Some Applications Of Fractional Calculus To The Constitutive Equations Of Viscoelastic Materials

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-...