Physical and Geometrical Interpretationof Fractional Derivatives in Viscoelasticity

and Transport Phenomena

Thesis submitted for the degree

of Philosophiae Doctor

by

Vikash Pandey

Faculty of Mathematics and Natural Sciences

University of Oslo

Autumn 2016

© Vikash Pandey, 2016

Series of dissertations submitted to the Faculty of Mathematics and Natural Sciences, University of Oslo No. 1781

ISSN 1501-7710

All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission.

Cover: Hanne Baadsgaard Utigard.Print production: Reprosentralen, University of Oslo.

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i

Preface

This thesis has been submitted to the Faculty of Mathematics and Natural Sciences at the Uni-

versity of Oslo in partial fulfillment of the requirements for the degree Philosophiae Doctor(Ph.D.). The work was carried out at the Digital Signal Processing and Image Analysis research

group in the Department of Informatics, University of Oslo. The thesis has been supervised by

Professor Sverre Holm and Dr. Sven Peter Näsholm. This work was financed through a PhD

grant by the University of Oslo, Norway.

Acknowledgements

It is reasonable to expect that a doctoral student acknowledges his supervisor, however hav-

ing Prof. Sverre Holm as a mentor makes the task quite easier. He is an extraordinary person

both at professional and personal level. I am deeply indebted to him for his constant guidance

and encouragement. I could not have asked for a more engaging and insightful supervisor. I

am grateful for his close collaboration, thorough discussions and reviews of my work, and for

pushing me forward whenever I needed it. His faith and devotion have always made me feel

the need of working harder to compensate for his support. Besides his professionalism, I have

also learned from his scientific thinking and writing skills, which certainly will influence me

throughout my future career. His art of gently massaging the manuscript for publication is re-

markable. Further, he has arranged several research stays abroad and conference participations

for me. I also need to specially thank my co-supervisor Dr. Sven Peter Näsholm for regularly

updating me with the latest advances related to my research. He has been actively involved in

broadening my research perspective which eventually will be more useful when I explore new

ideas. I was always motivated by the stimulating intellectual discussion sessions whenever I

had with both of my supervisors. With their great knowledge and open attitude, they created a

fertile, cross-disciplinary environment of which this thesis is the yield.

My sincere regards to Prof. Fritz Albregtsen since discussing physics with him besides being

joyful has always led to a deeper understanding. A pleasant and motivating work atmosphere is

of utmost importance to a foreign PhD student, and so I must convey my gratitude to the head

of the research group, Prof. Anne Schistad Solberg along with Assoc. Prof. Andreas Austeng

for sustaining a perfect social environment to ensure my natural comfort at the work place. I

am also thankful to Drift for providing the necessary technical support during my PhD studies.

It is also appropriate to acknowledge all those on whose scientific work I have built on, in

particular, Prof. Michael J. Buckingham of Marine Physical Laboratory, University of Califor-

nia, San Diego.

For the past four years, I have been fortunate to be working with brilliant colleagues from a

wide range of research areas. I thank them for the fruitful discussions, advice and encourage-

ment. Also, I wish them good luck in all their future endeavors.

Finally, my heartfelt thanks to my parents for supporting my decision of opting for an

academic-research career. Without their blessings this thesis could not have been conceived.

They have been the tower of strength and motivation for me. Their devotion for my betterment

will always be reflected in every milestone I achieve in my life.

iii

Abstract

Few could have imagined the vast developments made in the field of fractional calculus which

was first merely mentioned in the year 1695 in a correspondence between the pioneers of cal-

culus, Leibniz and L’Hôpital. However, since fractional order derivatives can easily be seen

as a “natural” generalization of the integer order derivatives, their studies have mostly been

limited to the mathematics community. This is evident from the fact that despite having more

than three hundred years of history, the order of the fractional derivative is mostly obtained by

curve-fitting the experimental data from the theoretically predicted curves. Besides, very few

works have demonstrated a deductive approach to fractional derivatives from the real physical

processes. The physical interpretation of the fractional order has remained an open question for

the fractional community as well as for those who use them to describe anomalous, complex,

and memory-driven physical phenomena. Consequently, despite its widespread applications in

the field of acoustics, rheology, seismology, and medical physics, fractional calculus has been

plagued as an “empirical only” mathematical methodology.

This thesis mainly aims at identifying the mechanisms which give rise to power law be-

havior, and hence a deductive approach to the fractional derivatives. This is first achieved by

mapping the grain-shearing model of wave propagation in marine sediments into the framework

of fractional calculus. The time-dependent viscosity of pore-fluid in the model is identified as

the non-Newtonian property of rheopecty. The wave equations from the grain-shearing model

turn out to be of fractional order.

In the pursuit to analyze the grain-shearing model, it is found that a linearly time-varying

Maxwell model yields a similar relaxation modulus as that of a fractional dashpot, and Lom-

nitz’s law as its creep compliance. Further, a linearly time-varying viscosity whose varying part

dominates over the constant part is established as the common physical mechanism underlying

both Nutting’s law and Lomnitz’s law. The fractional order in the Nutting law, and also the

terms in the creep law gain physical interpretation. This physical justification has been lacking

in both Nutting’s law and Lomnitz’s law since their inception in 1921 and 1956 respectively.

We have also investigated fluid dynamics in fractal media. It is shown that the fractional

Navier-Stokes equation and the fractional momentum diffusion equation naturally arise in such

a medium. These findings infer that fluid flow in fractal media lead to fractional derivatives

in time, the order of which is related to the fractal dimension of the medium. Thus, fractional

derivatives also gain a form of geometrical interpretation.

In addition, spatial dispersion of elastic waves in a nonlocal elastic bar is studied using

space-fractional derivatives. The nonlocal attenuation kernel is tempered in order to circumvent

the strong singularity encountered at the local point of stress application. Though the dispersion

behavior obtained numerically is unusual, it turns out to be physically reasonable.

The overall goal of this thesis is to show that fractional calculus is not just a mathematical

framework which can only be empirically introduced to curve-fit the experimental observations.

Rather, it has an inherent connection to real physical processes which needs to be explored more.

We hope that the results obtained here may benefit the scientific communities of fractional

calculus, seismology, non-Newtonian rheology, sediment acoustics, and fluid dynamics.

Publications

This thesis is based on the following publications, referred to in the text by their Roman numer-

als (I-IV):

I. Vikash Pandey, Sven Peter Näsholm, and Sverre Holm, “Spatial dispersion of elastic

waves in a bar characterized by tempered nonlocal elasticity”, Fractional Calculus andApplied Analysis, Volume 19, Number 2, Pages 498–515, April 2016.

DOI: http://dx.doi.org/10.1515/fca-2016-0026

II. Vikash Pandey and Sverre Holm, “Linking the fractional derivative and the Lomnitz creep

law to non-Newtonian time-varying viscosity”, Physical Review E, Volume 94, Number

3, Pages 032606-1–032606-6, September 2016.

DOI: http://dx.doi.org/10.1103/PhysRevE.94.032606

III. Vikash Pandey and Sverre Holm, “Connecting the grain-shearing mechanism of wave

propagation in marine sediments to fractional order wave equations”, The Journal of theAcoustical Society of America, Revised version submitted on August 15, 2016.

IV. Vikash Pandey, Sverre Holm, and Sven Peter Näsholm, “Relating fractional Navier-

Stokes equation to fractals”, Submitted for publication on September 27, 2016.

vii

Related publications

The following publications written in the context of the PhD study are related to the included

papers, but are not included in full text in this thesis.

