Fractional Calculus and Waves in Linear Viscoelasticity

The following sections are included:

• Preface

• Acknowledgements

• Contents

• List of Figures

The following sections are included:

• The fractional integral with support in ℝ⁺

• The fractional derivative with support in ℝ⁺

• Fractional relaxation equations in ℝ⁺

• Fractional integrals and derivatives with support in ℝ

• Notes

The following sections are included:

• Introduction

• History in ℝ⁺: the Laplace transform approach

• The four types of viscoelasticity

• The classical mechanical models

• The time - and frequency - spectral functions

• History in ℝ: the Fourier transform approach and the dynamic functions

• Storage and dissipation of energy: the loss tangent

• The dynamic functions for the mechanical models

• Notes

The following sections are included:

• The fractional calculus in the mechanical models
  • Power-Law creep and the Scott-Blair model
  • The correspondence principle
  • The fractional mechanical models

• Analysis of the fractional Zener model
  • The material and the spectral functions
  • Dissipation: theoretical considerations
  • Dissipation: experimental checks

• The physical interpretation of the fractional Zener model via fractional diffusion

• Which type of fractional derivative? Caputo or Riemann-Liouville?

• Notes

The following sections are included:

• Introduction

• Impact waves in linear viscoelasticity
  • Statement of the problem by Laplace transforms
  • The structure of wave equations in the space-time domain
  • Evolution equations for the mechanical models

• Dispersion relation and complex refraction index
  • Generalities
  • Dispersion: phase velocity and group velocity
  • Dissipation: the attenuation coefficient and the specific dissipation function
  • Dispersion and attenuation for the Zener and the Maxwell models
  • Dispersion and attenuation for the fractional Zener model
  • The Klein-Gordon equation with dissipation

• The Brillouin signal velocity
  • Generalities
  • Signal velocity via steepest–descent path

• Notes

The following sections are included:

• The regular wave–front expansion

• The singular wave–front expansion

• The saddle–point approximation
  • Generalities
  • The Lee-Kanter problem for the Maxwell model
  • The Jeffreys problem for the Zener model

• The matching between the wave–front and the saddle–point approximations

The following sections are included:

• Introduction

• Derivation of the fundamental solutions

• Basic properties and plots of the Green functions

• The Signalling problem in a viscoelastic solid with a power-law creep

• Notes

The following sections are included:

• Appendix A The Eulerian Functions
  • The Gamma function: Γ(z)
  • The Beta function: B(p, q)
  • Logarithmic derivative of the Gamma function
  • The incomplete Gamma functions

• Appendix B The Bessel Functions
  • The standard Bessel functions
  • The modified Bessel functions
  • Integral representations and Laplace transforms
  • The Airy functions

• Appendix C The Error Functions
  • The two standard Error functions
  • Laplace transform pairs
  • Repeated integrals of the Error functions
  • The Erfi function and the Dawson integral
  • The Fresnel integrals

• Appendix D The Exponential Integral Functions
  • The classical Exponential integrals Ei (z),
  • The modified Exponential integral Ein (z)
  • Asymptotics for the Exponential integrals
  • Laplace transform pairs for Exponential integrals

• Appendix E The Mittag-Leffler Functions
  • The classical Mittag-Leffler function E_α(z)
  • The Mittag-Leffler function with two parameters
  • Other functions of the Mittag-Leffler type
  • The Laplace transform pairs
  • Derivatives of the Mittag-Leffler functions
  • Summation and integration of Mittag-Leffler functions
  • Applications of the Mittag-Leffler functions to the Abel integral equations
  • Notes

• Appendix F The Wright Functions
  • The Wright function W_λ,μ(z)
  • The auxiliary functions F_ν(z) and M_ν(z) in ℂ
  • The auxiliary functions F_ν(x) and M_ν(x) in ℝ
  • The Laplace transform pairs
  • The Wright M-functions in probability
  • Notes

• Bibliography

• Index