Mathematical Modelling: Principle and Theory 9781470469641, 9781470473884

Citation preview

786

Mathematical Modelling Principle and Theory

Hemen Dutta Editor

Mathematical Modelling Principle and Theory

Hemen Dutta Editor

786

Mathematical Modelling Principle and Theory

Hemen Dutta Editor

EDITORIAL COMMITTEE Michael Loss, Managing Editor John Etnyre

Angela Gibney

Catherine Yan

2020 Mathematics Subject Classification. Primary 78A40, 78A97, 74-10, 26A33, 35J35, 35B38, 35J87, 76D05, 35B32, 65N99.

Library of Congress Cataloging-in-Publication Data Names: Dutta, Hemen, 1981– editor. Title: Mathematical modelling : principle and theory / Hemen Dutta, editor. Description: Providence, Rhode Island : American Mathematical Society, [2023] | Series: Contemporary mathematics, 0271-4132 ; volume 786 | Includes bibliographical references. Identifiers: LCCN 2023010266 | ISBN 9781470469641 (paperback) | 9781470473884 (ebook) Subjects: LCSH: Differential equations, Partial. | Mathematical models. | Engineering mathematics. | Mathematical physics. | AMS: Optics, electromagnetic theory – General – Waves and radiation. | Optics, electromagnetic theory – General – Mathematically heuristic optics and electromagnetic theory. | Real functions – Functions of one variable – Fractional derivatives and integrals. | Partial differential equations – Elliptic equations and systems – Variational methods for higher-order elliptic equations. | Partial differential equations – Qualitative properties of solutions – Critical points. | Partial differential equations – Elliptic equations and systems – Nonlinear elliptic unilateral problems and nonlinear elliptic variational inequalities. | Fluid mechanics – Incompressible viscous fluids – Navier-Stokes equations. | Partial differential equations – Qualitative properties of solutions – Bifurcation. | Numerical analysis – Partial differential equations, boundary value problems – None of the above, but in this section. Classification: LCC TA342 .M3545 2023 | DDC 620.001/51–dc23 LC record available at https://lccn.loc.gov/2023010266

Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2023 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

28 27 26 25 24 23

Contents

Preface

vii

Evanescence and evanescent waves D. N. Ghosh Roy and S. Mudaliar Constitutive fractional modeling Jordan Hristov

1 37

Normalized solutions for Schr¨ odinger type equations under Neumann boundary conditions Gaetano Siciliano 141 Well-posedness of steady-state Bingham type system by a quasi variational-hemivariational approach ´ rski and Sylwia Dudek Stanislaw Migo

185

Catastrophes of cylindrical shell Vasilii A. Gromov

215

v

Preface This book discusses several key principles and theories in the fields of waves, thermodynamics, electromagnetics, fluid dynamics, and catastrophes, as well as their potential applications in the development of various mathematical models having interdisciplinary and multidisciplinary relevance. The methodologies and techniques used in each chapter should also aid readers in gaining a better understanding of the underlying and related concepts. This book is expected to be useful to a wide range of readers, including researchers, professionals, educators, and students who are interested in new and evolving mathematical modelling principles and theory, as well as their various perspectives and applications. The book consists of five chapters, and some of the major aspects of each chapter are presented below. Evanescence is central to the physics of near-fields in electromagnetics, optics, photonics and acoustics, and is instrumental in a number of breakthrough technological innovations in high resolution microscopy, imaging and inverse reconstructions of unknown objects. The chapter “Evanescence and evanescent waves” aims to give an overall theoretical account of evanescence with emphasis on Green’s tensors and dipole emissions in near fields. Examples are presented to illustrate how evanescent waves arise under various physical situations. The characteristics of these waves are highlighted, such as their exponential decay and increased intensity around a scattering object. The importance of evanescence in today’s cutting-edge, high-resolution, near-field technology is discussed. The chapter “Constitutive fractional modelling” discusses constitutive fractional modelling based on fundamental thermodynamic principles, with an emphasis on fractional operators with singular and non-singular memory kernels. The developed models are based on the Boltzmann superposition and fading memory principles, which refer to the formulation of diffusion and viscoelastic phenomena. The models developed are illustrated with pro and con examples that highlight the key differences between correctly formulated models and those that have been formally fractionalized. The chapter “Normalized solutions for Schr¨ odinger type equations under Neumann boundary conditions” demonstrates how to use the Krasnoselski genus theory to prove the existence and multiplicity of solutions to some elliptic systems. The systems have a physical motivation and are concerned with the Schr¨odinger equation coupled with the electrostatic equation of two different electromagnetic

vii

viii

PREFACE

theories: the first is Maxwell’s classical theory and the second is Bopp and Podolsky’s generalized electrodynamics. The chapter “Well-posedness of steady-state Bingham type system by a quasi variational-hemivariational approac” discusses new results on the solvability, compactness, and upper semicontinuity property of solution sets to an abstract elliptic quasi variational inequality with constraint set-valued maps, as well as a quasi variational-hemivariational inequality with constraints. The results are applied to the Bingham type fluid model, yielding findings on its unique weak solvability and continuous dependence. It is obtained, in particular, that if the plasticity threshold approaches zero, the Bingham fluid behaves like a Newtonian fluid. The chapter “Catastrophes of cylindrical shell” examines the bifurcation paths for a cylindrical shell under external pressure and axial compression in terms of catastrophe theory. It reviews recent advances in tracing bifurcation paths of the non-linear boundary problem of the von Karman equations and presents the differential and variational formulations for this system of partial differential equations. It also discusses methods for solving the non-linear boundary problem and analyzing its bifurcation paths within the framework of catastrophe theory. I would like to thank the American Mathematical Society (AMS) for agreeing to publish this volume. I would also like to thank the authors, reviewers, and editors at the AMS for their contributions to the success of this volume. Hemen Dutta

Contemporary Mathematics Volume 786, 2023 https://doi.org/10.1090/conm/786/15810

Evanescence and Evanescent Waves D. N. Ghosh Roy and S. Mudaliar Abstract. In scattering and radiation experiments in electromagnetics, optics and acoustics, the scattered and radiated fields are frequently measured in the radiation zone or far-field, many wavelengths away from the source These fields are homogeneous or propagating with a 1/r fall-off. But in the near-field within a distance of the order of a wavelength from the source, the fields are quasi-static and are called inhomogeneous and non-propagating, decaying as 1/r 3 . It is the phenomenon of evanescence and the spatial spectrum of the near-field is dominated by evanescent waves. These waves are unobservable unless frustrated, i.e., thwarted by some object. They are and are not waves at the same time and have characteristics that seem to defy common sense. They can skim along a flat surface as regular propagating waves while exponentially decaying in the perpendicular direction. Yet, they are known to be present at infinity along some special directions. Whether their presence away from the surface is location or source dependent, is unclear. Moreover, how the propagating modes disentangle themselves from the quasi-static near-field and arrive at the radiation zone is also sub judice. Controversies around these questions abound. Having a purely imaginary wavenumber, the evanescent waves were once thought to be of mathematical interest only, with no particular physical imports. But in the physics of near-field electromagnetics, optics, photonics and acoustics, with a number of breakthrough technological applications, evanescent waves occupy central position. Owing to their close proximity to the source, they are carriers of its high frequency information and are instrumental in achieving resolutions far exceeding the diffraction-limited classical limit. Evanescent modes are, therefore, crucial in high resolution imaging and inverse reconstructions of unknown objects. They are also highly relevant to scattering involving different media, high resolution spectroscopy and even high density data storage. In quantum mechanics, the evanescent waves are thought to be virtual photons and the phenomenon of evanescence as quantum tunneling. It is also known that in non-relativistic physics, the evanescent modes obey the macroscopic principle of causality of cause preceding effect. Evanescence has been discussed in a number of monographs and journal articles. The primary focus is overwhelmingly on optics. Our objective in this Chapter is to give an overall account of evanescence including simple, physical, pictures of how it can arise in unrelated practical situations.

1. Introduction Consider a subwavelength aperture in an infinite screen, illuminated by a uniform plane light wave. The media involved are non-dissipative. The emergent light 2020 Mathematics Subject Classification. Primary 65M80, 35805, 35Q60, 78-02, 78-10, 78A40. c 2023 American Mathematical Society

1

2

D. N. GHOSH ROY AND S. MUDALIAR

is found to show distinctive spatial behaviors. Immediately behind the aperture is a proximity region of dimension a small fraction of λ where light remains constant and confined. It is followed by a second, near-field (NF) region, roughly of dimension λ/2, in which the field decreases very rapidly and hence, strong gradients exist. The rest of the space is the unbounded radiation zone where the rate of decrease of the field is 1/R, R being the distance from the source [dF01]. Such a picture is not confined only to subwavelength apertures in screens, but is typical of small compact bodies and point particles. The general picture is that their radiated or scattered fields show characteristic behaviors with distance. Immediately adjacent to the source, in a distance of roughly one λ, the field falls off as 1/R3 which then tapers off to 1/R2 and finally to 1/R as the distance increases. Accordingly, these may be called near field, intermediate or middle field (MF) and far-field (FF), respectively. All three modes are present in the near-field and have finite amplitudes before branching off into the other two. But of the three, only one has the 1/R3 decay, the evanescent or inhomogeneous wave. The others are propagating or homogeneous waves. The hallmark of an evanescent wave is that its amplitude and phase fronts do not coincide. The amplitude decays in the direction normal to the propagation vector. Thus evanescent waves are distinct from regular plane waves for which the amplitude and phase both remain constant on any wavefront perpendicular to the direction of propagation that implies real wavenumbers. Furthermore, unlike propagating waves, evanescent waves cannot transport energy. Although there is no energy flux in an evanescent wave, it can still transmit energy under proper conditions and are instrumental in preserving energy balance in reflection and propagation. The NF is an outward extension of the field existing inside the radiator or scatterer. The homogeneous nature of space-time imposes continuous variation of the field outside the boundary. It has been long known that the presence and symmetry reducing property of surfaces influence the properties of the field around it such as the generation of evanescent fields [Ros76]. As a matter of fact, evanescence is almost always associated with the presence of surfaces [CMD72]. We would like to point out that we have used standard terminologies of evanescent wave literature irrespective of its nature, that is, whether such designations truly reflect the actual physics or not. Also note that the terms homogeneous and inhomogeneous do not refer to the solutions of homogeneous and inhomogeneous Helmholtz equation, but to propagating and evanescent waves, respectively. Any solution of the Helmholtz equation contains both homogeneous and inhomogeneous waves. At this point, it would be interesting to mention that there exist waves that can be called weekly inhomogeneous which must be distinguished from evanescent waves. Consider the scalar Helmholtz equation, (Δ + k02 )ψ(x) = 0. Clearly, any plane   = k0 . That is, in Cartesian coordinates, wave eiK·x is a solution provided that |K|  2 2 2 2 2 . Let  Kx + Ky + Kz = k0 . Let K⊥ = (Kx , Ky )T ∈ R2 . Then Kz = k02 − K⊥ 2 2 2 k0 = ζ + iη be complex where ζ = Re k0 and η = Im k0 > 0 is a small quantity. 2 2 ) + iη is complex. For K⊥ < ζ, Kz ≈ If |K⊥ | is real, then k02 − |K⊥ |2 = (ζ − K⊥   2 2 2  ⊥ + zˆKz . ζ − K⊥ [1 + iη/2(ζ − K⊥ )]. As η → 0, Kz → ζ − K⊥ , and k0 → K That is, as dispersion  vanishes, the wave becomes homogeneous. However, when  2 2 − ζ[1 + η/2(K 2 − ζ)] → i K 2 − ζ. In this case, the > ζ, then Kz ≈ i K⊥ K⊥ ⊥ ⊥ wave does not tend to be homogeneous even when dispersion vanishes. The former is called weakly imhomogeneous, and the latter evanescent [Dev12].

EVANESCENCE AND EVANESCENT WAVES

3

ˆ iK|x| /|x|, x ∈ As |x| → ∞, the propagating mode ψpr (x) behaves as f (θ)e R3 , θˆ = {θ ∈ [0, π], φ ∈ [0, 2π]} ∈ S (2) , the unit sphere in R3 . ψpr is a radially propˆ agating spherical wavefront, amplitude modulated by the scattering amplitude f (θ).  Evanescent modes ψev are also described by plane wave forms ei(K·x−ωt) , but with  imaginary. For example, consider an at least one component of the wavevector K interface to which the z-axis is perpendicular, the interface being in a homogeneous  2 = k02 . space of wavenumber k0 . As earlier, conservation of energy dictates that |K|   ⊥ · iK x⊥ +Kz z  T 2 In reference to the interface, ψ = e , K⊥ = (Kx , Ky ) , Kz = k02 − K⊥ T and x⊥ = (x, y) . When K⊥ > k0 , Kz becomes imaginary in which case, ψ be comes ψev = eiK⊥ ·x⊥ e−Kz z . Im Kz > 0. ψev then decays exponentially along z. It should also be mentioned that there are situations where the decay can occur in arbitrary directions not aligned with the direction of propagation. Examples include Rayleigh, Lamb and Stoneley waves in mechanical elastic media [Woo15, Bre60]. Here, the wavenumber in any of these directions is complex. These are, however, not within the scope of this Chapter which is devoted to optics, electromagnetics and acoustics. The archetypical setting for evanescence is wave excitation of a flat interface dividing R3 into two half spaces, R+ (z > 0) and R− (z ≤ 0). All sources, apertures, etc., are assumed to be in one of the half spaces while the fields are sought in the other. In optics which is the traditional area of evanescent waves, this is the geometry of total internal reflection (TIR) of plane waves [BBP60, Lot68, Gri15]. Membranes and flat plates embedded in a fluid medium and under illumination provide further examples. Thus evanescence is defined with respect to a particular flat surface. A detector in the close proximity of the surface may see waves skimming along the surface while decaying in the normal direction. A coupling exists between the two media. For example, upon excitation by a line source or dipole, waves are set up in both media which then react upon each other. The greater the coupling, the greater is the reaction. Two different velocities exist in this case, one in the fluid (cf ) and the other in the thin interface (cs ). Depending upon their ratio, waves may be trapped by the interface which then propagate parallel to it and most of the energy is carried away by the medium adjacent to the interface. This gives rise to evanescent-like behavior [MI68]. The adjective evanescent or the verb evanesce (from the Latin word evanescere meaning vanishing away) appear to have a ring of mystery around them. Indeed, unless subjected to rigorous scrutiny, they seem to defy common sense. For example, they decay, but without dissipation of energy. Left to themselves, they do not transport energy and yet can transfer excitation across a dielectric surface. The presence of evanescent waves in TIR indicates that they can cross classically forbidden regions, analogous to quantum tunneling [BA03, Sch55]. If they are waves, then they are indeed strange waves when compared to one’s intuitive feeling of a wave. The fact is that whether an evanescent wave is a wave at all or just correlated vibration (as in a string) is debatable. In electromagnetics, evanescent waves seem to contradict the very laws of classical electrodynamics, especially, the fundamental properties of transversality and correct speed of propagation [Mil13]. Merely calling them inhomogeneous does not clarify things. Another peculiarity of evanescent waves is that in spite of their exponential decay, they can contribute to fields at infinity. Some [MA16, Xia99] assert that this contribution is at par with

4

D. N. GHOSH ROY AND S. MUDALIAR

that of the propagating modes, i.e., at a rate of 1/|x| in all directions while for others [WF98, SC99, SSL76, Arn01, Ber01], it can happen only in two special directions in reference to a flat surface. Some [WF98] consider this to be no more than a mathematical oddity, but others disagree [Arn01]. Indeed, whether evanescent fields are functions of observation points alone or of orientations of the dividing surfaces is of some controversy. To make matters worse, for a point source in R3 , these special directions can readily loose their meaning. It is because an interface passing through a point in free space is fictitious and its orientation entirely arbitrary unless it is physical, dictated by the physics of the problem. This coordinate dependence is an issue in interpreting evanescent waves. Evanescent waves cannot be observed physically unless they are frustrated, i.e., transformed into propagating modes by interaction with some other object (first demonstrated in [Bos94]). Experimentally, this can be achieved, for example, by letting a dielectric body approach the interface supporting the evanescent wave to within a distance of λ/2. If the dielectric body is a sharp optical fiber, then guided modes will propagate along it. This technique is known as photon scanning tunneling (PST) [dF01]. Unlike regular plane waves, evanescent waves do not occur in homogeneous media, but only in the presence of inhomogeneities such as interfaces. As a matter of fact, as already pointed out, these waves are always associated with surfaces. Invisibility, oddities in propagation, arbitrariness of reference surfaces and so forth, may tend to cast doubt about the physical reality of evanescent waves. Indeed, these waves, with their imaginary k, were originally thought as mere mathematical tools without any physical meaning. But as their understanding advanced in the last decades, their reality became undoubted. Evanescent waves can be manipulated experimentally, for example, manipulate surface plasmons (electron cloud) and their intensities. The near-field and the spatial-spectral picture of the close proximity of a radiating or scattering object, in particular, the high frequency part, are dominated by evanescent waves. As a matter of fact, evanescence can be considered to be a high-frequency phenomenon. They carry valuable information about rapidly varying spatial structures of objects. These waves are, therefore, highly important for studying matter on subwavelength scales and exert great influence on resolutions of imaging systems [dF01, BA03, GJG00]. As is known (see, e.g., [HS87, BW99]), the classical Rayleigh resolution is diffraction-limited. It is determined by homogeneous waves in the Fraunhofer region and is limited by the wavelength. But since evanescent waves carry detailed information of the object to be imaged, their inclusion offers the possibility of achieving diffraction resolution beyond the Rayleigh limit. For this reason, these waves are important in inverse problems of accurately reconstructing unknown objects with subwavelength resolution, as in near-field holography [Wil99]. They are highly relevant to scattering involving different media which has numerous technological applications. For example, in TIR, evanescence causes light beams to shift, i.e, travel a short distance along the interface before reflection and transmission can take place. This phenomenon is known as the Goos-H¨ anchen effect [dF01] which has many practical applications. Evanescent waves are instrumental in the rapid advancement of near-field physics, i.e., physics within the propagation distance (from the source) of a wavelength and underpin virtually every state of the art high resolution near-field inventions including scanning near-field optical microscope (SNOM), scanning tunneling optical microscope (STOM), atomic force microscope (AFM),