I. Vikash Pandey and Sverre Holm, “A fractional calculus approach to the propagation of

waves in an unconsolidated granular medium”, Abstract presented at the 170th ASA

meeting, Jacksonville, Florida. Published in The Journal of the Acoustical Society ofAmerica, Volume 138, Number 3, Pages 1766–1766, September 2015.

DOI: http://dx.doi.org/10.1121/1.4933584

II. Sverre Holm and Vikash Pandey, “Wave propagation in marine sediments expressed

by fractional wave and diffusion equations”, IEEE/OES China Ocean Acoustics (COA),Pages 1–5, IEEE Xplore, January 2016.

DOI: http://dx.doi.org/10.1109/COA.2016.7535803

III. Vikash Pandey and Sverre Holm, “Linking the viscous grain-shearing mechanism of wave

propagation in marine sediments to fractional calculus”, Abstract presented at the 171st

ASA meeting, Salt Lake City, Utah. Published in The Journal of the Acoustical Societyof America, Volume 139, Number 4, Pages 2010–2010, April 2016.

DOI: http://dx.doi.org/10.1121/1.4949910

IV. Vikash Pandey and Sverre Holm, “Connecting the viscous grain-shearing mechanism

of wave propagation in marine sediments to fractional calculus”, A four page abstract

presented at the 78th EAGE Conference and Exhibition, May 2016.

DOI: 10.3997/2214-4609.201600862

ix

Contents

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Challenges to fractional calculus . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Applications of fractional calculus 72.1 Some examples on interpretation of fractional derivatives . . . . . . . . . . . . 8

2.2 Applications studied in this thesis . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Nonlocal elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2 Fractional viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.3 Time-dependent non-Newtonian rheology . . . . . . . . . . . . . . . . 14

2.2.4 Fluid dynamics in fractal media . . . . . . . . . . . . . . . . . . . . . 19

3 Summary of publications 213.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Discussions and future work 234.1 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Bibliography 27

Paper I 31Spatial dispersion of elastic waves in a bar characterized by temperednonlocal elasticity

. . . . 31

Paper II 47Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity

. . . . 47

Paper III 61Connecting the grain-shearing mechanism of wave propagation in marinesediments to fractional order wave equations

. . . . 61

xi

Paper IV 89Relating fractional Navier-Stokes equation to fractals . . . . 89

xii

Chapter 1

Introduction

The thesis is organized into four chapters; an introduction, applications of fractional calculus,

summary of publications, as well as discussions and future work. The bibliography, and the

published/submitted papers referred to by their respective Roman numbers follow thereafter.

This chapter gives an overview of the mathematical framework of fractional calculus along

with its key historical developments. The applications discussed in Chapter 2 have eventu-

ally blossomed into the four papers which constitute the main body of the thesis. A deductive

approach is adopted to describe applications in sediment acoustics, rheology, seismology and

transport phenomena. On the one hand, Papers II and III are closely linked because a time-

varying Maxwell model is studied in both. In Paper II, the physical mechanism underlying

Nutting’s law and Lomnitz’s law is identified by investigating the stress-strain transient behav-

ior of the time-varying Maxwell model. The frequency dependent dynamic behavior of this

model is used in Paper III to describe the grain-shearing mechanism of wave propagation in

marine sediments. On the other hand, Paper IV investigates fluid dynamics in fractal media.

The Papers II-IV reflect the title of the thesis giving physical interpretation to the fractional dif-

ferential equations arising in the study of different physical phenomena. However, in Paper I,

spatial dispersion of elastic waves in the Eringen nonlocal elastic bar is investigated. The bar is

characterized by a tempered attenuation kernel which then leads to interesting results.

1.1 Background

A couple of the mathematical expressions introduced to middle school students are,

(i) xn = x · x · x · x · x · · · x︸ ︷︷ ︸n

,

(ii) n! = 1 · 2 · 3 · · · (n− 1) · n,

where x is a real number, and n is a positive integer. This concept of exponentiation and

factorial is quite easy to grasp and straight forward. But, the interpretation could become tricky,

and confusion is bound to arise when n is not necessarily a positive integer. However, even in

that case, one can still obtain the values of xn and n! from the following expressions:

1

(i) xn = en lnx,

(ii) n! = Γ (n+ 1),

where n is a real number, and Γ (·) is the Euler gamma function. Although the results are ver-

ifiable using a scientific calculator, the expressions are difficult to comprehend for non-integer

values of n. The problem further deepens if x is declared to be a regular physical quantity

such as length or time. Since quantities raised to non-integer exponents require non-integral di-

mensions, they seem counterintuitive to our accepted notion of physical dimensions. A similar

situation but of an even higher complexity and impact arises when the order of a derivative is

raised to a non-integer order, i.e., a fractional order.

Fractional Calculus as the name implies, is a field of mathematics that deals with derivatives

and integrals of arbitrary orders, including complex orders. In the literature, it is also referred

by several other names such as Generalized Integral and Differential Calculus, and Calculusof Arbitrary Order [1]. The name “Fractional Calculus” is a holdover, and accepted simply

for historical reasons [2] as explained in the subsequent paragraph. Fractional calculus has

applications in the field of material science, rheology, seismology, transport phenomena, nuclear

physics, medical physics and finance, just to name a few [3–9].

It is interesting to note that using the principle of mathematical induction, it is quite possible

to deduce and verify asymptotic behavior of fractional derivative at least for simple functions,

such as power function, exponential function and trigonometric functions. However, the birth

of fractional calculus is often accredited to an exchange of letter dated September 30th, 1695,

between the mathematical geniuses Leibniz and L’Hôpital. L’Hôpital referring to a particular

notation he had used in his publications for the nth-derivative of the linear function,

f (x) = y,dn

dxny, (1.1)

posed a question to Leibniz, “What if n be 1/2?”. Leibniz replied: “Thus it follows that d1/2xwill be equal to x

√dx : x. This is an apparent paradox from which, one day, useful conse-

quences will be drawn.” [10]. Here, “:” implies division, “/”. In this way, the seeds of fractional

derivatives were sown over 300 years ago. In the beginning, fractional calculus was primarily

reserved for the best minds in mathematics many of whom directly or indirectly contributed

to its development. Some of them who dabbled with it in chronological order are L. Euler,

P. -S. Laplace, S. F. Lacroix, J. -B. J. Fourier, N. H. Abel, J. Liouville, G. F. B. Riemann,

A. K. Grünwald, O. Heaviside, G. H. Hardy, and G. W. Scott Blair [3–5,10]. Although treating

it purely as a mathematical exercise, Lacroix was probably the first one to publish an article in

1819 which had fractional derivatives [11] mentioned as,

dn

dxnxm =

m!

