Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure

WAVELET AND WAVE ANALYSIS AS APPLIED TO MATERIALS WITH MICRO OR NANOSTRUCTURE SERIES ON ADVANCES IN MATHEMATICS FOR A...

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WAVELET AND WAVE ANALYSIS AS APPLIED TO MATERIALS WITH MICRO OR NANOSTRUCTURE

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Series on Advances in Mathematics for Applied Sciences -Vol. 74

WAVELET AND WAVE ANALYSIS AS APPLIED TO MATERIALS WITH MICRO OR NANOSTRUCTURE

Carlo Cattani University of Salerno, Italy

Jeremiah Rushchitsky Timoshenko Institute of Mechanics, Kiev, Ukraine

World Scientific N E W J E R S E Y • L O N D O N • S I N G A P O R E • HEIJING • S H A N G H A I • HONG KONG • T A I P E I • C H E N N A I

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Cattani, Carlo, 1954– Wavelet and wave analysis as applied to materials with micro or nanostructure / by Carlo Cattani & Jeremiah Rushchitsky. p. cm. -- (Series on advances in mathematics for applied sciences ; v. 74) Includes bibliographical references and index. ISBN-13 978-981-270-784-0 (hardcover : alk. paper) ISBN-10 981-270-784-0 (hardcover : alk. paper) 1. Wavelets (Mathematics) 2. Nanostructures--Mathematics. I. Rushchitskii, IA. IA (IArema IAroslavorich) II. Title. QC20.7.W38C38 2007 620.1'18015152433--dc22 2007016752

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1

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Preface

In this book three different areas of modern science are joined together and studied: two quite recent topics: − micro- and nano-mechanics of composite materials, and − the wavelet analysis as applied to physical problems (both started to be deeply investigated at the end of the 1980s), and a third new one − propagation of a new type of dispersive waves (solitary ones) in composite materials. Dispersive waves, i.e. waves with phase velocity depending on the phase, were actively explored in the last ten years within the classical theory of elastic waves propagation. They are nonlinear waves and show some unpredictable and outstanding nonlinear effects in the propagation. We found, as a common link of these three topics, the ability of wavelet analysis to perfectly describe some of the many nonlinear physical phenomena which arise in problems on composite materials, with micro- and nano-structure (and propagation therein). We begin by presenting each of the three parts in the simple and easily understood form, because the new area of the application of wavelet analysis, nano-materials and waves in nano-composites, can be very interesting for specialists working with wavelets, as well as the techniques from wavelet analysis can be useful for specialists working in mechanics of materials. The three parts concern with quite new fields of science and research and for this reason many conceptual facts are still in progress and under assessment. Thus we figure out a wide circle of readers and form the structure of the book in the understanding that some readers, maybe unprepared on all subjects, can nevertheless obtain a sufficient preliminary information. The specialists in each topic will find, instead, a full description of open problems and many suggestions for further new investigations. Our goal is to prepare readers to a v

vi

Wavelet and Wave Analysis

clear understanding of the procedures and techniques, which unify three different scientific areas into a new one and to inspire some readers to explore the more advanced aspects of this new promising scientific direction. The book consists of five chapters (Chapters 2 – 6). Chapter 2 has 11 sections and is thought by the authors as primary and introductory information about wavelet analysis. Two facts are shown as having a key role in wavelet theory: the link (similarities and distinctions) of wavelet analysis with Fourier analysis and the notion of frames, as in classical functional analysis. Chapter 3 presents the updated established understanding of materials with internal structure. Eight sections contain the primary information about materials from the notion of the internal structure of macro-, meso-, micro-, and nanolevels till the useful computer modeling data on real micro- and nanocomposites. We paid special attention to composite materials as the main representatives of materials having an internal structure. The basic models used in the modern studies of composite materials are shown and commented on. Chapter 4 is devoted to the analysis of waves in materials. It includes five sections, which cover fully the topic under consideration. The main purpose of this chapter is to give a sufficiently clear and simple representation of linear and nonlinear processes of elastic wave propagation in materials. Chapter 5 deals with the solitary waves in structured elastic materials. From this chapter we start with applications and form the skeleton of a new field in the analysis of material where: i) the object is defined as solitary waves, ii) the model is taken from wave analysis and mechanics of materials, and iii) the tools of analysis combine techniques from wave analysis and elastic wavelet analysis. Chapter 6 is the closest chapter to applications and computer simulation. It consists of six sections representing different stages of computer modeling. The regularities of propagation of solitary waves with different initial profiles are studied, by using the elastic wavelet technique. The main attempt is to show the ability of elastic wavelet technique to describe adequately the evolution of the wave initial profile as the fundamental phenomenon accompanying the solitary wave propagation. The mentioned technique is described in the most clear and open form, which ensures using all procedures in the new more advanced problems. C. Cattani J. Rushchitsky

Contents

Preface..................................................................................................................v 1. Introduction ......................................................................................................1 2. Wavelet Analysis............................................................................................13 2.1 Wavelet and Wavelet Analysis. Preliminary Notion ..................................13 2.1.1 The space L2 ( ℝ) ................................................................................15 2.1.2 The spaces Lp ( ℝ ) ( p ≥ 1) ..................................................................16 2.1.3 The Hardy spaces H p ( ℝ ) ( p ≥ 1) .....................................................17 2.1.4 The sketch scheme of wavelet analysis ..............................................18 2.2 Rademacher, Walsh and Haar Functions....................................................26 2.2.1 System of Rademacher functions ......................................................26 2.2.2 System of Walsh functions ................................................................28 2.2.3 System of Haar functions ..................................................................32 2.3 Integral Fourier Transform. Heisenberg Uncertainty Principle .................44 2.4 Window Transform. Resolution .................................................................52 2.4.1 Examples of window functions ........................................................54 2.4.2 Properties of the window Fourier transform......................................57 2.4.3 Discretization and discrete window Fourier transform......................59 2.5 Bases. Orthogonal Bases. Biorthogonal Bases ...........................................63 2.6 Frames. Conditional and Unconditional Bases..........................................71 2.6.1 Wojtaszczyk’s definition of unconditional basis (1997)................... 81 2.6.2 Meyer’s definition of unconditional basis (1997)............................. 82 2.6.3 Donoho’s definition of unconditional basis (1993) .......................... 82 2.6.4 Definition of conditional basis...........................................................82

vii

viii


2.7 Multiresolution Analysis ............................................................................83 2.8 Decomposition of the Space L2 ( ℝ ) ...........................................................95 2.9 Discrete Wavelet Transform. Analysis and Synthesis ............................109 2.9.1 Analysis: transition from the fine scale to the coarse scale .............111 2.9.2 Synthesis: transition from the coarse scale to the fine scale ............113 2.10 Wavelet Families ......................................................................................116 2.10.1 Haar wavelet ..................................................................................117 2.10.2 Strömberg wavelet.........................................................................120 2.10.3 Gabor wavelet................................................................................123 2.10.4 Daubechies-Jaffard-Journé wavelet...............................................123 2.10.5 Gabor-Malvar wavelet ...................................................................124 2.10.6 Daubechies wavelet .......................................................................125 2.10.7 Grossmann-Morlet wavelet ...........................................................126 2.10.8 Mexican hat wavelet......................................................................127 2.10.9 Coifman wavelet – coiflet..............................................................128 2.10.10 Malvar-Meyer-Coifman wavelet .................................................130 2.10.11 Shannon wavelet or sinc-wavelet ................................................130 2.10.12 Cohen-Daubechies-Feauveau wavelet .........................................131 2.10.13 Geronimo-Hardin-Massopust wavelet .........................................132 2.10.14 Battle-Lemarié wavelet................................................................133 2.11 Integral Wavelet Transform......................................................................137 2.11.1 Definition of the wavelet transform...............................................137 2.11.2 Fourier transform of the wavelet ...................................................138 2.11.3 The property of resolution .............................................................139 2.11.4 Complex-value wavelets and their properties................................141 2.11.5 The main properties of wavelet transform .....................................141 2.11.6 Discretization of the wavelet transform.........................................142 2.11.7 Orthogonal wavelets ......................................................................143 2.11.8 Dyadic wavelets and dyadic wavelet transform.............................144 2.11.9 Equation of the function (signal) energy balance ..........................144 3. Materials with Micro- or Nanostructure .......................................................147 3.1 Macro-, Meso-, Micro-, and Nanomechanics ............................................147 3.2 Main Physical Properties of Materials.......................................................156 3.3 Thermodynamical Theory of Material Continua ......................................160 3.4 Composite Materials..................................................................................168 3.5 Classical Model of Macroscopic (Effective) Moduli.................................174

Contents

ix

3.6 Other Microstructural Models ...................................................................181 3.6.1 Bolotin model of energy continualization .......................................182 3.6.2 Achenbach-Hermann model of effective stiffness...........................183 3.6.3 Models of effective stiffness of high orders ....................................184 3.6.4 Asymptotic models of high orders...................................................185 3.6.5 Drumheller-Bedford lattice microstructural models ........................186 3.6.6 Mindlin microstructural theory........................................................187 3.6.7 Eringen microstructural model. Eringen-Maugin model .................188 3.6.8 Pobedrya microstructural theory......................................................190 3.7 Structural Model of Elastic Mixtures ........................................................191 3.7.1 Viscoelastic mixtures.......................................................................210 3.7.2 Piezoelastic mixtures .......................................................................213 3.8 Computer Modelling Data on Micro- and Nanocomposites......................216 4. Waves in Materials .......................................................................................229 4.1 Waves Around the World ..........................................................................229 4.2 Analysis of Waves in Linearly Deformed Elastic Materials .....................232 4.2.1 Volume and shear elastic waves in the classical approach ................232 4.2.2 Plane elastic harmonic waves in the classical approach ....................237 4.2.3 Cylindrical elastic waves in the classical approach ...........................241 4.2.4 Volume and shear elastic waves in the nonclassical approach ............................................................................................244 4.2.5 Plane elastic harmonic waves in the nonclassical approach .............247 4.3 Analysis of Waves in Nonlinearly Deformed Elastic Materials ................253 4.3.1 Basic notions of the nonlinear theory of elasticity. Strains...............253 4.3.2 Forces and stresses...........................................................................260 4.3.3 Balance equations .............................................................................262 4.3.4 Nonlinear elastic isotropic materials. Elastic Potentials ..................267 4.4 Nonlinear Wave Equations........................................................................276 4.4.1 Nonlinear wave equations for plane waves. Methods of solving..............................................................................................276 4.4.1.1 Method of successive approximations ..................................281 4.4.1.2 Method of slowly varying amplitudes ..................................283 4.4.2 Nonlinear wave equations for cylindrical waves ..............................285 4.5 Comparison of Murnaghan and Signorini Approaches .............................308 4.5.1 Comparison of some results for plane waves ...................................308

x


4.5.2 Comparison of cylindrical and plane wave in the Murnaghan model ................................................................................................322 5. Simple and Solitary Waves in Materials .....................................................337 5.1 Simple (Riemann) Waves..........................................................................337 5.1.1 Simple waves in nonlinear acoustics ................................................337 5.1.2 Simple waves in fluids......................................................................340 5.1.3 Simple waves in the general theory of waves...................................344 5.1.4 Simple waves in mechanics of electromagnetic continua.................345 5.2 Solitary Elastic Waves in Composite Materials ........................................346 5.2.1 Simple solitary waves in materials ...................................................346 5.2.2 Chebyshev-Hermite functions ..........................................................347 5.2.3 Whittaker functions ..........................................................................349 5.2.4 Mathieu functions .............................................................................352 5.2.5 Interaction of simple waves. Self-generation...................................353 5.2.6 The solitary wave analysis................................................................359 5.3 New Hierarchy of Elastic Waves in Materials...........................................373 5.3.1 Classical harmonic waves (periodic, nondispersive) ........................374 5.3.2 Classical arbitrary elastic waves (D’Alembert waves) .....................374 5.3.3 Classical harmonic elastic waves (periodic, dispersive) ...................375 5.3.4 Nonperiodic elastic solitary waves (with the phase velocity depending on the phase) ...................................................................377 5.3.5 Simple elastic waves (with the phase velocity depending on the amplitude)..............................................................................379 6. Solitary Waves and Elastic Wavelets .........................................................381 6.1 Elastic Wavelets ........................................................................................381 6.2 The Link between the Trough Length and the Characteristic Length ......391 6.3 Initial Profiles as Chebyshev-Hermite and Whittaker Functions..............396 6.4 Some Features of the Elastic Wavelets......................................................410 6.5 Solitary Waves in Mechanical Experiments..............................................422 6.6 Ability of Wavelets in Detecting the Profile Features ..............................435 Bibliography.....................................................................................................443 Index ................................................................................................................. 455

Chapter 1

Introduction

The present scientific knowledge is very often successful in the so-called border areas when a few different and sometimes distant scientific theories and methods interlace and gather round new phenomena thus forming a new configuration of scientific models and approaches. Later on many of such configurations might transform into independent scientific branches. The first steps in the early stage formation of new directions are always the same, trying to collect physical objects, models, analytical methods, computer modeling, etc…, from different areas of science. In this book, the object of computer modeling are the new structured materials (micro- and nanocomposite materials), the object of analytical study is the new solitary waves in such materials. Both are studied with methods of wave analysis, but the main results obtained by a computer simulation are based on wavelet analysis. The first goal of this book is to give a full set of models and techniques, which is the essential tool for modeling wave propagation, in modern structured materials, by using wavelet and wave analysis. The second goal is to discuss each one of the main components of modeling: − the analysis of solitary waves in structured materials, − the analysis of mechanical behaviour of structured materials of micro- and nanolevel of the structure, − the wavelet analysis as a new tool in the modeling of materials.

1

2


Each component has enough resources for being developed as an individual self-consistent scientific branch. However, by developing wavelet analysis techniques as applied to composite nanomaterials (both in engineering, and non-engineering applications) and, in perspective, to nanomaterials and nano-formations of biological nature seems to be the most interesting task. The main goal of this book is to unify three new scientific directions – the analysis of new materials, – the analysis of solitary waves in materials, – the wavelet analysis and to develop the investigations of waves in new structured materials, which need the knowledge of the models and methods coming from all three directions. The first part of the book contains three chapters, on wavelet analysis (Chapter 2), theory of materials with internal structure (Chapter 3), and theory of waves in materials (Chapter 4). These chapters are selfsufficient, they don’t require any background on the specific subject, and they give all necessary information to understand the remaining two chapters, where all three directions are united for modeling the wave evolution in structured materials. The wavelet analysis can be roughly understood as a part of mathematics like mathematical analysis, functional analysis, harmonic analysis, or fractal analysis. It is widely accepted that wavelet analysis is one of the best achievements of mathematics in the twentieth century. However, wavelet analysis was developed and recognised mainly for its applications in applied mathematics, physics, engineering sciences, etc. Nowadays it is considered as an independent scientific branch placed on the border among mathematics, scientific computer modeling and applied theory of signals and images. Despite their hard mathematical tools, wavelets spread in many fundamental sciences other than mathematics, as medicine, biology, geophysics, physics, mechanics, economy etc. A special development of the wavelet analysis is achieved in the theory of information, coding theory, and the theory of signals and images.

3

Introduction

Wavelet analysis is the topic of Chapter 2 which consists of 11 sections. There exists a close link between Fourier (harmonic) analysis and wavelet analysis. Therefore very often wavelets are approached through Fourier analysis. Let us remember that the Fourier series of function f ( x) ∈ L2 ( −π , π )

by

the

{en ( x)} ∈ L2 ( −π , π ) , n ∈ ℤ

orthonormal

k

1 2π

functions

ek ≡ ∑ f , ek ek ,

k∈ℤ

f ,g =

of

is defined as the series

∑c where

system

k∈ℤ

+π

∫

f ( x) g ( x)dx is the inner product. So, the given

−π

function f ( x) which take an infinite number of values over some interval is replaced by a series which is characterized by given

functions

ek ( x) ( k ∈ ℤ ) and infinite number of coefficients ck = f , ek . For a fixed k we can say that the continuous function is discretized, since it is replaced by a discrete set of coefficients of Fourier series. We will see, in discrete wavelet analysis, that the following representation of function f (t ) ∞

f (t ) = ∑ c jo ,kϕ jo ,k (t ) + ∑ ∑ d j ,kψ j ,k (t ) , k∈ℤ

j = jo k∈ℤ

holds, where ϕ k ,m (t ) = 2k / 2 ϕ ( 2k t − m ) , ψ k ,m (t ) = 2k / 2ψ ( 2k t − m ) are a family of functions based on the mother ϕ (t ) and father ψ (t ) wavelet functions. Each term of the family is obtained by a simple operation of dyadic scaling and translation. They might form sets of different kind (including orthogonal ones and frames). The sets of wavelet transform coefficients

{c } j ,k

j , k ∈ℤ

, {d j , k }

j , k∈ℤ

are treated as system of numbers,

which in the definite way is set in accordance to the signal f (t ) (this signal is discretized) and which is called the discrete wavelet transform.

4


The first sum in wavelet representation gives the coarse approximation and the second sum gives the details of the signal in any necessary scale. Such, more detailed, representation into two separate parts of the signal seems to be impossible with Fourier integral transform and this explains why the wavelet transform is sometimes called the mathematical microscope. Wavelet is a special function, which vanishes everywhere except a “small” interval, where its profile looks like a piece of some wave-like function. There exist many different families of wavelets, each one reflects some special features, as the Mexican or French hats, or its discoverer like Daubechies or Shannon wavelets. Let us now consider the integral Fourier transform. If the complexvalued function f ( x) is given over the real axis and it is absolutely +∞

integrable over this axis, that is,

∫ | f ( x) | dx < ∞ ,

then the Fourier

−∞

integral transform of function f ( x) is the integral +∞

F f (ω ) =

∫

f ( x)e − iω x dx .

−∞

Wavelet analysis is based on a similar integral transform. The basic idea, in wavelet analysis, is the decomposition of the function (signal) into two families of functions constructed by means of the scaling 2 function (mother wavelet) ϕ ( x) ∈ L ( ℝ ) . The first family is built by

translation and scaling of the scaling function. The second one is produced by translation and scaling also, of the wavelet function (father 2 wavelet) ψ ( x) ∈ L ( ℝ ) . The wavelet family (set) is

{

s ψ ( s ( x − xɶ) )

}

s , xɶ∈ℝ 2

.

The wavelet transform of a function f ( x) ∈ L2 ( ℝ ) is conventionally defined as

Introduction

5

+∞

W f ( s, xɶ) =

∫

f ( x) s ψ ( s ( x − xɶ) ) dx .

−∞

Among the many similarities of Fourier and wavelet representations, the common representation property of isometry or the property of measure invariance is very important. This property is often commented as the existence of adequateness in the description of the signal and its Fourier or wavelet transforms. So, two facts are shown as having the deciding role in the explanation of the wavelet theory, the link of wavelet analysis with the classical Fourier analysis and the introduction of the new notion of frames into the wavelet analysis, which can be understood as some development of classical functional analysis. Therefore Chapter 2, especially the first 6 sections, should be considered as having a theoretical character also, which seems to be quite natural owing to the mathematical origins of wavelet analysis. In section 7, the multiresolution analysis (MRA) is described as the discrete variant of wavelet analysis. In section 10 the most known wavelet families are shown and section 11 is devoted to the integral wavelet transforms. Chapter 3 describe the theory of materials with internal structure. Eight sections contain the primary information about materials from the notion of the internal structure of macro-, meso-, micro-, and nanolevels till the useful computer modeling data on real micro- and nanocomposites. A special attention is drawn to composite materials as the main representatives of materials having an internal structure. The basic models (one structural model of the first order and different structural models of the second order) utilized in the modern studies of composite materials are shown and commented on. The new classification of internal structure of materials by the attribute of admissible size of particles of materials including nanomechanics is shown schematically below and commented on in chapter 3.

6


The last section seems important for the numerical modeling. It contains different data on granular, fibrous, and layered micro- and nanocomposites, which are the necessary component of modeling the waves in materials. Chapter 4 complete the representation of the introductory information on the three basic directions and is devoted to the wave analysis in materials. It includes five sections, which cover this topic in full. The main purpose here is to give the sufficiently transparent and simple representation of linear and nonlinear (free) elastic wave propagation in materials. First we give two basic theoretical models for linear elastic waves: the plane and cylindrical waves. Then the modern nonlinear theory of elastic materials is explained in the most comprehensible form. Finally the nonlinear wave equations and basic approaches to their solution (including two most often utilized procedures: the method of successive approximations and the method of slowly varying amplitudes – van der Pol method) are discussed. The object of united efforts of three mentioned above theories is described in Chapter 5: the solitary waves in structured elastic materials. The solitary wave is defined as the wave with initial profile equal to zero practically everywhere besides some finite interval where one or more humps form the wave profile. So, the shape of solitary wave is very

Introduction

7

similar to the wavelet shape. A typical solitary wave is the classical bellshaped signal. We study, in particular, two types of solitary elastic waves in materials: the waves with initial profiles in the form of one special function as Chebyshev-Hermite, Mathieu, Whittaker functions and the waves with profiles, which are observed in some experiments on wave propagation in materials. A new hierarchy of elastic waves in materials is considered: 1. Periodic and nonperiodic waves with the constant phase velocity: classical non-dispersive waves. 2. Periodic waves with the phase velocity depending on frequency: classical dispersive waves. 3. Nonperiodic waves with the phase velocity depending on the actual phase: new type of waves. 4. Nonperiodic waves with the phase velocity depending on the actual amplitude: classical Riemann waves. The third type of waves is studied in Chapters 5 and 6. From Chapter 6 we start immediately with applications and form the skeleton of a new field in the analysis of material where – the object is defined as solitary waves, – the model is taken from wave analysis and mechanics of materials, – the tools of analysis combine techniques from wave analysis and elastic wavelet analysis. Chapter 6 consists of six sections representing different stages of computer modeling and starts with Kaiser physical (optical and acoustic) wavelets, Newland harmonic wavelets, and elastic wavelets proposed by the authors. Optical wavelets own this name because they satisfy the linear wave equations of optics in the simplified form of Maxwell electromagnetic equations. The acoustic wavelets were proposed as those wavelets satisfying the linear wave equations in acoustics. These wavelets have a very special

8


shape, which correspond to the so-called chirp-signals, very common in acoustic antennas. These signals have the explicit analytical representation xα sin (1 x β ) and, in accordance with the name, they represent those sounds similar to the sounds emitted by dolphins or bats with very characteristic oscillations. Harmonic wavelets were suggested by Newland. The Newland harmonic wavelets can be referred to physical family of wavelets for many reasons, but mainly because they are especially proposed for the analysis of physical problems on oscillations. Although they have a slow decay in space variable they are very well localized in frequency domain. The base for this kind of physical wavelets family is formed by the Shannon wavelets:

sin

ψ T ,k ( x) =

π T

π T

( 2x − k )

.

(2x − k )

The mother harmonic wavelet is defined as follows

ϕ ( x) =

e i 2π x , i 2π x j

ei 2π (2 x − k ) − 1 ϕ j ,k ( x) = ϕ (2 x − k ) = i 2π (2 j x − k ) j

The corresponding family of wavelets has the form j

w( 2 x − k ) =

e

i 4π 2 j x − k

(

) − ei 2π ( 2

j

x−k

)

i 2π ( 2 j x − k )

with Fourier transform in the form of trapezoidal box

.

Introduction

9

j − iω k / 2  , ω ∈  2π 2 j , 4π 2 j  , (1 2π ) (1 2 ) e W (ω ) =  0, ω ∉  2π 2 j , 4π 2 j  .  j

We propose a new family of wavelets: elastic wavelets on the base of: 1. Kaiser’s idea of constructing the physical wavelets as (admissible) solutions of wave equations; 2. the theory of solitary waves (with profiles of the ChebyshevHermite functions) propagation in elastic dispersive media; 3. the theory and practice of using the Mexican hat wavelet family, the mother and father wavelets (and their Fourier transforms), which are analytically represented as the Chebyshev-Hermite functions of different indexes. The elastic wavelet is the solution of the wave equations for the linear elastic dispersive medium. In particular, we consider the two-phase elastic medium. The base system of plane wave equations describe the propagation of solitary waves, where the profiles of solitary waves are chosen as Chebyshev-Hermite functions. The Chebyshev-Hermite functions can be associated with the Mexican Hat (MH) wavelets. This is the main reason why the MH wavelets were appointed as the first candidate for the elastic wavelets. MH wavelet family is an example of continuous wavelets with infinite support (that is, it is defined over the whole real axis), whereas the most famous wavelet families are defined on a finite support (that is, a finite interval of the real axis). The MH wavelets form a frame, whereas many families of wavelets form an orthonormal basis. Further we study the regularities of propagation of solitary waves with different initial profiles. On this stage, the elastic wavelet technique is the basic tool for investigation. We show the ability of elastic wavelet technique to describe adequately the evolution of the wave initial profile as the fundamental phenomenon accompanying the solitary wave propagation.

10


As application of the elastic wavelets we describe in details the numerical modeling of solitary elastic waves with MH initial profile, the assumptions on weak nonlinearity of the material, the choice of the frame limit, the determination of resolution exactness and the choice of the limit value of scale order in the fine resolution. The propagation of solitary waves has revealed that there are at least two fundamental questions: – How can we establish correctly the relationship between the parameters of the wave (its trough as the most typical parameter) and of the material (the characteristic microstructural dimension as the most important parameter in any microstructural theory)? – At what distance from the starting point of motion or at what time from the beginning of motion are data on the change of the initial wave profile already incorrect? A solitary wave is defined by two parameters: profile and amplitude. Usually, the wave profile has the so-called trough (bottom), i.e., an interval of the abscissa axis within which the corresponding ordinates on the profile are considerably different from zero. The word “considerably” in the definition of trough implies that its length can be calculated differently. The trough of a solitary wave may be regarded as the length of a harmonic wave, i.e., it is possible to try to compare the trough with the characteristic structural dimension of medium (material) and to find out how, if at all, to account for the internal structure in the model of the medium where the wave propagates. For a solitary wave, it is obvious that a trough length that exceeds the characteristic structural dimension of material by an order of magnitude should also be considered a threshold value. This minimum trough length, corresponding, possibly, to the most intensive manifestation of profile evolution, is used in computer simulation of wave profile evolution. The maximum trough length, at which the evolution is still somewhat noticeable, can be found from computer simulation. The utilized elastic wavelet technique permits to describe many wave effects, considered in the last three sections of Chapter 6. For example, the effect of transition from one mode to two modes (breaking-up the

Introduction

11

primary wave into two modes) can be observed and in the process of evolution of the solitary wave three characteristic stages can be separated. On the first stage the profile moves in a finite time interval without significant distortions. The second stage consists in the separation (breaking-up) of the wave into two waves which propagate with different phase velocities. The profile distorts on the leading edge of an impulse and then the separation of the second impulse starts. The third final stage shows the separate propagation of two waves with distinguishing phase velocities and amplitudes. We tried to generalize the procedures of Chapter 6 in a such way, to be used also for studying other evolution problems arising in many different processes of applied science.