EVANESCENCE AND EVANESCENT WAVES

5

photon tunneling microscope (PTM), high-density data storage [dF01, Kow95], to name a few. These waves play a key role in diffraction from plane surfaces, cylinders, gratings and so forth. Evanescent waves can be generated in various ways including TIR of plane waves at a flat interface, dipole emission, a moving point charge [TdF60], a moving planar sheet of charge [KK21], vibrating structures in fluids and so forth. In a dipole emission, light remains confined near the dipole and does not propagate to any appreciable distance from it. In acoustics, evanescent waves have been generated by phased arrays of sources [TLP+ 90] using the method of spectral division [ATT13]. The recent avalanche of developments in nanophysics and technology [dF01, Oht04, CB94], near-field electromagnetics, optics and acoustics, have catapulted efforts to theoretically understand and practically apply these waves, if they can be called so. It appears that technological applications of evanescent waves have marched relatively ahead of their fundamental theoretical treatments. The conceptual problems of electromagnetic field and energy, energy storage and transfer in subwavelength or local regions surrounding a system such as an antenna are substantial. Evanescent waves are clear manifestations of local dynamics. Meshing the physics in the local system with that at a global or finite distance is a matter which is still sub judice. The characteristics of fields in close proximities of the source differ so significantly from those at finite distances that it is not unreasonable to ask if the local and global regions are parts of the same entire system or exist separately in their own rights. The vanishing of energy flux in evanescence causes conceptual problems of energy conservation. Careful considerations of local energy and excitation transfer are, therefore, of central importance when discussing near-fields. Our focus in this Chapter is not on evanescent technology which has been detailed in a number of specialized texts [dF01, MA16, Oht04, CB94], but rather on the fundamentals of their physics and mathematics, unique features and above all, their interpretations. Furthermore, although evanescent waves dominate the near-field, the two are not synonymous. The issues in near-field physics go beyond evanescence. The morphogenesis of non-propagating near-field modes to propagation in the far-field is an example. The discussion of near-field per se is not in the scope of this Chapter. The attention here is primarily on evanescent waves. 2. Examples of Evanescence Let us begin by presenting a few physical pictures of evanescence. The examples show how evanescent waves can arise in different physical situations. Example 1 (Total Internal Reflection (TIR)). An archetypical example of evanescence is a half-plane problem, shown in Figures 1a - 1b. An interface P at z = 0 divides R2 into R+ (refractive index n1 ) and R− (refractive index n2 ). Let  i · x ˆ i, K  i ) = eiK  i = Ki K ˆi n1 < n2 and R± lossless. Let a plane wave Ψi (x, K , K − the unit vector, be incident on P from the denser medium R at an angle θi to  i = (Kiy , Kiz ) = (Ki sin θi , Ki cos θi ). There exists a range of the z-axis. Then K θi ∈ [0, θc ) in which a (regular) triad of vectors, the incident wave Ψi ∈ R− , a reflected wave Ψr ∈ R− and a transmitted Ψt ∈ R+ exists at angles θi , θr and θt , respectively, to the z-axis (Figure 1a). Moreover, θi = θr . As θi → θc , the

6

D. N. GHOSH ROY AND S. MUDALIAR

transmitted wave Ψt ∈ R+ moves away from the z-axis toward y. At θi = θc , ψt exists only on P and decays along z (Figure 1b). The transmitted wave is said to have become evanescent. As θi increase beyond θc , the rate of decay also increases. θc is called the critical angle. No energy is transported away by the evanescent wave in R+ in which it was generated, The reflection is total. Hence the name total internal reflection. We have used Ψ as a generic symbol. In optics, Ψ can be electric and magnetic field vector with polarizations while in acoustics, it can be the velocity potential ψ. TIR is discussed in detail later in Sections 9.1 and 9.2.

Figure 1. Total Internal Reflection

Example 2 (A Mass-String-Spring System). Consider an infinitely long massless string studded with beads of mass m, uniformly separated by an interval d. A disturbance in the string displaces the beads transversely. We assume that the disturbance is small so that the beads move vertically from their equilibrium positions. It means that sin θ ≈ tan θ. Let hn denote the displacement of the n-th bead and T the tension (which has the unit of force). The equation of motion of the n-th bead is: (2.1)

dtt hn (t) = ω02 (hn−1 − 2hn + hn+1 ).

The constant ω02 is T /md. ω0 thus has the unit 1/t, i.e, of frequency. Dispersion vanishes if d a can be directly obtained from that on the boundary. Let kρ >> k in which case, kz is imaginary. Let Im kz > 0. Moreover, (1) H0 (iζ) = −iπ/2K0 (ζ), where K0 is the modified Bessel function of order zero

EVANESCENCE AND EVANESCENT WAVES

9

[AS04]. For large arguments, K0 (ζ) ≈ −(π/2ζ)e−ζ and we obtain:

(1) a −kρ (ρ−a) H0 (ρkρ ) e ≈ . (2.10) (1) ρ H0 (akρ ) Equations (2.9) and (2.10) clearly demonstrate that ψ decays exponentially with increasing radius and becomes evanescent. This result remains unaltered for n = 0. Example 4 (A Moving Vortex Sheet). In electromagnetics, a uniformly moving plane of uniformly distributed and sinusoidally oscillating electrons is known [Pur85, Kov00] to generate evanescent waves. The electric field is perpendicular to the moving surface and the magnetic field is perpendicular to the current, but parallel to the surface. The evanescent field is also sinusoidal along the plane of the charges, but its amplitude decays as one moves away perpendicular to the plane. An interesting analogue exists in acoustics where, instead of a uniformly moving plane of electrons, one has a fluctuating moving line source. A surface or sheet of trailing vortices is generated [MI68, RH21]. For example, when a fluid moves past a blunt object such as the nacelle of an aircraft, a spherical ball, a flag pole, and so forth, an oscillatory flow pattern develops in the wake. It is called vortex shedding. A surface of trailing vortices is generated. In order to generate the sheet, the velocity potential must be discontinuous along mean streamlines of the flow. Vortex shedding induces a fluctuating pressure on the surface from which the vortices are shed and then radiate away as sound and become evanescent.. An example is the famous aeolian sound [RH21, Sch21] emitted from a thin wire in the wind. (The word aeolian has its origin in Greek fables). We will briefly describe the phenomenon. We follow [RH21] in the main. Assuming linearized acoustics, the flow equations are:  0, ρ0 Dt Φ + p = (2.11) Dt p + ρ0 c20 ΔΦ = 0. Φ → ∞, |z| → ∞. Φ is the velocity potential, ρ0 , c0 are equilibrium mass density and wave speed and Dt = ∂t + U0 ∂x is the material derivative [Eri67], U0 being the mean flow speed. For a discontinuous velocity potential, the eigensolutions of Eqs. (2.11) consist of:  p(x, t) = 0, (2.12) Φ(x, t) = u(x − U0 t, z), Δu = 0. From Eqs. (2.11) and (2.12), the solution is [Eri67]:  R1 F (α) sgn (z) e−α|z|+iα(x−U0 t) dα. (2.13) Φ(x, t) = sgn is the signum distribution [Jac99]. For a sinusoidal oscillation, i.e., for Von Karman vortex stream [Sch21], Eq. (2.13) reduces to: (2.14)

Φ(x, t) = F0 sgn (z) eiω(t− U0 − U0 |z| . x

ω

The quantity ω/U0 is called hydrodynamic wavenumber and the parameter ωL/U0 is the Strouhal number [MI68, Sch21] where L is a typical length scale of the system. The exponential decay is clearly indicated by Eq. (2.14). Moreover, the larger the Strouhal number, greater is the decay. These are the same characteristics that were found in the previous examples. For further details of vortex shedding, we refer to [RH21].

10

D. N. GHOSH ROY AND S. MUDALIAR

It is necessary to point out that evanescence calculations are considerably more involved than what was presented in the examples. Our objective was merely to demonstrate how evanescence can arise under different physical contexts. The above four examples well demonstrate the point. The examples clearly shows that evanescence is associated with high frequency.

3. Green’s Functions and Tensors We now focus attention on the two principal components of the theory of evanescent waves or for that matter of near-field physics and mathematics per se. These are Green’s functions and dipole emission. We will use the term Green’s function generically to include scalar and tensor Green’s functions and operators. In modern, state-of-the-art, high resolution (resolution far exceeding the conventional diffraction-limited Rayleigh criterion [HS87, BW99]) near-field technology, emissions from infinitesimal and point sources, in interaction with physical interfaces, play crucial roles. These are best formulated in terms of Green’s functions. Of particular interest is their angular spectrum representation and the celebrated Weyl’s integral [Ros76, Dev12, MA16, BW99, Wil99]. It is intuitive to think of a small (e.g., subwavelength) particle such as an atom or a molecule as pointlike. A typical radius on the atomic scale is merely 1-10 A or 10−8 - 10−7 centimeter. Intuitively, so small a particle is almost a point, especially, from the far field point of view. It may, of course, be necessary to take field variations within the particle into account and consider higher-order (> 2) multipole expansions [Jac99, Jen17, Kel99]. But in an overwhelming majority of cases, a pointlike entity is a good approximation for a sub-λ object [Kel99]. An archetypical example in physics is the representation of an infinitesimal charge or current distribution as a point dipole, used in an immense number of investigations. Indeed, a point dipole, electric or otherwise, is the work horse of near-field physics. The field of a point emitter is called Green’s function or tensor depending upon the nature of the problem [Bar17,GR20,Sta98,Fri56]. In this text, the scalar Green’s function will be denoted by G0 (x, x ) and the tensor by G0 (x, x ), the superscript 0 indicating free space (not necessarily vacuum). x, x refer to the observation or field point and source location, respectively. In this Chapter, the tensor G0 is the electromagnetic Green’s dyad. The equations for Green’s functions are: (3.1a)

AG0 (x, x ) = −δ (3) (x − x ) : scalar

(3.1b)

AG0 (x, x ) = −Iδ (3) (x − x ) : tensor.

A is a scalar and A a vector differential operator. I is the unit tensor. Note that Eqs. (3.1a) and (3.1b) are distributional equations. Their physical and spectral or reciprocal space representations in three dimensions are presented below. 3.1. The Scalar Green’s Function G0 (x, x ). In scalar problems, A in Eq. (3.1a) is the Helmholtz or reduced wave operator Δ + k02 . k0 = ω/c0 . c0 is the speed of the wave. The expression for the outgoing 3-D G0 (x, x ) is well known: (3.2)

G(0) (x, x ) =

eik0 R  = |x − x |, x, x ∈ R3 . , R = |R| 4πR

EVANESCENCE AND EVANESCENT WAVES

11

Equation (3.2) is the physical space representation of G(0) . Its spectral or Fourier space representation is:    eiK·(x−x ) 1  dK. lim (3.3) G(0) (x, x ) = (2π)3 →0 R3 k02 − K 2 ± i ˆ the integral in Eq. (3.3) can be considered as the spherical  = K 2 dKdθ, Since dK coordinate representation of G(0) . But when a flat interface is present, whether physical or mathematical, it is more appropriate to use Cartesian coordinates. Fig with respect ures 4 shows, in a self-explanatory way, the decompositions  of x and K 2 2 2  to the interface. Since K = |K| = k0 , we have Kz = k0 − K⊥ , K⊥ = Kx2 + Ky2 .  Let p be the direction cosine vector. We will also use direction cosines of K. p = x ˆi pi . It is customary to denote p1 , p2 , p3 by p, q, m. The dispersion relation is p2 + q 2 + m2 = 1. Moreover, p⊥ = p2 + q 2 . Integration is 2-D and the area  ⊥ = dKx dKy or d differentials are dK p⊥ = dpx dpy = dpdq. Because of the square root function, there is a branch cut in which case, the convention is: ⎫ 2, K ≤ k ; p = 2 , p ≤ 1,⎪ ⎬ Kz = k02 − K⊥ 1 − p ⊥ 0 z ⊥ ⊥ (3.4) 2 − k 2 , K > k ; p = i p2 − 1, p > 1⎪ ⎭ = i K⊥ ⊥ 0 z ⊥ 0 ⊥   2 − k 2 and K⊥ p2⊥ − 1 are assumed positive. Note that there are no restrictions 0 on the values of Kx , Ky or on p, q. They can take all values from −∞ to ∞. In what  to represent K  or p in Eqs. (3.4). follows, we use a generic symbol U Weyl’s integral can now be stated.    eiU ·R  iα eik0 R (0)  = dU⊥ G (x, x ) = 4πR 8π 2 R2 Uz     ei(U⊥ ·R⊥ +Uz |z−z |)  iα (3.5) = dU⊥ . 8π 2 R2 Uz  ⊥ = x⊥ − x . If U  = K,  α = 1. But when U  = p, α = k0 . Equation (3.5) is the R ⊥ (0) spectral representations of G . The integrand is its angular spectrum or simply spectrum (of plane waves).  ⊥ = 0. We Remark 1. Note that Weyl’s integral diverges if z = z0 and U  ⊥ = 0. When z = z0 , the integrand is neither L1 nor L2 . The will assume that U integrals are to be evaluated first on a finite scan plane S0 , circular or rectangular. The dimensions of S0 are then allowed to recede to infinity. In order to facilitate the calculations, the integral is expressed in cylindrical coordinates and the properties of the Bessel function J0 is used. An asymptotic analysis is then performed to establish convergence. An in-depth discussion is in [HY99]. The limits of integration can be broken down into a finite interval I< and a semi-infinite interval I> = R2 \ I< . Therefore:        eiU ·R  eiU ·R  iα (0)  dU⊥ + dU⊥ . (3.6) G (x, x ) = 8π 2 I< Uz I> Uz  = K,  I< = |K  ⊥ |2 ≤ k02 and I> = |K  ⊥ |2 > k02 . For U  = p, I< = | For U p ⊥ |2 ≤ 1 2 p⊥ | > 1. The first interval represents propagation whereas the second and I> = |

12

D. N. GHOSH ROY AND S. MUDALIAR

evanescence. Accordingly, Eq. (3.6) gives: (3.7a) (3.7b)

G(0) x, x ) pr (

=

x, x ) = G(0) ev (

iα 8π 2 iα 8π 2

 I


eiU ·R  dU⊥ . Uz



 

 

Weyl’s integrals in Eqs. (3.7a) and (3.7b), therefore, yield the propagating and evanescent components separately. Remark 2. Weyl’s representation of G0 (x, x ) is with reference to a flat interface. However, the orientation of the interface is entirely arbitrary. One can, therefore, refer the source to a Cartesian frame (x, y, z) and the observation point to a second (primed) frame which is rotated around the origin through an arbitrary angle. In this case, expressions for the vector potential and fields will have complicated dependence upon the rotation matrix. For us here, the interface is given and the rotation matrix is the unit matrix. 3.2. Electromagnetic Green’s Tensor G0 (x, x ). We begin by introducing Maxwell’s equations.  x) = iω B(  x) : Faraday’s Law, (3.8a) ∇ × E(  x) + μ0 Je (x) : Amp´  x) = −iω0 μ0 E( (3.8b) ere’s Law ∇ × B( 1  x) = (3.8c) ρe (x) : Gauss’ law of electrostatics, ∇ · E( 0  x, t) = 0 : Gauss’ law of magnetostatics (3.8d) ∇ · B( with harmonic (e−iωt ) time oscillation. Equations (3.8a), (3.8b), (3.8c), and (3.8d) are microscopic Maxwell’s equations in free space in SI-mksA unit The fields are assumed to be continuous. The divergence equations (3.8c) and (3.8d) can be eliminated by a single equation of charge conservation or electromagnetic equation of continuity: (3.9) ∇ · Je = iωρe . They, however, supply the boundary conditions for the fields. The source region containing the electric charge density ρe and current density Je is assumed to be bounded in R3 . 0 is the free space dielectric constant and μ0 the free space magnetic permeability. It is interesting to note that Maxwell’s equations describe a single photon. The two curl equations in (3.8a) and (3.8b) can be manipulated to yield the wave equations for the fields.  + iωμ0 Je ,  = iω∇ × B  = ω 2 0 μ0 E (3.10a) ∇×∇×E (3.10b)

 ∇×∇×B

 + μ0 ∇ × Je . = ω 2 0 μ0 B

Using the vector identity ∇ × ∇× = ∇(∇·) − Δ, Eq. (3.10a) can be recast into:  − ∇(∇ · E)  = −iωμ0 Je . (Δ + k02 )E  Moreover, k0 = ω √0 μ0 = ω/c0 . c0 = 1/√0 μ0 is the speed Similarly, for B. of electromagnetic waves in vacuum. It can be shown rigorously [HY99] that the second order, uncoupled, wave equations (3.10a) and (3.10b) are completely (3.11)

EVANESCENCE AND EVANESCENT WAVES

13

equivalent to the four first order, coupled, Maxwell’s Eqs. (3.8a), (3.8b), (3.8c), and (3.8d). The operator Δ + k02 describes retarded (or advanced) propagation upon being  By Gauss’ law (Eq. (3.8c)), Fourier transformed in time. But not so for ∇ · E.  ∇ · E = ∇ρ/0 can be singular at the origin. For example, for a point dipole at xs , the singularity is ∇δ 3 (x − xs ). This term, therefore, represents the non-radiative part of the field outside the dipole and is of major interest in evanescence. As for Green’s dyadic, the vector operator A in Eq. (3.1b) is (Δ + k02 ) − ∇(∇·). The equation is, therefore: ((Δ + k02 ) − ∇∇·)G0 (x, xs ) = −Iδ 3 (x − xs ).