(m− n)!xm−n, m ≥ n. (1.2)

Surprisingly, Eq. (1.2) which Lacroix had derived using the technique of mathematical induction

is in accordance to the modern day definition of fractional derivatives [4, 5]. Finally, by letting

m = 1 and n = 1/2, he obtained,

d1/2

dx1/2x = 2

√x

π. (1.3)

2

1.2 Fractional Calculus

The correspondence between Leibniz and L’Hôpital motivated many mathematicians and physi-

cists to give a definition to the fractional derivatives. The most famous of these definitions that

has been popularized are due to Riemann and Liouville. Unfortunately Riemann’s work which

he had accomplished in his student days [10] was published only posthumously in 1876. He

had used a generalization of the Taylor series to derive a formula for integration of arbitrary

order [5]. Riemann further utilized the analytic continuation of Cauchy’s n-fold integral [3, 5],

(In f) (t) =1

(n− 1)!

t∫0

(t− τ)n−1 f (τ) dτ, (1.4)

by replacing its factorial with the Gamma function to yield the fractional integral as,

(Im f) (t) =1

Γ (m)

t∫0

(t− τ)m−1 f (τ) dτ. (1.5)

Here, n is a positive integer, and m is a real-valued number. And, the function f(τ) is expected

to be a continuous, causal, and well-defined in the given limits. Also, Γ (·) is the Euler Gamma

function defined for a complex variable z as:

Γ (z) =

∞∫0

xz−1e−xdx, � (z) > 0, (1.6)

where �(·) denotes the real part of a complex number. It is understood that Euler’s invention

of the gamma function for the non-integer values of the factorial was eventually a result of

his investigation of fractional integrals of monomials of arbitrary order [12]. On replacing the

order m by (n − m) and introducing a nth order derivative before the integral in Eq. (1.5),

Riemann-Liouville definition of fractional derivative [4, 5] is obtained as,

(Dmf) (t) =1

Γ (n−m)

dn

dtn

t∫0

(t− τ)n−m−1 f (τ) dτ, m ∈ (n− 1, n) , (1.7)

where, n ∈ N, is the set of positive integers. As observed, having an integral in its expression,

fractional differentiation is non-unique and non-local, just as is the regular integration. On the

contrary, integer order derivatives are both unique and local. The fact that the choice of limits of

integration affects the result of fractional derivative explains why Leibniz considered the subject

to be paradoxical. Leibniz must have been aware that the result of integrating a function depends

on how the function behaves over the range for which the integral is calculated. However, such a

generalization has a merit too because it essentially unifies integrals and derivatives into a single

operator. Thus, in order for the fractional generalizations to converge to the regular calculus

where both uniqueness and locality is guaranteed, it is expected that the theory involves some

sort of boundary conditions, involving information about the behavior of the function in the

3

given limits.

While the sheer number of actual definitions is quite numerous and are often variations of

the Riemann-Liouville definition, we have followed the Caputo definition [4,5] which is defined

as the convolution of the power law kernel Φm (t) with the ordinary derivative:

dm

dtmf (t) ≡ 0D

mt f (t) � Φm (t) ∗

(dn

dτnf (τ)

), (1.8)

where,

Φm (t) =tn−m−1

Γ (n−m). (1.9)

Using Eq. (1.9) in (1.8), we have

dm

dtmf (t) =

1

Γ (n−m)

t∫0

1

(t− τ)m+1−n

(dn

dτnf (τ)

)dτ. (1.10)

From Eqs. (1.5)–(1.10) it can be seen that the power law kernel is built into the fabric of frac-

tional calculus. The power law kernel which is also referred to as the “memory kernel” is the

key to the modeling of materials which remember their past deformations. Some examples of

such materials from daily life are cheese, clay, silly putty, rubber, and plastic. This application

aspect of fractional derivatives had initially led to the development of the theory of “heredi-

tary” solid mechanics. In the last 50 years, the theory has blossomed as the field of fractional

viscoelasticity which is discussed in more detail in Chapter 2.

As noticed, the Riemann-Liouville definition differs from the Caputo definition by the order

of convolution and differentiation. In the former, convolution is carried out first, in contrast,

differentiation is followed by the convolution in the latter. The fact that the Caputo fractional

derivative satisfies the relevant property of being zero when applied to a constant, but Riemann-

Liouville definition does not, often favors the use of the Caputo definition [4,5]. This allows for

the standard inclusion of traditional initial and boundary conditions in the Caputo formulation.

But, modeling of physical systems based on other definitions of fractional derivative may re-

quire the values of the fractional derivative terms at the initial time, which may be impractical.

This startling fact could be one of the reasons why fractional calculus historically had a difficult

time getting embraced by the scientific communities. Clearly, the Caputo fractional derivative

represents a sort of regularization in the time origin for the Riemann-Liouville fractional deriva-

tive. Therefore, we have followed Caputo’s definition in this thesis due to its better suitability

to the modeling and analysis of physical phenomena.

It may be even easier to see the extension to fractional derivatives from the Fourier trans-

form property for integer order derivatives. The Fourier transform of the power law, Φm (t) in

Eq. (1.9) is also a power law [4,5]. Since a convolution in time domain is a product in frequency

domain, we see from Eq. (1.8) that,

F[dm

dtmf (t)

]= (iω)m F (ω) . (1.11)

4

The spatio-temporal Fourier transform is defined as

F [f (x, t)] = F (ω) �∞∫

−∞

∞∫−∞

f (x, t) ei(kx−ωt)dx dt, (1.12)

where ω is the temporal angular frequency, and k is the corresponding spatial frequency.

A simple example of fractional derivatives of a quadratic function, f(t) = t2, is plotted

for different values of the fractional order in Figure 1.1. The values are obtained using the

expression,dm

dtmtp =

Γ(p+ 1)

Γ(p−m+ 1)tp−m, where p is a real number. (1.13)

0 1 2 3 4t

0

2

4

6

8

10

12

14

16

Fractional

derivative,

dm

dtmf(t)

f(t) = t2

m = 0.25m = 0.50m = 0.75m = 1.00

Fig. 1.1: Fractional derivatives dm

dtmf(t), of a quadratic function, f(t) = t2 (blue, solid line).

The fractional order, m has values 0.25 (magenta, dotted line), 0.50 (black, dashed line), 0.75

(green, dash-dot line), and 1 (red, thicker solid line).

It is reasonable to acknowledge that fractional calculus can be considered as a branch of

mathematical physics which deals with integro-differential equations, where integrals are of

convolution type and exhibit weakly singular kernels of power law type. Thus, the current

name of “Fractional calculus” is actually a misnomer, and the designation of “Calculus of ar-bitrary order” would be more appropriate. To summarize, the “integral” nature of the fractional

derivatives broadens the landscape which then requires some peripheral vision to cover the view.

5

1.3 Challenges to fractional calculus

Although almost as old as the classical Newtonian calculus, fractional calculus fell into an

oblivion for many years. The underlying reasons can be stated into four points. First, several

of the definitions proposed for fractional derivatives were inconsistent, implying they worked

in some cases but not in others. Oftentimes, the various definitions were not equivalent to

each other and led to a paradox [13]. Second, the mathematics appeared complex and also very

different compared to the well established integer order calculus. Third, since there were almost

no practical applications of this field yet, it was considered by many as merely an abstract

area for mathematical adventures with little or no use. Even at present there are very few

examples which directly relate fractional derivatives to real physical processes. And, the fourth

reason which upscaled the previous three challenges was the lack of an acceptable physical and

geometric interpretation of fractional derivatives. On the contrary, the integer order derivatives

have a clear physical and geometric interpretation, so their applications are easily understood.

However, a quite dedicated work to solve this problem was made by Podlubny [14] in 2002.

He tried giving a geometric interpretation of the Riemann-Liouville fractional integral based

on the projection of a very fascinating shadow of a fence on a wall. It may be concluded that

a convincing interpretation of fractional derivative has not been achieved yet, which is also

ascertained by a recent paper written by eminent researchers working in the field [15]. How

this limitation affects some of the present day applications of fractional calculus is explained in

more detail in Papers II and III.

Another possible drawback is the nonlocal nature of fractional derivatives which is con-

nected to the fact that the fractional operators are convolution integrals. Compared with the

numerical implementation of an integer order derivative, which only needs a few evaluations,

the numerical evaluation of a fractional derivative needs substantially more floating point op-

erations and data flow [13]. However, this aspect of fractional derivatives is not treated in this

thesis, besides there are ways to circumvent them partially, if not completely.