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Chapter 2

Wavelet Analysis

2.1 Wavelets and Wavelet Analysis. Preliminary Notions The wavelet analysis as a scientific discipline is placed on the border among Mathematics, Scientific Computer Modeling and Applied Theory of Signals and Images. Despite the complicated mathematical apparatus, wavelets found striking and wide applications in many fundamental sciences – biology, physics, mechanics, geophysics, mathematics, economy, medicine etc. A special development of the wavelet analysis is achieved in the theory of information, coding theory, and the theory of signals and images. It is widely accepted that the wavelet analysis, being brought to applied computer packages (for example,WaveLab in MatLab, LastWave in С for XII/Unix, Wavelet Explorer in Mathematica5, and others) represents one of the recent information technologies. The wavelet analysis is extremely novel from the point of view of history of mathematics, it arose in the 1980’s in France as a hybrid of abstract mathematics and applied science (from the point of view of mathematics) – the theory of geophysical signals. It was organized incredibly quickly into an independent scientific direction. It is now widely accepted that the wavelet analysis is one of the best achievements of mathematics in the twentieth century. Let us start by discussing the meaning of the word “wavelet”. This word has a special significance and should be associated beyond doubt with the word “wave”.

13

14


The notion of a wave underlies the scientific perception of outward things. Everyone sees waves, on water or on sand or somewhere else, and has their own criterion whether these are waves or not. Wave is defined as the space disturbance, observed at some welldefined place, which moves with a finite velocity to some other place. Observed in our life waves are, most often, conditionally periodic with alternating crests and troughs. The most common observation are the waves on a pond when, at calm weather, the slight breeze may cause the small waves on the water. Physicists know how to describe this phenomenon analytically. It is called the Helmholtz instability. From the point of view of mathematicians, the same phenomenon is a manifestation of violation of correctness by Hadamard of the mathematical problem stated by physicists. The simplest wave profile is associated with graph close to the sinusoid. As such it is not a very strong understanding of the wave is used for the basic term “wavelet”. The wavelet is assumed as the specially chosen function, which is equal to zero all over excluding the small interval, on which the graph of function is similar to the profile of some wave in the above mentioned everyday sense. Each wavelet is named after either the expression reflecting the feature of this wavelet (for example, Mexican hat) or the names of scholars proposed this wavelet (for example, Daubechies, Haar, Meyer wavelet, etc.). A simple definition of wavelet is based on the definition of a function ϕ ( x) called the mother wavelet , provided by the next properties: M1. ϕ ( x) is the function of real variable x . M2. ϕ ( x) is a localized function, that is different from zero in a small interval, as a rule, at neighbourhood of zero, and beyond this interval neither is equal to zero exactly, or is equal to zero practically (tends to zero sharply with increasing x ).

15

Wavelet Analysis

M3. Graph of the function in the neighbourhood of zero has the oscillatory character. M4. The first S moments of the function are equal to zero ∞

∫ ϕ ( x)dx =

∞

∞

−∞

−∞

∫

xϕ ( x)dx = ⋯ =

∫x

S −1

ϕ ( x)dx = 0 .

(2.1)

−∞

The functions considered conventionally in the wavelet analysis, belong to the Hilbert space L2 ( ℝ ) of Lebesgue

measure, square

integrable, and depending on one variable (possible, vector variable) functions f ( x) ∈ L2 ( ℝ ) . It should be noted that the condition to be measured by Lebesgue isn’t redundant. It allows the analysis of signals described by the defined on arbitrary sets of points functions. Just such signals are considered in the theory of signals and images.

2.1.1 The space L2 ( ℝ ) Note 1. Let us start with the strong definition of the space of functions L2 ( ℝ ) L1. Let us assume that these functions are the functions of real variable x ∈ ℝ , which can take the complex values f ( x) ∈ ℂ . L2. The functions f ( x) are measured by Lebesgue. L3. The scalar product of two functions f ( x), g ( x) ∈ L2 ( ℝ ) is defined by +∞

f ,g

2

L

=

∫

f ( x) g ( x) dx .

(2.2)

−∞

L4. The norm in L2 ( ℝ ) is f

+∞

2 2

L

=

∫

−∞

2

f ( x) dx < ∞ .

(2.3)

16


Note 2. The letter L, in the notation of the functional space, means the space of functions measured by the Lebesgue definition; the letter ℝ testifies the dependent variable, which takes all values over the real number axis; number 2 means that for each function from L2 ( ℝ ) the improper Lebesgue integral of squared module of function exists and is equal to a finite number. The last property of functions from L2 ( ℝ ) is a determining one and therefore this space is called the space of square integrable functions. Note 3. The same property means that the signals described by such functions have the finite energy. Note 4. The main distinction of the space L2 ( ℝ ) and any vector space consists in the infinite (in fact, countable) dimension of L2 ( ℝ ) ,that is, the space elements are not exhausted by linear combination of finite number of these elements. It must be noted moreover that L2 ( ℝ ) is the Hilbert space. For elements (functions) from L2 ( ℝ ) , the notions of the strong (on the average) f − f n

L2

→ 0 and weak

( fn , g )L

2

→ ( f , g ) L2 convergence

hold. An important aspect of this fact can be seen from an example. The functional sequence {sin nx}n∈ℕ converges weakly to zero and doesn’t converge on the average to any function.

2.1.2 The spaces Lp ( ℝ ) ( p ≥ 1) This space is introduced similarly to L2 ( ℝ ) by changing 2 on p.

17

Wavelet Analysis 1

Therefore the norm is defined as follows f

Lp

 +∞ p p =  ∫ ( f ( x)( dx  < ∞ .  −∞ 

To define the scalar product in Lp ( ℝ ) , it is necessary to consider, additionally, the space Lq ( ℝ )

( (1 p ) + (1 q ) = 1) .

Then the scalar product can be defined analogously to L2 ( ℝ ) +∞

f ,g

Lp

=

∫

f ( x) g ( x) dx .

(2.4)

−∞

It should be noticed that when we consider the Fourier series

and

p

wavelet series, then the spaces L [−π , π ) will be used as well. These

spaces p

form

a

q

L [−π , π ) ⊂ L [−π , π )

system

of

embedded spaces,

p > q . Whereas the spaces

that is,

Lp ( ℝ )

don’t

possess this property.

2.1.3 The Hardy spaces H p ( ℝ ) ( p ≥ 1) To define the space H p ( ℝ ) ( p ≥ 1) , let us take into consideration the open upper half-plane P , which will be defined as z = x + iy and y > 0 . The function f ( x + iy ) belongs to the Hardy space H p ( ℝ ) , if it

is holomorphic in the half-plane P and 1

 +∞ p p sup  ∫ f ( x + iy dx  < ∞ . y > 0  −∞ 

If this condition is satisfied, then the supremum taken over all y > 0 is also the limit when y → 0 . Further, f ( x + iy ) converges to the function, which is denoted as f ( x) when y → 0 . And this convergence should be understood by the norm in Lp ( ℝ ) .

18


Then the Hardy space H p ( ℝ ) can be considered as the closed subspace of the space Lp ( ℝ ) , and the notations are similar. Hardy spaces are a fundamental tool in the signal processing. There, the real signal f (t ) t ∈ ℝ with finite energy is associated with the analytical signal F (t ) ∈ ℂ, t ∈ ℝ , for which the real signal is the real part. +∞

The energy of the real signal

∫

2

f (t ) dt is written under the assumption

−∞

that the function belongs to L2 ( ℝ ) , which implies the analytical signal to belongs to the Hardy space H 2 ( ℝ ) .

2.1.4 The sketch scheme of wavelet analysis Let the function f (t ) be given. Often, in physics, a function it is better described, analyzed or studied, if it can be expressed in the form of expansion in terms of other functions

f (t ) = ∑ ak ψ k (t ) ,

(2.5)

k

where k is the index (integer) for the finite or infinite sum; ak ∈ ℝ are coefficients of the expansion; {ψ k (t )} ψ k (t ) ∈ ℝ is the set of functions. If the expansion is unique, then the set {ψ k (t )} is called the basis for the class of function, which can be expanded in this way. If the basis is orthogonal, that is

ψ k (t ),ψ m (t ) = ∫ψ k (t )ψ m (t )dt = 0

k≠m ,

(2.6)

then the coefficients can be evaluated using the scalar product ak = f (t ),ψ k (t ) = ∫ f (t )ψ k (t )dt .

(2.7)

19

Wavelet Analysis

If the basis is a non-orthogonal one, then one can try to introduce the biorthogonal basis, as well as, more general frames such as the Riesz basis and general frames, which will be discussed later. In particular, for trigonometric series, the orthogonal basis functions are sin kt , cos kt . For Taylor series, the basis is formed by the power functions, and this basis is not an orthogonal one. In the wavelet analysis we have for the representation of the function f (t ) ∞

f (t ) = ∑ c jo ,kϕ jo ,k (t ) + ∑ ∑ d j ,kψ j ,k (t ) , k∈ℤ

(2.8)

j = jo k∈ℤ

where the indexes j , k are integers, the functions ϕ j ,k (t ),ψ j ,k (t ) are the wavelets and can form bases of different kinds (including orthogonal ones and frames). Focusing on the analogy with the representation of a function by Fourier series, the set of coefficients of Fourier series

{ak }k∈ℤ

is the

system of numbers, which is, in a definite way, in accordance to the signal f (t ) (discretized signal) and which is called the discrete Fourier

transform. The sets of coefficients of wavelet transform (2.8) {c j , k }

j , k ∈ℤ

, {d j , k }

j , k∈ℤ

are treated as the system of numbers, which is, in a definite way, in accordance to the signal f (t ) (discretized signal) and which is called the discrete wavelet transform. Let the basis be orthonormal and form the set of wavelets on the father wavelet ψ ( x) as follows

ψ k ,m ( x) = 2k / 2ψ ( 2k x − m ) .

(2.9)

20


The function ψ ( x) ∈ L2 ( ℝ ) is called the orthogonal wavelet, when the family {ψ k , m ( x)}k , m∈ℤ forms the orthonormal basis in L2 ( ℝ ) and for all f ( x) ∈ L2 ( ℝ ) the representation (2.8) is valid. The series on the right of (2.8) are called wavelet series. The wavelet analysis is a new mathematical method (a tool) for studying first the signal and then, arbitrary enough, functions. For analysis or synthesis of the function, the wavelet analysis offers the expansion of function by special basis (frame), which is constructed on the base of the specially chosen wavelet. There exist many different wavelets, as well as bases (frames). The choice of the wavelet is dictated by the problem to which the wavelet analysis is applied. The idea of expansion of a function, using the given basis, underlies the classical theory of Fourier series and integrals. The wavelet analysis utilizes the bases very distinct from the Fourier analysis bases and gives therefore a row of unusual opportunities which permits us to say that the creation of the wavelet analysis is one of the most outstanding events in mathematics of twentieth century. Wavelet analysis seems to be easier when it is compared on many steps with Fourier analysis. It is very convenient both for understanding the nature of wavelet theory and the understanding of the procedures of wavelet analysis. Let us start therefore with a brief description of the Fourier analysis in L2 . Let us assume that we are working with functions f ( x) of real variable x ∈ ℝ , which can take the complex values, and these functions are measurable by Lebesgue and periodic with period 2π . Thus, the functions over the interval [ −π , π ] will be considered. The space L2 ( −π , π ) is the Hilbert space with the scalar (inner) product 1 f ,g = 2π

+π

∫

−π

f ( x) g ( x)dx .

21

Wavelet Analysis

Let the orthonormal basis in L2 ( −π , π ) be the set

{en ( x)} ∈ L2 ( −π , π ) , n ∈ ℤ . The orthonormality is understood as

ei , ek = δ ik , i, k ∈ ℤ . The Fourier series of the function f ( x) by the orthonormal system of functions

{en ( x)} ∈ L2 ( −π , π ) , n ∈ ℤ

∑c

k

k∈ℤ

is defined as a series of the form

ek ≡ ∑ f , ek ek .

(2.10)

k∈ℤ

So, the given function f ( x) , which take an infinite number of values on some interval, corresponds to a series of terms of known functions

ek ( x) k ∈ Z and coefficients ck = f , ek

and with an infinite number of

coefficients. Thus the continuous function is discretized, since it is taken into correspondence with the discrete set of coefficients of Fourier series. In this way by using the Fourier series, the decomposition of the function is realized in the infinite sum of orthogonal elements (basis functions). It should be noted also that the basis is formed by the operation of dilation (which is also called the integral dilation)

ek ( x) = eikx = e1 (kx)

k ∈ℤ .

Property of minimum This property marked out the Fourier series among other series. Let us denote the partial sum of Fourier series by Sn ( x) = ∑ ck ek ( x) and treat k 0 the finite subset Z 0ε ⊂ ℤ exists such that whichever will be the finite subset Z 0 ⊂ ℤ containing Z 0ε ⊂ Z 0 , the equality f −

∑c e

k k

≤ε

k∈Z 0

is valid. Let us remember a few facts associated with last theorem and Parseval formula. Fact 1. If, for certain orthonormal system, the Parseval formula is fulfilled, then the system is called the full one. The notion full is usually treated in the way that this system can’t be extended with saving the orthonormality. Fact 2. The Fourier series of function f ( x) ∈ L2 converges on average to

this function. Fact 3. On the left of Parseval formula, there is the norm of function (signal) in L2 whereas on the right – the norm of sequence in

24


l 2 . Therefore the Parseval formula sets the isometry between spaces L2 and l 2 , or between the signal and its discrete image. 2

Fact 4. It is known from definition f ( x) ∈ L2 that f

from the Parseval formula follows that

∑ (c )

< ∞ , therefore 2

1, c > 0) .

< ∞ is usually associated in wavelet analysis with

the concept of energy, so that only signals with finite energy are studied. The coefficients of the Fourier series give the full information on signal energy.

25

Wavelet Analysis

A classical example from physics. Let an electric circuit be given in a such way that the active unit resistance r = 1 and the voltage at clamps u (t ) is a periodic function u (t ) = u (t + 2π ) . The mean power P of the current i(t ) over one period can be found by the simple formula P =

i(t ) =

1 2π

+π

∫ u (t )i(t )dt .

The Ohm law

−π

u (t ) = u (t ) is valid, since the circuit is active. Then r P=

1 2π

+π

∫π [u(t )]

2

dt = u (t )

2 L2 [ − π ,π ]

.

−

If the voltage u (t ) is represented by a Fourier series with coefficients ck , then the Parseval formula gives P = ∑ ( ck ) . 2

k ∈ℤ

Thus, the mean power of current over one period is expressed through coefficients of the Fourier series for the current. Such a link of signals with their Fourier coefficients is often used and this energy terminology passed into the wavelet analysis. Let us note also that it is assumed that the point-wise convergence of Fourier series and the point-wise representation of the function by Fourier series don’t permit to represent well the best property of the Fourier series. There exist other representations and other kinds of convergence. Part of them is based on summation by Cesaro. A new partial sum is introduced by the formula sN ( x) =

∑ c e ( x) k

k

and then one more

|k | < N

function

σ N ( x) =

s0 ( x) + ... + sN ( x) . N +1

Let us remind the most commonly used criteria of Fourier series convergence.

26


∞ Dini-Lipschitz criterion: Let f ( x) ∈ C [−π , π ] and such that a

∫ 0

1 sup max f ( x − t ) − f ( x) dt < ∞ t τ ∈(0,t ) x∈[ −π ,π ]

(a > 0) ,

then the Fourier series of the function f ( x) converges uniformly to f ( x) , that is, lim f − S N f N →∞

L∞ [ − π ,π ]

= 0.

Here the function is assumed to have arbitrary oscillations. Dirichlet-Jordan criterion: Let f ( x) ∈V2π , that is, the function f ( x) is the 2π -periodic function of limited variation over [−π , π ] . Then the Fourier series of function f ( x) converges everywhere f ( x+ ) + f ( x− ) ∀x ∈ ℝ. N →∞ 2 Moreover, over all compact intervals [a, b] ⊂ [−π , π ] , that is, where the lim ( S N f ) ( x) =

function belongs to f ( x) ∈V2π ∩ C2π , the series converges uniformly.

2.2 Rademacher, Walsh and Haar Functions Other orthonormal systems, beyond trigonometric systems, include: Legendre polynomials, Chebyshev polynomials, Laguerre functions, Chebyshev-Hermite functions and many other classical functions of mathematical physics. Let us stop on three nonclassical orthonormal systems: Rademacher, Walsh, and Haar functions.

2.2.1 System of Rademacher functions It is defined as rad n ( x) = sign sin ( 2n +1π x ) , where

x ∈ [0;1], n ∈ N ∪ {0}

,

(2.13)

Wavelet Analysis

27

 +1 A > 0  sign A =  0 A = 0 .  −1 A < 0 

These functions are periodic with period 1 : rad n ( x) = rad n ( x + 1) . Let us construct the graphs of Rademacher functions rad n ( x) . For that, divide the interval [ 0;1] on 2n+1 equal parts and denote the arbitrary part  k k +1 ∆ k( n ) =  n +1 , n +1  , k = 0;1;...;2n +1 − 1 . 2  2

Then the Rademacher function can be represented in the form  +1 x ∈ ∆ (2nm) , m = 0;1;...;2n − 1,  k  rad n ( x) =  0 x = n +1 , k = 0;1;...;2n +1 − 1, (2.14) 2   −1 x ∈ ∆ (2nm)+1 , m = 0;1;...; 2n − 1. For convenience in the representation of the Rademacher, Walsh, and Наar functions the vertical axis was scaled (32 times). rad 0 ( x)

0 rad1 ( x)

rad 2 ( x)

½

1

28


rad 3 ( x)

Fig. 2.1 Plots of the first four Rademacher functions.

For the Rademacher functions the following properties hold. Property R1. The system of Rademacher functions is orthonormal over the interval [0;1] in the space L2 [0;1] . The orthonormality is proved by a graphical test. The analytical testing is not complicated, but testing on graphs is very convenient. Property R2. The system is not full. But when the unit will be added to the system, it will still be orthonormal. In addition, all Rademacher functions are odd and this restricts their application. Property R3. The Rademacher function rad n ( x) is closely linked with the binary expansion of a number x inside of the unit interval. The binary expansion of such number has the form of infinite fraction 0, a1a2 ...an ... (an ∈{0;1}) . It turns out that an = an ( x) =

1 [1 − rn ( x)] . 2

2.2.2 System of Walsh functions The Walsh function waln ( x) n ∈{0} ∪ N with arbitrary nonzero number is written as the product of Rademacher functions

29

Wavelet Analysis

waln ( x) = rad n1 ( x) rad n2 ( x)...rad ns ( x) . It is assumed that the function wal0 ( x) = 1 and

(2.15)

of zero index is defined separately

W1. The first number n in (2.15) is 1; such numeration is called the ordered (list) by Walsh one. W2. The number n is written as the binary number n = 2n1 + 2n2 + ... + 2ns (n1 > n2 > ... > ns ) .

(2.16)

This implies that when n = 2 p , then waln ( x) = rad p ( x) . The representation of the number of Walsh function in binary system calculus is used and it has such practical importance: this representation permits to distinguish the functions by the order and by the rank. The order is defined as the maximal number position of the corresponding binary number. The rank is defined as the number of units in the representation. For example, the binary representation of number 7 is 111, therefore it is characterized by the order 3 and the rank 3. 1. If the point x is the binary rational, that is, x= s

2n1 +1

( s = 1, 2,22 ,...,2n1 +1 ) ,

(2.17)

then the value of Walsh function is  s  1  s   s  waln  n1 +1  =  waln  n1 +1 + 0  + waln  n1 +1 − 0   . 2  2 2  2 

The system is orthonormal in L2 [0;1] , which can immediately. But it is yet the full system too. Let us show the graphs.

also be checked

30


wal0 ( x) = 1

wal1 ( x) = rad 0 ( x)

wal2 ( x) = rad1 ( x)

wal3 ( x) = rad1 ( x) rad 0 ( x)

3 = 21 + 20

wal4 ( x) = rad 2 ( x)

wal5 ( x) = rad 2 ( x) rad 0 ( x)

5 = 2 2 + 20

wal6 ( x) = rad 2 ( x) rad1 ( x)

6 = 22 + 21

Wavelet Analysis

wal7 ( x) = rad 2 ( x) rad1 ( x) rad 0 ( x)

7 = 22 + 21 + 20

wal8 ( x ) = rad 3 ( x )

wal9 ( x) = rad 3 ( x) rad 0 ( x)

9 = 23 + 2 0

wal10 ( x) = rad 3 ( x) rad1 ( x)

10 = 23 + 21

wal11 ( x) = rad 3 ( x) rad1 ( x) rad 0 ( x)

wal12 ( x) = rad 3 ( x) rad 2 ( x)

11 = 23 + 21 + 20

12 = 23 + 22

wal13 ( x) = rad 3 ( x) rad 2 ( x) rad 0 ( x)

13 = 23 + 22 + 20

31

32


wal14 ( x) = rad 3 ( x) rad 2 ( x) rad1 ( x)

14 = 23 + 22 + 21

wal15 ( x) = rad 3 ( x) rad 2 ( x) rad1 ( x) rad 0 ( x)

15 = 23 + 22 + 21 + 20

wal16 ( x) = rad 4 ( x)

Fig. 2.2 Plots of the first sixteen Walsh functions.

It should be mentioned that the graphs plotted above correspond to the Walsh ordering. Other types of ordering exist also - by Paley (R.E.A.C. Paley, 1892-1946) and by Hadamard (J. Hadamard, 1865-1963) types, in which the representation of number of the function in the Gray binary code is utilized. For the Paley ordering, the notation paln ( x) is used while for the Hadamard ordering the notation had n ( x) is used. The wide applications of Walsh functions is due to the fact that they take two values only: ±1 , which turned out to be very convenient in digital processing.

2.2.3 System of Haar functions It took about a century from the first Fourier’s publication (1807), in Fourier analysis, to the fundamental Haar’s publication (1909) in wavelet

Wavelet Analysis

33

analysis. Haar constructed the orthonormal system of functions, which, similar to functions in Rademacher and Walsh systems are not continuous (are the step-wise continuous functions). The basic function is as follows  1 x ∈ [0;1 2),  har( x) =  −1 x ∈ [1 2;1),  0 x ∉ [0;1). 

(2.18)

All other Haar functions are constructed according to the scheme which is actually called scaling (or binary scaling) (see (2.9)) j

harn ( x) = 2

2

har(2 j x − k )

( n ≥ 1, n = 2

j

+ k , j ≥ 0, 0 ≤ k ≤ 2 j ) .

(2.19)

Note H1. It is worthy to comment on all indexes. Numbers of functions n ∈ ℕ are changed sequentially from zero to infinity. The index j characterizes an amplitude and number of intervals (thereby the length of intervals also) on which the base interval is divided. The index k characterizes translation (shift) within the framework of the base interval. Note H2. The amplitude factor is needed for normalization. Note H3. This is the first opportunity to see the family of dyadic intervals, on which the basic interval is divided   k k + 1  .  ∆ jk =  j , j    2 2  k , j∈ℤ 

For each fixed scaling level j

the interval length 2− j (they don’t

intersect and divide the basic interval on 2 j equal parts) is given. Then for the function harn ( x) the support is supp harn ( x) = ∆ jk .

34


The Haar system is complemented by the function with zero index 1 har0 ( x) =  0

x ∈ [0;1), x ∉ [0;1),

(2.20)

and then the system becomes the full one. 1 0

½

har1 ( x) = ( j = 0, k = 0, ∆ 00 ) = har( x)

har2 ( x) = ( j = 1, k = 0, ∆10 ) = 2 har(2 x) 2

har3 ( x) = ( j = 1, k = 1, ∆11 ) = 2 har(2 x − 1)

har4 ( x) = ( j = 2, k = 0, ∆ 20 ) = 2har(4 x) 2

1

Wavelet Analysis

35

har5 ( x) = ( j = 2, k = 1, ∆ 21 ) = 2har(4 x − 1)

har6 ( x) = ( j = 2, k = 2, ∆ 22 ) = 2 har(4 x − 2)

har7 ( x) = ( j = 2, k = 3, ∆ 23 ) = 2 har(4 x − 3)

Fig. 2.3 Plots of the first seven Haar functions.

The next eight functions correspond to one and the same level – the third one will be similar (will differ by shift only). This property can be seen for the functions of the first and second levels also. 2 2

har8 ( x) = ( j = 3, k = 0, ∆ 30 ) = 2 2 har(8 x)

36


har9 ( x) = ( j = 3, k = 1, ∆ 31 ) = 2 2 har(8 x − 1)

har10 ( x)

har11 ( x)

har12 ( x)

Wavelet Analysis

37

har13 ( x)

har14 ( x) = ( j = 3, k = 6, ∆ 36 ) = 2 2 har(8 x − 6)

har15 ( x) = ( j = 3, k = 7, ∆ 37 ) = 2 2 har(8 x − 7)

Fig. 2.4 Plots of Haar functions with indices seven to fifteen.