(3.12)

As usual, x is the field point and xs the source point. A. Representation : Physical We now consider the representation of the free space (again, not necessarily vacuum) electromagnetic Green’s tensor or dyad G0 (x, x ) in three dimensions in the Lorenz gauge [Jac99, Jen17, HY02, Str41]. The gauge is formulated in terms  and an electric scalar potential Φe . It can be of the magnetic vector potential A summarized thus.   A(x) = μ0 (3.13a) G0 (x, x )Je (x ) dx : the vector potential, V

 = −μ0 Je : the wave equation for A,  (3.13b) (Δ + k02 )A ρ e (3.13c) (Δ + k02 )Φe = − : the wave equation for Φe , 0  ∇ · A = iω0 μ0 Φe , (3.13d) (3.13e)

 E

= = −∇φe + iω A

(3.13f)

 B

 = ∇ × A.

i ω0 μ0

 + iω A,  ∇(∇ · A)

The last equation (3.13f) follows from Faraday’s law. An Important Remark. In the literature, one frequently encounters the gauge being called the Lorentz gauge, after the Dutch physicist Hans Antoon Lorentz who is more than familiar in physics. But as pointed out by van Bladel [vB91] and later by Jackson in the 1999 (but not in the 1975) edition of his Classical Electrodynamics, the gauge should correctly be referred to as the Lorenz gauge after the Danish physicist Ludwig V. Lorenz. From the definition (3.13f) for the vector potential and the electric field expression in Eq. (3.13e), we have:    ∇(∇ · A) +  x) = iω A E( = iωμ0 G0 , Je . k02 in which: (3.14)

  ∇∇ G = I + 2 G0 . k0 0

G0 , as defined in Eq. (3.14), is the spatial representation of the free space, threedimensional, electromagnetic Green’s tensor. Many authors denote it by G0e in

14

D. N. GHOSH ROY AND S. MUDALIAR

order to emphasize its electrical character. Expanding ∇∇ in Eq. (3.14) yields:    ik0 R Δpr  1 e k 1 0 Δ − (3.15) G0 (x, x , k0 ) = − i iζ (iζ)2 (iζ)3 np where ζ = k0 R. Moreover: ˆR ˆ and Δ (R) = I − 3R ˆ R. ˆ Δpr (R) = I − R np

(3.16)

The subscripts pr, np indicate propagating and nonpropagating, respectively. Equation (3.15) is the most widely used form of G0 . Later in the Chapter, we will express Green’s tensor alternately in terms of spherical Hankel functions. It is important to note that any integral involving G0 is improper since G0 is (weakly) singular. Care must be exercised in their evaluations. We require cavity definitions [Str41, Yag85] of singular quantities. That is, a singular point in a volume V with boundary S is to be isolated by a cavity Vδ having boundary Sδ . Vδ is an exclusion zone of which δ is the maximal chord length. In other words, integrations must be performed as principal value (PV) integrals [Zor41, Zui88]:    ⎫ · · · dV = P V · · · dV = lim · · · dV,⎪ ⎪ ⎬ δ→0 V \V V V δ    (3.17) ⎪ ⎭ · · · dS = P V · · · dS = lim · · · dS.⎪ S

S

δ→0

S∪Sδ

The end result is that when a principal value integral such as in Eq. (3.17) is performed, a local dyadic, called depolarizing dyadic [HY02, Yag80] appears. It is generally dependent upon the shape of the cavity. The same result was independently derived by Frahm [Fra83] and was further developed by Hnizdo [Hni11]. B. The Spectral Representation Fourier transforming on both sides in Eq. (3.12) gives: (3.18)



ˆ K] ˆ · G0 (K,  x , ω) = −e−iK·x I. [(k02 − K 2 )I + K 2 K 

0  Let G0 be decomposed into its transverse resp. longitudinal component, G⊥ (K, ω)    x 0  0  0  0   −iK· resp. G (K, ω) so that G (K, x , ω) = e G⊥ (K, ω) + G (K, ω) . These are readily obtained from Eq. (3.18).   Δ⊥  x   0  −iK· 2  ω) = − (3.19) G0⊥ (K, e−iK·x . e , G ( K, ω) = − Δ /k 0 2 2   k0 − K

Δ⊥ = Δpr and Δ = Δnp , already defined in Eqs. (3.17). (See also [dVvCL89]).

Using Weyl’s integral for G0 (Eq. (3.5), the spectral representations of the radiating or propagating and the evanescent component of Green’s tensor is obtained. They are:  K   Ik02 − K  ik0  ⊥, (3.20a) G0pr (x, x ) = eiK·(x−x ) dK 2 8π K⊥ > 1 and G0rad is transverse to R. −1 R decay, the same energy flows through every sphere at any distance R. 0 and G0N F . As a matter of fact, The same, however, does not apply to GM F 0 ) exist only in the frequency domain, as was first pointed they (especially, GN F out by Keller [Kel99]. Indeed, the Fourier transform in time does not exist for G0N F whereas for G0M F , the transform exists, but is static. In any case, all three components are present in the space surrounding the point source, but only G0rad escapes to infinity. Thus G0M F and G0N F remain attached to the source while G0rad detaches itself and propagates toward infinity. The observation point is outside the source region and e−iωt monochromaticity was assumed.

EVANESCENCE AND EVANESCENT WAVES

17

Equations (3.15) and (3.16) show that the radiative part of G0 can be singled out if the attached field is subtracted from it. It is similar to Hadamard’s finite part regularization [Kan04] of a divergent integral. We, therefore, subtract ˆ R) ˆ from Eq. (3.15) and call the resulting Green’s operator G0 . (1/4πk02 R3 )(I − 3R T So regularized, the NF term reduces to: (4.2)

G0N F (x, x , k0 ) = −

1 eik0 R − 1 ˆ R). ˆ (I − 3R 4πk02 R3

By Taylor expanding the exponential in Eq. (4.2), the R−3 singularity can be made to vanish everywhere except at the dipole itself. Furthermore, the midfield contribution cancels out since both NF and MF terms have the same tensor coefficient, as was alluded in the previous Section. The resulting equation, which is G0T , decays as R−1 and is the radiative Green’s tensor. The radiative part, therefore, has been extracted out of the full Green’s tensor G0 . Remark 3. At this point, it would be interesting to point out an analogy between Eq. (4.2) and an RLC circuit model of an antenna system. The input impedance of the antenna is given by: Z(ω) = Prad + (XL − XC ). For the radiating antenna, Prad is the loss or the power radiated away while XL resp. XC are the inductive resp. capacitive reactance. In the antenna system, Prad is subtracted from Z in order to obtain XL − XC which is, in some sense, the attached field. In Eq. (4.2), it is the reverse. Here, the attached field is subtracted from G0 to yield the radiated field. We next consider the attached field. Its picture is best observed by looking at a point dipole radiating in its surrounding space. Here is the second principal component of evanescence that was pointed out in Section 3. 5. Dipole Emission In Free Space A few words about dipoles are in order here. One must distinguish between an infinitesimal and point distribution of charge or current. Point dipoles are small asymptotic limits of infinitesimal charge distributions. Let Je (x, t) be a harmonically oscillating infinitesimal current distribution around a position x0 and Je (x, t) as Je (x, t) = Je (x)f (t). In the point approximation:  Je (x, t) = δ 3 (x − x0 ) Je (x , t) dx = J0 (t)δ 3 (x − x0 ). Upon Fourier transforming: (5.1)

3  Je (x, ω) = δ 3 (x − x0 )J0 (ω) = d(ω)δ (x − x0 ).

Equation (5.1) is the electric point dipole approximation (PDA) of Je (x, t) and d is the electric dipole moment. The goal is to determine the radiation of the dipole in free space. The problem is one of a harmonically oscillating localized charge distribution.

18

D. N. GHOSH ROY AND S. MUDALIAR

5.1. Lorenz Gauge Calculation. We work in the Lorenz gauge and use only  The scalar potential is not needed. The gauge was summathe vector potential A. rized in Eqs. (3.13a), (3.13b), (3.13c), (3.13d), (3.13e), (3.13f). We begin with the well known spherical wave expansion [Jac99, Jen17, Kel99, Bar17, GR20, Sta98, HY02, HY99, Str41] of the scalar Green’s function G0 (x, x , ω) which is: ik0  (1) ˆ ˆ (5.2) G0 (x, x , ω) = h (k0 |x|> )j (k0 |x|< )YL  (θ> )Y L  (θ< ). 0 

 L

∞ 

= =0 m=− , |x|>((( θc , the reflected Fresnel coefficient R(p) becomes complex. This results in a phase shift between the reflected and transmitted waves which is responsible for the Goos-H¨ anchen effect. The effect can be appreciated thus. R(p) is given by [Jac99]: i ⊥ ) R(p) (K

(8.6)

δ

1 Kzi − 2 Kzt = e−2iδ , 1 Kzi + 2 Kzt   Ki  2 z . = tan−1 1 Kzt =

The upshot is that the reflected beam is retarded by the phase factor 2δ with respect to the incident beam at the interface. Pictorially, it is visualized by the horizontal

24

D. N. GHOSH ROY AND S. MUDALIAR

and vertical Goos-H¨ anchen shift of the center of the beam. As for the evanescence in p or TM- polarization, we have:  t eikty −ktz z e−iδ .  t = T (p) (ki⊥ )H (8.7) H The intensity of a p-polarized evanescent wave is maximum at the interface and can be greater (sometime substantially greater) than that of the incident wave. (Et /Ei )(z = 0) can be greater than unity. Spreading a thin layer of noble metal particles on the interface further enhances the transmitted intensity. This enhancement effect, an intense glow [MW83, Mor83, BC04] on the surface, is a hallmark of evanescence. 8.2. Total Internal Reflection: Acoustics. Consider an interface P at z = 0 in a fluid medium on which a pressure field p(z, 0) exists. As before, we would like to determine the pressure in the upper half space R2+ where z ≥ 0. Let us write:  ∞ pˆ(ky )eiky y dy. (8.8) p(y, 0) = −∞

pˆ(ky ) is the angular spectrum of p. In R2+ , p satisfies Helmholtz’s equation (Δ + k02 )p(y, z) = 0 and Eq. (8.8) can be generalized to:  ∞ (8.9) p(y, z) = pˆ(ky )ei(ky y+γ(ky )z) dy. −∞

γ(ky ) = k02 − ky2 . Im γ ≥ 0. Clearly, γ(0) = k0 . Equation (8.9) shows that the solution in the upper half-space is radiating iff ky ≤ k0 real and γ is also real positive. Otherwise, the solution will decay exponentially with z and no disturbance will be detected at large values of z. In summary, for the solution (of the wave equation) to be radiating, ky must be restricted to ky ≤ k0 = ω/c0 . ω positive and k real. Only within this range of ω and ky , a disturbance will produce sound. Otherwise, the disturbance will decay exponentially without an associated acoustic field. In short, evanescence will result. Therefore, for |ky | ≤ k0 , solutions are radiating or propagating, being evanescent otherwise. The picture is essentially the same as the optical TIR in the previous Section. At |ky | = k0 , singularity sets in. For ky > k0 , the field in the upper half plane is essentially hydrodynamic in nature and for ky >> k0 , we have virtually an incompressible flow field. It can be appreciated thus. Because of the strong decay, spatial gradients are also strong. Comparatively, time derivatives are weak. The reduced wave equation then tends toward Laplace’s equation for an incompressible potential flow. The regimes are distinguished by the so-called Helmholtz number He [RH21] which is a relative measure of the importance of time and space derivatives. In evanescence, He , c> wheny > 0. In the fashion of Jost solutions [AC91] in left incidence, the total potential can be written as:   −iKi cos θi z  , z≤0, (8.10a) Φ< = Φ0 eiKi sin θi y eiKi cos θi z+RL (ki )e (8.10b)

Φ>

=

 t ·  ty · x y +Ktz z  i )eiK  i )ei(K Φ 0 TL ( K = Φ 0 TL ( K , z > 0.

RL and TL in Eqs. (8.10a) and (8.10b) are the left reflection and transmission coefficient, respectively, and Φ0 is the amplitude of the potential. The same considerations in optics also apply here and we have the same evanescent picture. The coefficients RL and TL can be obtained from the conditions on the interface which consist of the continuities of the trace wavevectors (Snell’s law) and of particle  = ∇Φ. Thus: velocities normal to P. The later are the gradients U (8.11)

Kiz = Ktz : Snell’s law. ∇Φ< = ∇Φ> continuity of the potential.

Solving Eqs. (8.10a) and (8.10b) subject to the boundary conditions in Eq. (8.11) yield RL and TL . These are: Z> − Z< ρ< 2Z> RL = , TL = , Z> + Z< ρ> Z> + Z< ρ> c > ρ> c > , Z> = . Z< = sin θt sin θt Z>( = −iωρ> Φ> . The decay of evanescent waves was demonstrated in terms of spatial coordinates. However, the same results can be obtained if the angles are considered to be complex. Thus let θi in Figure 1 be complex and write θ in for θi .Then in in i in + iθ θ in = θR √I . We have replaced θ by θ in order to avoid confusion with the complex i = −1. Using standard trigonometric identities [AS04]: (8.12a)

in in sin θ in = sin θR cosh θIin + i cos θR sinh θIin ,

(8.12b)

in in cosh θIin − i sin θR sinh θIin . cos θ in = cos θR 

Using the identities (8.12a) and (8.12b), A monochromatic plane wave eik·x can be expressed as: 

eik·x

(8.13)

= = = = =

in

in

eikx x eiky y = eikx sin θ eiky cos θ in in in in in in in in eikx sin θR cosh θI e−kx cos θR sinh θI eiky cos θR cosh θI eky sin θR sinh(θI in in in in in in in eik(x sin θR cosh θI +y cos θR cosh θI ) e−k sinh θI (x cosR −y sinR ) in in in in in in in eik(x sin θR +y cos θR ) cosh θI +cosh θI ) e−k sinh θI (x cosR −y sinR ) in in ei(ˆeK ·x) cosh θI e−(ˆe⊥K ·x) sinh θI .

26

D. N. GHOSH ROY AND S. MUDALIAR

in in Equation (8.13) demonstrates evanescence. eˆK ˆ⊥K  = (sin θR , cos θR ) and e  is perin pendicular to eˆK . Thus the real component, θ , of the angle, gives the direction  R of propagation whereas the imaginary component, θIin , determines the sense of the decay. Furthermore, let k sinh θIin be given and denote it by γ. Then: γ  (8.14) θ in = sinh−1 . k Equation (8.14) allows us to determine the rate of decay with θ in . The picture does not change in three dimension.

8.3. Ewald Diagrams. The Ewald diagrams to be presented in this Section can be viewed as facilitating graphical constructions of evanescent waves. Consider   the plane wave eiK·x = ei(K⊥ ·x⊥ +Kz z) . When normalized by k0 , the dispersion 2 2 2 2 2 2 relation  is the sphere S described by p + q + m = 1. p + q = |K⊥ /k0 | and 2 2 m = 1 − (p + q ). In terms of the wavenumber, the radius of S is k0 . We will call S the Ewald sphere. The boundary of its circular, equatorial base B is the circle C of unit radius which will be designated as the Ewald circle. In the literature, C is also named coincident circle [Wil99, Pie81]. A point Q on the Ewald sphere gives a relation between the wavenumbers on B and the direction of the radiated or scattered plane waves. its projection on B (Figure 3) determines the nature of these waves. The projected points inside C such as Qh in the figure describe the propagating or homogeneous waves while those such as Qe that project outside the circle result in evanescence. The interior of C corresponds to wave speeds in excess (super) of phase speeds whereas those outside C have speeds less than phase speeds (sub). In optical TIR, the interior (exterior) of C corresponds to super (sub) critical angles and are called super (sub)-luminal regions. In acoustics, the designations are super (sub)-sonic speeds. Note that super (sub) refer to the trace waves and not to the vacuum speed of light in optics or Mach number in acoustics. Mach number is defined as V /c, where v is the actual speed and c is the speed of sound in the medium. As K⊥ increases in magnitude and crosses the Ewald circle, the wave goes from propagating to evanescent. The dominance of K⊥ over k0 in the near-field is the most important characteristic of electromagnetic radiation.

Figure 3. The Ewald diagram. The Ewald circle C is the boundary of the circular base B which is the projection of the Ewald sphere S on the horizontal plane. The other legends appear in the text.

