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E-book: Fractional Calculus And Waves In Linear Viscoelasticity: An Introduction To Mathematical Models (Second Edition)

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(Univ Of Bologna, Italy)
  • Format: 628 pages
  • Pub. Date: 16-Aug-2022
  • Publisher: Imperial College Press
  • Language: eng
  • ISBN-13: 9781783264001
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Fractional Calculus And Waves In Linear Viscoelasticity: An Introduction To Mathematical Models (Second Edition)Larger Image
  • Format: 628 pages
  • Pub. Date: 16-Aug-2022
  • Publisher: Imperial College Press
  • Language: eng
  • ISBN-13: 9781783264001
Other books in subject:

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Fractional Calculus and Waves in Linear Viscoelasticity (Second Edition) is a self-contained treatment of the mathematical theory of linear (uni-axial) viscoelasticity (constitutive equation and waves) with particular regard to models based on fractional calculus. It serves as a general introduction to the above-mentioned areas of mathematical modeling. The explanations in the book are detailed enough to capture the interest of the curious reader, and complete enough to provide the necessary background material needed to delve further into the subject and explore the research literature. In particular the relevant role played by some special functions is pointed out along with their visualization through plots. Graphics are extensively used in the book and a large general bibliography is included at the end.This new edition keeps the structure of the first edition but each chapter has been revised and expanded, and new additions include a novel appendix on complete monotonic and Bernstein functions that are known to play a fundamental role in linear viscoelasticity.This book is suitable for engineers, graduate students and researchers interested in fractional calculus and continuum mechanics.
Preface to the Second Edition vii
Preface to the First Edition xi
Acknowledgments xv
List of Figures
xxvii
List of Tables
xxxvii
1 Essentials of Fractional Calculus
1(46)
1.1 The Fractional Integral with Support in 1R+
2(3)
1.2 The Fractional Derivative with Support in M+
5(8)
1.3 Fractional Relaxation Equations
13(3)
1.4 The Law of Exponents
16(2)
1.5 Fractional Integral Equations of Abel Type
18(5)
1.5.1 Abel integral equation of the first kind
19(1)
1.5.2 Abel integral equation of the second kind
20(3)
1.6 Fractional Relaxation Equations of Distributed Order
23(10)
1.6.1 The two forms for fractional relaxation
23(1)
1.6.2 The fundamental solutions
24(2)
1.6.3 Examples
26(7)
1.7 Fractional Integrals and Derivatives with Support in 1R
33(3)
1.8 Tables and Plots of Riemann-Liouville Fractional Integrals and Derivatives
36(4)
1.9 Notes
40(7)
2 Essentials of Linear Viscoelasticity
47(56)
2.1 Introduction
47(4)
2.2 History in H+: The Laplace Transform Approach
51(2)
2.3 The Four Types of Viscoelasticity
53(3)
2.4 The Classical Mechanical Models
56(5)
2.5 The General Mechanical Models
61(5)
2.6 Remark on the Initial Conditions
66(3)
2.7 The Time -- and Frequency -- Spectral Functions
69(4)
2.8 History in 1R: The Fourier Transform Approach and the Dynamic Functions
73(10)
2.8.1 Storage and dissipation of energy: The loss tangent
75(5)
2.8.2 The dynamic functions for mechanical models
80(3)
2.9 The Viscoelastic Models of Becker and Lomnitz
83(6)