Let us discuss now the importance of the Haar system in revising the Fourier series theory. The Fourier trigonometric series are not always convergent and this fact was certainly a tragedy for many mathematicians. In 1873 Paul Du Bois-Reymond constructed a function (continuous, 2π-periodic) of real variable, with a Fourier series which, in contrast with the established theory, was divergent in a given point. In the following years many of such examples were given. Mathematicians had three possibilities to develop the Fourier series theory and all three were realized: 1. Modify the notion of function and find that one, which will be adapted in some sense to Fourier series.

38


2. Modify the notion of convergence for Fourier series. 3. Find other orthonormal systems, for which the counter example discovered by Du Bois-Reymond for trigonometric system, would not hold. Just according to the third direction, Haar put this question: Does there exist a system of orthonormal functions, different from trigonometric ones, har0 ( x), har1 ( x),..., harn ( x),... , defined over [ 0;1] , such that for any function f ( x) defined over

[0;1] ,

the series in terms

of this system of functions f , h0 har0 ( x) + f , h1 har1 ( x) + ... + f , hn harn ( x) + ... will converge to the function f ( x) uniformly over [ 0;1] ? 1

Here the scalar product is assumed as u , v = ∫ u ( x)v( x)dx . 0

Let us focus on two critical facts. Fact H1. By using the Haar system, consisting of atoms hn ( x) , we build the continuous function. But these atoms are not continuous functions. It could be more convenient to do this by using continuous atoms. This was done when the continuous wavelets were developed. Fact H2. Suppose that the function is not continuous over the interval

[0;1]

and belongs to C [ 0;1] , that is, it is continuous and has the

continuous first derivative. Then the approximation by step-wise functions is still more inadequate. Despite these critical facts, the Haar system is now adapted just to describe continuous and square integrable functions over [ 0;1] or, more

39

Wavelet Analysis

abstractedly, to functions with regularity index close to zero. The tool for measuring the regularity is known as the Lipschitz index. To avoid both facts, Faber and Schauder created a new system by taking those functions which are the primitive of the Haar functions. These functions are based on the so-called triangle function 2 x  ∆ ( x ) = 0 2(1 − x) 

x ∈ [ 0;1 2] x ∉ [0;1] x ∈ [1 2;1]

1

0

1/2

1

Fig. 2.5 The triangle function.

The first element is 1 and the second element is x . In general it is: ∆ n ( x) = ∆ ( 2 j x − k )

(n = 2

j

+ k , j ≥ 0, 0 ≤ k ≤ 2 j − 1) ,

∆ 0 ( x) = ∆ ( x), ∆1 ( x) = ∆ (2 x), ∆ 2 ( x) = ∆ (2 x − 1),... .

Fig. 2.6. Graphs of the first five functions of the Schauder-Faber basis.

Each of these functions is the primitive for the corresponding Haar function being additionally multiplied on 2 ⋅ 2 j 2 . The sequence

40


1, x, ∆ 0 ( x),..., ∆ n ( x),... is the Schauder basis for the Banach space of functions continuous over [0;1] . Thus, an arbitrary continuous function over [ 0;1] can be represented as ∞

f ( x) = a + bx + ∑ α n ∆ n ( x) . n=0

This series converges uniformly over [ 0;1] and its coefficients are unique. The Haar system doesn’t form a Schauder basis in Banach space of continuous functions because this basis must be formed from elements of the space, whereas the Haar functions are not continuous. However the Haar system is already, generally speaking, a wavelet system (see below). For proper understanding of the essence of wavelet theory, it would be very expedient to consider briefly the contribution of Littlewood and Paley. In the Fourier series theory, some complications can arise, when the localization of energy is considered. This can be explained as follows. 1 2π

In the signal theory, the integral

2π

∫

2

f ( x) dx is the mean value of

0

energy of periodic signal on one period. But it is important to know if the energy is concentrated near some separate points or it is distributed over the whole interval. To do this, the integral

1 2π

2π

∫

4

f ( x) dx must be

0

evaluated or, more in general, the integrals 1 2π

2π

∫

f ( x)

2p

dx

( p ≥ 1)

0

should be evaluated. When the energy is concentrated near few points, then this integral will be greater, by the mean value, than the mean value of energy. When the energy is uniformly distributed, then both integrals will be one and the same order.

41

Wavelet Analysis 1 p

 1 2π  p If we remember that f Lp [0,2π ] =  f x dx ( )  , then the norm can ∫  2π 0  be compared and this comparison enables us to estimate the energy distribution. But when the index is more than two, then the Fourier series coefficients can’t be used both for norm evaluation, and for estimating this norm. It turns out that the Fourier series coefficients still have the information about the energy distribution but in the hidden form. For this reason, the series should be transformed, as was done by Littlewood and Paley. They took the Fourier trigonometric series ∞

a0 + ∑ ( ak cos kx + bk sin kx ) k =1

and defined the dyadic blocks ∆ N f ( x) as follows ∆ N f ( x) =

∑ (a

k

2 N ≤ k < 2 N +1

cos kx + bk sin kx ) .

It should be noted that the feature of this representation is the separation of dyadic level 1

2,3

4,5,6,7

8,9,10,11,12,13,14,15 16-31

32-63 .

(2.21)

Then the series can be represented in the form ∞

f ( x) = a0 + ∑ ∆ N f ( x) .

(2.22)

N =1

Littlewood and Paley have proven this theorem

Theorem 2.4 (Littlewood-Paley) For any 1 < p < ∞ , there exist two constants C p > c p > 0 , such that

42


cp f

p

 2 ∞ 2 ≤  a0 + ∑ ∆ N f ( x)  N =1  

1

2

≤ Cp f

p

.

(2.23)

The structure of dyadic blocks and inequality (2.23) turned out to be very important for wavelet theory. Just now we can see similarity in (2.21) and the scale level of Haar functions, which will be further treated as simplest wavelets. Let us define the mother wavelet quite afresh (but in actual fact almost equivalent to the old definition). The mother wavelet is the infinite differentiable and strongly n decreasing function ψ ( x) , defined in Euclidean space x ∈ ℝ , whose ∞

Fourier transform ψˆ (ξ ) = ∫ ψ ( x)e − ixξ dx satisfies three conditions: −∞

1 1. ψˆ (ξ ) =  0

| ξ |∈ [1 + α ;2 − 2α ], | ξ |≤ 1 − α , | ξ |≥ 2 + 2α ,

1−α 1 1+α

0

2 − 2α

2

1 α ∈ (0; ] . 3

2 + 2α

Fig. 2.7 Graph of the function ψˆ ( ξ ) .

2. ψˆ (ξ ) is infinitely differentiable in ℝ n .

∑ ψˆ (2

3.

−j

2

ξ) =1

(2.24)

∀ξ ≠ 0 .

j∈ℤ

The third condition can be commented in a way that the first condition ensures that it satisfies almost automatically. Only the condition

2

2

ψˆ (ξ ) + ψˆ (2ξ ) = 1

must

be

accepted

on

pieces

43

Wavelet Analysis

(1 − α ≤| ξ |≤ 1 + α

and

2 − 2α ≤| ξ |≤ 2 + 2α ),

which

form

the

neighbourhoods of 1 and 2 . Actually the full analogy is seen between expansion in terms of Littlewood-Paley atoms (dyadic blocks) and expansion in terms of orthonormal system elements in wavelet analysis. The condition 3 is treated as the condition of energy balance of the transformed by Fourier signal. Let us consider this analogy. In the space ℝn , the wavelet systems of functions

{ψ

j

( x) = 2 j 2ψ (2 j x)}

j∈ℤ

should be introduced and the

Littlewood-Paley dyadic blocks are changed on such convolutions ∆ j ( f ( x) ) = ( f ( x) ) ∗ (ψ j ( x) ) .

(2.25)

The Littlewood-Paley function corresponding to the function f ( x) has the form 1

2  2 (2.26) g ( x) =  ∑ ∆ j ( f ( x) )  .  j∈ℤ  The main property of the Littlewood-Paley function is as follows:

If f ( x) ∈ L2 ( ℝ n ) , then g ( x) ∈ L2 ( ℝ n ) and the equality f

L2

= g

L2

is

valid (energy of real impulse is saved by the Littlewood-Paley function). The next theorem is very important. p n Theorem 2.5 For any 1 < p < ∞ and for all f ( x) ∈ L ( ℝ ) , there exist

two constants C p ≥ c p > 0 such that cp g

p

≤ f

p

≤ Cp g

p

.

In conclusion the Littlewood-Paley gives the method of analysis, in which the main role is played by the ability to change arbitrarily the scale. This ability is a determinative factor in the wavelet analysis.

44


2.3. Integral Fourier Transform. Heisenberg Uncertainty Principle Let us consider the Fourier integral transform as an operator on functions in L2 ( ℝ ) . If the complex-valued function f ( x) is given over the real axis and it is absolutely integrable over this axis, that is, +∞

∫ | f ( x) | dx < ∞ ,

−∞

then the Fourier integral transform of f ( x) is the integral +∞

F

{ f ( x)} = F (ω ) = ∫

f ( x)e− iω x dx .

(2.27)

−∞

We will show now the transition from Fourier series to Fourier transforms. Let the complex-valued function f ( x) be periodic, T = 2l and satisfy the Dirichlet conditions on the period. Assume, additionally, that it is absolutely integrable.

-2l

-l

0

l

2l

x

Fig. 2.8 Absolutely integrable function.

Let us write the Fourier series of f ( x) ∞

f l ( x) =

∑ cn e−iωnt , ωn = n =−∞

adopt the notations

nπ 1 , c = ∫ f (t )e −iωn t dt , l 2l −∞ ∞

(2.28)

45

Wavelet Analysis

ωn =

nπ nπ (n − 1)π π , ∆ωn = ωn − ωn −1 = − = →0 l l l l l →∞

and transform the Fourier series f l ( x) = f , e

∞

∑ce

− iωn x

iωn x

n

n =−∞

=

1 l  = ∑  ∫ f (t )e − iωnt dt  eiωn x = n =−∞  2l − l  ∞

l  iω ( x − t ) dt  ∆ωn .  ∫ f (t )e n ∑ n =−∞  − l 

1 l 2l π

∞

l

Consider the internal integral

∫ f (t )e

Φ l (ωn ) =

iωn ( x − t )

dt

and the

−l ∞

corresponding integral Φ (ω ) =

∫

f (t )eiω ( x −t ) dt which can be written as

−∞

l

−l

−l

−∞

Φ (ω ) = ∫ +

∫

∞

−l

∫

+∫ .

l

−∞

l

+ ∫ = Φ l (ω ) +

∞

This can be estimated by taking into account the absolutely integrability of the function +∞

∫

Φ (ω ) ≤

+∞

f (t ) eiω ( x −t ) dt = ∫ f (t ) dt < ∞ .

−∞

−∞

It follows from this estimate that Φ (ω ) = lim Φ l (ω ) < ∞ → Φ l (ω ) = Φ (ω ) + ε (l )

(ε (l ) → 0) .

l →∞

l →∞

Return now to the first representation (2.28) f l ( x) = =

1 2π 1 2π

∞

∑ Φ l (ωn )∆ωn = n =−∞ ∞

∑ Φ(ω )∆ω n

n =−∞

n

1 2π

∞

∑ [Φ(ω ) + ε (l )] ∆ω n

n =−∞

⌢ + ε (l ) .

n

46


Let us pass to the limit when

l → ∞ . Then f l ( x) → f ( x)

π

→ 0 and the last sum is the integral sum for Φ (ω ) , if l ⌢ lim ε (l ) = 0 is valid (which is indeed valid). ∆ωn = l →∞

So, we obtain 1 f ( x) = 2π

1 ∫−∞ Φ(ω )dω = 2π +∞

 +∞  iω ( x −ξ ) ∫−∞  −∞∫ f (ξ )e d ξ  dω , +∞

(2.29)

which is called the Fourier integral formula. +∞

∫

The integral F (ω ) =

f (ξ )e − iωξ d ξ is called the direct Fourier transform.

−∞

1 The integral f ( x) = 2π

+∞

∫ F (ω )e

iω x

d ω is the inverse Fourier transform.

−∞

If the direct and inverse Fourier transforms exist, then they form the pair f ( x) ↔ F (ω ) . The next properties of the Fourier transform are important because their structure is very close to the properties of the wavelet transform. Property F1. Linearity

α f ( x) + β g ( x) ↔ α F f (ω ) + β Fg (ω ) . Property F2. Symmetry f ( x) ↔ F (ω ), F ( x) ↔ 2πf (−ω ) . Property F3. Shift f ( x − x0 ) ↔ e − iω x0 F (ω ), e − iω0 x f ( x) ↔ F (ω − ω0 ) . Property F4. Scaling

47

Wavelet Analysis

f (α x) ↔

ω  F  ,α ∈ R . |α |  α  1

Property F5. Differentiation dn dn n n f ( x ) ↔ ( i ω ) F ( ω ), ( − ix ) f ( x ) ↔ F (ω ) . dx n dω n Property F6. Integration x

∫

f ( y )dy ↔

−∞

F (ω ) . iω

Property F7. Moments ∞

Mn =

∫x

n

f ( x)dx

n ∈ N ∪ {0} ,

−∞

(−i )n M n =

∂n F (ω ) |ω =ω0 ∂ω n

n ∈ N ∪ {0} .

Property F8. Convolution It is defined by ∞

( f ∗ g ) ( x) = ∫

f (t ) g ( x − t )dt .

(2.30)

−∞

The main properties of convolution are: 1. It is commutative ( f ∗ g ) ( x) = ( g ∗ f ) ( x) . 2. If f , g ∈ L1 , then ( f ∗ g ) ∈ L1 . 3. If f , g ∈ L2 , then ( f ∗ g ) ( x) is uniformly bounded over ℝ , that is,

( f ∗ g ) ( x) < C

∀x ∈ R .

Theorem 2.6 (Convolution) f ( x) ∗ g ( x) ↔ Ff (ω ) ⋅ Fg (ω ) . Theorem 2.7 (Product or modulation)

(2.31)

48


f ( x) g ( x) ↔

1 Ff (ω ) ∗ Fg (ω ) . 2π

(2.32)

Theorem 2.8 (Derivative of convolution) d d ( f ∗ g ) ≡ ∫ ( f ( x) g ( x − ξ ) ) dx dx dx −∞ ∞

 d   d  = f ( x)  ∗ g ( x) = f ( x) ∗  g ( x)  .  dx   dx 

(2.33)

Consider now the conditions for which the function can be represented by a Fourier integral. This is fixed by the classical theorem.

Theorem 2.9 If the function f ( x) : 1. is complex-valued and defined over the real axis; 2. satisfies the Dirichlet conditions over each finite interval

[ −l , l ] , 3. is absolutely integrable, then its Fourier integral converges to f ( x) at each point of continuity and to (1 2 ) [ f ( x + 0) + f ( x − 0) ] at the points of discontinuity. It must be noted, however that, the notion of convergence must be defined anew. The integral F (ω ) =

1 2π

+∞

∫

f (ξ )e − iωξ d ξ is called the convergent one in

−∞

2

the space L ( ℝ ) , if there exists a function g (ω ) ∈ L2 ( ℝ ) , such that lim ∫

N →∞

R

1 g (ω ) − 2π

2

+N

∫

f (ξ )e

− iωξ

dξ dω = 0 .

−N

The next theorem is often used.

Theorem 2.10 (Plancherel) For any function f ( x) ∈ L2 ( ℝ ) such that the Fourier transform fˆ (ω ) ≡ F (ω ) ∈ L2 ( ℝ ) exists

49

Wavelet Analysis L2 ( ℝ )

1

fˆ (ω ) =

+∞

2π

∫

(2.34)

−∞

the Parseval-Plancherel’s equality fˆ 2 = f L

f ( x)e− iω x dx

(2.35)

L2

is fulfilled. It should be noted that constants, polynomials, and Heaviside’s function don’t belong to Lp (ℝ ) . Then in the common sense the Fourier transforms don’t exist for them. The Parseval identity is given by: 2 Theorem 2.11 For all f , g ∈ L (ℝ) the relation

f , g = fˆ , gˆ holds

true. The proof is short and based on two facts: Parseval-Plancherel equality and polarization identity: f ,g =

f +g

2 L2

− f −g 4

2 L2

f − ig +

2 L2

− f + ig

2 L2

4i

.

Let us now fix three facts of physical comments of Fourier transform to be used in the following. Fact F1. If f (t ) is the signal defined over the time domain t ∈ ℝ , then fˆ (ω ) is called the spectrum of this signal defined over the frequency domain ω ∈ ℝ . In signal theory negative frequencies are unacceptable. If we use the Hardy space H 2 ( ℝ ) , then the characteristic property of its elements is that their Fourier transform is zero for negative frequencies f ( x) ∈ H 2 (ℝ ) → Ff (ω ) = 0

∀ω < 0 .

The function Ff (ω ) is also called the spectral density of function f ( x) . This can be explained by the fact that Fourier series have the discrete

50


spectrum of frequencies

{ωn }n∈ℤ ,

whereas in Fourier transforms this

spectrum is changed in the function F f (ω ) ω ∈ ℝ . Such a question arises often: What happens when the spectrum is uniformly spreading over all the axis or, in other words, when the function has the spectral density equal to 1 ? Such a function will have the representation f ( x) =

1 2π

+∞

∫

1 ⋅ eiω x d ω = δ ( x)

−∞

which is the usual Dirac function or impulse function. Fact F2. Let the so-called bell-shaped function (Gauss function, error function etc.) f ( x) = Ce −α x

2

,

( C ,α − const )

be given. Its Fourier transform can be evaluated directly F (ω ) = C = 2π

+∞

∫e

−∞

1 2π

iω   −α  ξ +   2α 

+∞

−αξ − iωξ ∫ Ce e dξ = 2

−∞

2

e

−

ω2 4α

dξ =

C 2π α

e

−

C 2π ω2 4α

+∞

∫e

−αξ 2 − iωξ

dξ

−∞

∫e (S )

− z2

dz =

C 2 πα

e

−

ω2 4α

,

(2.36)

where 2

iω  iω  ω 2   −αξ 2 − iωξ = −α  ξ 2 + ξ  = −α  ξ + ,  − α  2α  4α   iω   z = α ξ +  . 2α   It must be noted that since the last integral is taken over the straight line in the complex plane, then its evaluation isn’t trivial. It is equal to

π. Thus, the Fourier transform of the bell-shaped function is the bellshaped function. This is usually a known fact.

Wavelet Analysis

51

Let us discuss now a less known question. For this purpose, the bottom or width of the bell should be defined as those bounding values of argument, for which the function is decreased 2 e times. Thus the bell width is ∆x = , the width of its Fourier

α transform ∆ω = 4 α . Since ∆x∆ω = 8 , then change of parameter α is resulted in the narrowing of one bell bottom and as many times extension of other bell bottom. In signal theory, this situation is commented as follows: the more the signal is localized in time, the more its spectrum is smeared, and vice versa, the narrowing spectrum (it is said, selectivity increasing) is resulted in an extension of signal in time. This impossibility of simultaneous signal localization and increasing of its selectivity is the immanent property of Fourier transform. This shows also the so-called Heisenberg uncertainty principle. Maybe, it will be convenient to explain that in quantum mechanics the particle with energy E and frequency of wave function E ℏ ( ℏ is the Planck constant) is observed usually during small time ∆t . The Heisenberg principle consists in that it is impossible in principle to measure time and frequency, since they are linked by the inequality ∆t ⋅ ω ≥ c ∼ 1 . This means that if the particle has the frequency ω , then observation of the particle must be done at least during one period, that is ∆t ≥ 1 .

ω

Fact F3. Consider the convolution of two functions f , g ∈ L1 (ℝ ) +∞

h( x) = ( f ∗ g ) ( x) =

∫

f ( x − t ) g (t )dt .

−∞

Let us put this question: is there any function d ∈ L1 (ℝ ) , such that f ∗ d = f ? The answer is negative. However this equality cannot be written exactly. If the equality holds, then after transition to Fourier transforms, the next equality could also be valid

52


fˆ (ω )dˆ (ω ) = fˆ (ω ) . In this case, there should exist a function d ( x) , whose Fourier transform dˆ (ω ) is equal to 1. Thus, a sequence {dα ( x)} ⊂ L1 (ℝ ) exists such that dˆα (ω ) ≈ 1 α → 0, ω ∈ ℝ . This sequence can be normalized without loss of generality +∞

∫ dα (t )dt = 1 or in this way dˆα (0) = 1 .

−∞

The Gauss functions

dα ( x ) =

1 2 πα

e

−

x2 4α

are appropriate, since their

2 Fourier transforms are dˆα (ω ) = e−αω . With this sequence we have

+∞

∫

f ( x − t ) dα (t )dt ≈ f ( x − 0) = f ( x) α → 0+

−∞

which is equivalent to

( f ∗ dα ) ( x ) ≈

f ( x)

α → 0 . It will be rather

valid at all points where the function is continuous. Thus, the sequence of Gauss functions approximately tends to the Dirac function. 1 α Another example of sequence {dα ( x)} is dα ( x) = . π x2 + α 2

2.4 Window Transform. Resolution Let us start with two critical notes on the standard Fourier transform. Note F1 If the standard Fourier transform is applied to some function, then it will give by use of high harmonics coefficients an information on features of the function (discontinuities, local peaks, jumps etc), but it can’t indicate on the base of this information where exactly they are placed. That is, this information isn’t localized spatially. Note F2 Since the standard Fourier transform is defined by integral over

53

Wavelet Analysis

all number axis, then the spatial localization of features becomes even more impossible. In order to use the Fourier transform and at the same time to detect where in the space (in which points of the number axis) the features of signal are placed, the Hungarian scientist-physicist G. Gabor proposed to use “windows” in the space. The window is understood in the case of one variable as a certain interval of a given length. In the case of two variables, this will certainly be rectangular and it will really be like to a window. It is proposed to look at the signal through this window and to see the only given length fragment of signal. To look at the whole signal (it is said, to cover the whole signal), the window must be translated along the axis from −∞ to ∞ . Mathematically, the window is formed by multiplying the function f ( x) by a special window function g ( x) . Then the window Fourier transform of the function f ( x) ∈ L2 ( ℝ ) is +∞

G f (ω , xɶ ) =

∫e

− iω x

g ( x − xɶ ) f ( x)dx .

(2.37)

−∞

This strict definition for

g ( x)

is used: the nonzero function

g ( x) ∈ L2 (ℝ ) is called a window function if xg ( x) ∈ L2 (ℝ) . The center and the width of window function are defined by C gx =

1 g ( x)

∞

2

∫ x g ( x)

2

dx,

−∞

12

∞ 2   ∆ = x − C gx ) x g ( x) dx  . 2 ∫ ( g ( x)  −∞ 

x g

1

The width of the window is assumed as two radii 2∆ xg . For mathematical considerations (for normalization of the window function), it is assumed that

54


2

g ( x) =

+∞

∫

2

g ( x) dx = 1 .

−∞

Gabor chosen the window function in the form of well-known Gauss function

2

g ( x) = β e −α x (α , β > 0) . Later this approach has been

generalized on the arbitrary window function and it was named the window Fourier transform. The window transform gives the signal (function) decomposition through the system of functions g ω , xɶ ( x) = eiω x g ( x − xɶ ) . If the last formula is taken into account in the definition (2.37), then the window transform can be written as a scalar product G f (ω , xɶ ) = f ( x), gω , xɶ ( x) .

(2.38)

In contrast to the standard Fourier transform, which depends on one parameter, the window transform depends on two parameters. The window function can’t be arbitrary. It must be a real positive function and energy of its Fourier transform must be concentrated in the range of low frequencies. From the point of view of signal theory, the window function can be considered as the pulse characteristics of a low frequency filter.

2.4.1 Examples of window functions Window 1 The simplest window is the rectangular function 1 g ( x) =  0

-T/2

x ∈ [ − T 2, T 2] , x ∉ [ − T 2, T 2].

T/2

Fig. 2.9 The rectangular window.

55

Wavelet Analysis

This function is discontinuous and gives a bad localization in frequency. Therefore, usually the more smooth functions are used. Window 2 The triangle function is a slightly better window. It gives a good localization in frequency (the spectrum decreases as 1 ω 2 )  1 + 2 x x ∈ [ − T 2,0] ,  g ( x) =  1 − 2 x x ∈ [ 0, T 2] ,  0 x ∉ [ − T 2, T 2].  1

-T/2

T/2

Fig. 2.10 The triangle window.

For comparison, the next four windows will be shown together. Window 3 The Hamming window is given by (1 2 ) ( A + B cos ( 2π x T ) ) g ( x) =  0 

x ∈ [ − T 2, T 2] , x ∉ [ − T 2, T 2].

The following plot corresponds to the function 0.54 + 0.46cos 2π x .

56


1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0 -0.4

-0.2

0

0.2

0.4

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

-0.4

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

0 -0.4

-0.2

0

0.2

0.4

Fig. 2.11 The most known windows.

Window 4 The Blackman window is given by  A + B cos ( 2π x T ) x + C cos ( 4π x T ) g ( x) =  0 

x ∈ [ − T 2, T 2] , x ∉ [ − T 2, T 2] .

The plot corresponds to the function 0.42 + 0.5cos 2π + 0.08cos 4π x . Window 5 The Hanning window is given by  A cos 2 ( 2π x T ) g ( x) =  0 

x ∈ [ − T 2, T 2] , x ∉ [ − T 2, T 2].

57

Wavelet Analysis

The plot corresponds to the function cos 2 π x .

Window 6 The Gabor window is the best g ( x) = β e

−α x 2

(α , β > 0) ↔ G (ω ) = β

(π α )e

−

ω2 4α

.

2

The plot corresponds to the function e −18 x . The family of functions gω , ɶx ( x)

{

}(ω

, ɶx )∈ℝ 2

is formed by the translation

of the window function g ( x) in the space (parameter xɶ ) and in the frequency domain (parameter ω ). This fact of dependence on two parameters here is determinative. The basics for this family function and its Fourier transform are given by formulas

gωo , xɶo ( x) = eiωo x g ( x − xɶo ) , gˆωo , xɶo (ω ) = e − iωo ω gˆ ( x − xɶo ) .