Thus, it is not surprising that most scientists and engineers have remained unaware of this

broader field of fractional calculus since it is not taught in schools and colleges. Despite the fact

that fractional derivatives have the potential to describe natural phenomena of higher complex-

ity, the difficulty to introduce them even at a college level of education has contributed to the

skepticism prevalent in scientific communities. Furthermore, relating them directly to physical

processes has remained an open problem.

The challenges to fractional framework is self-evident from the fact that despite of being an

invention of the 17th century, the first major scientific conference on fractional calculus and its

applications was organized much later in June 1974 by Ross at the University of New Haven,

Connecticut [15]. Since then, a more intensive development of the field has taken place with

more focus towards its potential applications [3]. Consequently, a specialized international jour-

nal for fractional calculus, Fractional Calculus and Applied Analysis, was established in 1998.

An international conference series, Fractional Differentiation and its Applications, devoted to

bring the researchers on a common platform is being held every second year since 2004. At

present, fractional calculus is one of the most intensively developing areas of the mathematical

physics as a result of its intriguing leaps in engineering and scientific applications. This can

also be witnessed by the vast number of publications made in the last five decades [15, 16].

6

Chapter 2

Applications of fractional calculus

After the apparent dispute regarding the various definitions of fractional derivatives settled, a

paradigm shift in the last 50 years has taken place, and since then there has been more focus

towards its physical applications. The credit of this goes to Caputo, Mainardi, Podlubny, Kilbas

and Hilfer, among others. Probably, it were Caputo and Mainardi who popularized fractional

derivatives in the 20th century by first using it to describe the dissipative mechanism in earth

materials [17–19]. Surprisingly, the empirical use of the fractional calculus can be traced back

to almost 100 years ago when it was used for describing the power law behavior of complex

viscoelastic materials. Some of the key early contributions came from Gemant [20], Scott

Blair [21] and Gerasimov [5,22]. Gemant was possibly the first to suggest its use by introducing

a half-order derivative in the Maxwell element to explain the transient behavior of flour dough

like viscoelastic media [20]. While Scott Blair was primarily motivated by the experimental

results [21] on cheese and clay, Gerasimov’s treatment was mainly a mathematical exercise as he

had considered a power law kernel for the relaxation function in Boltzmann’s integral equation

for viscoelasticity [4]. Some valuable contributions were made by Rabotnov and Cole-Cole;

the former developed the theory of hereditary solid mechanics [4] using fractional derivatives,

and the latter connected the fractional derivatives to the relaxation phenomena in dielectrics

[23,24]. Such applications were inspired by two distinct features of fractional derivatives. First,

contrary to the exponential decaying laws obtained from the integer order differential equations,

fractional derivatives could give power law behavior which was observed in many complex

media. Second, it could also capture power law jumps which sometimes were exhibited by

the same material at different time scales. This was difficult to encompass using the integer

order derivatives. Even though, Gerasimov’s and Rabotnov’s work date back to 1948, the two

scientists from the former Soviet Union are apparently little known in the field because their

work were published in Russian. Further, Scott Blair may be the first one to establish a fractional

framework underlying Nutting’s power law [21]:

σ ∝ t−m, 0 < m < 1, (2.1)

where σ is the stress, t is the time. He had intuitively referred to the order m as the “coefficientof dissipation”, and considered it to give a measure of the firmness of the material in his earlier

work [25]. The Nutting law essentially describes material which exhibit memory. The study of

these materials fall into the category of fractional viscoelasticity discussed in Subsection 2.2.2.

7

2.1 Some examples on interpretation of fractional derivatives

The suitability of the fractional derivatives in describing biological and viscoelastic media has

been sufficiently demonstrated [3, 26]. They could describe both the transient stress-strain be-

havior as a function of time as well as the dynamic wave dispersion and attenuation as a function

of frequency [4]. Interestingly, the use of half-order derivatives has also lead to a formulation

of certain electro-chemical and heat-mass transfer problems which are found more suitable than

the classical Fick’s law of diffusion [27]. But, fractional derivatives have mostly been “induc-

tively” employed in these applications. The number of works which have “deduced” fractional

derivatives from physical processes are very few.

The honor of the first application [1] of fractional derivative belongs to Niels Henrik Abel in

1826. He demonstrated a deductive approach when he posed the solution of an integral equation

that arises in the formulation of the mechanical problem of the tautochrone. This problem,

sometimes also called the isochrone problem, is that of finding the shape of a frictionless wire

lying in a vertical plane such that the time required by a bead to slide down the wire under the

influence of gravity is independent of its initial position on the wire. It turns out that Abel’s

solution motivated by the use of Laplace transform comprises of fractional derivative of order

half, which corresponds to a cycloid as the isochrone, as well as the brachistochrone curve

[28]. The brachistochrone problem which deals with the shortest time of slide had also been an

exciting problem of the calculus of variations. It was probably Abel’s elegant solution which

attracted and encouraged Liouville who made the first major attempt to give a more acceptable

definition of a fractional derivative.

Another contribution bearing a deductive approach came from the study of Bagley and

Torvik on the Rouse model of polymer dynamics [29, 30] in 1980’s. They established a di-

rect connection to the fractional derivatives by deriving a fractional stress-strain relationship

of order half underlying the macroscopic mechanical properties of polymers. This finding was

further developed [31] by generalizing the Rouse relaxation time which then extended the order

of the fractional derivative to 0 and 1.

It should be noted that there could be more such studies which may have demonstrated a

direct connection between fractional derivatives and physical processes. However, the two ex-

amples of the isochrone and polymer dynamics have solely been included for their importance,

and to resonate with the goal of the thesis. Further, the present scenario in most of the applica-

tions is that the value of the fractional order is usually obtained by curve-fitting the experimental

data. This ambiguity has led to a lacking in the confidence of the use of fractional derivatives.

However, fractional derivatives naturally emerge in complex processes and the order is found

to be related to the physical parameters of the model as illustrated in this thesis through the Pa-

pers II and III. Besides, as shown in Paper IV, fluid transport in fractal media lead to fractional

order differential equations, where the order is linked to the fractal dimension of the media.

8

2.2 Applications studied in this thesis

2.2.1 Nonlocal elasticity

An interesting fact is that, the power law memory inherent in a material is not limited to the

time domain, but rather it could also exhibit in the space domain [32–34]. The study of such

materials falls into the realm of nonlocal elasticity which goes back to the works of Kröner

and Eringen [33, 35]. The study of nonlocal elasticity which is one of the branches of Nonlo-

cal continuum field theories, is built upon two assumptions. First, the molecular interactions

in a material are inherently nonlocal. Second, the theory assumes the energy balance law

to be valid globally for the entire body. The applications of nonlocal elasticity have already

been established in describing complex properties of liquid crystals, fluids, suspensions, and

dielectrics [34]. Some latest applications are in nanomechanics which integrates the study of

mechanical (elastic, thermal and kinetic) properties of physical systems at the nanometer scale,

e.g., combining solid mechanics with atomistic simulations to analyze the strength and disper-

sive properties of carbon nanotubes.

It is well established that, under the action of an applied stress, the constituent points of

a nonlocal elastic material display range-dependent nonlocal interaction between themselves.

Physically, it implies that the effect at a given point in space at a given time is dependent on

the causes at the same point for all preceding times. Consequently, the applied stress is not

confined to a local point, but rather distributed to all the interacting points of the material. An

illustration comparing the local and nonlocal elasticity is shown in Figure 2.1, where constituent

elements in a nonlocal material are shown connected to each other by means of springs of

varying stiffnesses.

Fig. 2.1: A schematic comparison of local and nonlocal elasticity.