EVANESCENCE AND EVANESCENT WAVES

27

The point can be better appreciated by considering the power flow. Let ψ = . For an evanescent wave for which Im Kz ≥ 0, the power flow P can be ψ e evaluated to give:  x 0 iK·

(8.15)

 ⊥ e−2Kz z , P = Im (ψ∇ψ) = |ψ 0 |2 K

 ⊥ , but decaying exponentially in the zshowing that power is flowing along K  direction. That is, only the homogeneous wave, eiK⊥ ·x⊥ is carrying power. As a physical picture of power evanescence in an evanescent wave, consider that two point sources are vibrating (assume unit amplitude) in an infinite flat plate, one at the origin (0, 0) and the other at (0, y0 ). Their combined velocity amplitude can be written as: v(x⊥ ) = δ(x)δ(y − y0 ) − δ(x)δ(y). The negative sign in v indicates that the point sources are vibrating in phase opposition. We will show later that the combined power can be expressed as: P (ω) = 2p(ω)(1 − sinc (ky0 )), where p(ω) is the power radiated by each source individually. The presence of the sinc factor is indicative of the interference between the two radiations, as is typical in physics. If ky0 >> 1, i.e., the sources are separated by many wavelengths, then the Sinc term is negligible and the power is roughly the sum of the two individual powers. However, as y0 diminishes, P decreases and when y0 → 0, no power is radiated, a complete evanescence. Thus when two point sources in an infinite flat surface (frequently called a baffle) vibrate in phase opposition, the radiated power diminishes as the separation between the sources decreases, finally becoming evanescent as this distance tends toward zero. A practical manifestation of such evanescence is an infinite flat plate vibrating in a standing wave mode with λx0 and λy0 as nodal line separations. The plate is trying to launch radiation into the upper half-space, z > 0. It is assumed that no sources exist in this space, all sources being below the 2 − k2 plate. When either or both λx0 , λy0 become less than λ/2, kz = k2 − kx0 y0 becomes imaginary and evanescence takes place. The dispersion relations can be formulated thus. ˆ zt = 1 − K ˆ i2 . ˆ t2 = 1 − (n2 /n1 )2 K K ⊥ ⊥ ˆ t and n2 k0 K ˆ i are conserved. For n2 = 1, K ˆ zt reduces to: Recall that n1 k0 K ⊥ ⊥ ˆ zt = K

ˆ i2 . 1 − (1/n1 )2 K ⊥

ˆ i , Re K ˆ zt = 1 when K ˆ i = 0 and is It can be seen that for n2 < n1 and for real K ⊥ ⊥ i i i ˆ z vs. K ˆ is a quadrant of a circle of unit ˆ = n1 /n2 . The plot of Re K zero when K ⊥ ⊥ ˆ i2 − 1. In ˆ i > n1 /n2 , Im K ˆ zt = (n2 /n1 )2 K radius. On the other hand, when K ⊥

⊥

ˆ zt is a hyperbola that rises at K ˆ i = n1 /n2 and goes to infinity. In this case, Im K ⊥ ˆ i = 1 and proceeds to infinity. ˆ zi is also a hyperbola rising at K the same vein, Im K ⊥ t i ˆ ˆ ˆ zi Kz vs. K⊥ curve is asymptotic to a straight line of slope n2 /n1 while the curve K i ˆ has the slope n1 /n2 which is steeper than n2 /n1 . vs. K ⊥ Having discussed the direct half-space TIR problems in optics and acoustics, the next step is to investigate the reversed half-space problem. The reverse problem is the backbone of modern, high resolution, near-field technology.

28

D. N. GHOSH ROY AND S. MUDALIAR

9. The Reversed Half-Space We have seen that for classical, homogeneous plane waves, no energy can be transported in a lossless medium by the components in the incidence above a critical angle. Evanescent waves, therefore, do not transport energy away from the interface where they are generated. In other words, there is no transport of energy by an evanescent wave in the space in which it exists. In Figures 1, this is the upper-half space. However, transport of energy must not be confused with transfer of energy. We have alluded to frustrating an evanescent wave, the process in which the wave transfers energy and excitation to a third dielectric medium. It turns out that if there are inhomogeneous components in the incident wave, then the transmitted wave can be both propagating and decaying. The surface normal trace wave can be real and energy can propagate away into the interior of the second medium. The transmitted wave, of course, must decay because the incident wave does so. This phenomenon takes place even for supercritical incidence. Let us then reverse the direct TIR and interchange the half-spaces. The denser medium now becomes the rarer and vice versa. Referring to Figures 1, n1 is now greater than n2 and incidence is from n2 -side. The evanescent wave now impinges on the interface and the task is to determine the ensuing fields. The conserved ˆ t and n2 k0 K ˆ i . In terms of U  , the wavevetors in the reversed quantities are n1 k0 K ⊥ ⊥ problem are as follows. =

(9.1b)

i U t U

(9.1c)

ˆzt U

=

(9.1a)

=

(9.1d)

=

(9.1e)

=

(9.1f) (9.1g)

ˆzi U

= =

ˆ i, k0 n2 U t i ˆ t, U ˆ⊥ ˆ⊥ k0 n1 U = (n2 /n1 )U , ˆ t |2 = 1 − (n2 /n1 )2 |U ˆ i |2 1 − |U ⊥ ⊥ 1 i ˆ i |2 , n2 |U ˆ⊥ n21 − n22 |U | < n1 , ⊥ n1 i i ˆ i |2 − n2 , n2 |U ˆ⊥ n22 |U | > n1 , 1 ⊥ n1 i ˆ i | 2 , |U ˆ⊥ 1 − |U | < 1, ⊥ i ˆ i |2 − 1, |U ˆ⊥ | > 1. i |U ⊥

ˆ i | into [0, 1] ∪ [1, (n1 /n2 )] ∪ [(n1 /n2 ), ∞). We can break down the range of |U ⊥  j , j = i, r, t, are real. So the waves are all In the first interval, all three vectors U ˆ i = sin θ i , θ i ∈ homogeneous and we are in subcritical regions. In terms of angles, U ⊥ ˆ t . In the [0, π/2). This situation was described in Figures 1. The same applies to U i r t  and U  are evanescent, but U  is real. It is the second interval [1, (n1 /n2 )], U region of frustration in which an incident evanescent wave results in a homogeneous transmitted wave and energy transfer takes place. It is the optical tunneling effect which is analogous to barrier penetration in quantum mechanics [Sch55]. In the last interval [(n1 /n2 ), ∞), all three waves are evanescent. The fields are attached to the surface and no energy is propagated away. The Ewald diagram in Figure 3 remains the same except that a second circle of radius (n1 /n2 ) > 1 now surrounds the circle C of radius 1 in the figure. The physics in the intervals I1 = [1, (n1 /n2 )] and I2 = [(n1 /n) , ∞) are shown in Figures 4a and 4b, respectively, for a point dipole. The dipole is on the plane z = 0 while the dielectric interface is at a height

EVANESCENCE AND EVANESCENT WAVES

29

t z0 above it. The solid line in the upper half space in Figure 4a indicates that E in this figure is real whereas the dotted lines represent evanescent eaves. The rest is self-evident. In the first interval, all angles are subcritical. θ i < θc . θ t < π/2.

Homogeneous transmission

Evanescent transmission

Figure 4. An evanescent wave from a point dipole d on the plane z = 0 is incident on a dielectric interface at a height z = z0 . n1 > n2 . (a) The reflected wave is evanescent, but the transmitted wave homogeneous. (b) The same as (a) except that all three waves are evanescent. The solid line indicates a homogeneous wave and the dotted lines evanescent waves.  t being real The angle of its incidence in In the second interval, θ t is subcritical, K this interval is, therefore, π/2 − iα, α > 0. The dipole fields are: ⎧ i r ⎪ ⎨E (x) + E (x) if 0 < z < z0 ,  x) = E  e (x) + E  r (x) if z < 0, (9.2) E( ⎪ ⎩t E (x) if z ≥ z0 . Our interest in this Chapter is in the second interval. Considering the space restriction, we will not go into the detailed computations and evaluations of the integrals, but focus on the basic physical results. The free space dipole emissions in the angular spectrum representation were calculated in Eqs. (6.5a) and (6.5b). We now replace free space Green’s function (1) (ik0 /4π)h0 (k0 x) in these equations by Weyl’s integrals (3.5), (3.6), (3.7a), (3.7b) and use the wavenumber relations in Eqs. (9.1a), (9.1b), (9.1c), (9.1d), (9.1e), (9.1f), and (9.1g) . For evaluating the transmission in the denser upper half-space when the incidence in the rarer lower half-space is evanescent, the program is as t = TE  i , T the  i using Weyl’s integrals; (2) determine E follows. (1) Calculate E t   t; Fresnel transmission coefficient; (3) find the magnetic induction B = (1/iω)∇× E t t∗ i  ×B  } for U ˆ ∈ [0, 1]∪[1, (n1 /n2 )]. The (4) evaluate the Poynting vector (1/2)Re{E ⊥ upshot of all these is that an exponential function appears in the resulting integral ˆ i ∈ [1, (n1 /n2 )]. It is this exponential that characterizes evanescence in the when U ⊥ transmission. 9.1. Power Transmission in Free Space. Let us first calculate the power flow in the emission of a point dipole located at x0 in free space in the interval ˆ⊥ | ∈ [0, 1]. The radiation from such a dipole was given in Eqs. (6.5a), (6.5b). |U

30

D. N. GHOSH ROY AND S. MUDALIAR

˜ Also, since we are For notational convenience, let us define (αk02 )/(8π 2 ω0 ) = β. ˆ   ˆ ˆ restricting |U⊥ | to [0, 1], we can write K for U and K for U . The emission is given by:  1 i  x) = −β˜ ˆ i ) · J0 (ω)eiK0 Kˆ i (x−x0 ) dK ˆ iK ˆ⊥ E( (I − K i ˆ 2 K R z which we write more compactly as:  1  ˆi i ⊥  x) = −β˜ J0⊥ (ω)eiK0 K ·(x−x0 ) dK (9.3) E( i ˆz R2 K ˆ i ) · J0 (ω) is the component of J0 (ω) perpendicular to ˆ iK in which J0⊥ (ω) = (I − K  of the magnetic field B  field is: ˆ i . The complex conjugate, B K  ˜ 1 ˆi  ˆi i  x) = − βK0  x) = 1 ∇ × E( ⊥ (9.4) B( (K × J0⊥ (ω))e−iK0 K ·(x−x0 ) dK . i ˆ iω ω 2 R Kz The corresponding complex Poynting vector follows from Eqs. (9.3) and (9.4) and reads:   1 ˆ i ˆ i 2 K0  i i ˜   i × J0⊥ ˆ⊥ (9.5) S(x) = β {J0⊥ × K }eiK0 (x−x0 )·(K −K ) dK dK⊥ . ω R2 R2 Kzi Kzi The power across a plane located at a distance Z from the dipole is then:  1  x⊥ (z0 ) + zˆZ) dx⊥ (z0 ). S( (9.6) P = zˆ · Re 2 R2 x⊥ (z0 ) is the location of a point on the plane z = z0 . The space integral over x⊥ (z0 ) ˆi −K ˆ i ). Inserting Eq. (9.5) into Eq. (9.6) and yields the delta function (1/K0 )δ(K simplifying obtains: ˜ 2 K0  1  2 ˆi |β| |J0⊥ | dK⊥ . (9.7) P = 2ω ˆ i ≤1 Kzi 0≤|K As expected, Eq. (9.7) shows that there is no contribution from the evanescent spectrum. The angular spectral interval considered excludes evanescent waves. The power is transmitted only by the homogeneous waves. The procedures outlined in this Subsection will be used in the calculations that follow. 9.2. Power Transimmison in Interval II. Interval II is the sum [0, 1] ∪ t t [1, n1 /n2 ]. The power transmitted is also the sum P = P t = P[0,1] + P[1,n . But 1 /n2 ] t   t (x) dx where S t is E  . The fields E t × B  and B  in the now P t = (1/2)Re R2 S i i  and B  . Furthermore, the exponent previous section are now the incident fields E x0 − x(0) in Eqs. (9.4) and (9.5) is now (x0⊥ − x⊥0 ) + zˆ(z − z0 ). x0 is a point  t. E  t is obtained on the plane at z0 and x(0) is on z = 0. We need to define E i  by multiplying E in the last Section by the Fresnel transmission coefficient T and ˆi ˆt ˆ ⊥ , p) where p replacing the wave term eiK K ·x by eiK K ·x . The coefficient T = T (K is the polarization vector. T is given by: TT E =

ˆ zi 2K ˆ zi + n1 K ˆ zt n2 K

, TT M =

ˆ zi 2K ˆ zi + n2 K ˆ zt n1 K

.

EVANESCENCE AND EVANESCENT WAVES

31

The transmitted fields are given by:  t   i (x⊥ (z0 ))T (K ˆ zi , p)eiK0 Kˆ t ·x dx⊥ (z0 ) E (x) = E 2 R  1  i ˆ zi , p)eiK0 Kˆ t ·x dK ⊥ J0⊥ T (K = −β˜ (9.8a) , ˆ zi II K  t i  (x) = − K0  (x⊥ (z0 ))e−iK0 Kˆ t ·x dx⊥ (z0 ). ˆt × E B (9.8b) (K ω R2 The power computations essentially follow the line of reasoning in the previous ˆ i ∈ [0, 1], the result Section. The delta function integrals remain the same. For K is only slightly different from Eq. (9.7) and is given by: ˜ 2 K0  |T (K ˆ zi , p)|2 |β| t i ˆ zt |J0⊥ |2 dK ˆ⊥ (9.9) PI = K . ˆ zi )2 2ω (K I I = [0, 1]. In the interval II = [1, n1 /n2 ], the calculations are as follows. From Eqs. (9.3), (9.4), (9.8a), and (9.8b), we obtain:  ˆ zi , p) T (K t i ˜  ˆ i , ω)eiK0 Kˆ t ·x eiK0 Kˆ i ·(x⊥0 +ˆzz0 ) dK ˆ⊥ J0⊥ (K (9.10a)E (x) = β , i ˆ Kz R2 ˜ 0  T (K ˆ zi , p) t βK  ˆ t × J0⊥ (K ˆ i , ω))e−iK0 Kˆ t ·x · B (x) = − (K i ˆ ω 2 Kz R ˆ i ·( −iK0 K x⊥0 +ˆ z z0 ) i ˆ⊥ (9.10b) dK . · e From Eqs. (9.10a), (9.10b): t

t × B  )(x) (E

=

˜ 2 K0 |β| ω



 R2

R2

ˆ zi , p) T (K ˆ i , ω)eiK0 Kˆ t ·x J0⊥ (K i ˆz K ! ! i i ˆ⊥ ˆ⊥ ˆ ⊥ i · dK dK dK

(9.11) × eiK0 K ·(x⊥0 +ˆzz0 ! · is the expression in the first curly bracket with  on the vectors. The power flowing through the plane at z0 into medium 1 is then:  t 1  )(x) dx⊥ (z0 ). t × B (E (9.12) P t = zˆRe 2 R2 ˆi

We next insert Eq. (9.11) in Eq. (9.12) and recall that in the interval under considˆ zi is purely imaginary. It leads to a factor e−2K0 Kˆ zi z0 to the exponentials. eration, K The final result is: ˜ 2 K0  |T (K ˆ zi , p)|2 |β| i ˆ zt |J0⊥ |2 e−2K0 Kˆ zi z0 dK ˆ⊥ K (9.13) PIt = . i 2 ˆ 2ω ( Kz ) II ˆ zi , p)|2 = T (K ˆ zi , p)T (−K ˆ zi , p). In this interval where kˆi is purely imaginary, |T (K To sum up. In view of Eq. (9.13), the transmitted waves into the upper medium ˆi are homogeneous, but the incident field is evanescent and is decayed by e−2k0 kz z0 as it impinges on the plane at z = z0 . The frustration of the incident evanescent wave by the upper dielectric medium is clearly demonstrated. The polarization effects can be incorporated by replacing the transmission coefficient T by its explicit expressions given in the above. It should be mentioned that the similar result also holds if a line source is used in place of the dipole [PS03].