(2.39)

Such a family of two sets of functions is called the family of coherent states in quantum physics. The last one is associated with HeisenbergWeil group, actually it is associated with the (ax + b) group. This group

is also called the group of wavelets.

2.4.2 Properties of the window Fourier transform Property W1. Isometry This is indeed an analog of the Parseval formula and is proved using the Parseval formula +∞

∫

2

f ( x) dx =

−∞

1 ∫ −∞∫ G f (ω , xɶ ) d ω dxɶ . 2π −∞ +∞ +∞

(2.40)

Property W2. Reconstruction This property can be considered as an analog of the inverse transform f ( x) =

1 2π

+∞ +∞

∫ ∫G

−∞ −∞

f

(ω , xɶ ) g ( xɶ − x)eiω x d ω dxɶ .

(2.41)

58


Let us draw attention to the fact that the direct transform is introduced by a simple integral, whereas the inverse one by a double integrals.

Property W3. Window sizes Let us introduce the standard deviations (dispersion) for functions, which define the family of coherent states +∞

σ x2ɶ =

∫

2

+∞

∫ω

x 2 gω , xɶ ( x) dx , σ ω2 =

−∞

2

2

gˆω , xɶ (ω ) d ω .

−∞

They can be calculated, when the window function is known. If it is Gaussian (as it is proposed by Gabor), then it will be quite easy. The window transform can be written in an expanded form ∞

G f (ωo , xɶo ) =

∫

−∞

∞

f ( x) gωo , xɶo ( x)dx =

∫

fˆ (ω ) gωo , xɶo (ω )d ω .

(2.42)

−∞

Since the function gω , xɶ ( x) is centered at the point with two coordinates: o o 1. spatial coordinate xɶo and standard deviation σ xɶ , 2. frequency coordinate ωo with standard deviation σ ω , then this means that the properties of the window are as follows:

Property W3.1. The first integral of (2.42) testifies that the transform G f (ω0 , ɶ x0 ) depends essentially by spatial coordinate on values of function f ( x) in the domain x ∈ [ xɶo − σ xɶ , xɶo + σ xɶ ] only. Property W3.2. The second integral of (2.42) testifies that the transform G f (ω0 , ɶ x0 ) depends essentially by frequency coordinate

on the values of the transformed function fˆ (ω ) in

the domain ω ∈ [ωo − σ ω , ωo + σ ω ] .

59

Wavelet Analysis

ɶ ω , the window (or Heisenberg Property W3.3. In the plane xO rectangle) is formed with sizes 2σ xɶ × 2σ ω , which don’t depend

on choice of the point (ωo , xɶo ) and has the, lower bounded, area 2σ ɶx × 2σ ω ≥ 2π .

(2.43)

That is, the observed in ordinary Fourier transform Heisenberg indeterminacy is saved in the window Fourier transform. We will have a formula like (2.43) also in the wavelet transforms.

2.4.3 Discretization and discrete window Fourier transform The window Fourier transform gives a redundant information about the function. Therefore, it is proposed not to evaluate the transform ɶ ω on a Gf (ωo , xɶo ) for all values (ωo , xɶo ) , but discretize the plane xO doubly periodic system of rectangles with sides parallel to coordinate axes and equal xɶo and ωo , and evaluate Gf (ωo , xɶo ) at the vertices of these rectangles only.

( 0,7ω0 )

( 0,0 )

( 2 xɶ0 ,0 )

( 4 xɶ0 ,0 )

( 6 xɶ0 ,0 )

Fig. 2.12 The system of rectangles in the procedure of discretization.

Now the discrete window Fourier transform can be defined strictly by the next formulas

60

Wavelet and Wave Analysis +∞

∫e

G df (m, n) = G f (mωo , nxɶo ) =

− imωo x

g ( x − nxɶo ) f ( x)dx m, n ∈ ℤ ,

(2.44)

−∞

G df ( m, n) = f ( x), eimωo x g ( x − nxɶo = f ( x), g mωo ,nxo ( x) ,

(2.45)

where we have the new family of window functions

{g

mωo , nxo

( x)

}

m , n∈Z

.

In order to construct the inverse transform to G df (m, n)

the condition

for the lattice xɶoωo > 2π must be necessarily fulfilled. This condition isn’t trivial, a strict mathematical analysis of this condition exists. The next theorem is very useful for the reconstruction of function by the coefficients of the window transform.

Theorem 2.12 We have a stable reconstruction of function f ( x) by the coefficients of the window transform +∞

f , g mω0 ,nx0 =

∫e

− imωo x

g ( x − nxɶo ) f ( x)dx

(2.46)

−∞

if the window functions

{g

mωo , nxo

( x)

form a frame, that is, there

}

m , n∈ℤ

exist two constants 0 < A ≤ B < ∞ such that ∞

A ∫ f ( x) dx ≤ 2

∑

2

f , g mωo ,nxo

m , n∈ℤ

−∞

{

{g

}

mωo , nxo

2

−∞

If the window functions g mω ,nx ( x) o o frame with elements

∞

≤ B ∫ f ( x) dx .

( x)

}

form the frame, then the dual

m , n∈ℤ

should be formed, and the m , n∈ℤ

reconstruction of function f ( x) ∈ L2 (ℝ ) is realized by

f =

∑ m , n∈Z

f , g mω0 ,nx0 g mω0 ,nx0 =

∑ m , n∈Z

f , g mω0 ,nx0 g mω0 , nx0 .

(2.47)

Wavelet Analysis

61

Thus, the frames and dual frames should be discussed together.

Critical remark. A disadvantage of the window Fourier transform is associated with the signal resolution, which is constant both for space and frequency domain. The Gabor transform and any arbitrary window transform give decomposition of the signal on intervals with width 2σ xɶ relative to spatial coordinate and intervals with width 2σ ω relative to frequency coordinate. Let us focus now on the space domain, where the resolution is determined by the standard deviation σ xɶ . If the signal has a simple discontinuity, by means of the window Fourier transform, this signal can be localized with exactness up to σ xɶ . But σ xɶ is a constant quantity, therefore, if the signal has many sharp transitions in sizes (as for images), then the window Fourier transform is unable to optimize the resolution. This resolution could be variable, but it is (by definition) constant. Let us consider the problem from another point of view and concentrate on the ability of standard Fourier transform to fix the local singularities of a function (as the peaks, in particular). The limit case of this singularity is the Dirac function δ ( x) . Since δˆ (ω ) = 1 and +∞

+∞

−∞

−∞

δ (t ) = ∫ δˆ (ω )eiω x d ω = ∫ eiω x d ω , then for the description of this function we need all the harmonics (all the frequencies) with unit amplitude. This corresponds to the limit case of Heisenberg principle, when ∆T = 0 and ∆ω = ∞ . In order to fix, by means of the standard Fourier transform, the peak δ ε (t ) of width ∆T ∼ ε , it is required to use the band of frequencies of finite width ∆ω ∼ 1 . To narrow the peak in time, the more high-

ε

frequency exponents must be taken into account in the standard Fourier transform. For narrow peaks, the long procedures of numerical calculations are needed. It is widely accepted that such an approach is

62


ineffective. This inefficiency of Fourier transform in fixing the local iω x

peak is caused by the nonlocal nature of this transform (exponents e are smeared over all the axis). The window Fourier transform solves this problem only partially. If the chosen window has the width T , then the function is reconstructed by means, no (complex) exponentials (as in the standard Fourier transform), but functions from a family of coherent states gω , xɶ ( x) . But these functions are also given on the chosen window of width T . If the peak has the width ∆x > T ), and small ( ∆x T > 0 , Then  π f ( x) = ∑ f  n  T n∈Z

 sin ( xT − nπ ) .   xT − nπ

(2.54)

To prove it, let us write the formula of inverse Fourier transform for the function f ( x) (it is possible according to theorem conditions) f ( x) =

1 2π

∫

R

1

Ff (ω )eiω x dx =

2π

+T

∫ F (ω )e

iω x

f

dx .

−T

⌢ Let us construct on the base of function Ff (ω ) , the new function F2T (ω ) , which will already be 2T-periodic and coincide with Ff (ω ) on the base period. For this function we write the Fourier series. It will be convergent to the function (provided that from theorem conditions there follows the differentiability of Ff (ω ) ) π ⌢ in ω F2T (ω ) = Ff (ω ) = ∑ c( F , n ) e T . n∈ℤ

Find the coefficients c( F ,n )

1 = 2T

+T

∫ F (ω )e f

−T

− iω n

π T

1 dω = 2T

1 = 2T

2π

∫

ℝ

F f (ω )e

1 2π

∫

ℝ

− iω n

π T

Ff (ω )e

dω =

− iω n

π T

dω =

2π  π  f  −n  . 2T  T 

Now take into account the expansion of the Fourier transform into the series and find the values of coefficients

70


f ( x) =

=

1 2π

+T

∫

Ff (ω )eiω x dx =

−T

 2π   ∑   n∈Z

1  1  2π  2T

1 2π

π in  T c e ( F ,n) ∫−T  ∑ n∈Z +T

π  π   in f  − n   ∫ e T eiω x dx  T   −T +T

T

=

1 2T



π

∑ f  −n T  n∈Z

1

π  i x + n  T 

e

 iω x  e dx 

π  iω  x + n  T 

=∑ n∈Z −T

π  sin  x + n  T  π f  −n   . π T   x+n    T 

Thus, the theorem is proven. The Shannon theorem has a version for the discrete Fourier transform: If the continuous periodic function f has only a finite number of nonzero Fourier coefficients, such that the positive integer K exists that c( f ,k ) = 0 for all k > K , then the finite number of values of the function is sufficient for finding all Fourier coefficients without integration. It seems to be logical to remember the existence of many theorems on discretization of functions. They are divided into classical and nonclassical ones. The first theorem is the classical Cauchy theorem on the cardinal series.

Theorem 2.15 (Cauchy) Let f (t ) =

∑ceπ

2 i nt

n

and N = 2 M + 1 . Then

n ≤M

sin π Nt f (t ) = N

N −1

∑ ( −1) m=0

m

f m

( T) . sin π ( t − m ) T

This theorem had a strong influence in the theory of function interpolation and in the number theory. In 1915 Whittaker introduced the notion of the cardinal function

71

Wavelet Analysis n

f c (t ) = sin 2π Ω t ∑ cn n∈ℤ

( −1) sin π ( t − nT )

in the class of functions X ( ℝ ) → f ( x) ∈ X ( ℝ) : Tf (nT ) = cn (T > 0, n ∈ ℤ ) which he called the cotabular set of functions. There exists a link between the cardinal function and the well-known Newton-Gauss interpolation formula

x( x − 1) 2 x( x 2 − 1) 3 ∆ f (−1) + ∆ f (−1) 2! 3! x( x 2 − 1)( x − 2) 4 + ∆ f (−2) +⋯ 4! ∆f (0) = f (1) − f (0),

f ( x) = f (0) + x∆f (0) +

∆ 2 f (0) = ∆f (1) − ∆f (0) = f (2) − 2 f (1) + f (0). The cardinal function and the function from the last formula converge to one and the same sum for some classes of functions. Further, the nonclassic theorems will be considered when the synthesis (decomposition) of the functions is realized by means of different kinds of wavelets forming frames in L2 (ℝ) .

2.6 Frames. Conditional and Unconditional Bases In this section we consider expansions formed by systems of functions, the number of which is either redundant, that is, the number is more than it is needed for forming the basis, or is such that the basis can’t be formed at all. A family of functions from Hilbert space { xk } ⊂ H is called a frame, if there exist two constants 0 < A ≤ B < ∞ such that the following condition is fulfilled

72

Wavelet and Wave Analysis 2

A f

H

≤ ∑ xk , f

2 H

≤B f

2 H

.

(2.55)

k

This condition is called the frame condition, the numbers A, B  are the frame bounds, x are the frame elements, the numbers fɶ = f , x are k

k

k

called the transform of function f ( x) relative to the frame. The frame operator is defined by Sf ( x) = ∑ f ( x), xk ( x) xk ( x) .

(2.56)

k∈ℤ

This key notion, in wavelet analysis, was introduced at 1952 by the mathematicians Schaeffer and Duffin while studying the nonharmonic Fourier series. The frame is called exact frame, when by extracting one arbitrary element the frame stops being a frame. If A = 0, B = ∞ , the frame is a loose frame. When A = B the frame is a tight frame. For the tight frame, the inequalities (2.55) are transformed into the equality

∑

xk , f

2 H

=A f

2 H

(2.57)

k

and then the function can be decomposed (discretized) by means of the series with countable numbers of coefficients

1 1 S f ( x) = ∑ xk , f H xk . (2.58) A A k Equality (2.57) corresponds to the Parseval equality for Fourier transforms and means: when the function is transformed by means of tight frame, then no information is lost. It should be noted that for arbitrary case of the frame А < B , the f =

function can also be reconstructed using the transform f , xk . But this procedure is essentially more complicated. The frame is normalized if all the functions in the frame have the unit norm.

Wavelet Analysis

73

If the tight frame is normalized, then the number A gives the redundant ratio. For example, if A = 2 , then the frame has elements in numbers two times more than is needed for forming the basis in the space. If the elements xk are linearly independent and the frame is normalized, then A ≤ 1 ≤ B . The frame becomes an orthonormal basis only, when A = B = 1,

xk = 1 ∀k .

The frame operator S maps the function f ( x) ∈ L2 (ℝ ) into the function fɶ (m) ∈ L2 (ℤ) ( m ∈ ℤ ) ( S f ) (m) = f , xm . The number

S f (m) = f , xm

2

characterizes, for fixed m , the

number of atoms or blocks used for this function analysis. Moreover, (2.58) is just the basic formula of analysis of functions. Therefore, the operator S is called the analyzing frame operator. The synthesis or reconstruction of function consists in the transition fɶ (m) → f ( x) . Then the frame can be interpreted as guaranteeing the fact that the inverse transform exists in the form of a bounded operator. For an arbitrary frame, the theorem holds similarly to the theorem on minimum property of the Fourier series coefficients.

Theorem 2.16

Among all possible coefficients of the operator S for

function f ( x) , the coefficients f , xk minimize the “energy”

∑

f , xk

2

.

k∈Z

When the function is reconstructed by a computer, then the tight frame guarantees the good convergence in approximations. Therefore, it seems useful to introduce the parameter δ , which characterizes to which extent the frame is tight.

74


If δ = 0 , then the frame is tight; when δ = ∞ , then the frame is the loose one. The frames, for which the condition 0 < δ 0 such that

x ≤ C ∀ε n = ±1 .

n n

n∈ A

Example 2.8 Each absolutely convergent numerical series

∑x

n

is also

n∈ A

the unconditionally convergent one.

Example 2.9 If {ψ n }n∈A is an orthogonal system in Hilbert space H and such

that

∑ψ

2

n

< ∞ , then the series

n∈ A

∑ψ

n

is unconditionally

n∈ A

convergent in H .

Example 2.10 The series

∞

1

∑ne

inx − x 2

e

is unconditionally convergent in

n =1

L2 ( ℝ ) , but it isn’t absolutely convergent in L2 ( ℝ ) . Consider further some definitions associated with wavelet analysis and different kinds of bases. The first one is the Wojtaszczyk definition of unconditional basis, for which we need to remind the definition of the Riesz sequence. A sequence of functions { xn }n∈A in a Hilbert space H is called a Riesz sequence, if there exist some constants 0 < c ≤ C such that the system of inequalities

80


  c  ∑ | an |2   n∈A 

12

≤

  an xn ≤ C  ∑ | an |2  ∑ n∈ A  n∈A 

(2.59)

is fulfilled for all sequences of numbers {an }n∈A . The Riesz sequence is called the Riesz basis, if in addition span { xn }n∈A = H . n

It should be noted that if c = C = 1 , then the system of inequalities (2.59) is transformed into the well-known Parseval formula for orthonormal systems. Therefore, the system of Riesz functions is a somewhat generalization of the notion of orthonormal system and the Riesz basis is a generalization of the notion of orthonormal basis. Two important statements relative to the Riesz basis are the following:

Statement 1 If

{ xn }n∈ℤ

is the Riesz basis in H , then there exist some

biorthogonal functionals { xn∗ } sequence { xn∗ }

n∈ℤ

n∈ℤ

from H , such that

xn∗ , xm = δ nm . The

is also a Riesz basis in H .

It must be noted here that this statement fixes only the existence. The system ( xn , xn∗ )

n∈ A

of elements xn from X and functionals xn∗ from

∗ X ∗ is called the biorthogonal system, if xn , xm = δ nm .

Statement 2

If

{ xn }n∈ℤ

is a Riesz basis in H , then there exist some

constants 0 < c ≤ C such that the inequalities are fulfilled

 c x ≤  ∑ x, xn  n∈Z

12

2

  

≤C x .

This system is the same as in the definition of the frame. Therefore, the frames and the Riesz bases are often identified. The important distinction between the frame and the Riesz basis consists in that in the frame the

81

Wavelet Analysis

system of functions isn’t necessary a linear independent one, whereas in the Riesz basis the independence is obligatory. If the Riesz basis is normalized, then c ≤ 1 ≤ C . This can be checked by substituting f = xm .

2.6.1 Wojtaszczyk’s definition of unconditional basis (1997) The biorthogonal system ( xn , xn∗ )

n∈ A

with countable set of indexes A is

an unconditional basis in Banach space X , if: W1. span { xn }n∈A = X ; W2. There exists a constant C such that for all f ∈ X , xn∗ ( f ) = f , xn∗

X

and for each finite set B ⊂ A the inequality

∑

f , xn∗ x =

n∈B

∑ x ( x) x ∗ n

n

≤C f

n∈B

is fulfilled. It seems important to remind that a Riesz basis (in particular, the orthonormal one) in the Hilbert space is an unconditional basis.

Theorem 2.22 (Wojtaszczyk) If ( xn , xn∗ )

n∈ A

is the unconditional basis

in X . Then: WB1. For each

f ∈X

the series

∑ x ( f )x ∗ n

n

converges

n∈B

unconditionally to f . WB2. There exists a constant C such that for each f ∈ X and each bounded sequence of numbers {an }n∈A the inequality

∑ a x ( f )x ∗ n n

n∈B

holds true.

n

≤ C sup an ⋅ f n∈A

82


2.6.2 Meyer’s definition of unconditional basis (1997) The set of functions { xk ( x)}k∈ℤ ∈ H1 (ℝ ) forms an unconditional basis, if it admits for all functions f ( x) ∈ H1 (ℝ) the decomposition ∞

f ( x) = ∑ ak xk ( x) , k =0

the coefficients ak ∈ ℝ are evaluated by the formula ak = f , xk

H1 (ℝ)

and the basis fulfils the strong condition of independence: ∞

if ∃C > 0 ∀k ∈ ℕ ak < bk , then

∑

∞

ak xk ( x)

≤ C

k =0

∑ b x ( x) k

k

.

k =0

2.6.3 Donoho’s definition of unconditional basis (1993) Let the class of functions M with the norm functions

{ fk }

be

given,

M

and the basic set of

and such that an arbitrary function

g ( x) ∈ M has the unique representation

g ( x) = ∑ ak f k ( x),

where

k

equality means the limit by the norm. Consider the infinite expansion g ( x) = ∑ mk ak f k ( x). k

If for all g ( x ) ∈ M this sum is convergent with all | mk |≤ 1 then the basis

{ fk }

is called the unconditional or the absolute basis.

2.6.4 Definition of conditional basis If the convergence of the sum depends on the choice of m k (that is, the series is convergent, but not for arbitrary choice of the coefficients and there always exists a variant of the choice when the series converges), then the basis is called a conditional basis. The conditional convergence means independence of convergence on rearrangement of series members and signs of coefficients.

Wavelet Analysis

83

It is known that the conditional basis is stable, in numerical calculations, when the coefficients are increasing rapidly. The main idea of bases or frames consists in the representation of continuous or slightly worse functions by means of the coefficients of the expansion with respect to a basis or a frame respectively. In the case of orthonormal bases and tight frames, the Parseval theorem establishes the relationship between the norm in L2 ( ℝ ) and the norm of coefficients in l 2 ( ℝ ) . When being passed to the other functional spaces, the norm will change, but the link will rest. When the basis or frame are unconditional ones, then the norm can be linked not only with the norms of coefficients in expansion of the function by the basis, but also the absolute values of coefficients give sufficient information to establish the link between the norms. This kind of bases or frames doesn’t require the conditions for signs or phases of coefficients (these conditions are absent according to definition of unconditional basis) – the norm of function should be known only. Therefore the basis is called the unconditional one. Wavelets are unconditional bases and for this reason they open wide possibilities for the analysis.

2.7 Multiresolution Analysis This subsection is devoted to the multiresolution analysis of functions (shortly MRA). This part of the wavelet analysis can be referred to as development of the approach in Fourier analysis which is associated with the series. In the idea of MRA, the key role is played by the scaling function, since just using this function the wavelet is defined and the wavelet analysis of functions is organized.

84


Let the function ϕ ( x) ∈ L2 ( ℝ ) , called scaling function, be given. The selection of such functions and restrictions in this choice will be discussed later. Use the operation of translation and form the set of scaling functions according to ϕk ( x) = ϕ ( x − k ) k ∈ ℤ . (2.60) Denote the closed subspace as V0 ⊂ L2 (ℝ ) and note that it is the linear span for the set of functions {ϕ k ( x) = ϕ0,k ( x)}k∈ℤ V0 = span {ϕk ( x)}k∈ℤ . k

This subspace and family of like subspaces is organized in MRA by means of the following theorem.

Theorem 2.23 Let the function ϕ ( x) ∈ L2 be given, and such that M1. {ϕk ( x) = ϕ ( x − k )}k∈ℤ is the Riesz sequence in L2 ( ℝ ) . M2. ϕ ( x 2 ) = ∑ hk ϕ ( x − k ) or ϕ ( x 2 ) = ∑ hk ϕ ( x − k ) and k∈ℤ

k∈ℤ

these series are convergent in the mean square sense. M3. The Fourier transform ϕˆ (ω ) is continuous at zero and not equal to zero at this point. Then the subspaces V j = span {ϕ (2 j x − k ) = ϕ j ,k ( x)} k

k∈ℤ

form the base of a

MRA.

From the definition of subspace V j follows that if the function f ( x) ∈V j , then it can be represented in the form f ( x ) = ∑ a j ,k ϕ (2 j x − k ) . k∈ℤ

In particular, there follows from f ( x) ∈V0 that f ( x ) = ∑ a0,k ϕ ( x − k ) . k ∈ℤ

85

Wavelet Analysis

Roughly speaking, the scaling function should be chosen in the form of “hump” or “bump” or “pimple” of width w and centered around zero. Then each ϕm ,n constructed by means of scaling function ϕ ( x) is also the “pimple” of width 2− m w and centered around x = 2− m n . The translated functions ϕm ,n , with fixed scale, are orthonormal

ϕm, n ,ϕm ,n = δ nn . *

ϕ ( x)

ϕ ( x − 1)

*

ϕ ( x − 2)

2ϕ (2 x) Fig. 2.13 View of scaling functions for two successive scales

It should be noted that in the definition of scaling functions the notions of Riesz sequence and basis were used. The base of MRA consists of the sequence of embedded closed subspaces ... ⊂ V−2 ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ V2 ⊂ ... with such properties: V1. Growing up completeness

∪V

k

= L2 ( ℝ ) .

k∈ℤ

V2. Growing down completeness

∩ V = {0} . k

k∈ℤ

V3. Invariance of scales f ( x) ∈Vk ⇔ f (2− k x) ∈V0 or f (2k x) ∈Vk ⇔ f ( x) ∈V0 .

86


V4. Invariance of translation f ( x) ∈V0 ⇔ f ( x − m) ∈V0

∀m ∈ ℤ.

V5. Existence of orthonormal basis There exists a function ϕ ( x) ∈V0 such that

{ϕ

0, m

= ϕ ( x − m)}m∈ℤ

is the orthonormal basis for V0 . *

V5 . Existence of Riesz basis There exists a function ϕ ( x) ∈V0 such that

{ϕ

0, m

= ϕ ( x − m)}m∈ℤ

is the Riesz basis for V0 . One of the basic notions of MRA is the orthogonal projection which is defined as follows: The orthogonal projection Pj f of function f ∈ L2 (ℝ) onto

the

subspace V j is Pj f ( x) = ∑ f , ϕ j ,k ϕ j ,k ( x) . k∈ℤ

The main idea of MRA consists in that: when these five properties are fulfilled, then in the space L2 ( ℝ ) there exists an orthonormal wavelet basis

{ψ } j ,k

j , k∈ℤ

with ψ j ,k = 2 j 2ψ ( 2 j x − k ) , such that for all functions

f ∈ L2 (ℝ) the orthogonal projections Pj +1 f , Pj f into two neighboring

subspaces V j +1 ,V j are linked by Pj +1 f = Pj f + ∑ f ,ψ j ,k ψ j ,k .

(2.61)

k∈ℤ

However, it should be noted that from the definition of orthogonal projection it follows that it doesn’t equal to the function, being only an approximation. The property V1 states that lim Pm f ( x) = f ( x) . m →−∞

87

Wavelet Analysis

So all subspaces V j are scaled versions of the central subspace V0 . Just in this consists the multi-resolutioness of this analysis. It must be stressed that the function ϕ ( x) from property V5 is the scaling function of MRA. It follows from properties V3-V5

{ϕ

0, m

that if

the functions

= ϕ ( x − m)}m∈ℤ form the orthonormal basis for V0 , then the system

of functions ϕ m ,k = 2m 2 ϕ ( 2m x − k )

{

}

k∈ℤ

form the orthonormal basis for

Vm : V j = span {ϕ j ,k } k

k∈ℤ

.

This also means that if the function f ( x) ∈V j , then it can be represented as:

f ( x) = ∑ ak 2 j 2 ϕ ( 2 j x − k ) . This concerns, in particular,

the

k∈ℤ

scaling function as well. If ϕ ( x) ∈V0 , where the linear span of the subspace is built on the base of this subspace, then it is contained in V1 , where the linear span is built on functions of the next scale ϕ (2 x) and then we have

ϕ ( x) = ϕ0 ( x) = ∑ hk 2 ϕ ( 2 x − k ) = ∑ hk ϕ1,k ( x) , k∈ℤ

(2.62)

k ∈ℤ

where the coefficients must be found from the scalar product hk = ϕ0 , ϕ1,k .