The nonlocal strain is expressed by the convolution integral of the strain in the neighbor-

hood ε (y) with the attenuation kernel g (x− y), where y is the Euclidean distance from the

local point x. For an infinitely long, one-dimensional, isotropic, linear nonlocal elastic bar, the

9

constitutive equation [36, 37] has the form,

σ (x) = E0

⎡⎢⎢⎢⎢⎢⎣ ε (x)︸︷︷︸

Local strain

+κ

∞∫−∞

ε (y) g (x− y) dy

︸ ︷︷ ︸Nonlocal strain

⎤⎥⎥⎥⎥⎥⎦ , (2.2)

where E0 is the elastic modulus at zero frequency, and κ is a material constant which gives a

measure of the strength of nonlocality. In contrast, stress in a local elastic medium is limited to

the point of impact, x.

From Eq. (2.2), it may seem that, in the dynamic case of wave propagation in a nonlocal

medium, the change in strain at the local position x, will instantaneously alter the stress at the

nonlocal position y. Therefore, the communication between the points x and y may appear to

be occurring at an infinite speed, and hence be non-causal and physically invalid. However, it

should be noted that, one of the axioms on which the nonlocal continuum field theory is built

upon, incorporates the axiom of causality [36]. Causality which is one of the basic principles

of physics, dictates that the cause has to precede its effect. This imposes restrictions on both

the time-domain and frequency-domain responses of a system. In the case of wave dispersion,

which connects the oscillations in space with the oscillations in time, the causality principle

enforces that the speed of a propagating wave cannot keep on increasing indefinitely. In order

to satisfy the causality principle, the real and imaginary parts of the wave propagation vec-

tor, which are frequency dependent, must obey the Kramers-Kronig relations [7]. Since the

Kramers-Kronig relations are essentially a Hilbert transform pair, it further implies that the

wave attenuation is always coupled with its velocity.

On the one hand, the causality principle is very well taken care of, when models are for-

mulated in the time domain and the resulting dispersion is “temporal” in nature. On the other

hand, a mechanical wave propagating in a nonlocal elastic medium exhibits “spatial” disper-

sion, such that the phase velocity and the wave attenuation are primarily dependent on the

spatial wavenumber. Further, similar to the case of the temporal dispersion, it is reasonable

to expect the validity of the causality principle in the spatial dispersion too. This aspect, in the

case of a nonlocal elastic medium, is governed by the choice of the attenuation kernel g (x− y),

which determines the extent of nonlocality present in the medium. This is investigated in Pa-

per I, where the attenuation kernel adopted is a power law in space and has a singularity [38]

at the local point. The kernel is therefore tempered [39, 40] to obtain the phase velocity dis-

persion and wave attenuation. It is shown that the elastic waves in such a nonlocal bar become

non-dispersive in high frequency regimes. The dispersion plots in Fig. 2 of Paper I show that

the phase velocity never exceeds the upper limit of lossless speed. Further, the attenuation in

the corresponding frequency regime is negligible. Thus, the causality principle is not violated

if an elastic wave undergoes spatial dispersion in such a nonlocal medium.

10

2.2.2 Fractional viscoelasticity

The characterization of media has been a long withstanding problem since the classical notion

of solids, liquids and gases is not sufficient to match the properties of the medium we encounter

in our daily life. This is better reflected by the viscoelastic bodies, which when deformed,

exhibit both viscous and elastic behavior through simultaneous dissipation and storage of me-

chanical energy [4, 41]. Consequently, stress-strain constitutive equations for the viscoelastic

behavior embody elastic deformation and viscous flow as special cases, and at the same time

also allow response patterns that characterize behavioral blends of the two. Accordingly, mul-

tiple viscoelastic models have been proposed to describe the mechanical as well as the wave

dispersive properties of biological materials and earth materials. The models traditionally in-

clude ideal springs and dashpots which are arranged in series and (or) parallel combinations

to depict the interplay between the elastic and viscous properties of a material [41, 42]. The

viscoelastic models; the Maxwell model and the Kelvin-Voigt model central to the theme of the

Papers II and III are illustrated in Figure 2.2 a) and b) respectively.

Fig. 2.2: A mechanical equivalent sketch of the classical viscoelastic models; a) the Maxwell

model, and b) the Kelvin-Voigt model. c) The fractional derivative element; a fractional dashpot.

Figure adapted from Mainardi [4].

The constitutive relation for the spring and the viscous dashpot follow from Hooke and

Newton [4, 42] respectively as,

σ (t) = E0ε (t) , (2.3)

and

σ (t) = ηdε (t)

dt, (2.4)

where ε is the strain, and η is the viscosity of a Newtonian fluid. The fact that stresses add

up in the parallel branches, and strains in series combinations, is used to obtain the stress-strain

relation of the models as desired. Linear viscoelasticity embodying the above two equations has

been a subject of intensive research mainly because it has the Boltzmann superposition principle

inherent in it [4]. The underlying assumption of the principle is that the creep in a specimen

11

is a function of the entire loading history, or simply the “memory”. And, each increment of

load makes an independent as well as an additive contribution to the total deformation. Further,

it should be emphasized that all these mechanical viscoelastic models are only “models”, and

are not explanations, however they are found quite useful when analyzing complex material

behavior.

There are also many complex media such as biological tissues, polymers and earth sedi-

ments which do not obey the Newtonian fluid property expressed by Eq. (2.4), and therefore

could not be satisfactorily described by the classical viscoelastic models. However, their frac-

tional order counterparts which are essentially obtained by replacing the Newtonian dashpot in

them by the fractional dashpot illustrated in Figure 2.2 c), have been found comparatively more

useful [4,7–9,43–45]. The fractional dashpot is also referred to as the Scott Blair element due to

historical reasons as mentioned in the first paragraph of this Chapter. The constitutive relation

of the fractional dashpot [4] is given as,

σ (t) = E0τmdmε (t)

dtm, (2.5)

where the characteristic retardation time τ gives a measure of the time taken for the creep strain

to accumulate. From Eq. (2.5), it can be seen that as m goes from 0 to 1, the fractional dashpot

interpolates from a spring to a Newtonian dashpot, hence it is also termed a springpot.

The fractional dashpot has attracted a lot of interest which eventually led to the development

of the field of fractional viscoelasticity. This also brings into light the key difference between

the framework of viscoelasticity and fractional viscoelasticity [4, 26]. In viscoelastic studies,

the material is simply regarded as a combination of an elastic and a viscous element. On the

contrary, in fractional viscoelasticity, material properties can be represented by various states

between an elastic solid and a viscous fluid. Also, a noticeable advantage of fractional vis-

coelasticity is that it uses fewer viscoelastic elements which makes the parameter identification

procedure robust and easier. Besides, such anomalous behavior of the fractional dashpot has

been found to be thermodynamically consistent [46].

Moreover, the fractional viscoelastic models should not be seen as an entirely different class

of models, but rather an extension of their classical counterparts. This is evident from the find-

ings of Schiessel and Blumen [47] as they showed that the viscoelastic models with fractional

attributes can be realized through hierarchical arrangements of springs and Newtonian dashpots

in the form of trees, ladders and fractal networks [3]. As illustrated in Figure 2.3, a ladder net-

work of an infinite number of Maxwell models is equivalent to a fractional dashpot. Another

way of realizing a fractional dashpot is through the Maxwell–Wiechert model illustrated in Fig-

ure 2.4. The model is motivated from the study of Cole-Cole on dielectrics [23]. Also, it is more

intuitive than the ladder network, as it takes into account the fact that the relaxation does not

occur with a single time constant but with a distribution of time constants. These models may

also explain why fractional models are being increasingly used in the investigation of biological

tissues [8] in which the number of constituents is supposedly very large.