2.9.1 The creep laws
84(2)
2.9.2 The relaxation laws
86(1)
2.9.3 The retardation spectra
87(2)
2.10 Beyond the Jeffreys-Lomnitz Model of Linear Viscoelasticity
89(8)
2.10.1 The extended Jeffreys-Lomnitz laws of creep and relaxation
90(5)
2.10.2 Retardation spectrum for the extended Jeffreys-Lomnitz law of creep
95(2)
2.11 The Mean Relaxation and Retardation Times in Linear Viscoelasticity
97(2)
2.11.1 The fluids and the mean relaxation time
98(1)
2.11.2 The solids and the mean retardation time
98(1)
2.12 Notes
99(4)
3 Fractional Viscoelastic Models
103(40)
3.1 The Fractional Calculus in Linear Viscoelasticity
104(4)
3.1.1 Complex modulus, effective modulus and effective viscosity
104(1)
3.1.2 Power-law creep and the Scott-Blair model
105(3)
3.2 The Fractional Operator Equation and the Correspondence Principle
108(17)
3.2.1 Fractional Kelvin-Voigt model
110(3)
3.2.2 Fractional Maxwell model
113(2)
3.2.3 Fractional Zener model
115(3)
3.2.4 Fractional anti-Zener model
118(3)
3.2.5 Fractional Burgers model
121(4)
3.3 Fractional Viscoelasticity with History Since --∞ and the Dynamic Functions
125(2)
3.4 Analysis of the Fractional Zener Model
127(10)
3.4.1 The material and the spectral functions
127(3)
3.4.2 Dissipation: Theoretical considerations
130(4)
3.4.3 Dissipation: Experimental checks
134(1)
3.4.4 The physical interpretation of the fractional Zener model via fractional diffusion
135(2)
3.5 Notes
137(6)
4 Waves in Linear Viscoelastic Media: Dispersion and Dissipation
143(40)
4.1 Introduction
143(1)
4.2 Impact Waves in Linear Viscoelasticity
144(8)
4.2.1 Statement of the problem by Laplace transforms
144(4)
4.2.2 The structure of wave equations in the space-time domain
148(2)
4.2.3 Evolution equations for the mechanical models
150(2)
4.3 Dispersion Relation and Complex Refraction Index
152(16)
4.3.1 Generalities
152(3)
4.3.2 Dispersion: Phase velocity and group velocity
155(3)
4.3.3 Dissipation: The attenuation coefficient and the specific dissipation function
158(1)
4.3.4 Dispersion and attenuation for the Zener and the Maxwell models
159(1)
4.3.5 Dispersion and attenuation for the fractional Zener model
160(1)
4.3.6 The Klein-Gordon equation with dissipation
160(8)
4.4 The Brillouin Signal Velocity
168(9)
4.4.1 Generalities
168(1)
4.4.2 Signal velocity via steepest-descent path
169(8)
4.5 The Energy Velocity
177(3)
4.5.1 Introduction
177(1)
4.5.2 The energy velocity for linear viscoelastic waves
177(3)
4.6 Notes
180(3)
5 Waves in Linear Viscoelastic Media: Asymptotic Representations
183(32)
5.1 The Regular Wave-Front Expansion
183(7)
5.2 The Singular Wave-Front Expansion
190(10)
5.3 The Matching Between the Wave-Front Solution and the Long-Time Solution
200(4)
5.4 The Saddle-Point Approximation
204(8)
5.4.1 Generalities
204(1)
5.4.2 The Lee-Kanter problem for the Maxwell model
205(4)
5.4.3 The Jeffreys problem for the Zener model
209(3)
5.5 The Matching Between the Wave-Front and the Saddle-Point Approximations
212(2)
5.6 Notes
214(1)
6 Diffusion and Waves via Time Fractional Calculus
215(36)
6.1 Introduction
216(2)
6.2 Derivation of the Fundamental Solutions
218(6)
6.3 Basic Properties and Plots of the Green Functions
224(1)
6.4 The Green Functions as Probability Density Functions
225(4)
6.5 The Signaling Problem in a Viscoelastic Solid with a Power-law Creep
229(3)
6.6 Locations and Velocities for the Fractional Diffusive Waves
232(11)
6.6.1 Location and evolution of the maximum of the Green functions