For the Fourier transform it is ϕˆ (ω ) =

ϕˆ (ω ) = m0 (ω 2 ) ϕˆ (ω 2)

,

1 2

∑ae

− ik ω 2

k

ϕˆ (ω 2 ) or

k ∈ℤ

m0 (ω 2 ) =

1 2

∑a

k

e− ik ω 2 .

k∈ℤ

Let us stop on the basic iterative equation

ϕ ( x) = ∑ hk 2 ϕ ( 2 x − k ) k ∈ℤ

(2.63)

88


owing to the high importance of this equation for the wavelet analysis. This is called the refinement equation or dilation equation or MRA equation. The coefficients hk are the scaling coefficients. Let us formulate a list of theorems which impose some conditions on the scaling coefficients.

Theorem 2.24 If ϕ ( x) ∈ L1 (ℝ ) additionally

∫ ϕ ( x)dx ≠ 0 ,

is the refinement equation and

then the scaling coefficients fulfill the

condition

∑h

k

= 2.

(2.64)

k

Condition (2.64) is the weakest condition on the scaling coefficients.

Theorem 2.25 additionally

If ϕ ( x) ∈ L1 (ℝ ) is the refinement equation and

∫ ϕ ( x)dx = 1

and the scaling function also

fulfills the

following condition relative to translation

∑ϕ ( x − k ) = ∑ϕ (k ) = 1 , k

(2.65)

k

then the scaling coefficients fulfill the condition

∑h

2k

k

= ∑ h2 k +1 .

(2.66)

k

And vice versa, when (2.66) is fulfilled, then (2.65) is valid. Condition (2.66) is called the fundamental condition. It is stronger than (2.64) and weaker than the condition of orthogonality of coefficients.

Theorem 2.26 If ϕ ( x) ∈ Lp ( ℝ) is a solution of the refinement equation and {ϕ ( x − k )}k∈ℤ form the Riesz basis in the space linear span of which they are (see V5*), then the scaling coefficients fulfill the condition (2.66)

Wavelet Analysis

∑h

2k

= ∑ h2 k +1 .

k

Theorem 2.27 refinement

{ϕ ( x − k )}k∈ℤ

89

k

If ϕ ( x) = ϕ0 ( x) ∈ L2 (ℝ ) ∩ L1 (ℝ ) is a solution of the

equation and the set of translated scaling functions is orthonormal with the basic scaling function

∫ ϕ ( x)ϕ ( x − k )dx = δ

0k

, then the scaling coefficients fulfill the condition

∑h h

n n−2k

= δ 0k .

(2.67)

n

The set of coefficients fulfilling condition (2.67) is called the quadrature mirror filter or conjugate mirror filter. Condition (2.67) is usually called the quadratic condition.

Corollary 2.1 (from theorem 2.27) The scaling coefficients as a whole have the unit norm, that is

∑h

2

k

=1.

(2.68)

k

Corollary 2.2 (from theorem 2.27) The scaling coefficients fulfill not only the condition (2.64)

∑h

k

= 2 , but also the condition

k

∑h

2k

= ∑ h2 k +1 =

k

k

1 . 2

(2.69)

Thus, within the conditions of theorem 2.27 all three conditions on the scaling coefficients are fulfilled simultaneously.

Theorem 2.28 If the scaling function ϕ ( x) has a compact support over 0 ≤ x ≤ N − 1 and the set of translated scaling functions {ϕ ( x − k )}k∈ℤ forms the linearly independent system of functions, then the scaling coefficients also have the compact support over 0 ≤ x ≤ N − 1 that is hk = 0

∀k ∉ [ 0, N − 1] .

90


The number N is called the support or length of the sequence {hk } . From these theorems follows that when the scaling function has a compact support and the set of translated scaling functions is orthogonal to the base function, both the conditions (2.64)

∑h

k

= 2 and (2.67)

k

∑h

k

hk − 2 n = δ 0 k should be fulfilled. The total number of conditions

k

is N 2 : N −1

N −1

∑

n=0

n =1

N n = −1 2

∑ hn hn = δ 00 , ∑ hn hn−2 = δ 01 ,... ,

N −1

hn hn − N + 2 = δ

N  0  −1 2 

.

Then the support of scaling functions must be an even number and the independent coefficients will be ( N 2) − 1 . In the case of compact support, two matrices are used. The first matrix M 0 is called the refinement matrix. This is the discrete variant of refinement equation. It has the form for N = 6  h0   h2  h4 2  0 0  0

0 h1 h3 h5 0

0 h0 h2 h4 0

0 0 h1 h3 h5

0 0 h0 h2 h4

0

0

0

0

0  0 0  h1  h3   h5 

 ϕ0   ϕ0       ϕ1   ϕ1   ϕ2   ϕ2    =    ϕ3   ϕ3  ϕ  ϕ   4   4   ϕ5   ϕ5 

This representation shows that the components ϕ k (k = 0,...,5) of the scaling function vector are eigenvectors of the scaling coefficients matrix M0 . The second matrix links the scaling function vector with the same shift on 1/2 vector

.

91

Wavelet Analysis

 h1   h3 h 2  5 0 0  0

h0 h2 h4

0 h1 h3

0 h0 h2

0 0 h1

0 0 0

h5 0 0

h4 0 0

h3 h5 0

0  0 h0   h2  h4   0 

 ϕ0   ϕ1/ 2       ϕ1   ϕ3 / 2   ϕ2   ϕ5 / 2    =    ϕ3   ϕ7 / 2  ϕ  ϕ   4   9 / 2  ϕ  5  ϕ11/ 2 

This matrix is usually denoted as M 1 . In the general case, the refinement matrix M is an infinite matrix and both matrices M 0 , M 1 are its parts. Among other matrices, the most important one is the transition matrix T , which is built using the auto-correlative (correlative) matrix of 1 scaling coefficients H : T = HH T . 2 The positive defined matrix A is correlative or auto-correlative, if its elements are built by the formula aij

(

aii a jj . All diagonal elements

)

of correlative matrix are equal to 1 and all other elements are bounded from above modulo 1. To complete the picture, we consider the theorems on sufficient conditions relative to the scaling coefficients (the prior theorems have dealt with the necessary conditions).

Theorem 2.29 If the condition (2.64)

∑h

k

= 2 is fulfilled and the

k

scaling coefficients have the finite support, then there exists a unique scaling function, which satisfied the refinement equation. It should be noted that since this theorem uses the weakest condition, then the final result is weakest too. The next theorem is stronger.

.

92


1 is 2 k k fulfilled and the scaling coefficients have the finite support, then there exists a unique scaling function which satisfies the refinement equation and is correctly defined on the set of dyadic rational numbers. Moreover the following equality

Theorem 2.30

∑h

If the condition (2.69)

2k

= ∑ h2 k +1 =

∑ϕ ( x − k ) = 1

(2.70)

k

is valid. This last condition is called the fundamental condition. Let us show now two theorems on the transition matrix.

Theorem 2.31 If the transition matrix has one eigenvalue equal to 1 and all other eigenvalues are less than 1, then the scaling function belongs to the space of square integrable functions ϕ ( x) ∈ L2 (ℝ) . The essentially stronger theorem exists, but with stronger conditions.

Theorem 2.32 If the scaling coefficients have the finite support and the conditions (2.64)

∑h

k

k

= 2

and (2.67)

∑h h

k k −2n

= δ 0 n are fulfilled,

k

then there exists a scaling function ϕ ( x) ∈ L2 (ℝ) which generates the system of wavelets, and forms a tight frame in L2 (ℝ) . Consider a few simple cases, when the scaling function has a compact support. Case 1. Let N = 2 . Then h0 + h1 = 2 , (h0 ) 2 + (h1 ) 2 = 1 . This case corresponds to the Haar scaling function and the Daubechies scaling function of length 2  1 1  hHa = hD 2 = {h0 , h1} =  , .  2 2

93

Wavelet Analysis

Case 2. Let N = 4 . Then h0 + h1 + h2 + h3 = 2 , (h0 ) 2 + (h1 )2 + (h2 ) 2 + (h3 )2 = 1 , h0 h2 + h1h3 = 0 . When additional parameter α being used, the scaling coefficients can be written in the form h0 =

h2 =

1 2 2 1 2 2

(1 − cos α + sin α ) , (1 + cos α − sin α )

h1 =

, h3 =

1 2 2 1 2 2

(1 + cosα + sin α ) , (1 − cos α − sin α ) .

This case corresponds to the Haar scaling function and Daubechies-2 π 3π scaling function only for values α = 0; ; . The Daubechies-4 2 2 scaling function is realized for α = ( π 3) . In this case the scaling coefficients are as follows 1 + 3 3 + 3 3 − 3 1 − 3  hD 4 = {h0 , h1 , h2 , h3 } =  , , , .  4 2 4 2 4 2 4 2  Case 3. Let N = 6 . Then h0 + h1 + h2 + h3 + h4 + h5 = 2 , (h0 ) 2 + (h1 )2 + (h2 ) 2 + (h3 )2 + (h4 ) 2 + (h5 ) 2 = 1 , h0 h2 + h1h3 + h2 h4 + h3 h5 + h0 h4 + h1h5 = 0 . By means of the additional parameters α and β the scaling coefficients are written in the form h0 = (1 + cos α + sin α )(1 − cos β − sin β + 2sin β cos α )  4 2 , h1 = (1 − cos α + sin α )(1 + cos β − sin β − 2sin β cos α )  4 2 , h2 = (1 + cos(α − β ) + sin(α − β ) ) 2 2 , h3 = (1 + cos(α − β ) − sin(α − β ) ) 2 2 ,

94


h4 = 1

(

2 − h0 − h2 ,

)

h5 = 1

(

2 − h1 − h3 .

)

This case corresponds to the Haar scaling function and Daubechies-2 scaling function for all values α = β . Coefficients for Daubechies-4 scaling function can be obtained for α = ( π 3) and β = 0 only. The Daubechies-6 scaling function corresponds to

α = 1,35980373244182, β = −0,78210638474440 . The next two examples show the refinement equations and scaling coefficients for two simplest families of wavelets.

Example 2.11 Let the scaling function be the box function (that is, the Haar function har0 ( x) ). Then

ϕ H ( x) = har0 ( x) = har0 (2 x) + har0 (2 x − 1) , and the refinement equation and scaling coefficients are as follows h0 =

1 1 1 1 , h1 = , ϕ ( x) = 2ϕ (2 x) + 2ϕ (2 x − 1) . 2 2 2 2

Example 2.12 Let the scaling function be the triangle function (that is, the spline of the first order or the linear spline). Then 1 1 ϕ ∆ ( x) = ∆ ( x) = ∆ (2 x) + ∆(2 x − 1) + ∆(2 x − 2) 2 2 that is, the scaling coefficients and the refinement equation have the form h0 =

2 2

, h1 =

1 1 , h2 = , 2 2 2

1 1 2ϕ (2 x − 1) + 2ϕ (2 x − 2) . 2 2 2 2 2 The widely used MRA notion of resolution can be now commented on this way: when the index j increases, then the linear span of subspace

ϕ ( x) =

1

1

2ϕ (2 x) +

Wavelet Analysis

95

V j increases also, the support of each function ϕ ( 2 j x − k ) decreases and it is translated on lesser step. If the index is being decreased, then the support becomes wider. The scaling function is able to characterize the chosen space by more fine or more coarse details of the function. Here certain analogy can be seen with the accepted physics notion of resolution of a camera or a microscope to distinguish better the image details. That is why sometimes the wavelet analysis is treated as the mathematical microscope.

2.8 Decomposition of the Space L2 ( ℝ ) Let us start with the embedding subspaces V j . The embedding property means that if the function belongs to certain space V j , then it belongs also to all spaces with greater indexes. This is shown schematically on Fig. 2.14. Thus, each subsequent subspace contains all elements of the prior subspace and additionally some elements allow us to distinguish

Fig. 2.14 Scheme of embedded spaces Vm

the details which were not visible for the prior level of scaling. Such considerations permit to build explicitly the series

96


Pj +1 f = Pj f + ∑ f ,ψ j ,k ψ j ,k . k∈ℤ

It is clear from the form of this equality that the new elements are just the wavelets. Let us discuss now the orthogonality of the scaling functions and the wavelets. Let us define the space W j as the orthogonal complement of the space V j in the space V j −1 . This is written as V j −1 = V j ⊕ W j and means that all elements {ϕ j , k ( x)} all elements {ψ j , k ( x)}

j , k∈ℤ

j , k∈ℤ

of the space V j are orthogonal to

of the space W j (it includes the wavelets of

the resolution level j )

ϕ j ,k ( x), ψ j ,k ( x ) = ∫ ϕ j ,k ( x) ψ j , k ( x) dx = 0. A

Fig. 2.15 Scheme of spaces Wm .

Some properties of the spaces W j : Property W1. The spaces W j are orthogonal with each other W j ⊥ Wm

97

Wavelet Analysis J − j +1

WJ − k ( j ≠ m ) , so that the representation V j = VJ ⊕ ⊕ k =0

holds:

V1 = V0 ⊕ W0 , V2 = V1 ⊕ W1 = V0 ⊕ W0 ⊕ W1 , V3 = V2 ⊕ W2 = V0 ⊕ W0 ⊕ W1 ⊕ W2 , and finally L2 = V0 ⊕ W0 ⊕ W1 ⊕ W2 ⊕ ⋯ (in

particular,

for

high

level

of

resolution

2

L = V7 ⊕ W7 ⊕ W8 ⊕ W9 ⊕ ⋯ and for low level of resolution L2 = V−7 ⊕ W−7 ⊕ W−6 ⊕ W−5 ⊕ ⋯ ). Property W2. A decomposition of the space L2 ( ℝ ) through mutually orthogonal subspaces L2 ( R ) = ⊕W j is valid. j∈Z

Property W3. The spaces W j save property 3 of invariance in scaling the spaces V j : f ( x) ∈Wk ⇔ f (2− k x) ∈W0 or f (2k x) ∈Wk ⇔ f ( x) ∈W0 . Since the functions ψ 0,k ( x) = ψ ( x − k ) , from the space W0 belong also to the next (by the scale) space V1 , which has the basis

{ϕ

1, k

( x) = 2ϕ ( 2 x − k )

}

k ∈ℤ

,

⌢ then they should be represented by means of some set of coefficients hk

and using the above basis

⌢

ψ ( x) = ∑ hk 2 ϕ ( 2 x − k ).

(2.71)

k∈ℤ

This formula should be considered, first of all, as the representation of the mother wavelet through the scaling function. In this connection, the scaling function is sometimes called the father wavelet.

98


The

link

between

the

coefficients

ϕ ( x) = ∑ hk 2 ϕ ( 2 x − k ) and

ψ ( x) = ∑ k∈ℤ

k ∈ℤ

follows

of the representation ⌢ hk 2 ϕ ( 2 x − k ) , is as

⌢ hk = (−1)k h1− k ,

(2.72 )

and, more in general, the link between scaling and wavelet coefficients is given by

Theorem 2.33 If: S1. The scaling coefficients fulfill the condition of existence (that is, the weak condition (2.64)

∑h

k

= 2 or the stronger

k

1 are fulfilled and the 2 k k scaling function ϕ ( x) has either the finite support or decreases

condition (2.67)

∑h

2k

= ∑ h2 k +1 =

quickly enough) and the scaling functions are orthogonal

∫ ϕ ( x)ϕ ( x − k )dx = δ

0k

.

W1. The wavelet is defined by the scaling function according to the formula (2.71). Then: S2. The translation of this wavelet ψ ( x − k ) forms the system of wavelets, which is the linear span for the subspace W0 = span {ψ ( x − k )} k ∈ℤ

S3. V1 = V0 ⊕ W0 , that is, the wavelets are orthogonal to the scaling functions ϕ ( x − k ),ψ ( x − m) = 0 , if and only if there exists the ⌢ hk , hk

link between the scaling and wavelet coefficients

⌢ hk = ±(−1) k hN − k ,

(2.73)

99

Wavelet Analysis

where N is an arbitrary odd integer, it can be 1, as shown in (2.72). The main result of this theorem is that the coefficients in formula (2.71) ⌢ ψ ( x) = ∑ hk 2 ϕ ( 2 x − k ) can be computed through the known k∈ℤ

coefficients from formula (2.63) ϕ ( x) = ∑ hk 2 ϕ ( 2 x − k ) . k ∈ℤ

Theorem 2.34 If the conditions (S1 and W1) of theorem 2.33 are fulfilled and additionally: S1*. The translation of the wavelet ψ ( x − k ) forms the system of wavelets, which is the linear span for space W0 = span {ψ ( x − k )} . k ∈ℤ

S1**. V1 = V0 ⊕ W0 , that is, the wavelets are orthogonal to the scaling functions ϕ ( x − k ),ψ ( x − m) = 0 , then the coefficients ⌢ hk , hk fulfill the condition

∑h

k

⌢ hk − 2 n = 0 .

(2.74)

k∈ℤ

The last theorem completes the characterization of coefficients.

the

wavelet

Theorem 2.35 If the scaling coefficients hk satisfy the linear condition (2.64) and the quadratic condition (2.67), then the wavelet coefficients ⌢ ⌢ hk fulfill the condition ∑ hk = 0 and the wavelet fulfills the condition k ∞

∫ ψ ( x)dx = 0 .

−∞

Now we can give a list of the main general properties of the scaling and wavelet coefficients and their corresponding functions. The general properties of scaling functions are:

100


Property S1. The basic system of scaling functions is orthonormal and ∞

the basic scaling function is also normed

∫

ϕ ( x) dx = 1 .

−∞

Property S2. The basic scaling function ϕ ( x) can be represented not only through the set of scaling functions of the next scale

ϕ ( x) = ∑ hk 2 ϕ ( 2 x − k ) , k ∈ℤ

but also through the set of scaling functions of any other scale

ϕ ( x) = ∑ hk( j ) 2 j 2 ϕ ( 2 j x − k ) . k ∈ℤ

(1) k

Here h

= hk and hk( j +1) = ∑ hn( j ) hn( −j )2 k . n∈ℤ

Property S3. The formula for dyadic discretized scaling functions m = 2j . j  

∑ ϕ  2

m∈Z

Property S4. The prior property gives for zero index j = 0 the property of unit partition

∑ ϕ ( m) = 1 . m∈ℤ

Property S5. If the scaling function is continuous, then the more arbitrary property of unit partition is valid

∑ ϕ ( x − m) = 1 . m∈ℤ

The general properties of scaling functions, which depend orthogonality condition are:

on the

2

∞ ∞  2 Property S1 .  ∫ ϕ ( x)dx  = ∫ [ϕ ( x) ] dx .  −∞  −∞

*

∞

Property S2*.

∫ ψ ( x)dx = 0 .

−∞

It should be noted here that the conventional mother wavelet is normed

101

Wavelet Analysis ∞

∫ [ψ ( x)]

2

dx = 1 .

−∞

Property S3*. Not only the wavelets, formed by translation with given scale level, are orthogonal, but also the wavelets of different scales are orthogonal ∞

∫ (2

j 2

ψ ( 2 j x − k ) 2i 2ψ ( 2i x − m ) dx = δ kmδ ij

)(

∀i, j , k , m.

)

−∞

Property S4*. The wavelet and scaling functions of different scales and different translations are also orthogonal ∞

∫ (2

j 2

ψ ( 2 j x − k ) 2i 2 ϕ ( 2i x − m ) dx = 0

)(

)

∀i, j , k , m.

−∞

Property S5*. The Parseval theorem holds true

∑ k∈ℤ

2

ϕˆ (ω + 2kπ =

∞

∫

2

ϕˆ (ω ) d ω =

−∞

∞

∫ ϕ ( x)

2

dx .

−∞

Property S6*. If the scaling functions form the orthonormal basis or the tight frame, then the scaling coefficients can be evaluated by ∞

hk = ϕ ( x), ϕ1,k ( x) = 2 ∫ ϕ ( x)ϕ (2 x − k )dx . −∞

Property S7*. If the scaling functions form the orthonormal basis or tight frame, then the wavelet coefficients can be evaluated by ∞

hk = ψ ( x), ϕ1,k ( x) = 2 ∫ ψ ( x)ϕ (2 x − k )dx . −∞

When the mother wavelet is given, we can then build the family of j 2 j wavelets ψ j ,k ( x) = 2 ψ ( 2 x − k ) .

Thus the explicit algorithm can be used to find the wavelets by the known (chosen at the beginning of the analysis) scaling function.

102


Let us return now to the previous two examples, where the scaling functions were considered for cases of Haar function and the linear spline function, and find the mother wavelets.

Example 2.13 If the scaling function is the Haar function har0 ( x) , then

ψ H ( x) = har0 (2 x) − har0 (2 x − 1) = har1 ( x), ⌢ ⌢ 1 ⌢ 1 that is, the coefficients hk are equal to h0 = h1 = , h1 = −h0 = − . 2 2

Therefore, ψ ( x) =

1 1 2ϕ (2 x) − 2ϕ (2 x − 1). 2 2

Example 2.14 If the scaling function is the triangle function (the linear spline), then 1 1 ψ ∆ ( x) = ∆(2 x) − ∆(2 x − 1) − ∆(2 x + 1) , 2 2 ⌢ that is, the coefficients hk are equal to ⌢ ⌢ ⌢ 1 1 1 h0 = −h2 = − , h1 = h1 = , h2 = −h0 = − . 2 2 2 2 2 Therefore, ϕ ( x) =

1 1 1 2ϕ (2 x) − 2ϕ (2 x − 1) − 2ϕ (2 x + 1) . 2 2 2 2 2

Let us continue with example 1 with the father wavelet in the form of Haar function ϕ ( x) = har0 ( x) , the mother function in the form of Haar function

ψ ( x) = har1 ( x) and build the subspaces of the next scales

V j ,W j ,

as

well

as

the

corresponding

scaling

functions

ϕ j ,k ( x) = 2 j 2 ϕ ( 2 j x − k ) and wavelets ψ j ,k ( x) = 2 j 2ψ ( 2 j x − k ) . Let us remember that for the scale j = 0 the subspace V0 is formed by the functions ϕ ( x − k ) = ϕ0, k ( x) = har0 ( x − k ) , where har0 ( x) is the

103

Wavelet Analysis

Haar function at the interval [ 0;1] , and har0 ( x − k ) are its translations with step 1 to the left and right. har0 ( x) 0

½

1

Fig. 2.16 The plot of har0 ( x) .

Analogously, the subspace W0 is formed by the functions

ψ ( x − k ) = ψ 0,k ( x) = har1 ( x − k ) . The Haar function har1 ( x) , at the interval [ 0;1] , is then translated with a unitary step to the left and right. har1 ( x)

Fig. 2.17 The plot of har1 ( x) .

Consider now the scale j = 1 . The subspace V1 is built here by the procedure V1 = V0 ⊕ W0 on functions of spaces V0 ,W0 . It consists of the functions ϕ1,k ( x) = 2ϕ ( 2 x − k ) = 2 har0 ( 2 x − k ) . They are the Haar functions

2 har0 ( 2 x ) , 2 har0 ( 2 x − 1) at the interval [ 0;1] , which are

translated with the step 1 to the left and right. 2 har0 ( 2x )

2 har0 ( 2 x − 1)

Fig. 2.18 The plots of

2 har0 ( 2x ) and

2 har0 ( 2 x − 1) .

104


Thus the spaces V j ,W j are made on the functions

ϕ j ,k ( x) = 2 j 2 ϕ ( 2 j x − k ) , ψ j ,k ( x) = 2 j 2ψ ( 2 j x − k ) which have the amplitude factors 2 j 2 . But the amplitudes can’t be introduced into this representation, since these amplitudes have no influence on the space structure (it is built by the linear combinations only). Therefore, sometimes the amplitudes aren’t introduced into representation and such representation of linear spans V j = span ϕ ( 2 j x − k ) , k

{

}

W j = span ψ ( 2 j x − k ) k

{

}

can be met. Remember now that the formula V j +1 = V j ⊕ W j has the equivalent representation span 2( k

{

j +1) 2

ϕ ( 2 j +1 x − k )

}

= span 2 j 2 ϕ ( 2 j x − k ) ⊕ span 2 j 2ψ ( 2 j x − k ) . k

{

}

k

{

}

Thus each function from V j +1 can be represented in the form of the direct sum of linear combinations ⌢     f ( x) =  ∑ ln ⋅ 2 j 2 ϕ ( 2 j − n )  ⊕  ∑ ln ⋅ 2 j 2ψ ( 2 j − n )  .  n∈ℤ   n∈ℤ 

This is valid, in particular, for the basis functions 2 har0 (2 x) =

) ( 2 ⋅ har ( x) ) , 2 har (2 x − 1) = ( 2 ⋅ har ( x) ) ⊕ ( − 2 ⋅ har ( x) ) . 0

(

2 ⋅ har0 ( x) ⊕ 0

1

1

The subspace W1 consists of the functions

ψ 1, k ( x) = 2ψ ( 2 x − k ) = 2 har1 ( 2 x − k ) .

Wavelet Analysis

105

At the interval [ 0;1] , it will be two mutually orthogonal Haar functions 2 har1 ( 2 x ) = har2 ( x), 2 har1 ( 2 x − 1) = har3 ( x) , which are translated with the step 1 to the right. 2 har1 ( 2x )

2 har1 ( 2 x − 1)

Fig. 2.19 The plots of

2 har1 ( 2x ) and 2 har1 ( 2 x − 1) .

Consider the next scale j = 2 . The subspace V2 is built by the procedure V2 = V0 ⊕ W0 ⊕ W1 = V1 ⊕ W1 on functions of spaces of more coarse scales. It consists of functions

ϕ2,k ( x) = 2ϕ ( 4 x − k ) = 2 har0 ( 4 x − k ) At the interval [ 0;1] , there

will be four mutually orthogonal Haar

functions 2 har0 ( 4 x ) , 2 har0 ( 4 x − 1) , 2 har0 ( 4 x − 2 ) , 2har0 ( 4 x − 3) , which are translated with the step 1 to the right. 2 har0 ( 4x )

2 har0 ( 4 x − 1)

106


2 har0 ( 4 x − 2 )

2 har0 ( 4 x − 3)

Fig. 2.20 The plots of 2 har0 ( 4 x ) , 2 har0 ( 4 x − 1) ,2 har0 ( 4 x − 2 ) ,2 har0 ( 4 x − 3) .