32

D. N. GHOSH ROY AND S. MUDALIAR

The reversed half-space problem demonstrates that energy of an evanescent wave can be recovered. It is similar to the available energy in thermodynamics. Since energy in the space immediately surrounding the dipole is nonpropagating (not necessarily static), the reversed solution implies that evanescent energy is localized around the dipole and can be recovered. A part of the energy may escape ˆ i ∈ [n, ∞]. This implies that to infinity unless we are in the third interval where U ⊥ the localized energy is not synonymous with the stored energy, but a part of the total energy. The propagated energy can never be stored. In a thermodynamic system, one would like to maximize available energy. In a radiative or scattering system, on the other hand, it is not necessarily the case. In these problems where one generally works in the far field, maximization of the radiated or scattered field is desired. 10. Summarization We conclude with a summarization of the discussions. We have presented the fundamentals of evanescence and evanescent waves and their spatial an spectral behaviors. The concepts of near (NF), middle (MF) and far-field (FF) or radiation zone are introduced and explained. The unique features of each are pointed out including attached and radiated electromagnetic fields. Several seemingly odd behaviors of evanescent waves are also mentioned. Examples are presented to illustrate how evanescent waves arise under various physical situations. The hallmarks of these waves such as their exponential decay and enhanced intensity around a scattering object are highlighted. The crucial role of evanescence in today’s stateof-the-art, high resolution, near-field technology, is elaborated. It is pointed out that the main theoretical underpinnings for applications are: (1) Green’s functions and tensors and (2) radiations from infinitesimal charge distributions and point dipoles. The concepts of transversality (T) and longitudinality (L) are introduced along with T and L delta currents and Green’s propagators. Detailed calculations of radiations are presented both in the Lorenz gauge and in angular spectrum representations. They include emissions in both free space and in the presence of an interface separating two dielectric media. The celebrated Weyl’s integral for 3-D scalar Green’s function is discussed in depth. The near-field implications of these calculations are discussed. The archetypical direct half-space problems of optical TIR and its acoustic analogue are presented at length. The reverse half-space problem where the incident and reflected waves are evanescent, but the transmitted wave is homogeneous, is also presented. The reverse problem embodies the phenomenon of frustration of evanescence the importance of which is pointed out. Finally, a number of future tasks are suggested which include the nature and storage of evanescent energy and electromagnetic energy in general in classical electromagnetics. 11. Future Tasks The question of energy, evanescent energy in particular, is paramount in the analysis of evanescence. The exact nature of energy in an evanescent wave is not entirely transparent. What exactly is the nature of energy in an evanescent wave, where and how is it stored? Given that this energy can be frustrated, the conventional idea of a purely static, attached field, does not hold. The attached field is clearly transferable and hence movable. What precisely is an attached wave? Does  2 exhaust the energy of  2 + (1/2)|H| it mean static or is it localized? Does (1/2)|E|

EVANESCENCE AND EVANESCENT WAVES

33

an evanescent wave? Poynting’s theorem does not suffice here since the theorem is concerned only with the propagation of total energy and not about what this energy may be. These and the question of storage of energy are not specific to evanescence, but of classical electrodynamics in general. The physics of evanescence, however, has an added complication, a morphogenetic inquiry. In other words, how a static, localized or attached wave can develop into a dynamic, propagating wave in the radiation zone? What is the mechanism? The Wilcox expansion shows that we can trace the far-field all the way down to radiator. In this series, the far-field is the zeroth-order term. In the region surrounding the radiator, a large number of terms must be retained. The interactions between them can, in principle, be evaluated. Can the Wilcox expansion provide a morphogenetic picture that is desired? In a sense, it is analogous to the development of Fraunhoffer diffraction in the far-field from the Fresnel region in the near-field region. The questions of energy and the mechanism of morphism of the near-field to the far-field seem to be interesting as future tasks

References M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series, vol. 149, Cambridge University Press, Cambridge, 1991, DOI 10.1017/CBO9780511623998. MR1149378 [Arn01] H. F. Arnoldus, Representation of the near-field, middle-field, and far-field electromagnetic green’s functions in reciprocal space, J. Opt. Soc. Am. B 18 (2001), 547–555. [AS04] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, Delaware, 2004. [ATT13] J. Ahrens, M. R. Thomas, and L. Tashev, Gentle acoustic crosstalk cancelation using the spectral division method and ambiophonics, Proc. IEEE Wkshp. on Sig. Process. to Audio and Acoustics (New Platz, New York), vol. 1, 2013, pp. 1–4. [BA03] Y. Ben-Aryeh, Nonclassical high resolution optical effects produced by evanescent waves, J. Opt. B: Quantum Semiclassical Opt. 58 (2003), no. 4, S553–m556. [Bar17] G. Barton, Elements of Green’s functions and propagation: Potentials, diffusion, and waves, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1989. MR1011248 [BBP60] J. J. Brady, R. O. Brick, and M. D. Pearson, Penetration of microwaves into the rarer medium in total reflection, J. Opt. Soc. Am. 50 (1960), no. 11, 1080–1084. [BC04] P. W. Barber and R. K. Chang, Optical Effects Associated with Small Particles, vol. 1, World Scientific, Singapore, 2004. [Ber01] M. V. Berry, Asymptotics of evanescence, J. Modern Opt. 48 (2001), no. 10, 1535– 1541, DOI 10.1080/09500340110055656. MR1851519 [Bos94] J. C. Bose, On the influence of the thickness of the air-space on total reflection of electric radiation, Proc. Roy. Soc. London 62 (1894), 300–300. [Bre60] L. M. Brekhovskikh, Waves in layered media, Applied Mathematics and Mechanics, Vol. 6, Academic Press, New York-London, 1960. Translated from the Russian by David Lieberman; translation edited by Robert T. Beyer. MR0112392 [BW99] M. Born and E. Wolf, Principles of Optics, Cambridge University Press, Cambridge, UK, 1999. [CB94] D. Courjon and C. Bainier, Near field microscopy and near field optics, Rep. Prog. Phys. 57 (1994), 989–1028. [CMD72] C. K. Cargniali, L. Mandel, and E. H. Drexhage, Absorption and emission of evanescent photons, J. Opt. Soc. Am. 62 (1972), no. 4, 479–486. [CTDRG89] C. Cohen-Tannoudji, J. Dupont-Roc, and D. Grynberg, Photons and Atoms, Introduction to Quantum Electrodynamics, Wiley, New York, 1989. [AC91]

34

[Dev12]

[dF01] [DGR03] [dVvCL89] [Eri67] [Fra83] [Fri56] [GJG00] [GR20]

[Gri15] [Hni11] [HS87] [HY99]

[HY02] [IH04]

[Jac99] [Jen17] [Kan04] [Kel99]

[KK21] [Kov00] [Kow95] [Lot68] [MA16] [MI68] [Mil13] [Mor] [Mor83] [MW83]

D. N. GHOSH ROY AND S. MUDALIAR

A. J. Devaney, Mathematical foundations of imaging, tomography and wavefield inversion, Cambridge University Press, Cambridge, 2012, DOI 10.1017/CBO9781139047838. MR2975719 F. de Fornel, Evanescent waves, Springer, New York, 2001. D. K. Dacol and D. G. Roy, Wave scattering in waveguides, J. Math. Phys. 44 (2003), no. 5, 2133–2148, DOI 10.1063/1.1563847. MR1972768 P. de Vries, D. V. van Coevorden, and A. Lagendjik, Point scatterers for classical waves, Rev. Mod. Phys. 70 (1989), 447–466. A. C. Eringen, Mechanics of Continua, John Wiley, New York, 1967. C. P. Frahm, Some novel delta-function identities, Am J. Phys. 51 (1983), 826–829. B Friedman, Principles and Techniques of Applied Mathematics, Wiley, New York, 1956. C. Girard, C. Joachim, and S. Gauthier, The physics of the near-field, Rep. Prog. Phys. 62 (2000), 893–938. D. N. Ghosh Roy, Mathematics of wavefields, Applied mathematical analysis: theory, methods, and applications, Stud. Syst. Decis. Control, vol. 177, Springer, Cham, 2020, pp. 187–231. MR3930651 D. J. Griffiths, Introduction to Electrodynamics, Cambridge University Press, Cambridge, UK, 2015. V. Hnizdo, Generalized second-order derivatives of 1/r, Eur. J. Phys. 32 (2011), 287–297. E. Hecht and A. Sejac, Optics, 2 ed., Addison-Wesley, Reading, MA, 1987. T. B. Hansen and A. D. Yaghjian, Plane-wave theory of time-domain fields: Nearfield scanning applications, IEEE Press Series on Electromagnetic Wave Theory, IEEE Press, New York, 1999, DOI 10.1109/9780470545522. MR1767418 G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics, Springer, New York, 2002. T. Inoue and H. Hori, Quantum theory of radiation in optical near field based on quantization of evanescent electromagnetic waves using detector mode, Progress in Nano-Electro-Optics IV (M Ohtsu, ed.), Springer, Springer Series in Optical Sciences, New York, 2004, pp. 127–200. J. D. Jackson, Classical electrodynamics, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1975. MR0436782 U. D. Jentsura, Advanced Classical Electrodynamics, World Scientific, Singapore, 2017. R. P. Kanwal, Generalized functions: Theory and applications, 3rd ed., Birkh¨ auser Boston, Inc., Boston, MA, 2004, DOI 10.1007/978-0-8176-8174-6. MR2075881 O. Keller, Attached and radiated electromagnetic fields of an electric point dipole, J. Opt. Soc. Amer. B Opt. Phys. 16 (1999), no. 5, 835–847, DOI 10.1364/JOSAB.16.000835. MR1699674 Y. Kalkal and V. Kumar, Understanding energy propagation during reflection of an electromagnetic evanescent wave, Am. J. Phys. 89 (2021), 877–883. A. Kovetz, Electromagnetic Theory, Oxford University Press, Oxford, UK, 2000. M. W. Kowatz, Homogeneous and evanescent contributions in scalar near-field diffraction, Appl. Opt. 34 (1995), no. 17, 3055–3063. H. K. V. Lotsch, Reflection and refraction of a beam of light at a plane interface, J. Opt. Soc. Am. 58 (1968), no. 4, 551–561. S. Mikki and Y. Antar, New Foundations for Applied Electromagnetics, Artech House, Boston, 2016. P. M. Morse and K. U. Ingaard, Theoretical Acoustics, Princeton University Press, Princeton, NJ, 1968. M. Milosovic, On the nature of the evanescent wave, Appl. Spectros. 67 (2013), no. 2, 126–131. D. Morin, [email protected]. A. Moroz, Depolarization field of spheroidal particles, J. Opt. Soc. Amer. B Opt. Phys. 26 (2009), no. 3, 517–527, DOI 10.1364/JOSAB.26.000517. MR2544107 W. Meier and A. Woukun, Enhanced fields on large metal particles: dynamic depolarization, Opt. Lett. 8 (1983), no. 11, 581–583.

EVANESCENCE AND EVANESCENT WAVES

[Oht04] [Pie81] [PS03] [Pur85] [RH21] [Ros76]

[SC99] [Sch55] [Sch21] [SSL76]

[Sta98]

[Str41] [TdF60] [TLP+ 90]

[vB91] [WF98] [Wil56] [Wil99] [Woo15] [Xia99] [Yag80] [Yag85] [Zor41] [Zui88]

35

M. Ohtsu (ed.), Progress in Nano-Electro-Optics IV, Springer Series in Optical Sciences, San Diego, 2004. A. D. Pierce, Acoustics, McGraw-Hill, New York, 1981. L. E. R. Petersson and G. E. Smith, Role of evanescent waves in power calculations for counter propagating beams, J. Opt. Soc. Am. A 20 (2003), no. 12, 2378–2384. E. M. Purcell, Electricity and Magnetism, McGraw-Hill Inc, New York, 1985. S. W. Rienstra and A. Hirschberg, An Introduction to Acoustics, Eindhoven University of Technology, Netherlands, 2021. M. Roseau, Asymptotic wave theory, North-Holland Series in Applied Mathematics and Mechanics, Vol. 20, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. MR0433015 A. V. Shchegrov and P. S. Carney, Far-field contribution to the electromagnetic green’s tensor from evanescent modes, J. Acoust. Soc. Am. 16 (1999), 2583–2584. I. M. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1955. H. Schlichting, Boundary layer theory, McGraw-Hill, New York; Pergamon Press, London; Verlag G. Braun, Karlsruhe, 1955. Translated by J. Kestin. MR0076530 ´ Lalor, Asymptotic approximations to angularG. C. Sherman, J. J. Stamnes, and E. spectrum representations, J. Mathematical Phys. 17 (1976), no. 5, 760–776, DOI 10.1063/1.522975. MR406088 I. Stakgold, Green’s functions and boundary value problems, 2nd ed., Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1998. A WileyInterscience Publication. MR1487078 J. A. Stratton, Electromagnetic Theory, McGraw-Hill, New York, 1941. ˇ G. Toraldo di Francia, On the theory of some Cerenkovian effects (English, with Italian summary), Nuovo Cimento (10) 16 (1960), 61–77. MR121054 D. H. Trivett, L. D. Luker, S. Petrie, A. L. van Buren, and J. E. Blue, Understanding energy propagation during reflection of an electromagnetic evanescent wave, J. Acoust. Soc. Am. 87 (1990), no. 6, 2535–2540. J. van Bladel, Lorenz or lorentz?, IEEE Antennas Propag. Mag. 33 (1991), no. 2, 69. E. Wolf and J. T. Foley, Do evanescent waves contribute to the far field?, Opt. Lett. 23 (1998), 16–18. C. H. Wilcox, An expansion theorem for electromagnetic fields, Comm. Pure Appl. Math. 9 (1956), 115–134, DOI 10.1002/cpa.3160090202. MR89000 E. Williams, Fourier Acoustics, Academic Press, San Diego, 1999. D. C. Woods, On the use of evanescent plane waves for low frequency energy transmission across material interfaces, J. Acoust. Soc. Am. 138 (2015), 2062–2078. M. Xiao, Evanescent fields do contribute to the far field, J. Modern Opt. 46 (1999), no. 4, 729–733, DOI 10.1080/095003499149773. MR1692664 A. D. Yaghjian, Electric dyadic green’s functions in the source region, Am J. Phys. 68 (1980), 248–263. A. D. Yaghjian, Maxwellian and cavity electromagnetic fields within sources, Am J. Phys. 53 (1985), 859–863. V. A. Zorich, Mathematical analysis. II, Universitext, Springer-Verlag, Berlin, 2004. Translated from the 2002 fourth Russian edition by Roger Cooke. MR2033095 C. Zuily, Problems in distributions and partial differential equations, North-Holland Mathematics Studies, vol. 143, North-Holland Publishing Co., Amsterdam; Hermann, Paris, 1988. Translated from the French, DOI 10.1016/S0304-0208(08)70020-3. MR972505

Email address: [email protected] Sensors Directorate, Air Force Research Laboratory, Wright-Patterson AFB, Ohio 45433 Email address: [email protected]

Contemporary Mathematics Volume 786, 2023 https://doi.org/10.1090/conm/786/15795

Constitutive fractional modeling Jordan Hristov Abstract. The chapter addresses constitutive fractional modeling based on basic thermodynamic principles with emphasis on applications of fractional operators with singular and non-singular memory kernels. The Boltzmann superposition and the fading memory principles form the fundament of the developed models and refer to the formulation of diffusion and linear viscoelastic phenomena.

Contents 1. Constitutive equations of dynamic systems with memories 1.1. Causality 1.2. Constitutive modelling 1.3. Restrictions on Constitutive Relationships 1.4. Fading memory concept 2. Fractional Calculus 2.1. Main functions in fractional calculus 2.2. Fractional operators with a singular memory 2.3. Fractional operators with non-singular kernels 2.4. Fractional Taylor series with derivatives with singular kernels 3. Shared functions in fractional calculus and the relaxations 3.1. Exponential function 3.2. Power-law function 3.3. Kohlrausch function 4. Approximations of functions by exponential sums 4.1. Prony’s method 4.2. Exponential sum approximation for t−β 4.3. Exponential sum approximation of Mittag-Leffler function 4.4. Exponential sum approximation of the Kohlrausch function 4.5. Some briefs on exponential sum approximations 5. Fractional models build-up: a systematic approach 5.1. Fractional diffusion models through Taylor’s series expansions 5.2. Application of the fading memory concept to constitutive diffusion flux relationships 5.3. Extended fading memory concept 5.4. Extended fading memory concept: Effect of the memory kernel Key words and phrases. Constitutive models, fading memory, Boltzmann superposition, diffusion, viscoelasticity. c 2023 American Mathematical Society

37

38

JORDAN HRISTOV

6. Diffusion problems with memory:Fractional models 6.1. Diffusions with singular memories 6.2. Fractional diffusion 6.3. Diffusion models with non-singular memories 6.4. Dodson’s diffusion equation 6.5. Diffusion equation with a Mittag-Leffler memory 6.6. Some final briefs on the fractional diffusion models 7. Linear Viscoelasticity: Models with memories 7.1. Caputo-Mainardi approach: Models with singular kernels 7.2. Systematic approach based on the fading memory principle 7.3. Implementation of the Caputo-Fabrizio operator in viscoelastic rheology 7.4. Discontinuous relaxation spectrum by Mittag-Leffler function 8. Viscoelasticity: Interconversion of relaxation and creep 8.1. Interconversion of power-law relaxation and creep 8.2. Interconversion of Prony series of relaxation and creep:Examples 9. Viscoelasticity: Constitutive equations with non-singular kernels 9.1. Caputo-Fabrizio operator in constitutive equations 9.2. Fractional non-isothermal linear viscoelastic models 9.3. Final remarks on linear viscoelastic models 10. Final Comments References

Although a good deal of research over the last century and the first two decades of the 21st one has been devoted to fractional calculus the art of modeling real physical problems employing fractional operators is far from the complete and corrected application. About 15 years ago when I started working step by step with fractional calculus I was surprised by the fact that no new models appear and the dominating publications are on fractional paraphrases of already existing models from mathematical physics. Precisely, the common approach applied is the formalistic fractionalization of already existing models by simple replacements of integer-order derivatives by fractional counterparts. Many people who profited from the enormous proliferation of publications with factional operators do not create models but only apply different techniques to solve fractional paraphrases with missing physical analyzes of the results. As mentioned by Hilfer [1] the existing fractional models could be formally divided into two groups: 1) Approximate models. In these models natural laws are approximately described 2) Questionable models. This term encompasses all models obtained by simple replacement of the differential operators with fractional. Hilfer’s term “questionable “ is too polite since most of these models are paraphrases of existing models and in the dominating cases are physically incorrect, and therefore the results cast doubt since at this moment cannot be verified. This study, arranged as a chapter of a book, is just the initial step of a systematic analysis and application of fractional operators with different memory kernels in the modeling of physical phenomena. Precisely, we stress the attention on the main problems in the modeling with fractional operators, mainly with fractional in

CONSTITUTIVE FRACTIONAL MODELING

39

time derivatives, related to the causality of the models and their accurate derivations, applying correct constitutive equations and fundamental balance laws. We skip the problems related to the properties of the fractional operators from a mathematical point of view, which for now is a hot topic in the literature [2–4] since the main task is the construction of fractional constitutive equations and adequate (reasonable) fractional models of real-world phenomena thereof. Many moments and results used are developed by other researchers and the basic ideas of how mathematical models should be constructed exist for years. Thus, we may stay on the shoulder of giants of classic knowledge and apply accurate rules in model building the task is to obtain wonderful and right results with fractional operators. In this way, the ideas developed here will provide a framework allowing seeing how mathematical models with fractional operators must be constructed. The main contribution here is the compilation of basic rules about model-buildup, fractional operator applications, and natural laws. And, last but not least, to take into account the physics of the modeled processes because the astonishing publications on fractional paraphrases indicate forgotten physics behind the model solved. It is impossible, at this initial stage, to encompass all physical phenomena and the relevant models involving fractional operators. Because of this, the main line developed here passes through diffusion models, such as anomalous diffusion, heat conduction, and simple rheological models of the linear viscoelasticity This chapter addresses some principal points in fractional modeling, namely: a) Constitutive equations and from where they come from b) Causality of the models, precisely their constitutive equations, c) Thermodynamic approach in formulation of constitutive equations with memories , d) Observability the constitutive equations with memories, e) Fractional operators with singular and non-singular kernels (memories) as well as applications in f) Fractional diffusion models and g) Fractional linear viscoelastic models We realize that all work with results in the discussed here areas cannot be encompassed and analyzed but we will do attempts to present the main results by applying a unified point of view and allowing us to see real applications of fractional operators and how models have to be constructed.