On the one hand, the linear, time invariant systems exhibit exponential decay in time, imply-

ing no memory. On the other hand, power law decay inherent in the fractional derivatives can

be obtained when several simultaneously decaying processes have closely spaced exponential

decay rates. This hints about the statistical origins of fractional viscoelasticity [49].

12

Fig. 2.3: Ladder arrangement of the classical Maxwell models. Figure adapted from Hilfer [3],

and Schiessel and Blumen [47].

Fig. 2.4: The Maxwell–Wiechert model. Figure adapted from Näsholm and Holm [48].

13

2.2.3 Time-dependent non-Newtonian rheology

Apart from the viscoelasticity described earlier, fluids departing from the Newtonian behavior

could also be classified into another two categories; time independent and time dependent. The

time-independent fluids can be further classified into three types; Bingham plasticity, dilatancy

and pseudoplasticity [50,51]. Since the inherent physical mechanism underlying such complex

behavior is difficult to understand at the level of molecular dynamics, many empirical relations

have been put forward for their investigation. One of such mathematical relations which has

been employed extensively to describe various types of time-independent fluids stems from the

Herschel-Bulkley fluid model as,

σ = σth +K(ε̇)q, (2.6)

where σth is the threshold stress required for a Bingham plastic like fluid-flow, e.g., tomato-

ketchup and toothpaste, and the constant K is the consistency index. However, some fluids do

not require a threshold stress to flow. Then, letting σth = 0, reduces Eq. (2.6) to the Ostwald

power law equation [50] which describes the remaining two types of time independent non-

Newtonian fluids. The flow index q > 1 corresponds to the shear-thickening dilatant fluids, e.g.

a mixture of cornstarch and water. And, q < 1 corresponds to the shear-thinning pseudoplastic

fluids such as blood and ice. The Newtonian fluid property is maintained, if q = 1. The various

time-independent behavior of the non-Newtonian fluids and their dependence on the shear rate

is illustrated in Figure 2.5.

Fig. 2.5: Various types of time-independent non-Newtonian fluids; Bingham plastic (orange,

fine dash-dot line), Bingham pseudoplastic (magenta, fine-dashed line), Dilatant (red, dash-dot

line), and Pseudoplastic (blue, dotted line). The Newtonian (black, solid line) is included as a

reference. Figure adapted from Deshpande [50].

Contrary to the time-independent non-Newtonian fluids, their time-dependent counterparts

have not received significant interest owing to their complexity. The two types of the time de-

pendent fluid behavior are called thixotropy and rheopecty [51]. On the one hand, thixotropic

14

fluids are very common such as yogurt, coal-water slurries and paints, and they display de-

creasing viscosity with respect to time. On the other hand, rheopectic fluids are comparatively

rare and they display the opposite, i.e., increasing viscosity with respect to time, when agi-

tated. A simple example of rheopectic behavior is a whipped cream; the longer one whips, the

thicker it gets. The apparent changes in the viscosity is attributed to the structural build-up and

breakdown of the internal components of a fluid which is determined by its kinematic shearing

history [50, 51].

It should also be noted that time-dependent properties rheopecty and thixotropy lead to sim-

ilar shear thickening and shear thinning behavior as those from the time-independent properties

of dilatancy and pseudoplasticity, respectively. This is evident from the “nature” of the curves

illustrated in Figure 2.6 a) and b), and therefore may lead to a confusion. However, on noting the

abscissa of the plots, the key difference underlying the two categories of non-Newtonian fluids

becomes clear. Moreover, the time-independent non-Newtonian fluids are dependent on strain

rate as mentioned earlier, in contrast, the time-dependent properties exhibit at a constant stress

or strain. Further, the experimental investigation of such non-Newtonian properties is rather

difficult since oftentimes material exhibit a blend of the time-independent and time-dependent

properties. Thus, even though the two contrasting properties may coexist in a material, time-

dependent behavior is rarely incorporated in the constitutive equation.

The transient behavior of rheopectic materials under a constant shear stress and a constant

shear strain is investigated in Paper II. There it is shown that if a linearly time-varying viscosity

as illustrated in Figure 2.7, and mathematically expressed as,

η (t) = η0 + θ · t, (2.7)

when replaces the Newtonian dashpot in Figure 2.2 a), yields a time-varying Maxwell model

whose relaxation modulus is,

G(t) =

(1 +

t

τ

)−m

, τ =η0θ, m =

E0

θ, t ≥ 0, (2.8)

which on a further approximation, θ · t η0, is similar to that of a fractional dashpot illustrated

in Figure 2.2 c). Here, η0 is the constant part of the viscosity, θ > 0 is the rheopectic coefficient,

and τ is the time constant of the material.

The creep compliance of the time-varying Maxwell model yields the famous logarithmic

creep law of Lomnitz [52] which he had experimentally induced in 1956 during his PhD studies,

J(t) = 1 +m ln

(1 +

t

τ

). (2.9)

We may be the first to obtain a derivation of this law. The creep law although motivated by

experimental observations on the igneous rocks, has widespread applications in fault dynamics,

plate tectonics, rock physics, and in the understanding of damping of Chandler wobble [53].

15

Fig. 2.6: Comparison of the two types of non-Newtonian fluids: a) time-independent; dilatant

(red, dash-dot line) and pseudoplastic (blue, dotted line), and b) time-dependent; rheopectic

(red, dash-dot line) and thixotropic (blue, dotted line). Newtonian (black, solid line) behavior is

included in both subplots as a reference. Figure adapted from Deshpande [50] and Sochi [51].

16

Fig. 2.7: A mechanical equivalent sketch of the time-varying viscosity.

The dynamic behavior of the time-varying Maxwell is used to connect to Buckingham’s

grain-shearing model in Paper III. The motivation behind the study is the rheopectic property

of the pore-fluid manifesting from the shearing of the sediment grains in a marine environment

(see, Figure 1 in Paper III.). The interesting derivation of the Kelvin-Voigt fractional derivative

wave equation and diffusion-wave equation and their dispersion analysis for compression wave

and shear wave propagation in marine sediments can be found in the paper. This discovery has

only been possible after realizing that what Buckingham had referred to as “strain hardening”

behavior of the intergranular sliding [54], is actually due to the the rheopectic behavior of the

pore-fluid. In order to complete this argument, it should also be noted that the term strain

hardening or work hardening is more commonly used in the the field of material science to

explain the increase in material strength when strained beyond the linear elastic region [55],

i.e., the non-Hookean behavior (see, Figure 2.8).

As expected, the non-Hookean behavior of strain-hardening leads to nonlinear elasticity

which is also evident from the mathematical expression used to describe it,

σ = Sεp (2.10)

where S is the strength coefficient of the material, and p is the strain hardening exponent, often

less than unity for metals and alloys. On the contrary, the framework of fractional viscoelasticity

is inherently linear.

It can be seen that, Papers II and III intend to bridge the time-dependent behavior with the

fractional viscoelasticity. Interestingly, as illustrated in Figure 2.9, the relaxation response G(t)

of a viscoelastic material to a unit step in strain, owing to its partly elastic behavior, will differ

from that of a medium characterized by time-dependent properties alone.

17

Fig. 2.8: Strain hardening (blue, dotted line) of a material when stretched beyond its Hookean

region (black, solid line). On further stretching, the material undergoes necking, and even

further strain leads to a fracture. Figure adapted from Callister [55].

Fig. 2.9: Comparison of relaxation moduli, G(t) of viscoelastic material (black, solid line), and

time-dependent non-Newtonian rheopectic material (red, dash-dot line) and thixotropic material

(blue, dotted line). Figure adapted from Sochi [51].

Further, it should be noted that the classifications of media stated earlier are by no means

distinct or sharply defined. However, we also have the opinion that the framework of fractional

viscoelasticity may bring all fluid properties under one umbrella.