In accordance with the formula V2 = V0 ⊕ W0 ⊕ W1 = V1 ⊕ W1 , we have 2 har0 (4 x) =

(

2 har0 (4 x − 1) =

2 ⋅ 2 har0 (2 x) ⊕

( 2 har (4 x − 2) = ( 2 har (4 x − 3) = (

2⋅

0

2⋅

0

2⋅

) ( 2 ⋅ 2 har ( x) ) , 2 har (2 x) ) ⊕ ( − 2 ⋅ 2 har ( x) ) , 2 har (2 x − 1) ) ⊕ ( 2 ⋅ 2 har ( x) ) , 2 har (2 x − 1) ) ⊕ ( − 2 ⋅ 2 har ( x) ) . 2

0

2

0

3

0

3

The subspace W2 consists of the functions

ψ 2,k ( x) = 2ψ ( 4 x − k ) = 2 har1 ( 4 x − k ) . At the interval

[0;1] ,

there will be four mutually orthogonal Haar

functions 2 har1 ( 4 x ) = har4 ( x),

2 har1 ( 4 x − 1) = har5 ( x),

2 har1 ( 4 x − 2 ) = har6 ( x), 2 har1 ( 4 x − 3) = har7 ( x), which are translated with the step 1 to the right. har4 ( x)

107

Wavelet Analysis har5 ( x)

har6 ( x)

har7 ( x)

Fig. 2.21 The plots of 2 har1 ( 4 x ) ,2 har1 ( 4 x − 1) , 2 har1 ( 4 x − 2 ) ,2 har1 ( 4 x − 3) .

Finally, consider the scale j = 3 and the space V3 only. It is built by the procedure V3 = V0 ⊕ W0 ⊕ W1 ⊕ W2 = V1 ⊕ W1 ⊕ W2 = V2 ⊕ W2 on functions of more coarse scale spaces and consists of the functions

ϕ3,k ( x) = 2 2ϕ ( 8 x − k ) = 2 2 har0 ( 8 x − k ) At the interval

[0;1] ,

.

there will be eight mutually orthogonal Haar

functions 2 2 har0 ( 8 x ) , 2 2 har0 ( 8 x − 1) , 2 2 har0 ( 8 x − 2 ) , 2 2 har0 ( 8 x − 3) , 2 2 har0 ( 8 x − 4 ) , 2 2 har0 ( 8 x − 5 ) , 2 2 har0 ( 8 x − 6 ) , 2 2 har0 ( 8 x − 7 ) , which are translated instances with step 1 to the right.

108


2 2 har0 ( 8x )

2 2 har0 ( 8 x − 2 )

2 2 har0 ( 8 x − 4 )

2 2 har0 ( 8 x − 6 )

Fig. 2.22 The plots of 2 har1 ( 4 x ) ,2 har1 ( 4 x − 1) , 2 har1 ( 4 x − 2 ) ,2 har1 ( 4 x − 3) .

In accordance with the formula V3 = V2 ⊕ W2 2 2 har0 (8 x) = 2 2 2 2 2

( 2 ⋅ 2 har (4 x) ) ⊕ ( 2 ⋅ 2 har ( x) ) , 2 har (8 x − 1) = ( 2 ⋅ 2 har (4 x) ) ⊕ ( − 2 ⋅ 2 har ( x) ) , 2 har (8 x − 2) = ( 2 ⋅ 2 har (4 x − 1) ) ⊕ ( 2 ⋅ 2 har ( x) ) , 2 har (8 x − 3) = ( 2 ⋅ 2 har (4 x − 1) ) ⊕ ( − 2 ⋅ 2 har ( x) ) , 2 har (8 x − 4) = ( 2 ⋅ 2 har (4 x − 2) ) ⊕ ( 2 ⋅ 2 har ( x) ) , 2 har (8 x − 5) = ( 2 ⋅ 2 har (4 x − 2) ) ⊕ ( − 2 ⋅ 2 har ( x) ) , 0

0

4

0

0

0

0

0

0

0

0

0

4

5

5

6

6

109

Wavelet Analysis

2 2 har0 (8 x − 6) = 2

( 2 har (8 x − 7) = ( 0

2 ⋅ 2 har0 (4 x − 3) ⊕

) ( 2 ⋅ 2 har ( x) ) , 2 ⋅ 2 har (4 x − 3) ) ⊕ ( − 2 ⋅ 2 har ( x) ) . 7

0

7

Thus, the choice of Haar functions for mother and father wavelets enables us to build two systems of orthonormal functions, one of which consists of scaled and translated Haar zero index functions, whereas the other system is simply the system of Haar functions.

2.9 Discrete Wavelet Transform. Analysis and Synthesis The sets of functions {ϕk ( x)}k∈ℤ (scaling function based on the father wavelet) and {ψ j , k ( x)}

j , k∈ℤ

(wavelet functions based on the mother

wavelet) are the linear span for all functions from L2 ( ℝ ) . By using these sets, an arbitrary function f ( x) ∈ L2 ( ℝ ) can be represented as the sum of two series expansions f ( x) = ∑ ck ϕk ( x) + k∈ℤ

∑ ∑d

jk

ψ j ,k ( x) .

(2.75)

j∈ℕ ∪{0} k∈ℤ

If both systems of functions are orthonormal or form a tight frame, then the coefficients in expansions (2.75) can be evaluated by ∞

ck = f ( x),ϕ k ( x) =

∫

f ( x)ϕk ( x)dx ,

(2.76)

f ( x)ψ j ,k ( x)dx .

(2.77)

−∞ ∞

d jk = f ( x),ψ j ,k ( x) =

∫

−∞

First, it should be noted that when the systems aren’t orthogonal and don’t form the tight frame, then other approaches are needed. Second, representation (2.75)-(2.77) are here declared only, but the strict proof exists.

110


Third, these three formulas are final and basic in MRA. The first formula isn’t the most general. It is assumed here that the initial resolution level corresponds to the value j = 0 . The general formula should begin from the arbitrary value j0 and corresponding to this index resolution ∞

f ( x) = ∑ c j0 ,k ϕ j0 ,k ( x) + ∑ ∑ d j ,k ψ j , k ( x) k∈ℤ

(2.78)

j = j0 k∈ℤ

or ∞

f ( x) = ∑ c j0 ,k 2 j0 2 ϕ (2 j0 x − k ) + ∑ ∑ d j , k 2 j 2ψ (2 j x − k ). k∈ℤ

(2.79)

j = j0 k∈ℤ

The first sum is the projection of function on the space V j0 . It gives the coarse approximation of the function or, in other words, the representation of the function based on low resolution. The second sum starts with the first index j0 , and the more members j are taken into account, the more higher resolution level will be reached. The set of sum members with fixed index j is the projection of function on the details space W j . The size of details of function, which will be seen and taken into account in the representation of the two sums can be decreased by increasing the level of resolution in the second sum. But at different intervals in the time domain (at different fragments of the signal) the different resolution level can be chosen. This is one of the basic features of the wavelet transform. Some analogy with the Fourier series can be seen when allowance for more and more high harmonics enables fixing the finer details of the signal. The totality of coefficients c j0 ,k , d j ,k

{

wavelet transform of the function f ( x) .

}

j , k∈ℤ

is called the discrete

111

Wavelet Analysis

The wavelet coefficients fully characterize the function and like the Fourier series coefficients (the discrete Fourier transform) can be used first of all for analysis, description and filtering the signals. Daubechies has shown that the wavelet systems form an unconditional basis for wide class of signals. This means, in particular, that coefficients in the wavelet expansions are becoming very small by increasing the indices and for adopted approximation of the signal using a small enough number of coefficients. For this reason, the discrete wavelet transform is very effective for compression of signals and images. The similarity between discrete wavelet and Fourier transforms is very high, but wavelets are more flexible, informative and in addition they are able to localize the many features of the signal (function). Let us consider now, more in details, the methods of analysis and synthesis of a function by means of the coefficients c j ,k , d j ,k .

2.9.1 Analysis: transition from the fine scale to the coarse scale From equation (2.63), which links the scaling function (father wavelet) and the scaling functions of the finer scale,

ϕ ( x) = ∑ hk ϕ1,k = ∑ hk 2 ϕ ( 2 x − k ) , hk = ϕ ,ϕ1,k k ∈ℤ

k ∈ℤ

let us change x into 2 j x − n to obtain

ϕ (2 j x − n) = ∑ hk 2 ϕ 2 ( 2 j x − n ) − k k∈ℤ

(

)

= ∑ hk 2 ϕ ( 2 j +1 x − 2n − k ) . k ∈ℤ

Rename the translation index 2n + k = m ,

112


ϕ (2 j x − n) = ∑ hm − 2 n 2 ϕ ( 2 j +1 x − m ) . m∈ℤ

Remember now that V j = span 2 j 2 ϕ ( 2 j x − k ) , W j = span 2 j 2ψ ( 2 j x − k ) k

{

}

k

{

}

and respectively

span 2( j +1) 2 ϕ ( 2 j +1 x − k ) k

{

}

= span 2 j 2 ϕ ( 2 j x − k ) ⊕ span 2 j 2ψ ( 2 j x − k ) . k

{

}

k

{

If the function is projected into the space V j +1 of the scale

}

( j + 1) , then

the result has the form of expansion by the scaling functions only and the wavelets are absent f ( x) ≈ PjV+1 f ( x) = ∑ c j +1, k 2( j +1) 2 ϕ (2 j +1 x − k ).

(

)

k∈ℤ

When we go to a finer scale, the wavelets should be taken into account. Then the details will be available, which are absent on this scale level

( P f ) ( x) = ∑ c V j +1

j ,k

2 j 2 ϕ (2 j x − k ) + ∑ d j ,k 2 j 2ψ (2 j x − k ) .

k ∈ℤ

k∈ℤ

Evaluate the coefficients c j ,k , d j ,k in both sums through coefficients c j +1, k from the prior sum c j , k = f ( x),ϕ j , k ( x) = ∫ f ( x) 2 j 2 ϕ ( 2 j x − k ) dx A

= ∫ f ( x) 2 A

j 2

∑ hm−2k m∈ℤ

2 ϕ ( 2 j +1 x − m ) dx

113

Wavelet Analysis

=

hm − 2 k ∫ f ( x) 2( j +1) 2 ϕ ( 2 j +1 x − m ) dx ∑ m∈ℤ A

=

∑ hm−2k c j +1,m → c j,k = m∑∈ℤ hm−2k c j +1,m .

(2.80)

m∈ℤ

Formula (2.80) gives the necessary relationship for recalculating the scaling coefficients of the coarse level, when the scaling coefficients of the next fine level are known. A similar relationship will be held for the wavelet coefficients ⌢

d j ,k =

∑ hm−2k c j +1,m . m∈ℤ

(2.81)

2.9.2 Synthesis: transition from the coarse scale to the fine scale Let us apply anew the formula for the projection of a function onto the space V j +1 in the form where the wavelets are used

∑c

j +1, k

2( j +1) 2 ϕ (2 j +1 x − k )

k∈ℤ

= ∑ c j ,k 2 j 2 ϕ (2 j x − k ) + ∑ d j , k 2 j 2ψ (2 j x − k ) . k∈ℤ

k∈ℤ

Then we change (in the scaling functions and wavelets) to the scale on j unit more by changing the variable x on 2 x − n :

ϕ (2 j x − k ) = ∑ hn 2ϕ 2 ( 2 j x − k ) − n = ∑ hn 2ϕ ( 2 j +1 x − 2k − n ) ,

(

n∈ℤ

)

n∈ℤ

⌢

⌢

ψ (2 j x − k ) = ∑ hn 2ϕ 2 ( 2 j x − k ) − n = ∑ hn 2ϕ ( 2 j +1 x − 2k − n ) . n∈ℤ

So that

(

)

n∈ℤ

114


c j +1,k 2( j +1) 2 ϕ (2 j +1 x − k ) ∑ k∈ℤ   = ∑ c j ,k 2 j 2  ∑ hn 2 ϕ ( 2 j +1 x − 2k − n )  k∈ℤ  n∈ℤ  ⌢   + ∑ d j ,k 2 j 2  ∑ hn 2 ϕ ( 2 j +1 x − 2k − n )  k∈ℤ  n∈ℤ    = ∑ c j , k  ∑ hn 2( j +1) 2 ϕ ( 2 j +1 x − 2k − n )  k∈ℤ  n∈ℤ  ⌢   + ∑ d j ,k  ∑ hn 2( j +1) 2 ϕ ( 2 j +1 x − 2k − n )  ⇒ k∈ℤ  n∈ℤ  ⌢ c j +1, k = ∑ c j ,m hk − 2 m + ∑ d j ,m hk − 2 m . m∈ℤ

(2.82)

m∈ℤ

Thus formula (2.82) gives the necessary relationship for recalculating the scaling coefficients of the fine level, when the scaling and wavelet coefficients of the prior coarse level are given. If the scaling functions and wavelets form the orthonormal basis or the tight frame, then the Parseval formula is valid in the form f ( x) = ∑ cm0 ,n + 2

2

n∈ℤ

∞

∑∑d

2 m,n

.

(2.83)

m = m0 n∈ℤ

This formula permits us to separate the signal energy: for each fixed scale level (the index m is fixed) and for each fixed time interval or each local fragment of function (the index n is fixed). We should comment on the fact that the discrete wavelet transform saved all features of the window Fourier transform, but it uses additionally the variable scale. The window Fourier transform links the resolutions by time and frequency and the increasing of one causes the decreasing of the other.

Wavelet Analysis

115

The discrete wavelet transform doesn’t involve this restriction. Let us continue the analysis of functions separately by the scale level and the localization by time and represent formula (2.75) in the new form, where the initial scale index j0 is fixed ∞

f ( x) = f j0 ( x) +

∑ j = j +1

f j ( x),

(2.84)

0

f j0 ( x) = ∑ c j0 ,k 2 j0 2 ϕ (2 j0 x − k ),

(2.85)

k∈ℤ

f j ( x) = ∑ d j ,k 2 j 2ψ (2 j x − k ).

(2.86)

k∈ℤ

Formulas (2.84)-(2.86) show the procedure of approximate evaluation of the function, when the function is evaluated more exactly owing to allowance for more fine details of this function. Formula (2.85) explains in which way the coarse approximation appears, by the choice of the level j0 . As it can be seen from (2.84) and (2.86), the quality in resolution of details will depend on the number used in levels of fine resolutions. Let us represent now formula (2.75) in another form, where the localization index k is fixed f ( x) = ∑ f ktransl ( x) ,

(2.87)

k∈ℤ

∞

f ktransl ( x) = c j0 ,k 2 j0 2 ϕ (2 j0 x − k ) + ∑ d j ,k 2 j 2ψ (2 j x − k ) .

(2.88)

j = j0

Formulas (2.87)-(2.88) show the procedure of evaluating the function when this function is being translated from one interval (one window) to

116


another interval (another window). The observed pictures in different windows can be very different. This can be explained by the different choice of the initial (coarse) resolution and the number of fine levels of resolution. It seems worthy to note the next two facts. Fact 1 Transition to higher levels of resolution enables the distinguishing of finer details of the function. This is likely to be the window Fourier transform and the novelty consists here in the possibility of localization by coordinate. Fact 2 As it was fixed in many computer evaluations of the functions (signals, images), the use of detail spaces gives the special information about the function. For example, in the image processing for the low level of resolution, the detail space gives the approximate image of houses and shows their width with accuracy. When the level of resolution is being increased, the detail spaces fix very exactly the house features in the space (angles, discontinuities etc.).

2.10 Wavelet Families In this subsection some of the most known existing wavelets will be considered. Each of them generates its own system (family) of wavelets. At the beginning, a list of common properties for all wavelets will be commented. It is worthy to add at once that, as a rule, some of them don’t fulfill each concrete wavelet. 1. Orthogonality – It is accepted that such a demand is redundant and often this condition is weakened.

117

Wavelet Analysis

2. Presence of the compact support – Only part of wavelet families has this property. This is a favourable property, which simplifies both the theoretical and numerical analysis. In the signal theory this is defined by the filter length. 3. Rational coefficients – This is very welcome in computer applications. 4. Symmetry – An absence of this property causes some complications. For example, when the symmetry is absent, then the mirror image of the Fourier transform of the function isn’t the Fourier transform of the mirror image of the function. 5. Smoothness – It is defined by the number of zero moments of the wavelet function. For example, the first moment defines the smoothness of function reconstruction. 6. Interpolation – This property sometimes appears, when it is needed that some values of the function can be exactly interpolated.

2.10.1 Haar wavelet The scaling function for the Haar wavelet is the Haar function (2.18)

 1  har( x) =  −1  0 

x ∈ [0;1 2), x ∈ [1 2;1), x ∉ [0;1) .

By using this function we can define the Haar wavelets j

har jk ( x) = 2

2

har(2 j x − k )

 k k + 1 supp har jk ( x) =  j ; j  . 2 2 

(k, j ∈ ℤ) .

(2.89)

118


The index j defines the scale level of a given wavelet, on which the unit interval is divided into 2 j equally spaced intervals. Dyadic intervals (of close levels) have two important properties: Property 1. These intervals either don’t intersect or one interval is a part of the other. Property 2. If one interval is included into the other one, then it is either the right half or the left half of the last one. j

Theorem 2.36 The system

{

har jk ( x) = 2

2

har(2 j x − k )

of Haar

}

k , j∈ℤ

wavelets is orthonormal in L2 ( ℝ ) . The proof is direct and consists in the analysis of the scalar product ∞

har jk , harˆjkˆ

L2 ( ℝ )

=

∫2

j 2

ˆ ˆ har ( 2 j x − k ) ⋅ 2 j 2 har 2 j x − kˆ dx.

(

)

−∞

When both indexes are equal (wavelets are identical), it will be 1. When the scale levels are equal j = ˆj , but translations are different, the supports don’t intersect and integral is equal zero. And so on. j

Theorem 2.37 The system

{

har jk ( x) = 2

2

har(2 j x − k )

}

of Haar

k , j∈ℤ

wavelets is an orthonormal basis in L2 ( ℝ ) . To prove this it is necessary to introduce the family of closed spaces L2 ( ℝ ) :

{S n }

S n = span {har jk } j < n, k∈ℤ ,

... ⊂ S −1 ⊂ S0 ⊂ S1 ⊂ ... ,

f ( x) ∈ S n ⇔ f (2 x) ∈ Sn +1 , f ( x) ∈ S0 ⇔ f ( x + k ) ∈ S0 , k ∈ ℤ ,

  span ( S ) = ∑ ak xk , ak ∈ ℝ, xk ∈ S  k 

119

Wavelet Analysis

⌢ ⌢ (this is the subspace S of the space S , S ⊂ S , which consists of linear

combinations of all elements from S ). Further it can be proven that these subspaces include all functions, which are constant on dyadic intervals, and

∞

∪S

n

is dense in L2 ( ℝ ) .

n =−∞

Moreover the intersection of all subspaces is the empty set. So that the decomposition of the function f ( x) can be written through the Haar basis f ( x) = ∑

∑

f ,har jk har jk ( x)

(2.90)

j∈ℤ k ∈ℤ

and this series is convergent to f ( x) in L2 ( ℝ ) . The main disadvantage of the Haar wavelet is its low regularity. Plots of Haar functions ϕ ( x),ψ ( x) , already shown in prior subsections, are as follows

Fig. 2.23 The functions

ϕ ( x),ψ ( x) for Haar wavelets.

120


2.10.2 Strömberg wavelet It is described mathematically in a more complicated form and isn’t defined on a compact support. To define the wavelet, it is necessary to introduce four sets on the real axis ℝ : ℤ + = {1, 2,3,...}

,

ℤ − = −ℤ + ,

1  A1 = A0 ∪   . 2 V, the space

1 A0 = ℤ + ∪ {0} ∪ ℤ − , 2 Further

on

the

discrete

set

S (V ) of

all

2

functions f ( x) ∈ L ( ℝ ) is given. These functions are: S1. Continuous on ℝ . S2. Linear on all intervals I ∈ ℝ , I ∩ ℝ = ∅ . According to this definition, S ( A0 ) ⊂ S ( A1 ) , both spaces are closed subspaces of the space L2 ( ℝ ) . Then an arbitrary function f ∈ S ( A1 ) can be represented as the sum f = g + 1( x). Here g ∈ S ( A0 ), f = g ∀x ∈ A0 ,

 1 x = 1 2, 1( x) =   0 x ∈ A0 . The plot of this function can be as follows 1( x) ∈ A1 ,

-2

-1

0

½

1

Fig. 2.24 The plot of an arbitrary function f ∈ S ( A1 ) .

2

121

Wavelet Analysis

The Strömberg mother wavelet is the function S ( x) fulfilling these three conditions: MS1. S ( x) ∈ S ( A1 ) . MS2. S ( x)

L2 ( ℝ )

=1.

(2.91)

MS3. S ( x) is orthogonal to S ( A0 ) . It should be noted that the first two conditions are evident and simple, whereas the third one isn’t trivial.

Theorem 2.38 Strömberg wavelets

j

{

S jk ( x) = 2 2 S (2 j x − k )

}

, based

k , j∈ℤ

on (2.91), form an orthonormal basis in L2 ( ℝ ) . The proof is similar to the one used for the Haar system and it is based on the direct analysis of the scalar product ∞

S jk , S ˆjkˆ

L2 ( ℝ )

=

∫2

j 2

S ( 2 j x − k ) ⋅ 2 j 2 S 2 j x − kˆ dx. ˆ

ˆ

(

)

−∞

With the change u = 2 j x − k , s = ˆj − j, kɶ = 2 s k − kˆ , we have ∞

S jk , S ˆjkˆ

L2 ( ℝ )

= 2s 2

∫ S ( x) S (2

s

x − kɶ dx.

)

−∞

Two cases are usually considered: equal scales and equal translations. In both cases, the second multiplier under the integral has the property S 2 s x − kɶ ∈ S ( A0 ) , which in accordance with the third property from

(

)

definition (2.91) provides the orthogonality.

122


Let us show three additional facts, which can be accurately proven. Fact 1.

∪ S (2

−n

ℤ ) is dense in L2 ( ℝ ) .

∩ S (2

−n

ℤ) = 0 .

n∈ℤ

Fact 2.

n∈ℤ

Fact 3.

S ( 2− n −1 Z ) = S ( 2− n Z ) ⊕ span {Snk }k∈ℤ .

There follows that the Strömberg system of wavelets forms an orthonormal basis in L2 ( ℝ ) .

Theorem 2.39 (Wojtaszczyk) The Strömberg wavelet takes at points of the set A1 such values: S ( k ) = S (1)

(

3−2

k −1

)

, k ∈ ℤ+

1 1  S   = − S (1)  3 +  , S ( 0 ) = S (1) 2 3 − 2 2 2  k  k S  −  = − S (1) 2 3 − 2 3 − 2 , k ∈ ℤ+ .  2

(

(

Comment S1

Since

)(

)

)

3 − 2 < 0 , then the Strömberg wavelet changes

sign between two consecutive points of the set A1 and describes as if the damped oscillations. Comment S2 It can be shown that the Strömberg wavelet is the exponentially decreasing function, that is, such constant C and such α > 0 exist, that 1 S ( x) ≤ C e −α | x| ∀x ∈ ℝ α = − ln 2 − 3 ≈ 0.658 . 2

(

)

Comment S3 The Strömberg wavelet has the next smoothness property S ( x) ∈ Lp (ℝ ),

1 ≤ p ≤ ∞.

123

Wavelet Analysis

Comment S4 The Strömberg wavelet has some property of symmetry

 k S  −  = 10 − 6 3 S (k ),  2

(

k ∈ ℤ+ .

)

We will only discuss briefly the next 13 wavelet families.

2.10.3 Gabor wavelet This wavelet is formed by means of the Gabor window function. The function f ( x) is considered in the window of width b only. If the window function is Gaussian, then it will be the Gabor’s original approach 2

wk ,l (t ) = eiα kt w(t − bl ) = eiα kt e− ( t −bl ) , k ∈ ℕ, l ∈ ℤ .

(2.92)

This wavelet might cause some complications in calculations. Moreover, well-known theorem states that if the window function is regular enough, well localized, and more accurate, and if the next two integrals are finite ∞

2 2 ∫ (1+ | t | ) | w(t ) | dt ,

∞

−∞

−∞

∫ (1+ | ω | ) | wˆ (ω ) | 2

2

dω ,

then the Gabor wavelets will never form an orthonormal basis in L2 ( ℝ ) . Inconveniences in using the Gabor wavelets were avoided by proposing new wavelets.

2.10.4 Daubechies –Jaffard–Journé wavelet In this case the window function w(t ) and its Fourier transform decrease exponentially. These wavelets form (in contrast to the Gabor wavelets) an orthonormal basis in L2 ( ℝ ) :

124


k  uk ,l (t ) = 2w(t − 2lπ ) cos  t  , l ∈ 2ℤ, k ∈ ℕ, 2  u0,l (t ) = w(t − 2lπ ) , l ∈ 2ℤ , k = 0 , k  uk ,l (t ) = 2w(t − 2lπ )sin  t  , 2 

(2.93)

l ∈ 2ℤ + 1, k ∈ ℕ.