1. Constitutive equations of dynamic systems with memories 1.1. Causality. First of all, we have to answer the basic question: Do all physic laws are causal relationships? This question has, to a greater extent, a philosophical nature since it is in the fundament of the human attempt to understand nature and formulate its laws. In dynamical laws, modeled by differential equations, the description of the initial state representing the cause is related to the final state, i.e. the effect. It is noteworthy that with statistical laws such interpretation is impossible. However, in many cases, the statistical laws are genuine laws of nature [5], and therefore not in all cases when we are speaking about causality this property exists in such models since they are not causal. Therefore, if every law of nature represents a definite relationship, precisely dependency, of the final state on the earlier state, then it is causal relationship [5, 6].

40

JORDAN HRISTOV

1.1.1. The philosophical background of causality. In the metaphysics of Aristotle, the question of the cause of events is defined and several causes are outlined [5, 6],namely: • Causa materialis, i.e. material cause encountered in the reactions of chemical elements or the basic laws of the live of plants and animals. • Causa efficiens, i.e. efficient cause explaining who is the drive (who cause the motion) the process from the initial to the final sate. • Causa finalis, i.e. the final cause, which can be simply explained by the simple question “why? ” . Hence, we need answer to the motivation of change from the initial towards the final state, that is the reason of change. The causal relations obey some basic properties, among them: • The causal relation is asymmetric. That is, the asymmetry of the causal relation indicates its irreflexivity stressing the attention on the fact that no element is related to itself, i.e. a cannot be cause of a. Therefore, the cause cannot match the effect caused by it. • Concerning the time relations it follows that the cause and effect does not happen simultaneously, that is there is always a time-shift between them. This refers to the reality that there are no phenomena developing with infinite speed and sources of energies with infinite powers. Thomas Aquinas refers to the causality and the causal relations as proofs for the existence of God [5] and formulated tree basic axioms: A1:Something causes something else. A2:Nothing causes itself, that the property of irreflexivity holds mandatory. A3: If there is no first cause (causa efficiens) then there is no intermediate (transition state) and there is no causa finalis. Now, we refer to Leibniz’s philosophical interpretation due to his strong influence, and to a greater extent, his original contributions, to calculus. Following Leibniz, the causality is formulated by its principle of sufficient reason required to be obeyed by every complete scientific system built-up a basic axiom: Nothing happens without sufficient reason. Leibniz’s standpoint is that logic, mathematics, and metaphysics created on a basis of axioms are complete. Actually, the principle of sufficient reason expresses a universal completeness of causes [5, 6]. Further, we address the standpoint of Newton, the father of calculus and modern experimental physics that the causality and the consequences of the causal relations (In terms of forces and displacements) formulated in Principia are: N1: Causal change takes place in the background of a noncausal inertial movement (quoted by [7]). That the motion, which is the background of the inertial motion when the cause (force) application ceases, is due to an initially applied cause (force) to the body. N2:The change of position is causeless. This standpoint is mainly related to the motion of the planets. N3: Cause (force) and effect (acceleration) are simultaneous. Hence, Newton does not consider time-shift between the cause (force) and consequence (effect), in the context of the subject of this chapter no memories in the transport phenomena exist.

CONSTITUTIVE FRACTIONAL MODELING

41

1.1.2. Some remarks in the light of the modern knowledge. Here we like to stress the attention on some principal relationships in a dynamic system, such as the diffusion and viscoelasticity involving memory formalisms and modeled by the tools of fractional calculus. Following the analysis of Mittelstaedt and Weingartner [5], all laws of nature are causal relationships if the following conditions are satisfied: • a) A causal relation is defined in different ways for dynamic and statistical laws. Hence, it cannot be postulated as one single concept of causality since some causality features may appear in laws differently, and in some cases would not be satisfied. • b) In all applied cases the causal relation satisfies the chronological condition (i.e. the time shift between cause and effect), that is no closed time curves exist and always the cause precedes the effect. The main outcome of this condition is that only a dynamical law describing the time evolution of a certain physical system is a causal relation. As mentioned by Mittelstaedt and Weingartner [5] (page 219) under physical realistic situations, the causality of the realistic solutions and the chronological conditions are equivalent. In the light of the Newtonian mechanics, for instance, (see [5]page 235) every effect {xk (t1 ) , x˙ k (t1 )} at time t1 has exactly one cause {xk (t0 ) , x˙ k (t0 )}. In addition, since in classical mechanics the existence of absolute ( and universal) time is taken for granted no doubts about the objectivities of the causal relations exist. To complete this point, we have to say some words about the statistical laws (relationships) obtained with natural data. First of all, such relationships are incomplete since they do not relate data relevant to individual processes. Moreover, the incompleteness of the statistical relationships increases with an increase in the complexity of the described systems. For a more detailed analysis of this problem we refer to the book of Mittelstaedt and Weingartner [5]. 1.1.3. Mathematical aspects of causality. Following the above mentioned philosophical concepts, now we may address mathematical models of physical processes, mainly diffusion models, dispersion relations and constitutive rheological equations. Let us consider the latter case when particles are moving from an initial state to a final stage and we focus on the interactions between them. The interactions (collisions) between them can be considered to take place in three consequent regimes (states) [6]: • Initial stage, when the incoming particles are moving towards one another but their interaction could be neglected since the distance between them are enough larger than their free paths. • Intermediate stage, when the particles are closer and the distances between them are lesser than the free parts so the particles interact. • Final stage, when the particles leave the contact domain and the distances between them allow considering them as non-interacting. Actually, the basic problem engaging the attention in this case is the causality condition. We will formulate several conditions which are related to this concept, among them [6]: • C1. Primitive causality: The effect cannot precede the cause. In such situations the cause and the effects should be correctly defined.

42

JORDAN HRISTOV

• C2. Relativistic causality: No signal can propagate with velocity greater than the speed of the light in the vacuum. It could be considered as a macroscopic causality condition. However, the primitive causality condition is more fundamental and general than the relativistic causality condition. The causality principle implies that some functions describing transients in dynamical problems should obey some properties: to vanish over a range of values of its arguments (see further about the causality of the relaxation functions used in the memory integrals). Let us consider a physical system with a time-dependent input (cause) x (t) and the corresponding output (effect) y (t) , and satisfying the following conditions: • Linearity. That is, it obeys the superposition principle in its simple version implying that the output is a linear functional of the input. ∞ (1)

y (t) =

g (t, τ )x (τ ) dτ −∞

where y (t) , g (t) and x (t) may represent distributions. • Time-translation invariance. The system is time-translation invariant if the input is shifted (forward or backward) by some time interval τ and x (t + τ ) corresponds toy (t + τ ). In this case the function g (t, τ ) should depend only on the difference of the arguments, that is g (t, τ ) = g (t − τ ) and the linear functional can be expressed as ∞ (2)

g (t − τ )x (τ ) dτ = g (t) ∗ x (t)

y (t) = −∞

The relation (2) is a convolution between the input (cause) x (t) and the output (effect) y (t) and the correlation function (named also memory or kernel) allows to model the time-shift, i.e. the output at time y (t) corresponds to an earlier moment of the input x (t − τ ) • 3) Primitive causality condition. The input cannot precede the output. Therefore, the input x (t) vanishes for t < T that means that the same is valid for y (t), meaning without loss of generality that T can be assumed as zero (the moment when the input is applied). As consequence, we obtain that g (τ ) = 0 if τ < 0 . Since the functions involved in the convolution (2) can be considered as distribution (and we see further that is in a greater extent relevant to the memory functions involved in the models discussed in this chapter. Note: Before proceeding further, please, bear in mind that causality of model and causality of function (see in the definitions of the fractional operators and the memory functions) are two distinct issues with different definitions. 1.2. Constitutive modelling. The philosophy of writing adequate constitutive equations applicable to the description of experimental results has been developed through years piece by piece, by a generalization of the concepts of mechanics to continua and other transport phenomena, by a permanent interchange of knowledge from theory and experiments and vice versa [9](chapter 4). For a viscoelastic

CONSTITUTIVE FRACTIONAL MODELING

43

fluid of macromolecules, for instance, the extra stress is related to the second moment of the end-to-end vector of a macromolecule, and for a turbulent flow, the turbulent stress is related to the second moment of the fluctuating velocity field [9] (chapter 4). The creation methodology in constructing constitutive equations of continuous media contains some basic principles, among them: material frameindifference, objectivity, and determinism [9]. 1.2.1. Frame-Indifference. The principle of material frame-indifference (MFI) is related to the notion of the frame that is different from that of a simple coordinates system. In an inertial reference frame, the classical laws of Newtonian mechanics are valid. More precisely, in a such frame, the body momentum is constant over a finite time interval, then this corresponds to a vanishing force during that interval [9]. From another viewpoint, the physical situation in this frame consists in observing that the center of mass of the body moves along a straight line with a uniform speed if there is no force exerted on the body [9]. This is the definition of an inertial frame. If a frame is related to an observer, then MFI postulates that the material properties do not depend on the choice of the observers, that is, “the response of a material is the same for all observers [8, 9]. The principle of MFI imposes constraints on the form of the constitutive equations. For a given material, for instance, MFI is valid if the stress tensorial functional is invariant for a change of frame [8, 9]. 1.2.2. Transformations and objectivity. As is well known, the physical properties of fluid should not be dependent on the frame of the observer. The consequence of having such a property is that measurements of fluid properties made in one reference frame will apply to all other reference frames that are in rigid motion relative to one another. It then follows to formulate equations describing such property variations through variables that are independent of the motion of the observer, that is, objective variables. For example, if there are two frames, defined in the space and time, and characterized byF (x, t) and, respectively.F  (x , t ). The two frames are in relative time-dependent motion such that the following mapping relationship takes place [9]

(3)

x (t ) = Q (t) (x − x0 ) + c (t) ,

t = t − t s

Here, Q (t) is proper orthogonal tensor with time t as a parameter, c(t) is translation vector, and ts is a constant time shift. The relation (3) is a Euclidean transformation because the structure of the Euclidean vector space remains unchanged. In particular, in this case, vector lengths do not change in the frame transformation [9]. The transformation (3) is representative of the extended Galilean group[9]. If c (t) is assumed linear in time, that is c(t) = c0 + V t we get a transformation of the Galilean group. Equation (3) implies that both the length between two points and the angle between two directions are independent of the rigid motion of the coordinate frame [9]. The transformations can be objective or non-objective, but only objective transformations, i.e. independent of the motion of the observer are allowed when the material frame indifference principle holds [9]. The principle of material objectivity (3) states that the stress tensor must be objective under a change of frame. This means, as a postulate, that the material

44

JORDAN HRISTOV

has no inertial properties at the micro-level (micro-scale) and is assumed as a continuum. Truesdell and Noll [8] postulated that the response of the material should be independent of the observer. 1.2.3. Principle of determinism. This principle states that the current stress state in the material is determined by the past history of the motion. The future state of stress cannot be predicted by the current state, and the material has no properties to predict its future behavior [8, 9]. From the first two principles commented above, the principle of determinism requires that the stress σ (x, t) be determined by the history of the deformation of the body ε (t) through the functional t (4)

Rσ (t − s)ε (s) ds,

σ (x, t) =

t > 0,

s>0

−∞

where ε (s) is the deformation function whose influence is restricted to all past times up to the present time t by the effect of the function of influence (memory function) Rσ (t). Here this relationship is presented in its simplest form since it is used many times in this chapter. The class of functionals satisfying these first two principles can be rather large and because of that, it will be restricted to include constitutive equations discussed in this chapter. For this purpose, the discussion will focus on simple materials and, in particular on linear viscoelastic solids where the functional (3) is widely applied as a fundamental template. 1.3. Restrictions on Constitutive Relationships. Both the exact functional form and material (model) coefficients associated with the various constitutive equations are subject to restrictions imposed by the various principles discussed earlier in this chapter it is noteworthy to mention briefly in the next subsection the reasons for such a constraint originating in an entropy inequality. 1.3.1. A Thermodynamic Constraint for Constitutive Relationships: Dissipation principle. From a thermodynamic point of view, the constitutive equation must satisfy the Clausius-Duhem inequality because it is well known that the evolving physical events are irreversible and irreversibility is measured by the production of process entropy [9]. The Clausius-Duhem inequality requires the rate of change of entropy to be greater than the heat received by the material volume V divided by the (absolute) temperature θ [9].    QV q·n d dV − dS (ρs)dV ≥ (5) dt θ θ V

V

∂V

where ρs is the entropy density, q is the heat flux, and QV is a volumetric source of energy resulting from chemical reactions, heat sources, etc. In the case of heat conduction, for instance, with the Fourier law q = −λT ∇θ this inequality can be presented as   1 λT Dθ Ds ≥ − [ΣD] − 2 (∇θ)2 (6) ρ ρCV Dt θ Dt θ Equation (5) can be transformed to (applying the Reynolds transport theorem, convergence theorem) eliminating the volumetric contribution QV by the internal

CONSTITUTIVE FRACTIONAL MODELING

45

energy equation [9]

  q Ds QV Ds 1 De   1 (7) ρ ≥ − div ⇒ρ ≥ − ρ D + 2 (q · ∇θ) Dt θ θ Dt θ Dt θ  where [ D] is the stress tensor. For a more comprehensive explanation of these thermodynamic principles in material rheology, the book of Astarita and Marrucci [11] is highly recommended. 1.3.2. Simple materials. According to Noll [10] in a simple material (solid or fluid) the current stress is functional of the history of the deformation gradient. Simply, this means, in any different cases such as heat conduction, the heat flux is proportional to the history of the temperature gradient (the same with diffusion, where the mass flux is proportional to the history of the concentration gradient). 1.4. Fading memory concept. The fading memory concept incorporates the idea of what happened in the past should be encountered, but only recent past, while the past for a large time concerning the moment of interest has negligible influence. This is a very fruitful concept in the description of dissipative phenomena and will be briefly presented next. 1.4.1. Fading memory concept: General approach. Following Coleman [12] the theories of the dynamic of continua, such as fluid flow, heat conduction, and the dissipative effects can be described in several ways. The oldest idea is to relate the viscous stress to the rate of strain as in the Navier-Stokes equation. Another approach is the use of internal state variables influencing the stress and resulting in equations concerning the strain. The concept of fading memory, in polymer physic, for instance, assumes that the entire past history of the strain influences the strain, in a way compatible with the general postulate of this concept. Coleman and Noll [13] have formulated a systematic procedure for interpretation of the explicitly defined restrictions on constitutive equations of dissipative materials which are compatible with the second law. Further, the idea was successfully implemented in the description of constitutive equations of the viscous-stress type [13–15] and in the models with internal variables [16]. In materials with fading memory , the main logic is that configurations experienced in the recent past have a stronger influence on the present values (of stress and free energy) than configurations experienced in the distant past. The fading memory principle implies that response functionals, precisely the functions of influence (in this chapter referred to as material response function, memory functions, or memory kernels) cannot be chosen arbitrarily because the dissipation principle must be satisfied (see section 1.3.1), namely: For every admissible thermodynamic process in a body (medium) the Clausius-Duhem inequality must hold, that is the production of entropy should be positive. For viscoelastic materials, discussed in the second part of this chapter, the fading memory concept, expressed as a functional relationship, is defined as a singleintegral law [17] ∞ (8)

[m (ε) , ε (t − τ )]τ dτ

σ (t) = M ε (t) + 0

giving the stress σ (t) at time t when the strain ε (tλ ) is known for all past times tλ ≤ t.

46

JORDAN HRISTOV

In a deforming body the energy balance takes the form [17] de (9) = S · F + rq dt where e is the internal energy, S is the stress (usually Piola-Kirchhoff stress), F is the deformation gradient, and rq is the heat supply. Then, accounting for the absolute temperature θ, the following constitutive equations for stress and internal energy, upon assumptions of small strains, as single-integral laws can be expressed [17]

(10)

S (t) =



∞ {σ [F (t) , θ (t)] , F (t − τ ) , θ (t − τ ) , τ }dτ

[F (t) , θ (t)] + 0

∞ (11)

e {[F (t) , θ (t)] , F (t − τ ) , θ (t − τ ) , τ }dτ

e (t) = E [F (t) , θ (t)] + 0

Then the entropy ηe (t) in a single-integral form [17] ∞ (12)

h [F (t) , θ (t) , F (t − τ ) , θ (t − τ ) , τ ]dτ

ηe (t) = H [F (t) , θ (t)] + 0

should obey the entropy growth condition dηe /dt ≥ rq /θ . Similar constitutive equation of single-integral type have also been proposed for heat conduction [15] The fading memory idea can be implemented in various ways and we will use it systematically through the following part of this chapter. A more simplified form of the fading memory concept, relevant to the diffusion process of energy (heat) and mass can be presented as an implementation of the Boltzmann superposition principle [18] (see section 5.2) ∞ (13) σ (x, t) = εE0 + G (t − τ )ε (x, τ ) dτ 0

where G(t) is a material function (known as relaxation stress modulus), playing role of “weighting” (memory) function, and E0 is the initial Young’s modulus of the material. It is clear that all constitutive equations mentioned above have similar forms involving convolutions with influence functions which have to fade for t → ∞ , but influence the present status with the past history of recent times. Remark 1: It is important to mention that all constitutive equations based on the fading memory principle satisfy the conditions of the objectivity principle and MFI [12]. This is important since further in this chapter the fading memory equations are presented in terms of fractional derivatives. Hereafter, in this chapter, we will consider transport processes with dissipations in simple materials (media) where the flux is related to gradient though hereditary integral. The most important feature of these hereditary integrals is that they use as influence (memory) functions used in fractional calculus as singular and nonsingular kernels. This allows demonstration constructions of fractional constitutive equations in two areas: diffusion and linear viscoelasticity.

