分数阶微积分在粘弹性材料本构方程中的某些应用
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摘要
本论文由彼此相关而又独立的四章所组成。第一章为序言，简要介绍了本文所需的数学工具，也即分数阶微积分的基本概念、发展历史及应用。在§1．1节中，简要介绍了分数阶微积分的发展历史及其最近的应用，给出了Riemann-Liouville型分数阶积分算子_0D_t~(-α)(0＜R(α)＜1)、微分算子_0D_t~β(0＜R(β)＜1)和局部分数阶导数D~qf(y)的定义及主要性质，并讨论了分数阶积分和微分算子的Laplace变换。在§1．2节中，给出了广义Mittag-Leffler函数E_(α，β)(z)的定义及其某些重要公式。在§1．3节中，给出H-Fox函数H_(p，q)~(m，n)[(?)]的定义、级数表达式、渐近性态及其基本性质，并讨论了H-Fox函数的特例，如广义Mittag-Leffler函数E_(α，β)(z)和H_(1，2)~(1，1)(z)，Fox-H函数是求解分数阶微分方程的有力工具。在§1．4节，将分数阶微积分理论应用在粘弹性材料的本构方程中，分别讨论了整数阶粘弹性模型和分数阶粘弹性模型的发展及其应用。本章是以后各章的基础。
在第二章用分数阶的St．Venant模型研究了人颅骨的粘弹性。首先将标准的整数阶St．Venant模型推广至如下的分数阶形式：然后应用离散求逆Laplace变换的方法，根据Boltzmann迭加原理，可得在如下的准静态加载过程
This paper is composed of four chapters, which are independent and correlative to one another. In chapter 1, i.e. introduction, the history and applications of fractional calculus are introduced. In section §1.1, the development history and recent applications of the fractional calculus are introduced concisely, the definitions and the main properties of the Riemann-Liouville fractional integral operator _0D_t~(-α)(0 < R(α) < 1) and differential operator _0D_t~β (0 < R(β) < 1) and local fractional derivative D~qf(y) are given, and the Laplace transforms of fractional integral and derivative operators are discussed. In section §1.2, the definitions and some important formulae of the generalized Mittag-Leffler function E_(α,β)(z) are given. In section §1.3, the definition, series expression, asymptotic behavior and some basic properties of H-Fox function The special cases of the Fox function are discussed, such as the generalized Mittag-Leffler function E_(α,β)(z) and H_(1,2)~(1,1)(z). H-Fox function is a powerful tool for the solving of the fractional differential equations. In section §1.4, the fractional calculus theory is applied to the constitutive equations of viscoelastic materials. The developments and applications of the integer-order viscoelastic models and the fractional viscoelastic models are discussed respectively. This chapter is the basis for the following chapters.
In chapter 2, we investigate the viscoelasticity of human cranial bone by fractional St. Venant model. Firstly the standard (integer-order) St. Venant model is generalized to the fractional-order form as follows:
Applying the discrete inverse Laplace transform method and the Boltzmann superpo-
在第二章用分数阶的St．Venant模型研究了人颅骨的粘弹性。首先将标准的整数阶St．Venant模型推广至如下的分数阶形式：然后应用离散求逆Laplace变换的方法，根据Boltzmann迭加原理，可得在如下的准静态加载过程
This paper is composed of four chapters, which are independent and correlative to one another. In chapter 1, i.e. introduction, the history and applications of fractional calculus are introduced. In section §1.1, the development history and recent applications of the fractional calculus are introduced concisely, the definitions and the main properties of the Riemann-Liouville fractional integral operator _0D_t~(-α)(0 < R(α) < 1) and differential operator _0D_t~β (0 < R(β) < 1) and local fractional derivative D~qf(y) are given, and the Laplace transforms of fractional integral and derivative operators are discussed. In section §1.2, the definitions and some important formulae of the generalized Mittag-Leffler function E_(α,β)(z) are given. In section §1.3, the definition, series expression, asymptotic behavior and some basic properties of H-Fox function The special cases of the Fox function are discussed, such as the generalized Mittag-Leffler function E_(α,β)(z) and H_(1,2)~(1,1)(z). H-Fox function is a powerful tool for the solving of the fractional differential equations. In section §1.4, the fractional calculus theory is applied to the constitutive equations of viscoelastic materials. The developments and applications of the integer-order viscoelastic models and the fractional viscoelastic models are discussed respectively. This chapter is the basis for the following chapters.
In chapter 2, we investigate the viscoelasticity of human cranial bone by fractional St. Venant model. Firstly the standard (integer-order) St. Venant model is generalized to the fractional-order form as follows:
Applying the discrete inverse Laplace transform method and the Boltzmann superpo-
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