2.10.5 Gabor – Malvar wavelet Independently on prior authors (Daubechies – Jaffard – Journé) Malvar proposed some wavelets having the same algorithmic nature

  1  wk ,l (t ) = w(t − l ) cos π  k +  (t − l )  , 2   

k ∈ ℕ, l ∈ ℤ,

(2.94)

or

k  wk ,l (t ) = 2 w(t − 2π l ) cos  t  , 2  k  wk ,l (t ) = 2 w(t − 2π l )sin  t  , 2 

k ∈ ℕ , l ∈ 2ℤ , k ∈ ℕ, l ∈ 2ℤ + 1,

but the window choice seems to be simpler and clearer. Malvar proposed three conditions for the choice of the function w(t ) : M1. w(t ) = 0 t ≤ −π , t ≥ 3π . M2. 0 ≤ w(t ) ≤ 1, w(2π − t ) = w(t ). M3. w2 (t ) + w2 (−t ) = 1 t ∈ [−π , π ]. These wavelets form an orthonormal basis in L2 ( ℝ ) . The window function in the Gabor-Malvar wavelet can be very regular (for example, it can be infinite differentiable), but its Fourier transform can’t have the exponential decay. This is forbidden by condition M1.

125

Wavelet Analysis

2.10.6 Daubechies wavelet These wavelet families are distinguished by the indices and are built by the general formula

ψ j ,k (t ) = 2 j 2ψ ( 2 j t − k ) j , k ∈ ℤ . Theorem 2.40 (Daubechies)

If ψ ( x) is the wavelet with p zero

moments, which generates the orthonormal basis in L2 ( ℝ ) , then it has the support with the size greater than or equal to 2 p − 1 . The Daubechies wavelet has the support minimal size equal to [− p + 1, p ] . The support of the corresponding scaling function ϕ ( x) is equal to [0, 2 p − 1] . In particular, the wavelet with p = 1 is the Daubechies-2 wavelet or the Haar wavelet. N=4 φ

ψ

1

1 0.8 0.6

0.5

0.4

0 0.2 0

-0.5

-0.2 0

1

2

3

4

5

6

-3

7

-2

-1

0

1

2

3

4

N=6 φ

ψ

1

1

0.8 0.5

0.6 0.4

0 0.2 0

-0.5

-0.2 -1

-0.4 0

2

4

6

8

10

-4

-2

0

2

4

6

126


N=8 φ

ψ 1

1 0.8

0.5 0.6 0.4 0 0.2 0

-0.5

-0.2 -1

-0.4 0

2

4

6

8

10

12

14

-6

-4

-2

0

2

4

6

0

2.5

5

7.5

8

N=10 φ

ψ

1 0.75 0.8 0.5 0.6 0.25 0.4

0

0.2

-0.25

0

-0.5

-0.2

-0.75

-0.4

-1 0

2.5

5

7.5

10

12.5

15

-7.5

17.5

-5

-2.5

10

Fig. 2.25 Daubechies wavelets.

Thus, the Daubechies wavelets are different by the chosen number of zero moments of the scaling function or, what is simpler, by the indices which can only be even. The most commonly used wavelets correspond to indices 6, 8, 10, 12. Figure 2.25 shows the scaling function and wavelets (mother and father wavelets) for D 4, D 6, D8, D10.

2.10.7 Grossmann-Morlet wavelet This wavelet family is a system of complex value non-orthogonal wavelets. The Fourier transform of the wavelet is formed as the shifted Gaussian with condition ϕˆ (0) = 0 :

ψ ( x) =

1 4

(e π

− iξ o x

2

− e −ξo

2

)e

− xo2 2

,

(2.95)

127

Wavelet Analysis

ψˆ (ξ ) =

1 4

2

(e π

− (ξ −ξ o ) 2

2

2

− e −ξ 2 e −ξo

2

)e

− xo2 2

.

For the wavelet choice, a special procedure exists, which gives the value of parameter ξ o ≈ 5,3364... . Sometimes the value ξ o = 5 is taken. 0.6

0.6

0.4

0.4

0.2

0.2

0

0

-0.2

-0.2 -0.4

-0.4 -0.6

-0.6 -4

-2

0

2

4

-4

-2

0

2

4

Fig. 2.26 Grossmann-Morlet wavelets.

2.10.8 Mexican hat wavelet This wavelet will be described in more details in chapter 6. The Mexican hat wavelet is obtained as the second derivative of the Gaussian and is given by

ψ ( x) =

 2 x2  2 2 π −1 4 1 − 2  e − x / 2σ . 3σ  σ 

(2.96)

−ψˆ ( x )

−ψ ( x ) 1 0.8

1.5

0.6

1

0.4 0.2

0.5 -4

-2

2

4

-0.2

-4

-2

2

-0.4

-0.5

4

128


ϕ ( x)

ϕˆ ( x)

0.8

1.5 1.25

0.6 1 0.75

0.4

0.5

0.2 0.25

-4

-4

-2

2

-2

2

4

4

Fig. 2.27 Mexican hat mother and father wavelets and Fourier transform.

2.10.9 Coifman wavelet – coiflet This is a family of wavelets (like Daubechies family), for which the first S moments of the wavelet ψ ( x) vanish ∞

∫ ψ ( x)dx =

∞

−∞

−∞

∫

∞

xψ ( x)dx = ⋯ =

∫x

ψ ( x) dx = 0

(2.97)

ϕ ( x)dx = 0 .

(2.98)

S −1

−∞

and similarly for the scaling functions ∞

∫ ϕ ( x)dx = 1,

∞

−∞

−∞

∫

∞

x ϕ ( x)dx = ⋯ =

∫x

S −1

−∞

S is called the order of Coifman wavelet. On the next plots the Coifman wavelets of the order 2, 4, 6, 8, 10 are shown.

129

Wavelet Analysis

L=2 φ

ψ

1.5 2 1.5

1 1 0.5

0.5 0 -0.5

0

-1

-2

-1

0

1

2

-3

3

-2

-1

0

1

2

L=4 φ

ψ

1.2

1.5

1 1 0.8 0.5

0.6 0.4

0 0.2 -0.5

0 -0.2 -4

-2

0

2

4

6

-6

-4

-2

0

2

4

L=6 φ

ψ

1 1 0.8 0.5

0.6 0.4

0 0.2 -0.5

0 -0.2 -5

-2.5

0

2.5

5

7.5

10

-7.5

-5

-2.5

0

2.5

5

7.5

L=8 φ

ψ

1 1 0.8 0.5

0.6 0.4

0 0.2 -0.5

0 -0.2 -5

0

5

10

15

-10

Fig. 2.28 Coifman wavelets.

-5

0

5

10

130


2.10.10 Malvar – Meyer – Coifman wavelet This wavelet is given by the formula

u j ,k (t ) =

π 2 w j (t ) cos  lj  l j

 1   k +  (t − a j )  , 2  

k ∈ 0 ∪ ℕ, l ∈ ℤ.

(2.99)

To choose the function w(t ) , it is necessary to fulfill five conditions. Mother and father wavelets ϕ ,ψ have the next plots.

Fig. 2.28 Malvar-Meyer-Coifman mother and father wavelets.

2.10.11 Shannon wavelet or sinc-wavelet This wavelet was analyzed partially in prior subsections. The basic formula for the Shannon wavelet is as follows sin

ψ T ,k ( x) =

π T

π T

( 2x − k )

(2x − k )

.

(2.100)

131

Wavelet Analysis

φ

ψ

1

1 0.75

0.8

0.5 0.6

0.25 0.4

0 0.2

-0.25 -0.5

0

-0.75

-0.2 -10

-5

0

5

10

-10

-5

0

5

10

Fig. 2.29 Shannon mother and father wavelets.

Shannon wavelet is characterized by a slow decay in the time domain. More general representation in the form developed by Newland harmonic wavelets will be described in chapter 5. 2.10.12 Cohen-Daubechies-Feauveau wavelet This is a biorthogonal system consisting of two scaling functions and two wavelets

ψ j ,k (t ) = 2 j 2ψ ( 2 j t − k ) , j , k ∈ ℤ,

(2.101)

ψɶ j ,k (t ) = 2 j 2ψɶ ( 2 j t − k ) , j , k ∈ ℤ .

(2.102)

Plots of these functions ϕ , ψ , ϕɶ , ψɶ are as follows:

ϕ (t )

ϕɶ ( t )

132


ψɶ ( t )

ψ (t )

Fig. 2.30 Cohen-Daubechies-Feauveau mother and father wavelets.

2.10.13 Geronimo-Hardin-Massopust wavelet This system of wavelets is based on the fractal interpolation of functions. This maybe the most complicated procedure in the organization of scaling and wavelet functions. Here the plots of scaling functions ϕ1 ,ϕ 2 and wavelets ψ 1 ,ψ 2 are shown.

ϕ1

ϕ2

133

Wavelet Analysis

ψ1

ψ2

Fig. 2.31 The plots of Geronimo-Hardin-Massopust mother and father wavelets.

2.10.14 Battle-Lemarié wavelet The Battle-Lemarié family of wavelets is based on the B-spline. The mother wavelet is defined according to the condition 1 1 ϕ ( x) = ϕ ( 2 x + 1) + ϕ (2 x) + ϕ ( 2 x − 1) . (2.103) 2 2 If the spline is linear, then the scaling function has the form

 1 − x 0 ≤ x ≤ 1,  ϕ ( x) = 1 + x − 1 ≤ x ≤ 0,  0 x ∉ [−1;1] .  If the spline is quadratic, then the scaling function is

(2.104)

134

Wavelet and Wave Analysis 2  (1 2 )( x + 1) − 1 ≤ x ≤ 0,  2  ( 3 4 ) − ( x − (1 2 ) ) 0 ≤ x ≤ 1, ϕ ( x) =   (1 2 )( x − 2 )2 1 ≤ x ≤ 2,  0 x ∉ [−2; 2] . 

(2.105)

If the spline is cubic, then the scaling function has the form 3  (1 6 )( x + 2 ) − 2 ≤ x ≤ −1,  2  ( 2 3) − x 1 + ( x 2 )  − 1 ≤ x ≤ 0,  ϕ ( x) =  ( 2 3) − x 2 1 − ( x 2 )  0 ≤ x ≤ 1,  3 1 ≤ x ≤ 2, (1 6 )( x − 2 )   0 x ∉ [−2; 2] . 

(2.106)

The plots of scaling functions for linear, quadratic and cubic splines are shown in the next figure.

1

0.7 0.8

0.6

0.6 0.5 0.5 0.4

0.6

0.4 0.3 0.3

0.4

0.2

0.2

0.1

0.1

0.2

0 -2

-1

0

1

2

0 -2

0 -1

0

1

2

3

-3

-2

-1

0

1

2

3

Fig. 2.33 Spline scaling functions.

The main disadvantage of the spline family is that they don’t form orthogonal wavelets. The later discussed Battle-Lemarié wavelets correct this disadvantage, but they no longer have the compact supports. Like the spline wavelets, the Battle-Lemarié wavelets form a family, based on

135

Wavelet Analysis

linear, quadratic, cubic, etc, scaling functions. Usually they are written in terms of Fourier transforms. N=1 φ

ψ

1.2

1.5

1

1 0.8

0.5

0.6 0.4

0

0.2

-0.5 0 -0.2 -10

-5

0

5

10

-4

-2

0

2

4

N=2 φ

ψ 1

1 0.8

0.5

0.6

0 0.4 0.2

-0.5

0

-1 -0.2 -10

-5

0

5

-4

10

-2

0

2

4

0

2

4

N=3 φ

ψ

1 1

0.8 0.5

0.6 0.4

0

0.2 -0.5

0 -0.2 -10

-5

0

5

10

-4

-2

136


N=4 φ

ψ

1

1

0.8 0.5

0.6 0

0.4 0.2

-0.5

0 -1

-0.2 -10

-5

0

5

10

-4

-2

0

2

4

N=5 φ

ψ

1 1

0.8 0.5

0.6 0.4

0

0.2 -0.5

0 -0.2 -10

-5

0

5

10

-4

-2

0

2

Fig. 2.34 The plots of different Battle-Lemarie mother and father wavelets.

It should be noted that if for the linear spline it is

ϕˆ (ξ ) =

2 4 sin (ξ 2 ) , ξ2 2π

then the linear Battle-Lemarié scaling function has the form

ϕˆ (ξ ) =

sin 2 (ξ 2 ) 4 3 . 2π ξ 2 1 + 2 cos 2 (ξ 2 )

4

137

Wavelet Analysis

The Battle-Lemarié wavelets are defined by the same formulas as the spline wavelets.

2.11 Integral Wavelet Transform Let us start with some general formulas and definitions of the theory of wavelet transforms, based on Morlet and Grossman wavelets.

2.11.1 Definition of the wavelet transform The basic idea in wavelet analysis is the decomposition of the function (signal) into two families of functions constructed by means of the scaling function ϕ ( x) . The first family is built by translation and dilation of the scaling function ϕ ( x) . The second one is produced by translation and dilation of the wavelet function ψ ( x) , represented mathematically in the form

{ The

wavelet

s ψ ( s ( x − xɶ) )

transform

}

of

the

s , xɶ∈ℝ 2

.

function

(2.107) f ( x) ∈ L2 ( ℝ )

is

conventionally defined as follows +∞

W f ( s, xɶ) =

∫

f ( x) s ψ ( s ( x − xɶ) ) dx .

(2.108)

−∞

Both the function (signal) f ( x) and the wavelet ψ ( x) are assumed to be functions with values from a real number domain. It is usually assumed that the energy of the wavelet is equal to unit. Let

ψ s ( x) = s ψ ( sx ) denote the function formed by scaling the wavelet.

(2.109)

138


This function is the same type as the wavelet, but its graph is located on the support which is s times narrower or wider in dependence that will be s times less or greater than the unit. Let us remember that the graph of any wavelet should be practically distinguished from zero in the finite area in the neighborhood of coordinate origin – to have a (practically) finite support. Now the wavelet transform (2.108) can be written as the scalar product in the space L2 ( ℝ ) or as the decomposition of the function (signal) f ( x) on the family of functions { ψ s ( x − xɶ)}s , xɶ∈ℝ 2 Wf ( s, xɶ) = f ( x),ψ s ( x − xɶ) .

(2.110)

2.11.2 Fourier transform of the wavelet Let us denote it by ψˆ (ω ) . From some mathematical considerations on the function reconstruction, this condition +∞

Cψ =

∫ 0

2

ψˆ (ω ) d ω < +∞ ω

(2.111)

should be fulfilled. There follows, that: 1. ψˆ (0) = 0 . 2. The function ψˆ (ω ) is sufficiently small in the neighborhood of zero. Both the wavelet and its Fourier transform have in the signal theory some interpretation in terms of this theory. First of all, the wavelet can be interpreted as the impulse characteristic of the band filter. To explain this fact, first rewrite the wavelet transform Wf ( s, xɶ ) at the point xɶ and the scale s as the convolution of the signal with the scaled wavelet ψ s ( x)

139

Wavelet Analysis

Wf ( s, xɶ) = f ∗ψ s ( x) .

(2.112)

Now the wavelet transform can be commented as filtering the signal by the band filter with impulse characteristics ψ s ( x) .

2.11.3 The property of resolution Let us write the Fourier transform for scaled wavelet (2.109) in the form

ψˆ s (ω ) =

1 ω  ψˆ   . s s

This formula testifies that the wavelet transform (2.112)

Wf ( s, xɶ )

already has the essential distinction as compared with the window Fourier transform Gf (ωo , xɶo ) . In fact, the resolution of the window Fourier transform is constant in the space and frequency domains, whereas the wavelet transform changes resolution with the changing of the scaling parameter s . Since the wavelet is a function taking real values then its Fourier transform has the property of symmetry

ψˆ (ω ) = ψˆ (−ω ) . Denote the center of bandwidth ψˆ (ω ) through ωo , then +∞

∫ (ω − ω ) ψˆ (ω ) o

2

dω = 0 .

0

Denote also the mean square deviation (the standard deviation) of the bandwidth from its center through σ ω +∞

σ ω2 =

∫ (ω − ω ) o

0

2

2

ψˆ (ω ) d ω .

140


Thus for the Fourier transformed scaled wavelet ψˆ s (ω ) the bandwidth center will be sωo and the mean square deviation will be sσ ω . It seems to be important to comment that in logarithmic scale the standard deviation will be constant for all s ∈ ℝ + . Thus, the wavelet transform decomposes the signal (function) into the set of bandwidths which have the identical size in logarithmic scale. Denote now the standard deviation of the function ψ ( x)

2

from zero

(at neighbourhood of zero) by σ xɶ . It can be shown then that the energy of translated wavelet ψ s ( x − xɶo ) is concentrated around the point xɶo with the standard deviation σ xɶ s from this point. It was already noted that in the frequency domain the energy is concentrated around the point sωo with the standard deviation sσ ω from the point. Let us describe in which way (by what) the resolution of wavelet transform is defined. In the plane ɶxOω the wavelet transform resolution is characterized by the Heisenberg rectangle

σ ɶx σ xɶ    ɶxo − s , xɶo + s  × [ sωo − sσ ω , sωo − sσ ω ] .   Thus, in contrast to the window Fourier-Gabor transform, the size of window characterizing the resolution changes with the change of the scale parameter. When the scale parameter s is small, the resolution is coarse in the space area and vice versa in the frequency area the resolution of details is very exact. If the values of scale parameter increase, then the resolution in the space area increases, also whereas it decreases in the frequency area.

141

Wavelet Analysis

2.11.4 Complex-value wavelets and their properties Such complex wavelets, taking values from the set of complex numbers, are convenient in analysis, when the phase and module can be separated in the wavelet transform components. Morlet and Grossman defined the complex wavelets in a such way that their Fourier transform ψˆ (ω ) vanishes for negative values of frequency. Functions with this property belongs to the so-called Hardy space. For these functions, the wavelet transform Wf ( s, xɶ ) is a complex number. It is enough here to understand that the Hardy space H 2 ( ℝ ) is a closed subspace of the space L2 ( ℝ ) : H 2 ( ℝ ) ⊂ L2 ( ℝ ) . When the scale parameter is fixed and coordinate xɶ is used then the wavelet transform Wf ( s, xɶ ) as a function of the space coordinate xɶ is also a function from the Hardy space. For any point xɶ and arbitrary scale s , the module and the phase can be separated in the wavelet transform of the complex-value wavelet.

2.11.5 The main properties of wavelet transform Property of isometry +∞

Cψ

∫

1 ∫ −∞∫ Wf (s, xɶ) dsdxɶ . 2π −∞ +∞ +∞

2

f ( x) dx =

−∞

(2.113)

Here the constant Cψ is given by the formulas +∞

Cψ =

∫

ψˆ ( sω ) s

0

2

+∞

ds = ∫ 0

2

ψˆ (ω ) d ω =< +∞ . ω

(2.114)

Property of reconstruction

f ( x) =

1 Cψ

+∞ +∞

∫ ∫ Wf (s, xɶ)ψ

−∞ 0

s

( x − xɶ)dsdxɶ .

(2.115)

142


Property of reproduction +∞ +∞

∫ ∫ Wf (s, x) K ( s, s , xɶ, xɶ )dsdxɶ .

Wf ( s* , xɶ* ) = Given

the

points

K ( s, s* , xɶ, xɶ* ) =

1 Cψ

*

*

(2.116)

−∞ 0

( s, xɶ ) ∈ ℝ + × ℝ,

( s , xɶ ) ∈ ℝ ∗

∗

+

×ℝ

the

integral

+∞

∫ψ

s

( x − xɶ)ψ s* ( x − xɶ* )dx is the reproducing kernel.

−∞

2.11.6 Discretization of the wavelet transform Let us cover the plane by points with frequency and time line coordinates. Choose the sequence of scales {α k }

k∈ℤ

, where α is the

length of the elementary discretized scaling step. Formula (2.112) can be rewritten as the convolution of the function with the discretized wavelets Wf (α k , xɶ) = f ∗ψ α k ( xɶ) .

(2.117)

ω

2α 3σ ω

2α 2σ ω 2σ t α 3 2ασ ω 2σ t α 2 2σ ω

2σ t α О

2σ t Fig. 2.35 The scheme of discretization of wavelet transform.

t

143

Wavelet Analysis

Formula (2.117) can be interpreted as follows: − This is the decomposition of the function (signal) f (t ) into the set of frequency band centered around the point α k ωo with the mean square deviation of frequency band α kσ ω . Let us try to characterize the decomposed function (signal) in separate bands and define the discrete wavelet transform. It is necessary for this purpose to equally divide the bands in a ratio proportional to α k . Let

α k β be just that ratio for the fixed scale α k . The discrete wavelet transform is defined then according to nβ  Wd f ( k , n ) = Wf  α k , k α  +∞

=

∫

−∞

nβ  f ( x)ψ α k  x − k α 

  = f ( x),ψ α k 

nβ   x− k  α  

  nβ  dx = f ∗ψ α k  k  α

 . 

2.11.7 Orthogonal wavelets Orthogonal wavelets are characterized by the

Theorem

2.41

(Meyer-Strömberg)

There

exists

 n  ψ ( x) ∈ L2 ( ℝ ) such that the set of functions ψ 2  x − k   2 

k



a

wavelet

 ( k ,n )∈ℤ2

forms

2 an orthonormal basis in L ( ℝ ) .

In fact, the case considered above is the discrete wavelet transform when α = 2, β = 1 . Dyadic wavelets are defined as wavelets with dyadic scales k

{2 }

k∈ℤ

. This case brings more simple algorithms of decomposition and

is commonly used.

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The formula of reconstruction for orthogonal wavelets is f ( x) = ∑

∑

k∈ℤ n∈ℤ

f ( xɶ), ψ 2k ( x − n 2− k ) ψ 2k ( x − n 2− k ) .

2.11.8 Dyadic wavelets and dyadic wavelet transform Denote the wavelet transform in the scale 2k as W2k f ( x) = f ∗ψ 2k ( x) . The wavelet transform of this wavelet transform is given by the formula Wˆ2k f (ω ) = fˆ (ω )ψˆ (2k ω ) . It is accepted, for some mathematical considerations, that the following condition is fulfilled +∞

∑ ψˆ (2 ω ) k

2

=1 .

k =−∞

Wavelets, for which this condition is fulfilled, are called dyadic wavelets. The set of functions

{W

2k

f ( x)}

k∈ℤ

is called the dyadic wavelet

transform.

2.11.9 Equation of the function (signal) energy balance This equation for the case of the dyadic wavelet transform has the form

f ( x)

+∞

2

=

∑ k =−∞

W2k f ( x)

2

.

Wavelet Analysis

145

This last formula concludes the short introduction into the wavelet analysis. In the author’s opinion, the information given in this chapter information is sufficient for understanding the following chapters with applications of wavelet analysis for studying waves in structured materials.


Chapter 3

Materials with Micro- or Nanostructure

3.1 Macro-, Meso-, Micro-, and Nanomechanics

The shortest way, to correctly outline the, based on modern physics, present-day knowledge of materials is to introduce the system of definitions with short comments. The physical substance is defined as the aggregate of discrete formations, which have the rest mass (atoms, molecules, and more complicated formations of them). They are distinguished by the state of aggregation and the state of phase of the substance. Four states of aggregation are known: gaseous, plasmic, liquid, solid. −

The gaseous state is characterized by translatory, rotational, and oscillation motions of molecules. Distances between molecules are large, that is, the density of molecule packing is not high.

−

The plasmic state is differed from the gaseous one by that it is an atomized gas with the equal concentrations of positive and negative charges. It singles out only for this purpose that, as a lot people believe, the substance in Universe consists of just plasma.

−

The solid state is characterized by only oscillatory motions of molecules near immovable centers of equilibrium with 13 14 frequencies 10 ÷ 10 oscillations per second. Translatory and rotational motions are absent. Distances between molecules are small, that is, the packing density is high. 147

148


−

The liquid state is close to solid by character of packing, but it is close to gaseous one by character of molecule motions.

The phase states are distinguished by order in the reciprocal placement of molecules. There are three such states: crystalline, liquid, gaseous. −

The crystalline phase state is characterized by the “far” order in the placement of molecules, when the order is kept on distances, which exceed the molecule dimensions 102 ÷ 103 times.

−

The liquid phase state is the state with the “near” order in the placement of molecules, when the putting in order is observed only in immediate “nearness”, that is, on distances of few molecules. And further the placement is unpredicted.

Often, for this state the term amorphous state is used. Solid amorphous substances are called glasslike ones. The glasslike state differs essentially from the liquid amorphous state, and it is marked out sometimes as the isolated state. The gaseous state of aggregation and the gaseous phase state coincide practically. The solid state of aggregation corresponds to two different phase states: crystalline and glasslike.

Materials are defined as substances in the solid state of aggregation. The materials include the traditional machine-building and building materials, polymer and composite materials etc. The solidity mentioned is treated in mechanics as the property of any body to have some configuration, for which the body gives preference. A change of the body shape relative to the configuration is measured by the deformation. Within the framework of axiomatic procedure in constructing the mechanics of materials, these two notions (configuration, deformation) are defined exactly. This accuracy is reached within the framework of thermodynamics of material continua.