18

2.2.4 Fluid dynamics in fractal media

The regular geometrical structures taught in schools such as a sphere, cube, cone etc, are less

encountered in nature. Most of the structure we observe in our daily life are actually complex

and apparently irregular [56]. Fractal geometry has been applied to study such structures fol-

lowed by recent attempts to connect them to the fractional derivatives [57–61]. The impetus for

this is from the fact that self-similarity in the form of spatial power laws inherent in the fractal

objects could also be incorporated in the fractional framework.

Most media through which fluids flow in nature are often characterized by fractal networks,

such as plasma flow during lightening, underground water networks, blood vessels in human

body, and branching in trees and its root network [56]. A simplified example of such a network

relevant to the theme of the Paper IV is illustrated in Figure 2.10.

Fig. 2.10: A schematic representation of a fractal network observed in blood vessels and tree

roots. Figure adapted from Mandelbrot [56].

Further, just as fractional derivatives can have an arbitrary order, fractal geometrical struc-

tures can have an intermediate dimension, i.e., non-integral dimension. This idea can better be

understood from the definition of fractal dimension D given by Mandelbrot [56] as,

D =log N

log ε, (2.11)

where N is the number of self similar pieces, also called as the branching factor, and ε is the

inverse of the scaling factor. For example, the square and triangular Sierpinski carpets illustrated

in Figure 2.11 a) and b) respectively, have the fractal dimension of, Dsqr = log 8/ log 3 ≈1.892, and Dtri = log 3/ log 2 ≈ 1.585. Although unusual, the values of the fractal dimensions

19

make sense, since the carpets are definitely more than a one-dimensional object, but then they

do not completely fill up the two-dimensional space either.

Fig. 2.11: a) Square Sierpinski carpet, and b) Triangular Sierpinski carpet. Figure adapted from

Mandelbrot [56].

The fractal nature of the flow network naturally favors the fractional framework since power

law is inherent in both. On the one hand, some studies have fractionalized one of the main equa-

tions of the fluid flow, the Navier-Stokes equation, and proposed solutions to it [62]. However

the motivation is sheer mathematical. On the other hand, there are also works which have tried

to obtain closed-form expressions from the physics of the fluid flow in fractal media [57–59].

A recent work by Butera and Di Paola [60] falls in that category, where they have elegantly

tried to model fluid flow in a medium whose cross-sections are given by the Sierpinski carpets.

But, despite an elegant approach, a closed-form expression of a governing fractional differential

equation was not obtained. Their intermediate result relating velocity of fluid flow to temporal

power law is extended in Paper IV to naturally arrive at the fractional Navier-Stokes equation.

Further, a minor mathematical manipulation of this derivation leads to the fractional momentum

diffusion equation which to the best of our knowledge may be its first derivation. Moreover, the

order of the obtained transport equations is found to be related to the fractal dimension of the

medium, giving fractional derivatives also a geometrical interpretation.

20

Chapter 3

Summary of publications

3.1 Paper I

“Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity”Fractional Calculus and Applied Analysis, Volume 19, Number 2, Pages 498–515, April

2016.

In this paper, wave propagation in an infinitely long, one-dimensional, nonlocal elastic bar

is investigated. The measure of the nonlocality given by the attenuation kernel is considered

to be a power law, which has a singularity, and therefore tempered. The resulting stress-strain

constitutive relation is characterized by an integration implying strong nonlocal elasticity. The

spatial dispersion relation is obtained from which it is difficult to extract the expression of the

spatial frequency. Consequently, Müller’s method of root extraction is employed to obtain a

numerical set of solutions relating spatial frequency to the temporal frequency. It is found that

for low frequencies, elastic waves in such a medium exhibit strong phase velocity dispersion,

but become non-dispersive in the high frequency regime. Further, the unusual wave attenuation

in the nonlocal bar is explained to be physically valid. A bonus from this work is that we

have shown how tempering can be employed in a fractional framework when a singularity is

encountered.

3.2 Paper II

“Linking the fractional derivative and the Lomnitz creep law to non-Newtonian time-varying viscosity”

Physical Review E, Volume 94, Number 3, Pages 032606-1–032606-6, September 2016.

We have modified the classical Maxwell model by substituting its Newtonian dashpot by

another dashpot which is characterized by a linearly increasing viscosity. The new dashpot

then corresponds to a rare property of time-dependent non-Newtonian fluids, called rheopecty.

The transient behavior of the time-varying Maxwell model is investigated, and its relaxation

modulus and creep compliance are derived. It is found that its relaxation modulus is actually an

asymptotic Nutting’s law which is similar to that from a fractional dashpot. This then bridges the

two disparate fields of time-dependent non-Newtonian rheology and fractional viscoelasticity.

21

An offshoot of this work is witnessed by obtaining Lomnitz’s logarithmic creep law as the

creep compliance of the time-varying Maxwell model. In this way, both Nutting’s law and

Lomnitz’s law gain physical interpretation by identifying the common rheopecty mechanism

underlying them. An even more interesting outcome is that, both laws are found to share the

same assumption which is, the linearly time-varying part of the viscosity dominates over its

constant part. Further, the order of the fractional derivative gains a physical interpretation. It

is shown to give a measure of the interplay between the elastic and viscous properties of the

material, which also is in accordance to the intuitive observation of Scott Blair.

3.3 Paper III

“Connecting the grain-shearing mechanism of wave propagation in marine sediments tofractional order wave equations”

The Journal of the Acoustical Society of America, Revised version submitted on August 15,2016.

The findings of the Paper II are extended in this paper to investigate the grain-shearing (GS)

mechanism of wave propagation in marine sediment. This has been possible because the be-

havior of the pore-fluid mediating the sliding grains turns out to be rheopectic. This simple

observation then links time-dependent non-Newtonian rheology to sediment acoustics. Conse-

quently, fractional derivatives are shown to naturally arise in this study. It is shown that the

compressional wave propagation obtained in the original grain-shearing model turn out to be

the Kelvin-Voigt fractional derivative wave equation. And, the shear wave equation takes the

form of a diffusion-wave equation. Besides, in the fractional framework, the equations are much

easier to analyze and interpret. The dispersion relations obtained using fractional calculus are

shown to be exactly the same as those from the original GS model [54], which then estab-

lishes the equivalence of the wave equations obtained in the two different frameworks. Further,

asymptotic behavior of the dispersion curves are easily extracted in the fractional framework.

3.4 Paper IV

“Relating fractional Navier-Stokes equation to fractals”Submitted for publication on September 27, 2016.

Lately, some studies have proposed solutions to the fractional Navier-Stokes equation which

is one of the fundamental equations of fluid dynamics having applications in various areas.

However, the motivation behind that is purely mathematical. In this paper, we have derived the

fractional Navier-Stokes equation without modifying either the law of conservation of mass, or

the conservation of momentum. This has been possible by extending the findings of a recent

work by Butera and Di Paola [60]. A natural consequence of this derivation is that the fractional

momentum diffusion equation follows from the fractional Navier-Stokes equation. This con-

nects the fluid dynamics in fractal space to diffusion in momentum space. Further, fractional

derivatives also gain a geometrical interpretation because the order of the obtained transport

equations is found to be related to the fractal dimension of the medium.