232(5)
6.6.2 Centers of gravity and medians of the Green functions Qc and Qs
237(4)
6.6.3 Medians of the Green functions for the Cauchy and signaling problems
241(2)
6.7 Box Evolution of Fractional Diffusive Waves
243(3)
6.8 Notes
246(5)
7 Diffusion and Waves via Space-Time Fractional Calculus
251(40)
7.1 Fourier and Mellin Transforms
251(4)
7.1.1 The Fourier transform and pseudo differential operators
252(1)
7.1.2 The Mellin transform
253(2)
7.2 The Riesz-Feller Space-Fractional Derivative
255(2)
7.3 The Space-Time Fractional Diffusion-Wave Equation
257(5)
7.3.1 Scaling and similarity properties of the Green function
258(4)
7.4 Particular Cases of the Space-Time Fractional Diffusion-Wave Equation
262(5)
7.5 Composition Rules for the Green Function with 0 > β ≥ 0 1
267(2)
7.6 Mellin-Barnes Integral Representations for the Space-Time Fractional Diffusion-Wave Equation
269(6)
7.7 Computational Representations for the Green Function
275(14)
7.8 Notes
289(2)
Appendix A The Eulerian Functions
291(22)
A.1 The Gamma Function: Γ(z)
291(11)
A.2 The Beta Function: B(p, q)
302(4)
A.3 Logarithmic Derivative of the Gamma Function
306(2)
A.4 The Incomplete Gamma Functions
308(2)
A.5 Notes
310(1)
A.6 Exercises
310(3)
Appendix B The Bessel Functions
313(20)
B.1 The Standard Bessel Functions
313(7)
B.2 The Modified Bessel Functions
320(6)
B.3 Integral Representations and Laplace Transforms
326(2)
B.4 The Airy Functions
328(3)
B.5 Notes
331(1)
B.6 Exercises
332(1)
Appendix C The Error Functions
333(14)
C.1 The Two Standard Error Functions
333(2)
C.2 Laplace Transform Pairs
335(3)
C.3 Repeated Integrals of the Error Functions
338(1)
C.4 The Erfi Function and the Dawson Integral
339(1)
C.5 The Fresnel Integrals
340(4)
C.6 Notes
344(1)
C.7 Exercises
345(2)
Appendix D The Exponential Integral Functions
347(12)
D.1 The Classical Exponential Integrals Ei(z), (z)
347(1)
D.2 The Modified Exponential Integral Ein(z)
348(2)
D.3 Asymptotics for the Exponential Integrals
350(1)
D.4 Laplace Transform Pairs for Exponential Integrals
351(3)
D.5 The Volterra Functions
354(1)
D.6 Notes
355(1)
D.7 Exercises
356(3)
Appendix E The Mittag-Leffler Functions
359(32)
E.1 The Classical Mittag-Leffler Function Ea(z)
359(6)
E.2 The Mittag-Leffler Function with Two Parameters
365(3)
E.3 Other Functions of the Mittag-Leffler Type
368(3)
E.4 The Laplace Transform Pairs
371(5)
E.5 Derivatives of the Mittag-Leffler Functions
376(2)
E.6 Summation and Integration of Mittag-Leffler Functions
378(1)
E.7 Asymptotic Approximations to the Mittag-Lefler Function eα(t); = Eα(-tα) for 0 > a > 1
379(4)
E.7.1 The two common asymptotic approximations
379(4)
E.8 Notes
383(4)
E.9 Exercises
387(4)
Appendix F The Wright Functions
391(34)
F.1 The Wright Function Wλμ(z)
391(3)
F.2 The Auxiliary Functions Fv(z) and Mv(z) n C
394(3)
F.3 The Auxiliary Functions Fv(x) and Mv(x) in R
397(3)
F.4 The Laplace Transform Pairs
400(11)
F.4.1 The four sisters
407(4)
F.5 The M-Wright Functions in Probability
411(8)
F.6 The M-Wright Function in Two Variables
419(1)
F.7 Notes
420(2)
F.8 Exercises
422(3)
Appendix G Complete Monotone and Bernstein Functions
425(8)
G.1 Basic Definitions and Properties for Completely Monotone and Bernstein Functions
426(4)
G.2 Some Basic Examples of Completely Monotone and Bernstein Functions
430(1)
G.3 Notes
431(1)
G.4 Exercises
431(2)
Bibliography 433(150)
Index 583
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