CONSTITUTIVE FRACTIONAL MODELING

47

2. Fractional Calculus Here, we will avoid large explanations of fractional differential operators and their properties which can be found in many textbooks on fractional calculus. The main task of this section is to present fractional differential operators using relaxation (memory) kernels that coincide with the basic functions approximating material viscoelastic responses. 2.1. Main functions in fractional calculus. Here we present briefly some functions forming the basis of the fractional calculus and used in the development of this chapter. 2.1.1. Mittag-Leffler functions with one parameter. The Mittag-Leffler functions is defined as [19] Eμ (z) =

(14)

∞  k=0

zk Γ (μk + 1)

Eμ (z) is entire function of order ρ = 1/μ and type 1. If μ ∈ C, then ρ = 1/Re μ . The Mittag-Leffler function is a generalization of the exponential functions and from the definition (14) it follows that ∞ 

(15)

E1 (z) =

(16)

  E2 z 2 = coth z,

k=0

 (17)

(18)

E1/2

±z 1/2



zk = ez Γ (k + 1)   E2 −z 2 = cos z

  = ez erf c ∓z 1/2 ,

2 erf c (z) = 1 − erf (z) = 1 − π

z

z∈C

e−u du, 2

z∈C

0

For 0 ≤ μ ≤ 1, the function Eμ (z = −x) is completely monotonic on the axis x ≥ 0, so that dn Eμ (−x) ≥ 0, x ≥ 0, 0 ≤ μ ≤ 1 dxn 2.1.2. Mittag-Leffler with two parameter. The Mittag-Leffler function with two parameters is defined as [19] (19)

(20)

(−1)n

[Eμ,β (z) =

∞  k=0

zk , Γ (μk + β)

μ, β > 0

Moreover, from the definition (20) it follows (21)

(22)

Eμ,1 (z) = Eμ (z) Eμ,β (z) =

1 + zEμ,μ+β (z) Γ (β)

48

JORDAN HRISTOV

 (23)

d dz

m

" β−1 # z Eμ,α (z μ ) = z β−m−1 Eμ,β−m (z μ )

2.1.3. Wright function. The special function of Wright function is related to Mittag-Leffler function and defined as [20]  ∞  −λ dσ zn 1 = eσ+zσ (24) Wλ,γ (z) = n!Γ (λn + γ) 2πi σγ n=0 Ha

For λ ≥ 0, Wλ,γ (z) is entire function of order 1/(1 + λ) of exponential type. The Laplace transforms is   (25) L [Wλ,γ (z)] = s−1 Eλ,γ s−1 2.1.4. Mainardi function. The Wright function (24) for λ = −ν, γ = 1 − ν and 0 < ν < 1 is named Mainardi’s function [21] (26)

M (z; ν) = W−ν,1−ν (−z) =

1 W−ν,0 (−z) , νz

0 0], and F (s) = L [f (t)] are Laplace transforms.

CONSTITUTIVE FRACTIONAL MODELING

49

2.2.2. Riemann-Liouville fractional derivative. The Riemann-Liouville timefractional derivative is defined as  t f (z) dm 1 α m m−α f (t) = dz, 0 D f (t) =0 D 0 I Γ(m − α) dtm 0 (t − z)α+1−m (32) m − 1 < α < m, m ∈ N From this definition it follows that D0 = I 0 = I, that is Dα I α = I for α ≥ 0 . Moreover, the fractional integrals and derivatives of the power-law function and a constant are (33)

(34)

0D

−α β

t =

0D

Γ(β + 1) α+β t , Γ(β + 1 + α)

α β

t =

0D

Γ(β + 1) β−α t , Γ(β + 1 + α)

−α

C=

0D

α > 0,

α

C tα , Γ(1 + α)

α = 1, 2, . . .

t−α , Γ(1 − α) β > −1, t > 0

C=C

In addition, the Laplace transform of the Riemann-Liouville fractional derivative is  m  m−1  d m L f (t); s = s F (s) − sm−k−1 f (m) (0+) = dtm k=0 (35) m  = sm F (s) − sk−1 Dtα−k f (0+) k=1

2.2.3. Caputo time-fractional derivative. The Caputo derivative of a casual function f (t) is defined [19] as (36) C

Dtα f (t) =0 I m−α

dm 1 −(m−α) (m) f (t) =0 Dt f = tm Γ(m − α)

 0

t

f (m) (z) dz, (t − z)α+1−m m−1 a ) defined as ⎡ ⎤ x m 1 f (t) ˆ sα+n f (x) = d ⎣ ⎦ m−1 0

0

where α0 , α1 , . . . , αm is an increasing sequence of real numbers such that 0 < αk − 1−(αk −αk−1 ) 1+αk−1 D0 f. αk−1 ≤ 1 , k = 1, . . . m , and D(αk ) f = J0 2.4.2. Fractional Taylor expansion through Riemann-Liouville derivatives. A Taylor series expansion was developed by Trujillo et al.[62] as (81) with cm

n 

cm (m+1)α−1 + Rn (x, x0 ) , (x − x0 ) Γ [(m + 1) α] m=0   1−α ˆ mα Ds (x) x+ = Γ (α) (1 − x0 ) 0 we have

Tf

(x) =

0 0 allows modeling a set of completely monotone (CM) decaying functions. The Kohlrausch function can be presented as a Laplace transformation for a known stable distribution [84]. Andersen et al.[85] demonstrated that (125) is completely monotone and the conditions

(126)

(−1)

n nd f dtn

≥ 0,

n = 1, 2, · · ·

are obeyed. The common approach to the identification of the three parameters A, α and β is the double logarithmic transform resulting in linear relations such that [85]

(127)

    y (t) exp −αtβ ⇔ A = y (0) ⇔ ln − ln = α + βt y (0)

The starting point for the exponential sum approximations of KWW is Pollard’s work [84] with help of the following Laplace transform [85] (128)

  exp −tβ =

∞ φ (β, p) exp (−pt) dp,

0 < β ≤ 1,

t≥0

0

1 φ (β, t) = π (129)

∞

∞ exp (−pu)ϕ (u) du,

0

φ (β, p) = 1, 0

φ (β, pˆβ ) = maxp φ (β, p) = Kβ " β # " # ϕ (u) = exp −u cos (πβ) sin uβ sin (πβ) The function φ(β, t) is a stable distribution in the sense of Levy [89]. Then, for fixed β there exists a unique pβ such that φ (β, p) is strictly monotonically decreasing when p > pβ ; this is a property following from the unimodality of this function. Skipping detailed expressions (see for details the references sources mentioned above) and using the fact that φ (β, p) is a non-negative function, it is possible to

CONSTITUTIVE FRACTIONAL MODELING

65

express (128) as (130)

N    exp −tβ = γ0 +

pi+1

φ (β, p) exp (−pt) dp + γ∞

i=1 p i

and pi+1 − pi = [p∗∗ (β) − p∗ (β)] N −1 , i = 1, 2, . . . , N, . With a stipulated ε > 0, p∗ = (pi + pi+1 )/2 , i = 1, 2, . . . , N, it is possible to present (130) in a compact form as [85] (131)

N    a∗i exp [−p∗ (i, t)] exp −tβ ≈ i=1

4.5. Some briefs on exponential sum approximations. The three examples of widely applied functions modeling monotonic decays in physical experiments demonstrate that they can be successfully approximated by sums of exponentials. Certainly, such approximations depend on the desired accuracy but in general, they allow numerical procedures evaluating convolution integrals with these functions as memory kernels to be carried out successfully. Moreover, as we will see further in this chapter that there are relationships between the classical Caputo derivative α (with a kernel t− , 0 < α < 1), the ABC derivative [56] and the Caputo-Fabrizio operator [43, 44] through the exponential sums approximations of the kernels. The examples given in the remaining part of this chapter stress the attention on this fact and demonstrate how this could be done. 5. Fractional models build-up: a systematic approach Now we will draw some principal lines of development of diffusion models with memories and particularly models involving fractional derivatives 5.1. Fractional diffusion models through Taylor’s series expansions. From the lessons learned we may suggest the following approaches: 5.1.1. Flux damping by application of the fractional Taylor series expansion (first order approximation). Namely ∂ α j (x, t) ∂tα in fact, this is a fractional version of the Maxwell-Cattaneo approach [90]. It is noteworthy to mention that this is the simplest way to achieve model fractionalization (excluding the formal replacement of the derivatives by fractional ones, which in many cases results in unphysical models). However, there are no restrictions to applying high order approximation of the flux as (132)

(133)

j (x, t) = j0 + λj

j (x, t) = j0 +

K  k=1

λkj

∂ αk j (x, t) ∂tαk

5.1.2. Both flux and gradient damping by application of the fractional Taylor series expansions (first order approximations). This approach is a fractional version of the Dual-Phase Lag (DPL) concept [91, 92] and we will term it Fractional DualPhase Lag (FDPL) concept [90]. The time-fractional version of the constitutive equation relating the heat flux and the temperature gradient is

66

(134)

JORDAN HRISTOV

j (x, t) = j0 + λj

  ∂ α j (x, t) ∂ β ∂C (x, t) = −k 1 + λ T ∂tα ∂tβ ∂x

Obviously, the transport coefficients are λj = λC and there is no mandatory condition α = β, but following the causality principle we should have α = β. Precisely, the values of the fractional orders in both sides of the constitutive equation (134), in general, should be different since the relaxation responses of the medium are different, taking into account that λj = λC . It is attractive to use the case α = β, but it could be considered only as a mathematical exercise since the causality principle requires the relaxation times (implicitly represented by α and β ) to be different: the retardation time of the flux (output, response) should be greater than the relaxation time of the gradient (output, cause). A similar equation, termed Jeffrey’s type heat conduction model, was discussed and solved by [94] and analyzed in [93]. In its derivation, it is assumed that fractional derivatives are of Riemann-Liouville type [94] (in the original notations) (135)

(1 + τq Dtα ) q (x, t) = −k (1 + τT Dtα ∇T (x, t))

It was used as a basic construction and analyzed in two cases: diffusion regime for τq < τT , i.e. for τq /τ T < 1 (the causality condition is obeyed), and for a propagating regime when τq > τT ,i.e. for τq /τ T > 1 (the causality condition is not obeyed). For both cases the memory kernel is expressed in terms of the Mittag-Leffler function: the kernel becomes singular in the diffusion regime when α = 1 and regular in the propagating regime [93]. 5.1.3. Thermal gradient damping in a viscoelastic manner by applying timefractional derivatives. The idea for damping of the heat flux by high order time derivatives comes from the viscoelastic models of non-Newtonian general secondgrade fluids where the relationship between the shear stress and strain is presented as [95] (136)

τshear (t) = μεstrain (t) + λe Dtγ [εstrain (t)] ,

τ (t) = με (t) + E

dε (t) dt

Here μ is a fluid dynamic viscosity, and λe is the first normal stress modulus (analog to the heat conduction relaxation time in the expansions by the Taylor series commented above). In terms of fluid velocity, the 1-D equation of motion is

(137)

ρ

∂2u ∂2u ∂u = μ 2 + α1 Dtγ 2 ∂t ∂y ∂y

Alternatively, for γ = 1 in (136) the result is an integer-order equation (non-local) of the unidirectional flow of a second grade fluid [95]

(138)

∂2u ∂3u ∂u = ν 2 + β1 ∂t ∂y ∂t∂y 2

Therefore, using an analogy between the fluid velocity field and the temperature (concentration) field distribution in the body (medium) it is possible to formulate

CONSTITUTIVE FRACTIONAL MODELING

67

a constitutive relationship between the flux (of heat or mass) and temperature (concentration) gradient as (in terms of heat conduction model) [90]   ∂ β ∂T (x, t) ∂T (x, t) = −k 1 + λeT β (139) ∂x ∂t ∂x (x, t) = − ∂q(x,t) we get [90] ∂x   ∂T (x, t) ∂ 2 T (x, t) ∂ β ∂ 2 T (x, t) = −k − λeT β , β 0, etc., that is the derivatives alternate the sign [149]. Moreover, G (t) should be finite and integrable. Now, we stress again (see section 4.2 at the beginning) the attention that the weakly singular kernels can have discrete relaxation spectra [150] in terms of Dirichlet series [151]

(402)

G (ξ) =

∞  i=0

gi exp (−μi ξ),

μi < μi+1 ,

μi → ∞,

i→∞

CONSTITUTIVE FRACTIONAL MODELING

115

The example of Renardy [152] (see also [150]), related to polymer rheology, shows that a discrete relaxation spectrum with an accumulation point at zero behaves as a power-law for short times, namely G (ξ) =

(403)

∞ 

exp (−iγ ξ) → t− γ , 1

t → 0,

γ>1

i=0

and the singularity of G (ξ) follows from the inequality [150] ∞ (404)

∞      exp −y γξ dy < exp −iγξ < i=0

1

∞

  exp −y γξ dy

0

Therefore, the viscoelastic relaxation functions can be presented (approximated) by discrete relaxation spectra as Prony series φP (t) with N φ terms [153, 154] with rate constants βi , namely φ

(405)

φP (t) = φ∞ +

N 

φ

−βi t

φi e

= φ∞ +

i=1

N 

− τt

φi e

i

,

βi =

i=1

1 ≥0 τi

or through normalized weights (amplitudes or normalized relaxation moduli) λi [155] as φ

(406)

N    φP (t) λ (t) = =1+ λi e−βi t − 1 , φ∞ i=1

λi =

φi φ∞

In (405) and (406) the parameters φ∞ and φi are equilibrium (at large times) and relaxation moduli (stiffness), respectively, constrained according to [155] φ

(407)

φ∞ +

N 

φi = 1

1

The first derivative of the Prony series is Nφ

(408)

dφ d  = φi e−βi t dt dt i=1

It is easy to prove that for t = 0 the first derivative of the Prony series is finite, a fact irrespective of the number of terms used in the approximation. That is, this a bounded distribution for t → 0+ . The discrete relaxation spectrum [156, 157] corresponds to the generalized Maxwell-Wiechert model : with N φ spring-dashpot elements in parallel [158] and a parallel spring controlling the equilibrium behavior (see Figure 4). The relaxation time τi of a particular element element is a ratio of its viscosity and elastic modulus, that is τi = φi /λi [159]. The parameter estimation is important and the first step was done in the seminal work of Prony [66], and there exist a variety of algorithms, such as : log − log plots [160, 161], least squares method [162], nonlinear optimization methods [163] ( see the comments in [22–24]). The number N φ of terms (405) is determined by the adequate accuracy of the data fitting and commonly fourth-order Prony series

116

JORDAN HRISTOV

was found to fit adequately the stress-relaxation data in case of non-linear and viscoelastic behavior [164, 165]. According to Drozdov [166] the short-term relaxation tests can be approximated by 2-4 terms of truncated Prony series while for long-term relaxation tests 10-15 terms are needed. A common step in the approximation of such sums is to select preliminary stipulated decay rates βi , in a logarithmic scale [167] because the corresponding fractional parameters αi can be easily calculated. It is noteworthy to mention, in the context of the recent discussions about the appearance of the Caputo-Fabrizio derivative, that presentations of the stress relaxation spectra as a series of exponentials are not new: a very comprehensive explanation can be found in the book of Astarita and Marrucci [11] published in 1974. The new moment, toward the enrichment of fractional calculus applications, is the incorporation of these spectra into Caputo-Fabrizio operators [22–24] as it is explained next (section 7.3 ). 7.3. Implementation of the Caputo-Fabrizio operator in viscoelastic rheology. 7.3.1. Stress modulus approximation. Thus, applying the Prony approximation of the relaxation curve and substituting (405) in the convolution integral the following approximation is obtained t (409)

σ=

  t − s dε ds Ei exp − τi ds

0

Since σ (t) is assumed as a finite sum of elements, then it is possible to invert the summation and the integral [168] that leads to the expression [22] ⎤ ⎡ t t   N N  (t−s) dε (t−s) dε − − ds = ds⎦ Ei e τi Ei ⎣ e τi (410) σ (t) = ds ds i=0 i=0 0

0

Thus, the memory effect from the convolution integral can be straightforwardly incorporated in each term of the Prony series [22–24]. The approximation (409) can be compactly expressed as [22, 168]

(411)

σ (t) =

N 

t Ei ηi (t),

− (t−s) τ

ηi (t) =

i=0

e

i

dε ds ds

0

Hence, we may represent σ (t) as a finite sum of N + 1 terms σi (t) associated to each element of the generalized Maxwell model, that is, at any time t we have t (412)

σi (t) = Ei εi (t) ,

εi (t) =

− (t−s) τ

e

i

dε ds ds

0

is a product of the spring modulus Ei and its current strain εi (t) at a given time t (expressed as a hereditary integral with an exponential kernel), as in the convolution integral appearing in the Caputo-Fabrizio operator.