22

Chapter 4

Discussions and future work

4.1 Discussions

The results presented in this thesis can be summarized into five points. First, it is demonstrated

that fractional derivatives naturally arise in materials which are characterized by linearly time-

varying viscosity. The order of the fractional derivative is found to be a ratio of the physical

parameters of elasticity modulus and the rheopectic coefficient. Second, one of the problems

in the modeling of the time-dependent non-Newtonian fluids has been an ambiguity in the esti-

mation of its relaxation time constant. We found the time constant to be a ratio of the constant,

and the time-varying part of the viscosity. These two achievements have been possible after a

similarity between the relaxation modulus of a fractional dashpot and a time-varying Maxwell

model is established. A bonus of the linking is that the fractional viscoelasticity underlying

Nutting’s law gets a physical interpretation. The application of these findings has also been

demonstrated by formulating grain-shearing mechanism of wave propagation in marine sedi-

ments using fractional calculus. The obtained wave equations which are comparatively easier to

analyze and manipulate on a fractional framework, are shown to be equivalent to Buckingham’s

“convolved” wave equations. Thus, a connection between time-dependent non-Newtonian rhe-

ology, sediment acoustics and fractional viscoelasticity is found. Moreover, dual properties

such as, both wave and diffusion like behavior, are easily identified in the fractional framework.

Third, the creep compliance of the time-varying Maxwell model is found to be the same as

the Lomnitz creep law which imparts a physical interpretation to the creep law just as in the

case of Nutting’s law. Surprisingly, Nutting’s law and Lomnitz’s law share the same underly-

ing physical mechanism of rheopecty; viscosity linearly increasing with time. An important

observation which can be made here is that the connection between rheopecty and Lomnitz’s

creep law is not an independent one, but rather it also connects to the fractional dashpot. As

shown in Figure 4.1, the creep compliances of the time-varying Maxwell model (or, equiva-

lently the Lomnitz law expressed by Eq. (2.9)) and that of the fractional dashpot (see, Eq. (3.1)

in Mainardi [4]),

JFD(t) =1

Γ(1 +m)

(t

τ

)m

, (4.1)

despite being mathematically dissimilar, are actually numerically equivalent for very low values

of the order.

23

10-3 10-2 10-1 100 101 102 103 104 105 106

Time, t (seconds)

1

1.01

1.02

1.03

1.04

1.05

Creep

complian

ce,J(t)

Time-varying Maxwell modelFractional dashpot

Fig. 4.1: Numerical equivalence of the creep compliances of the fractional dashpot (red, dot-

ted line) and the time-varying Maxwell model (blue, solid line). The values of the plotting

parameters are taken from Lomnitz’s study [52] on igneous rocks, m = 0.002, and τ = 10−3

seconds.

Further, the convergence exists over a wide range, from milliseconds to weeks.

Fourth, the time-fractional Navier-Stokes equation and the fractional momentum diffusion

equation are found to be naturally governing the fluid transport mechanism in a fractal medium.

An interesting observation is that even without including the classical losses of any kind, the

fractality of the medium alone gives rise to a fluid flow with a power law form of velocity field.

The order of the derived transport equations is found to be related to the fractal dimension of

the medium, thus yielding a form of a geometrical interpretation to the fractional derivatives.

We believe this contributes to filling the gap between fractal geometry and fractional calculus.

Fifth, the wave dispersive properties of a nonlocal elastic bar is studied, the result of which

are anomalous, yet physically valid. In addition, we have also demonstrated how a strong sin-

gularity, which is sometimes encountered in the physical modeling using fractional derivatives,

can be countered by tempering the attenuation kernel. The results from this work may benefit

the material science industry where materials with unusual properties are artificially engineered.

Until now the disparate fields of fractional viscoelasticity and rheology have not shared any

significant connection with each other which is also evident from the lack of cross-referencing

between the two fields. We encourage the respective scientific communities to come together

for a symbiotic relationship, and experimentally test the findings reported in this thesis. The

blend of the two fields then besides increasing the confidence in the use of fractional derivatives

24

will have even better applications in the field of elastodynamics, rheology, seismology, transport

phenomena, biomedical acoustics, and sediment acoustics.

As illustrated through the problems investigated in this thesis it is evident that the study of

fractional calculus could lead to new possibilities by filling the voids of the classical calculus.

We have the opinion that results obtained in this thesis may lead a step closer in establishing

fractional calculus as a natural outgrowth of the integer order classical calculus, in much the

same way as the gamma function has outgrown from the factorials. Perhaps, fractional calculus

may serve as an efficient language to describe the natural phenomena. In that case, the interpre-

tation of the physical phenomena in the framework of fractional derivatives must be as robust as

in the framework of “classical” integer order derivatives. This would then necessitate a bridging

of the two frameworks of mathematical physics as illustrated in Figure 4.2. Moreover, a broader

fractional framework could possibly meet the demands of the physical reality without losing the

classical calculus, since the latter is an inherent subset of the former. This is evident, as in the

limiting conditions, results from the fractional calculus asymptotically converge to those from

the classical calculus.

Fig. 4.2: An illustration of building a bridge between the two isolated “islands” of fractional

calculus and classical calculus. Such a bridging may help in a better exploration of the vast

ocean of physics.

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4.2 Future work

A general notion accepted within the scientific community is that a problem when supposedly

solved often leads to the discovery of new problems, to which this thesis is no exception. An

obvious goal would be to analyze the viscous grain-shearing mechanism of wave propagation is

marine sediments. However, in that case, a direct test of the fractional framework would be in-

tended by fitting it to the experimental sediment data. Another interesting study could be to add

a fractional dashpot element to the established models of time-independent non-Newtonian flu-

ids, and then analyze their transient as well as their dynamic response. Further, an investigation

could be made to model the thixotropic properties of fluids.

The observation that the creep compliance of the time-varying Maxwell model numerically

converges to that from a fractional dashpot for very low values of the order leaves room for

further investigation. This could be to find the underlying physical mechanism which could

encompass the entire spectrum of creep compliance exhibited by the fractional dashpot.

Regarding fluid dynamics of fluids in fractal networks, an investigation could be made to

obtain a closed-form governing equation which could include both the apparent losses from

the fractality of the media as well as the classical Darcy losses. Further, analyzing the flow of

time-dependent non-Newtonian fluids in fractal networks could have direct implications to the

oil and gas industry. The problem can further be extended to find the dynamic response of a

porous fractal frame containing time-dependent non-Newtonian fluids. These studies may help

in a better understanding of the transport phenomena occurring in the fractal space.

I wish to also extend the application of fractional calculus to the fields of quantum mechan-

ics [63], and nonlinear dynamics and chaos [64], where not much work has been done. The

latter may also be verifiable in an electronics laboratory. An another potential application of

fractional derivatives could be much awaited in the field of Cosmology. The spectrum of cos-

mic microwave background radiation is influenced by the integrated Sachs-Wolfe effect which

essentially describes gravitational potentials that vary with time while the photons are moving

through them [65, 66]. The importance of the effect can be understood as, if the depth of the

potential had remained constant, the energy gained by the photon while falling into the poten-

tial would have exactly canceled its losses while coming out of the potential. In simple words,

the blue shift would then exactly cancel the red shift which could have led to a uniform back-

ground radiation on large scales. The observation of enhanced anisotropies on large angular

scales is due to this integrated Sachs-Wolfe Effect, and therefore a holistic approach is required

to investigate it, which might be possible in the larger framework of fractional calculus.

In order to justify the plan of future studies mentioned here, I would like to mention an inci-

dent. In 1831, somebody asked Michael Faraday to talk about the usefulness of his discovery:

the electromagnetic induction phenomenon. Faraday simply answered: What is the usefulnessof a child? He grows into a man! In 2016, I would like to conclude this thesis thinking of frac-

tional calculus as a young cricket player full of energy, who has just entered the field and can

bat, bowl, and field equally well, and therefore demonstrates promising qualities of becoming

an “all rounder” of the game in future.

26

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