到损耗正切的极限由应变与应力求导阶数的差确定：这也与实验结果一致.
第三章讨论了一类具有记忆的固体材料模型σ+aσ(α)+bσ(2α)=E(ε+cε(β)+dε(2β))(0.0.5)运用Laplace变换方法,得到了松弛与蠕哟变的表达式
拟合结果显示,α=0.2685,β=0.2710.a=0.130,b=0.900.c=0.187,d=1.000使实验结果同时达到与松弛与蠕变的拟合.通常松弛与蠕变试验只能单独进行,尚无证据表明二者存在必然联系[112].本文拟合的最佳结果表明,α,β,a,b,c与d在上述二拟合中取相同值.这印证了本文模型的有效性.在频率域内,高频率时高分子阻尼材料的损耗因子的理论结果由应变与应力关于时间导数阶的差决定：这与实验结果一致.
第四章讨论了具有三个分数阶导数参数的Bloch方程组利用Laplace变换技术,得到了该方程组的解析解(β≥γ时的表达式,β≤γ时的表达式与此类似,详见第四章)该解析解对核磁共振研究具有指导意义.图形显示,当α=β=γ=1时,即得经典Bloch方程的解.由图4.1还可看出,γ,β值越小,磁化强度在xy平面即横截面上的的分量衰减越快.
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: and
Fitting 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.
第二章讨论应力与应变的分数阶导数阶不相同的分数阶5参数广义Zener模型.§2.1节为引言,介绍了Zener模型的物理背景及4参数分数阶Zener模型.§2.2节引入分数阶5参数广义Zener模型σraσ(a)=Eε+Ebε(β),(0.0.1)在随后的两节里讨论了该模型的应力松弛及应变蠕变.运用离散求Laplace逆变换技术,得到了应力松弛及应变蠕变的解析表达式及分数阶4参数Zener模型为分数阶5参数广义Zener模型的特例.在这两节中,还分别将实验数据与得到的松弛与蠕变表达式进行拟合.与参数广义Zener模型相比,该解与实验数据更好地吻合.值得注意的是,通常松弛与蠕变试验只能单独独进行,尚无证据表明二者存在必然联系.本文拟合的最佳结果农明,β,α,b,a在上述二拟合中取相同值.这印证了本文模型的有效性.本文还讨论了该模型频率域上的性态,得到损耗正切的极限由应变与应力求导阶数的差确定：这也与实验结果一致.
第三章讨论了一类具有记忆的固体材料模型σ+aσ(α)+bσ(2α)=E(ε+cε(β)+dε(2β))(0.0.5)运用Laplace变换方法,得到了松弛与蠕哟变的表达式
拟合结果显示,α=0.2685,β=0.2710.a=0.130,b=0.900.c=0.187,d=1.000使实验结果同时达到与松弛与蠕变的拟合.通常松弛与蠕变试验只能单独进行,尚无证据表明二者存在必然联系[112].本文拟合的最佳结果表明,α,β,a,b,c与d在上述二拟合中取相同值.这印证了本文模型的有效性.在频率域内,高频率时高分子阻尼材料的损耗因子的理论结果由应变与应力关于时间导数阶的差决定：这与实验结果一致.
第四章讨论了具有三个分数阶导数参数的Bloch方程组利用Laplace变换技术,得到了该方程组的解析解(β≥γ时的表达式,β≤γ时的表达式与此类似,详见第四章)该解析解对核磁共振研究具有指导意义.图形显示,当α=β=γ=1时,即得经典Bloch方程的解.由图4.1还可看出,γ,β值越小,磁化强度在xy平面即横截面上的的分量衰减越快.
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: and
Fitting 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.
引文
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