CONSTITUTIVE FRACTIONAL MODELING

117

7.3.2. Creep compliance approximation. By analogy of the stress relaxation expression, for the creep compliance, we have [22, 23] JN (t − s) = (413)

 /   .  N t−s βi ji exp − ji exp − (t¯ − s¯) , = λi 1 − βi i−0 i−0

N 

¯ i = λi = 1 − βi λ t0 βi

Then, the convolution integral becomes t (414)

dσ ⇒ J (t − s) dt

0

t  N 0

ji

i−0

t (415)

.

ci (t) = 0

/ N  dσ (¯ s) βi ¯ d¯ s= (t − s¯) ji ci (t) exp − 1 − βi d¯ s i−0 

  dσ (¯ s) βi d¯ s exp − (t¯ − s¯) 1 − βi d¯ s

Thus, for the strain, we have [22, 23]    N N  1 (416) ε (t) = ci (t) = ji (1 − βi ) ji (1 − βi ) Dtβ σ (t) 1 − β i−0 i−0 as well as (417)

ε (t) = J∞ −

N 

ji (1 − βi ) ci (t) = J∞ −

i=1

N 

ji (1 − βi )Dtβi σ (s)

i=1

where ⎧ ⎨

(418)

1 Dtβi σ (s) = ⎩ (1 − βi )

t 0

⎫ ⎬ βi dσ (s) exp − (t − s) ⎭ ds 1 − βi 

defines a Caputo-Fabrizio operators with respect to σ (t) . 7.3.3. Fractional parameters and the relaxation spectrum relationships. Taking into account the relationship between the fractional parameter α and the scaled relaxation time of a single exponential kernel α = 1/(1 − τ /t0 ) we may see that for each element of the generalized Maxwell model the spectrum of relaxation times is defined as 1 (419) αi = 1 − τi /t0 Now, introducing εi (t) from (412) we go towards the Caputo-Fabrizio operator as t (420)

εi (t) = (1 − αi )

1 (t ¯−¯ − 1−α s) dε α

e

i

d¯ s

d¯ s = (1 − αi ) Dtαi ε (t)

0

Thus, the constitutive equation (412) can be presented as (421)

σ (t) =

N  i=0

Ei (1 − αi ) Dtαi ε (t)

118

JORDAN HRISTOV

After the development of this result, we turn on the determination of the spectrum of fractional orders αi = f (τi /t0 ) (419) through data fitting of a limited number of numerical values of relaxation times τi . The main problem now is the determination of the characteristic time t0 . This is the macroscopic time scale of the experiment since it lasts a limited time, that is t0 equals the elapsed time te of the experiments. The analysis in [22–24] reveals that there are two groups of relaxation times: relaxation times less than the elapsed time τi < te ⇒ τi /te ≤ 1 and relaxation time greater than the elapsed time τi > te ⇒ τi /te ≥ 1 . This forms two groups of fractional orders αi , namely: i) For τi /te ≤ 1 we have αi ∈ [0.5 − 1) related to fast relaxations, and ii) for τi /te ≥ 1 we have αi ∈ (0 − 0.5] modeling slow relaxations. 7.4. Discontinuous relaxation spectrum by Mittag-Leffler function. The kernel of the constitutive model may be expressed as [169]

φM L (t) = 1 +

n 

 φi

i=1

(422) φi > 0,

N 

     k t i Eki − −1 , τi

φi < 1,

ki > 0,

t≥0

i=0

where Ek is k -order Mittag-Leffler function defined as ∞  zj (423) Ek (z) = Γ (jkj + 1) j=0 The models involving the Mittag-Leffler relaxation function are commonly termed fractional models [169] (in the sense of the classical fractional calculus). In this case, N additional constitutive parameters kj are introduced concerning the discrete kernel approximation by the Prony series. It is clear that when ki → 1 the kernel (422) reduces to the Prony series approximation (405) since Ek=1 (z) = ez , this is only an apparent similarity. 7.4.1. The background of the Mittag-Leffler function as a memory kernel. Here we stress the attention on some cases appearing for a long time in the literature and forming an invisible background supporting the application of the Mittag-Leffler function as a memory kernel. For this purpose we consider two simple cases: With the Maxwell model. For the sake of simplicity, let the model selected consist of two fractional elements arranged in series [170]. Then, applying the Scott-Blair constitutive model we get dα1 G1 dα1 −α2 σ (t) = G ε (t) , α1 > α2 1 G2 dtα1 −α2 dtα1 where the fractional operators are of Caputo type with singular kernels. From here, the creep function of the Maxwell model becomes (424)

(425)

σ (t) +

J (t) =

1 tα1 tα2 1 + G1 Γ (1 + α1 ) G2 Γ (1 + α2 )

That is, we have a sum (a truncated series) of power-laws as a small example of a polynomial fractional series.

CONSTITUTIVE FRACTIONAL MODELING

119

With the Kelvin-Voigt model . Now, the elements are in parallel and the constitutive equation is [170]

(426)

σ (t) = G1

dα1 dα2 ε (t) + G ε (t) . 2 dtα1 dtα2

As a result, the creep-strain function is expressed through the Mittag-Leffler function as tα1 J (t) = Eα −α ,1+α1 G1 1 2

(427)



tα1 −α2 − G1 /G2



From the asymptotic expansions of the Mittag-Leffler functions we may find the short-time and long-time time approximations of J (t) [170], namely

(428)

Jshort−time () ≈

1 tα1 , G1 Γ (1 + α1 )

t  (G1 /G2 ) (α1 −α2 )

(429)

Jlong−time (t) ≈

1 tα2 , G2 Γ (1 + α2 )

t  (G1 /G2 ) (α1 −α2 )

1

1

Hence, the Mittag-Leffler function can model not only these two limiting cases but also sections of the response function between them. 7.4.2. The background of the Mittag-Leffler function as a memory kernel: Polynomial operators. The main idea of the so-called polynomial fractional operators which in general can be presented as

Pc (t) =

(430)

N 

an Dtαn [f (t)]

0

Dtαn

where [f (t)] are fractional derivatives with any type of memory kernels relevant to the modeled relaxation process. The idea of this type of polynomial fractional operator (PFO) begins from the work of Koeller [33] but was provoked by the studies of Bagley and Torvik on the application of fractional calculus in viscoelastic models, as mentioned above. With the power-law linear viscoelastic behavior, the creep compliance is taken as a Riesz distribution (431)

Rn (t) =

tn , Γ (n + 1)

Rn (t) = 0,

t ∈ (−∞, 0)

while for Rn (t) ≡ t−n we have (432)

Rn (t) =

t−n , t ∈ (0, ∞) Γ (1 − n)

used as a memory kernel in the singular fractional derivatives. Integrating the Riesz distribution, to obtain the Stieltjes integral representation of the fractional integral, we may integrate the Riemann-Liouville Integral (432) by

120

JORDAN HRISTOV

parts. Then, for α ≥ 0 we get

(433)

D

−α

t x (t) =

α

tα (t − τ ) dx (τ ) + x (0) Γ (1 + α) Γ (1 + α)

0

In terms of Riesz distribution, we may present this result as

(434)

D

−α

t R(−α) (t − τ ) dx (τ ) + x (0) R(−α) (t) =

x (t) = 0

  = R(−α) ∗ dx (t) + x (0) R(−α) (t) ,

α ∈ [0, 1]   where R(−α) ∗ dx (t) is a Stieltjes convolution. Similarly, applying the Leibniz rule to the definition of the Riemann-Liouville fractional derivative [33] t α

D x (t) = 0

(t − τ )−α t−α dx (τ ) + x (0) = Γ (1 − α) Γ (1 − α)

t

t R(α) (t − τ ) dx (τ ) + x (0)

(435) 0

t =

R(α) (t) = 0

  R(α) ∗ dx (t) + x (0) R(α) (t)

α ∈ [0, 1]

0

The polynomial operators of Koeller are directly applicable to the viscoelastic constitutive equations presented simply as (436)

P (D) σ(t) = Q (D) ε(t)

where P (D) and Q (D) are polynomial operators defined as

(437)

P (D) =

N 

pn Dαn ,

Q (D) =

n=0

N 

qn Dβn

n=0

with fractional (memory) parameters (orders) αn and βn When αn and βn are positive integers, then (436) is the standard differential operator constitutive law. Further, when σ (t) and ε (t) are specified, then (436) is a fractional differential equation without jump initial conditions. Hence, the solutions of (436) for any action as input shear stress (or input shear strain) require knowledge of the entire history of the shear stress (shear strain). The general formulation (436) can be developed as a linear hereditary law if we consider the properties of the Stieltjes convolution and the Riesz distribution, namely (438)

N  n=0

pn R(αn ) ∗ dσ =

N 

qn R(βn ) ∗ dε

n=0

Then, we may define fractional polynomials B (t) and D (t) as

CONSTITUTIVE FRACTIONAL MODELING

(439)

B (t) =

N  n=0

(440)

D (t) =

N 

pn R(αn ) (t) =

pn

t−αn Γ (1 − αn )

qn

t−βn Γ (1 − αn )

n=0

N 

N 

qn R(βn ) (t) =

n=0

n=0

121

and the constitutive law (436) can be presented in two forms B ∗ dσ = D ∗ dε

(441) t

t B (t − τ )dσ (τ ) =

(442) −∞

D (t − τ )dε (τ ) −∞

If B −1 and D−1 are defined as Stieltjes inverse of B and D, then applying the associative property of the Stieltjes convolution we have σ = G ∗ dε,

(443)

ε = J ∗ dσ

where G = B −1 ∗ D and J = D−1 ∗ B are the relaxation modulus and the creep compliance, respectively. Koeller example of a polynomial operator The Koeller example developed in [33] selects only one memory parameter β (which is a violation of the causality principle, since the input and output should have different time delays). Anyway, the following expansion was considered (three component Kelvin-Voigt model) (444)

    p0 + p1 Dβ + p2 D2β σ = q0 + q1 Dβ + q2 D2β ε

which possesses symmetry, that is no preference is given to the stress or strain [33]. The solution of (444) [33] by Laplace transforms yields expressions in terms of Mittag-Leffler functions

(445)



1 1 + J (t) = E0 E1

 (446) G (t) = E0 − E0 R1

         β β t t 1 1 − Eβ − + 1 − Eβ − τ1 E2 τ2

         β β t t 1 − Eβ − − E0 R2 1 − Eβ − τ1 τ2

where E0 , E1 and E2 are the moduli of the springs, and τ1 and τ2 are relaxation times (see also Figure 4 for visual understanding of the problem), and (447)

Eβ =

∞  n=0

n

(−1)

xn , Γ (1 + βn)

t > 0,

0 0, λ, λ0 ∈ Λ. Let u0 (x) be the solutions to the system (4.7), for λ = λ0 , from Ξ that satisfies (4.10). Let also Ψ (u0 , λ) and gradu Ψ (u0 , λ) be continuous in a neighbourhood of (u0 , λ0 ), and (4.13)

detgradu Ψ (u0 , λ) |(u0 , λ0 ) = 0.

Then there exist positive constants α and β, such that for any λ : λ − λ0 ≤ β, the combination of solutions to problems (4.3),(4.4) gives the only close solution to ux1 (x1 ) × ux2 (x2 ), coinciding with it at λ = λ0 .

Figure 1. Typical dependence of the discrepancy norm upon the iterations. To solve the one-dimensional boundary value problems, we use the method of reduction of a one-dimensional nonlinear boundary value problem to the equivalent initial value problem. The method implies that one seeks for a vector u0 for which an implicitly specified finite-dimensional map (4.12) vanishes. To calculate the

224

VASILII A. GROMOV

components of the map we integrate equations (4.7) using the Runge-Kutta method of the 4th order. To find the zeros of the functions (4.12), we use the iterative formula of Newton method:   (l+1) (l) (l) = u0 − J −1 Ψ u0 , λ ; u0       (4.14) sk u01 , ..., u0j + Δu0j , ..., u0N − sk u01 , ..., u0j , ..., u0N ∂sk ∂sk . , 0 ≈ J= ∂u0j ∂uj Δu0j The J matrix is a numerical analogue to the Fr´echet matrix gradu Ψxi (u0 , λ). In order to obtain an initial approximation close to the solution, we utilise the continuation method. For the computations, the accuracy of the Newton method εn was equal to 10−4 , in comparison with dimensionless quantities of order 1; the accuracy of the Runge-Kutta, to εr = ε10n = 10−5 . If the Newton method did not converge in mn = 5 iterations, then the iterative process was interrupted. Similar restrictions were imposed on the number of steps in the Runge-Kutta method: if the number of steps in the method exceeded mr = 100mn , where mn is the maximum number of steps of the Newton method, then the Runge-Kutta method as well as the Newton method, was interrupted. The following theorem [24] allows one to fix the critical points of the solution: Theorem 3. Let the vector functions describing the linearised system (4.7) be continuous in the neighborhood (u0 , λ0 ) together with the linearised boundary conditions. Then point (ux0 i , λ0 ) is a critical point of the problem (4.3), (4.4), if (4.15)

gradu Ψxi (u, λ) |(u0 , λ0 ) = 0,

for any system (4.4) and (4.6). The Newton method, applied to solve problem (4.7), suggests that one calculates numerical analogue to the Fr´echet matrix J (4.14). This makes it possible to utilise, in actual practice, the condition detJ = 0 instead of (4.15), where J are the numerical analogues to the Fr´echet matrices calculated at the last two iterations of the IGKM. To determine the type of critical point localised by condition (4.15), we consider an augmented Fr´echet matrix Jλ , with an appended column of derivatives of boundary conditions with respect to the continuation parameter q and a set of square matrices Jk obtained from Jλ by removing the k -th column. Then, at the bifurcation point (where the new solution branches out), the following condition is satisfied: rankJ = rankJk < N, ∀k = 1, N ; at the limit point the following condition is satisfied: rankJ < N, rankJk = N, ∀k = 1, N . These conditions can be written in a form convenient to implement algorithmically (for bifurcation and limit points, respectively): (4.16)

det J = det Jk = 0, ∀k = 1, N , det J = 0, det Jk = 0.

The singularity order of the matrix J at a critical point is an important characteristic to analyse bifurcations: the value is defined as the difference between the dimension of the matrix and its rank corankJ = N − rankJ. To calculate the corank, the singular values of the matrix considered are calculated, i.e., the eigenvalues of the product of the matrix J by the transposed J T .

CATASTROPHES OF CYLINDRICAL SHELL

225

Furthermore, when the algorithm switches, at a bifurcation point, from one branch of the solution to another, the variability of the solution functions can change, either along one of the coordinate directions, or along both directions simultaneously. This is reflected in the singularity of either one of the Fr´echet matrices, or in the singularity of both matrices simultaneously:  0; (1) det J x1 = 0, det J x2 = (2) det J x1 = 0, det J x2 = 0; (3) det J x1 = 0, det J x2 = 0, for the Fr´echet matrices corresponding to (4.3), (4.4). The type of the critical point, in this case, is also determined by the conditions (4.16). The above relations allow us to solve the problem of fixing and analysing the bifurcation state, i.e., establishing the structure of a bifurcation set for a nonlinear boundary value problem for partial differential equations. For nonlinear elliptic equations of the Karman type, we reveal, as a result of a large-scale simulation, that for any specific equations of this type, there exist a basic bifurcation structure, corresponding to a certain combination of parameters from . For other values of the parameters, the basic structure is destroyed in order to give rise to limit curves and isolated branches of the solution. The solutions corresponding to the limit points of these curves are similar to the solutions corresponding to the critical points of the post-bifurcation branches for the basic case. As a result, we identify several characteristic regions in the parameter space, each of these regions corresponds to a solution of a certain type. The set of descriptions of the boundaries of the indicated regions makes it possible to establish subsets of the bifurcation set that have the desired properties. Overcoming solution limit points. When the continuation algorithm approaches the limit critical point while moving along the parameter, we apply the technique of changing the continuation parameter: the fastest growing component of the vector is determined u0 : i∗ = argmax |uoi (λj+1 ) − uoi (λj )|, which is further used as the i=1,..,N

continuation parameter, the previous parameter is then included in the number of unknowns and is determined by the IGKM. The continuation parameter is changed when its change at two consecutive steps becomes too small: |Δλ| < Δλmin . Branching equations and solution branches at the bifurcation point. For a bifurcation point, the algorithm allows us to construct small solutions of the system (solutions belonging to a small neighbourhood of the bifurcation point), branching off at the considered point, namely find, for each branch a point belonging to it, which can be used as a starting position to move (within the framework of the parameter continuation method) along this branch. To find small solutions, the branching equations for one-dimensional nonlinear boundary value problems of the last iterations of the IGKM are numerically constructed. Similarly, to one-dimensional nonlinear boundary value problems solution routine, the residuals of the branching equations are represented as implicitly defined functions of their arguments; the corresponding equations are finitedimensional, and their dimension is determined by the order of singularity at the corresponding bifurcation point. The rank of the Fr´echet matrix r of the system (4.7) at the bifurcation point (u∗ , λ∗ ) is strictly less than its order r < N ; we will assume that the bifurcation

226

VASILII A. GROMOV

Figure 2. Typical behavior of the branches of the solution in the vicinity of the bifurcation point. point is isolated, i.e. there is a neighbourhood of it in which there are no other singular points of the system. Then, small solutions of the system admit an expansion: 0 0 0 00 0 0 0 0 0 0J11 J12 0 0u 0 0 0 < + 0Δ1 0 = 0. 0 0 0