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So, classical physics thinks of a solid body as a system of a great number of coupled and interacted particles, which has been previously called the discrete formations. It turns out that the description of the changing form of a body by taking into account the motion of each particle is to complicate the problem. Besides that, this description is inexpedient in classical macro-mechanics, since knowledge of an individual motion of the particle (their number in 1cm3 has the order 1022 ) gives a picture of the micro- or nanoscopic motion, whereas in many cases the changing of a body form can by studied successfully as a manifestation of the macroscopic motion. The macro-description of materials was predominant in mechanics of materials up to 20 century, when meso-description and micro-description were proposed and developed (the first one is mainly the in-depth analysis of metals; the second one is the wide fabrication and application of composite materials in the second half of 20 century). Both new descriptions are based on understanding the materials as having the internal structure of meso- and microlevel substance and on assumption that this structure cannot be neglected in mechanical processes studying in meso- and micromechanics. Actual mechanics of materials is divided on macromechanics, mesomechanics, micromechanics, and nanomechanics. Let us stop on nanomechanics as the very new and attractive part of mechanics of materials. It can be noticed that in the general system of knowledge on materials, the mechanics of nanomaterials can be related to the structural mechanics of materials. Structural mechanics of materials is understood as that section of mechanics of materials in which the basic relationships include the parameters characterizing the internal structure of materials. Nanomechanics arose as a result of formation and developing nanophysics and nanochemistry. First, recall that nano- (from the Greek word for “dwarf”) means one thousand millionth of a particular unit. The prefix “nano” in the words

150


“nanotechnology” and “nanomechanics” pertains to a length of 1 nm ( 1 ⋅ 10−9 m ). Richard Feynman was the first to predict the development of nanotechnology. In his well-known lecture There’s Plenty of Room at the Bottom, read at a meeting of the American Physical Society in 1959, Feynman formulated the basic principle of nanotechnology: “The principles of physics, as far as I can see, do not speak against the possibility of maneuvering things atom by atom”. Today we may state that at that time there were no tools to analyze the nanostructure of substance. Electronic microscopes, the main tool to deal with nanomaterials, have been invented fairly recently. The first scanning electronic microscope was developed in 1942 and became available in the 60s. The scanning atomic-force microscope and the scanning tunneling microscope, used to study nanomaterials, were created in the ’80s (the former, by Binnig and Rohrer (IBM Zürich) in 1981 and the latter, by Binnig, Quate, and Gerber in 1986; the inventors of both microscopes were awarded the Nobel Prize in Physics in 1986). Through these microscopes, the surface of a material can be seen at a nanometer scale. That is what favored the success of many experiments on nanomaterials. Eric Drexler is reckoned the second predecessor of nanotechnology. He once organized a new division of technology and wrote that nanotechnology is the principle of manipulating atoms by controlling the structure of matter at molecular level and that “this road leads toward a more general capability for molecular engineering which would allow us to structure matter atom by atom”. Atom-by-atom construction is now called molecular nanotechnology. Nanotechnology as a whole can be understood as research and technology development at the atomic, molecular or macromolecular levels in the length scale of approximately 1 – 100 nanometer range, to provide a fundamental understanding of phenomena and materials at the nano-scales and to create and use structures, devices and systems that have novel properties and functions because of their small and/or intermediate size. In some particular cases, the critical length scale may be under 1 nm or be larger than 100 nm. The last case includes the


151

composites (reinforced by nanoformations polymers), which have the unique feature at 200-300 nm as a function of the local bridges or bonds between nanoformations and the polymer matrix. The primary concept in theoretical interpretation of nanomaterials includes the idea that all materials are composed of granules, which in turn consist of atoms. This concept coordinates well with the classical concept. The next statement – these particles may be visible or invisible to the naked eye, depending on their size – introduces something novel into the classical understanding of the materials. The structural mechanics of materials assumed the size of granules from hundreds of microns to centimeters. Nanomechanics of materials introduces some changes into structural mechanics of materials. Now, in dependence on sizes of granules in the internal structure of materials, the structural mechanics can be divided into macromechanics, mesomechanics, micromechanics, and nanomechanics. Taking into account the results of numerous publications, the following classification of the admissible range of changing the characteristic size of inhomogeneities (particles) in the internal structure of materials 1. 2. 3. 4.

macro: 10−2 − 10−5 m (from 1 cm till 10 µm); meso: 10−3 − 10−8 m (from 1 mm till 10 nm); micro: 10−4 − 10−8 m (from 100 µm till 10 nm); nano: 10−7 − 10−9 m (from 100 nm till 1 nm).

The new classification of materials including nanomechanics is shown schematically in Fig. 3.1.

152


Fig. 3.1. Classification of internal structure of materials by the attribute of admissible size of particles.

Many people believe that nanomaterials as materials whose internal structure has nanoscales dimensions are something new to science. However, it was relatively recently to realize that some formations of oxides, metals, ceramics, and other substances are nano-materials. For example, ordinary (black) carbon was discovered at the beginning of 1900. Fumed silica powder – a component of silicon rubber – is a nanomaterial too. It came into commercial use in 1940. However, only recently it becomes clear that the particles constituting these two substances have nanoscale dimensions. The particle size is not the only characteristic of a nanoparticle, nanocrystal, or nanomaterials. A quite important and specific property of many nanomaterials is that the majority of their atoms localize on the surface of a particle, in contrast to ordinary materials where atoms are distributed over the volume of a particle. It should be discussed here especially the carbon nanoparticles as components used in the next numerical modeling nanocomposites. Science has long been aware of three forms of carbon: amorphous carbon, graphite, and diamond. The highly symmetric molecule of carbon C60 was discovered in 1985. It has a spherical form, resembling a football, with carbon atoms on the surface and contains 60 atoms in five-atom rings separated by six-atom rings. These molecules were named fullerenes and have come to be studied fruitfully. Scientists who studied fullerenes were awarded the Nobel Prize in Chemistry in 1997. Since


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then the number of discovered kinds of fullerenes has increased considerably, reaching many thousands to date. What is more important is that fullerene molecules form carbon nanotubes, which may be considered relatives of graphite. Nanotubes can be thought of as graphite lattices rolled up into a tube – they are the molecules with a very large number of atoms, C10000 − C1000000 . Nanotubes differ in length, diameter, and the way they are rolled. The internal cavities may also be different and tubes may have more than one sheet. Atoms at the ends of a fullerene molecule form the “hemispherical caps”. Sheets may be rolled differently, forming zigzag, chiral, and armchair structures. Two types of nanotubes are distinguished: single-wall and multi-wall nanotubes. It should also be noted that nanotubes are technologically advantageous over ordinary carbon fibers: the former are produced from colloidal solutions at room temperatures, whereas the latter need high temperatures. So, we can write the common experience that the uniting property of all known nanoparticles is their dimensions; and their internal structure may vary considerably. Not only does the mentioned have a high level of surface localization, but also various features in the chemical-physical structure of nanoformations – their intermediate position between macroworld and atomic world – manifest themselves as their peculiar mechanical properties. Their mechanical characteristics exceed considerably those of traditional materials. Today’s studies into the mechanical behaviour of nanoparticles, nanoformations, and nanomaterials are at an early stage; i.e., only external manifestations of mechanical phenomena are detected, but their mechanisms are not studied. In closing this necessary introduction into nanomechanics, it seems pertinent to recall a discussion on mechanical properties of new materials organized in June 6-7, 1963, and published in the Proceedings of Royal Society in 1964. In the concluding remarks, Bernal, one of the organizers, said: “Here we must reconsider our objectives. We are talking

154


about new materials but ultimately we are interested, not so much in materials themselves, but in the structures in which they have to function”. The nanomechanics faces the same challenges that micromechanics did 40 years ago and that Bernal described so eloquently. Mechanics of materials has dealt mainly with continuum models of materials. As it was mentioned above, both classical and modern physics assume the materials as having the discrete structure system of formations. A transition from the discrete system to the continual system is reached using the procedure of continualization – a change of the volume occupying by the discrete system on the same volume occupying by the continuum with certain continually distributed physical properties. So, the continualization establishes a correspondence between the real solid body, which occupies the volume V and has a complicated discrete internal structure and fuzzy external boundary, and the fictitious 3 body of the same volume V ⊂ R (and, of course, the same configuration with now fixed external boundary), to each point of which the set of averaged physical characteristics is attributed.

The first of these characteristics, which according to the definition form the fields and therefore are called the field of thermodynamical characteristics, is the mass density ρ . 3 The geometrical area V ⊂ R (finite or infinite), in which the field of mass density is given, is called in physics the material continuum or continuum.

The notion of material continuum only is not sufficient for the description of the deformation process of solid bodies. A notion of the body is defined as the material continuum in the regular area of a space. Usually, the continuum is equipped, that is, the scalar field of mass is complemented by three fields: vector field of displacements and tensor


155

fields of strains and stresses. Within the framework of axioms of rational mechanics, these three notions (fields) are defined exactly. It can be noted that the procedure of continualization of discrete system in hand gives the continuum description of the piece of material in hand. This piece can be considered separately. In this case it can be treated as a homogeneous material. But the material can consist of many continuum pieces (for example, a granular composite material consists of the matrix with embedded granules). In this case the discrete system is extended to a piecewise homogeneous material. Two basic approaches are then used: the exact approach based on application of the equations of continuum mechanics to each separate homogeneous piece and then taking into account the interaction of pieces at interfaces; the approximate approach based on the procedure of averaging of mechanical parameters. The procedure of homogenization (averaging) consists in that usually, in the space area, which the inhomogeneous body (material) occupies, a cube, dimensions of which are many times less of the area, is chosen. This cube must include the sufficient great number of pieces (otherwise, the procedure of averaging becomes false). In that way the chosen cube (volume) is called the representative cube (volume). The center of this cube is usually the point, to which all averaged properties of the cube are attributed. As a result, the homogeneous material with continuum characteristics is considered. The important role of the characteristic size of inhomogeneities of the material should be mentioned. This quantity with the dimension of length is also called the characteristic size of internal structure. Two restrictions on this new parameter are the most known. For wave problems, the characteristic size of internal structure must be at least on one order more than the wavelength. For problems with varying surface loading, the characteristic length of variability must be at least one order more than the characteristic size of internal structure. These restrictions in continualization are the concrete displaying of the general requirement that the elementary volume should be a representative one. In other words, the characteristic size of internal

156


structure can’t be commensurable with the scale of averaging. This condition is called the condition of effective homogenization. It must be noted that the final goal of the averaging procedure is the effective description of composite material as the material continuum.

3.2 Main Physical Properties of Materials The deformation will be understood as a change of the form of a solid (relative to some initial configuration and as a result of some causes, very often as a result of an action of external forces). Properties of materials, which are displayed by the deformation, and constitutive features of the process of deformation are very diverse. Part of these properties underlies the particular classical theories of deforming the materials. Let us shortly describe these properties according to observations and experiments. Elasticity. The property of elasticity consists of that the body practically simultaneously takes the initial configuration after removing deformation causes. In other words, if deformations are elastic, then they simultaneously vanish after removing the action of forces, which it caused.

This property, as also other properties, though, is seldom displayed in the pure form, that is, it is accompanied in real solid materials by a number of other properties. Plasticity, elastoplasticity, rigidplasticity. These properties are considered as the most important technically. If by some causes a body changes its configuration and doesn’t go back to the initial configuration after removing of these causes, then it is said that the plastic deformation has taken place.

In this case the body displays only the property of plasticity. Such a property means in fact for the body an absence of the property to resist


157

external forces, that is, means the loss of the basic attribute of a solid. In constructional materials, the property of elastoplasticity is observed the most frequently. This property consists in that the deformation process is elastic up to some value of deformations, and by exceeding this value the process becomes the plastic one. The property of rigid-plasticity is the limit case of the elastoplasticity property, its idealization. It is displayed in that the body doesn’t change its configuration up to some value of a parameter, which fixes external action intensity (that is, the body is a rigid one), and when this value is exceeded the body becomes the plastic one. Viscosity. The property of viscosity is the most characteristic for fluids. On the everyday level, the solid is different from the fluid by the property of the solid to conserve its form and an absence of this property for the fluid (besides the particular case of a bulk compression or other cases, which are reduced to the last one), that is, by the property of the fluid to flow.

In theoretical descriptions, this distinction is displayed in that if the motion of solids is described by deformations, then the motion of fluids is described by a velocity of deformations. Besides them, fluids are differed by the property of an internal friction: if this property is displayed slightly, then the fluid is called the ideal or perfect one, if strongly, then the viscous one. So, the observation shows that solids possess also the property of a viscosity. It is displayed in the dependence of the arising during the body deformation internal forces not only on deformations (which is the characteristic for solids), but also on the deformation velocities. The property of viscosity is displayed essentially only for isolated classes of materials, specifically for polymer materials. Thermoelasticity, thermoplasticity. These properties are observed very easily, since the most part of solids are deformed when heated up, and they heat up when they are deformed. For description of this phenomenon, the notion of temperature is introduced as a

158


measure of the heat state. If the change of temperature (that is, values of temperature in each points of the body, the temperature field) causes the elastic deformation and vice versa, then in the body the property of thermoelasticity is displayed. If the change of the temperature field causes the plastic deformation and vice versa, then the property of thermoplasticity is displayed in the body. Viscoelasticity, viscoplasticity. Let us use for explanation of properties of viscoelastic and viscoplastic deformations the theoretical scheme of description of elastic and plastic strains. As it is adopted in mechanics, the notion of strains is supplemented by the notion of stresses. Stresses characterize the internal state of a body, physically they are linked with a field of acting inside the body forces and are some abstraction.

This description concern two phenomena, which are not peculiar to elastic strains and display the presence in materials of the viscoelasticity property. The first phenomenon, the creep, consists in that if a body is deformed with a constant rate up to certain values of stresses (generally speaking, arbitrary), and further this level is conserved, then strains will increase. The second phenomenon, the stress relaxation, consists in that if a body is deformed with a constant strain up to certain level of strains (generally speaking, arbitrary), and further these strains are conserved constant, then stresses will be decreased with time (relaxed). In both cases the main property of elastic strains, their reversibility, is kept. The property of viscoplasticity is displayed in materials in such a way that: the material possesses the creep property, the phenomenon of stress relaxation is absent, and the main property of plastic deformations, their irreversibility, is kept.


159

Viscoelasticity assumes that the material has simultaneously both the elasticity and the viscosity properties. The elastic materials are able to accumulate the energy without losses, that is, they are not able to dissipate energy. Whereas the viscous bodies (fluids) dissipate their energy and accumulate it only in the case of bulk compression. The property of viscoelasticity is such that the material possesses these two properties, to accumulate and to dissipate energy, simultaneously. The property of viscoplasticity can be commented in a similar way. Diffusional elasticity. This property is displayed in a body, when in this body diffusion processes occur.

The diffusion in physics is the movement of molecules due to the heat molecular motion. That is, diffusion is one from mechanisms of mixing of two or more substances. For example, the diffusion of gold in solid lead is a topic studied very well. It is said about the diffusional elasticity, when diffusion is a cause of body deformations and changes the stresses in the body, and vice versa a presence in the body of deformations and stresses cause the appearance in the body of diffusional fluxes. This property is slightly similar to thermoelasticity property, the coupling of deformations with temperature is similar to the coupling of deformations with the substance concentration (basic physical relations have the same structure). Electroelasticity. Properties of solids associated with the electromagnetic field effect on the deformation process and vice versa are to a certain extent differing from traditional mechanical properties: elasticity and plasticity. This distinction consists in that elasticity and plasticity are in any event peculiar to all materials, whereas electroelasticity and magneto-elasticity are peculiar to certain classes of materials (to dielectrics, for example) and in several materials are not displayed in principle. The effect of coupling of deformations and an electric field is called the piezoelectric effect. Respectively, the property of the electroelasticity is displayed in particular materials as piezoelectrics.

160


Magnetoelasticity. Coupling of strain field and a magnetic field is the essence of a property of magnetoelasticity. This property reflects the piezomagnetic effect, which consists in that the macroscopic magnetic moment arouses when a body is deformed. Piezomagnetic effect was for the first time observed in an antiferromagnet, which was the compound of cobalt and iron. As the electroelasticity property, the property of magnetoelasticity is displayed in some narrow class of materials.

3.3 Thermodynamical Theory of Material Continua In thermodynamics the material continua are studied. They are called thermodynamical systems. These systems are characterized by parameters of two kinds: intensive and extensive. The first kind doesn’t depend on the mass (an amount of substance) of the system; the second kind is proportional to this mass. Thermodynamical parameters are introduced as the collective characteristics of a system at large. They characterize the state of a thermodynamical equilibrium of the system, that is, they are parameters of this state. Physical state and number of state parameters are determined by the physical essence of the system. State parameters can have mechanical, electromagnetic, chemical and other nature. The notion of the state of thermodynamical system is the fundamental one; the state is described in full by parameters of the state. Three of them, absolute temperature T , entropy S , and internal energy U , are always given. It is said that a thermodynamical system is in the equilibrium state, if this state is not changed with time and actions on the system of external processes are absent. If, in a system, some changes occur, then it is said that this system is in the state of a thermodynamical process.


161

Equilibrium processes are marked out separately. These processes are peculiar by that the body goes slowly from one equilibrium state to other one. This gradual and slow transition permits to neglect the deviations from equilibrium, which are always present in real processes. Equilibrium thermodynamical processes have such a feature that a system can revert to the initial state, from which the process started. Such a process is called the reversible one. In all other cases, processes are called irreversible processes. The notion of state parameters is used in the study of reversible processes. For description of irreversible processes, the notion of a local equilibrium at each point of the system is introduced. The choice of state parameters is a difficult problem. The usual way in modern mechanics is called the phenomenological approach. According to the approach, parameters are strongly defined theoretically, further physical experiments for the determination of physical constants are described, and finally, the theory has to predict new phenomena. For fundamental sciences, the necessity of attention to experiments and practice had been formulated as far back by Leibniz in his statement theoria cum practica. Today, it is understood as the necessity for any theory to amplify with experimentations. 200 years later, Boltzmann, stated “nothing is so practical as the theory”. In 1926 in a talk between Werner von Heisenberg and Albert Einstein, Heisenberg stated that each theory, in its building, must correspond to only those observed by this time fact. Einstein answered that it could be wrong to try to build a theory only on observed facts. Really, it happens the vice versa. Theory determines what we can observe. Let us consider three basic functions-state parameters. They are strong defining. For example, absolute temperature exists according to the law of the heat equilibrium transitivity, and describes the heat equilibrium between being in heat contact bodies. Heat is transferred from a body with the greater temperature to a body with the lesser one.

162


The basic problem of thermodynamics is to study those processes, which are possible for this given system. Basic laws of thermodynamics, the energy law and the entropy law, form the base in this theory. The first law of thermodynamics can be written as follows dU = δ Q + δ A + δ Z .

(3.1)

The law (3.1) can be framed: 1. The internal energy of a system U is the function of parameters of a system state, and it consists of a sum of the heat amount received Q , the work done A , and the energy Z , which is introduced into a system by the mass exchange. 2. The increment of an internal energy is a total differential of parameters of a system state, and is a sum of increments of the above defined amount of heat, work, and energy. In the case of equilibrium processes these three increments can be represented as differential forms of state parameters, but they will not be total differentials. Further, differential forms δ Q, δ A, δ Z have to be represented in the explicit form by means of state parameters. Here, the second law of thermodynamics helps. It states for equilibrium processes: − the heat gotten by a thermodynamical process cannot be fully transformed into work. In other definitions this law can be formulated in the form: − for equilibrium processes the entropy is some function of the state and such a formula is valid

δ Q = TdS .

(3.2)

The two laws are often combined into the equation dU = TdS + δ A + δ Z .

(3.3)


163

The following problem consists in a certain particular representation of the last two differential forms in (3.3), and also of internal energy by means of thermodynamical parameters. The choice of these parameters means the choice of the particular model of a medium. Remember that the classical theory of elastic deformation can be constructed without the thermodynamical notions. Chronologically (historically) it was done in such a way. But as soon as we suppose that the cause of deformations can be temperature, electric or magnetic field, diffusion or something else, then the deformation process can be described only with the help of thermodynamics. So, in order to construct the thermodynamically substantiated theories of deformation, there arise the necessity to formulate the axiomatically different models and for each model to choose its set of thermodynamical parameters. Let us describe further this procedure for models of materials, which are based on the above items of basic properties. It is logical to start with the classical model of an elastic deformation caused by only forces of mechanical nature. The phenomenological procedure of construction uses the balance equations for mass, pulse (momentum), momentum of a pulse (momentum of momentum), and energy. One needs to say that the complication of a deformation process and the necessary address to thermodynamics will effect writing the first and last balance equations, only. The internal energy is namely that function, which needs the particular choice of a thermodynamical parameters system. In the classical theory of elasticity it is found to be sufficient the choice of only one parameter – the strain tensor. It is the symmetric tensor of rank two, that is defined by six components, and internal energy depends actually on all the six. This ascertaining is as now enough, since classical nonlinear theory of elasticity will be commented more precisely later when studying the wave propagation processes. However, it is necessary to say here that the feature of processes of elastic classical wave propagation is in the

164


absence of any dissipation of wave energy, and this is laid in the model of this medium. The whole procedure of selecting a material continuum and the necessary set of thermodynamical parameters is sometimes called the procedure of equipment of material continua. Without a doubt, the accompanied by heating elastic deformation of materials is studied in the most detail from the point of view of thermodynamics. This model is based on the property of thermoelasticity; the theory is called the theory of thermoelasticity. The thermodynamical parameters in the theory are the absolute temperature and the strain tensor. It is expediently to focus on the phenomenon of dissipation, which is absent in classical elasticity and is instead basic in thermoelasticity. The axiom of dissipation states that there exists for each solid the limit value of a rate, with which heat can be transformed into energy without the production of a mechanical work. This axiom is often written in the form of Clausius inequality

TSɺ ≥ Qɺ .

(3.4)

If the notion of free energy F is introduced by the formula

F = U − TS ,

(3.5)

then the reduced Clausius inequality for densities of functions F , A, T , S can be written

fɺ − aɺ + sθɺ ≤ 0 .

(3.6)

Usually, the notion of internal dissipation δ is also defined as

δ = θsɺ − uɺ + aɺ .

(3.7)


165

Then the statement that an internal dissipation cannot be negative, has a name the Planck inequality

δ ≥0 .

(3.8)

It has been proven that in the theory of thermoelasticity the internal dissipation is absent. But in all other theories of materials, which take into account not only the elasticity property, the dissipation is always present. The next theories take into account besides the deformation field not only temperature, but also some other thermodynamical parameters, and therefore they overstep the framework of classical thermoelasticity. We introduce thermodynamical parameters in the theory of viscoelasticity, which studies the process of deformation with a memory of the history of the deformation. The model having a memory of the deformation is called the model of hereditary elasticity or, more generally, the model of elasticity with a long-range memory in contrast to the classical model elasticity with an infinitesimal memory. We can assume that in this model the effect of temperature on strains is also taken into account. That is, let us consider the model of a thermoelastic deformation with long-duration memory, the model of thermoviscoelasticity. The basic thermodynamical parameters in this model will be the same as in thermoelasticity. But the long-range memory will be displayed first of all in the presence of the internal dissipation, which fully agrees with a physical sense of the property of viscosity. One of the attributes of a viscous deformation is the above mentioned energy dissipation. It is quite obvious that the material does not store more energy only owing to its property to remember the deformation history. The history of the material is shown not only by the increasing the energy, but also in its separation, part of the energy will be stored in the form of internal dissipation.

166


The models of electroelasticity and magnetoelasticity are similar in choosing the thermodynamical parameters. These models use the classical thermodynamical parameter of mechanical nature – the strain tensor – and new parameter of electromagnetic nature – the vector of electric intensity for an electro-elastic model and the vector of initial magnetic intensity for a magneto-elastic model. One of the features of elastic and electromagnetic properties is that they do not permit the energy loss. Therefore models with a set of such properties are similar to the classical model of elastic deformation. The model of diffusional elasticity is closely linked with thermodynamics. In some sense it is formed in the close interaction with thermodynamics. In fact, this model is the sole model of materials, which takes into account the mass exchange during a deformation process. The model therefore saves in writing of internal energy the classical thermodynamical parameter, strain tensor, but also needs the concretisation of the increment of energy of the mass exchange δ Z . It is known that in equilibrium processes the increment of energy of the mass exchange δ Z is proportional to the increment of mass

δ Z = µɶ dM .

(3.9)

The quantity µɶ is called the chemical potential. When a system contains n distinguished particles, the material point (representative volume) contains particles of all sorts, so that the representation (3.9) must be generalized n

δ Z = ∑ µɶ k dM k . k =1

When going on local quantities (densities) it becomes

(3.10)

Materials with Micro- or Nanostructure n −1

∑c

k =1

k =1

δ z = ∑ µɶ k dck ,

167

n

k

= 1.

(3.11)

The introduced quantities ck are called volume fractions or volume concentrations or simply concentrations, they are those supplements to the strain tensor thermodynamical parameters, which permit to describe the model of diffusional elasticity. One essential point in this theory is that the concentrations are symmetric tensors of rank two. The equipped material continuum forms the base for one-continuum models. All the above described models were the one-continuum ones. Besides the last model, which in the general case can be formulated as multi-continuum one. The continuum, in which geometrical point material particles with distinct densities are placed simultaneously, is called n -continuum or n phase, or n -component continuum. Together they form an enclosed one into other material continua. Each continuum is equipped with one and the same set of thermodynamical parameters. The multi-continual model is formed in such a way. One and the same parameter (for example, the strain tensor) takes on its value for each continuum (phase, component) and varies for each phase in a different manner. Since each parameter refers to a separated continuum and forms the partial field, it is often said that one field of strain is enclosed in another field of strain. Now it is appropriate to say about mixtures. Above mentioned n component continua are often called the n -component ( n -phase) mixture. Later elastic mixtures and corresponding models will be discussed in more detail.

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3.4 Composite Materials Classical mechanics of materials was used to divide materials into two classes: homogeneous and heterogeneous ones. The homogeneous materials were understood as materials with internal structure of atomic-molecular character (with characteristic size of the structure close to atoms or molecules). It means that such materials have the discrete molecular structure, which is changed using the procedure of continualization to the model representation by the homogeneous continuum. The heterogeneous materials were understood as materials with internal structure essentially more than molecular-kinetic sizes (sizes of molecules, crystal lattice etc.). It means that these materials consist of components (phases) and have the macroscopically inhomogeneous internal structure. As a rule heterogeneous materials are modeled by a piecewise homogeneous continuum, which assumes that each component of internal structure is also modeled by homogeneous continuum. Thus, as it was mentioned above, the procedure of continualization is applied in this case not to the material as a whole but to separate components of the material. The composite materials are the typical representatives of heterogeneous materials. It can be distinguished by the natural and artificial composites. The composite materials are conventionally defined as consisting of a few components (phases) with differing physical properties. As a rule, these components alternate many times in the space. The way of alternating, conditions on the interface, a geometrical form and physical properties of components define the internal structure of the composite. In real composites the internal structure is at best close to a periodic one. The most difficult in continuum description are processes taking the place at an interface. Macro-, meso-, and micromechanics considered these processes practically from one and the same point of view based on


169

the general physics conceptions. Nanomechanics introduces into this problem the new features associated with intermediate states of interface processes between general physics laws and quantum physics laws. In the continuum modeling, all problems of composite interface are reflected in formulations of boundary conditions between matrix and fillers. Thus, the novel problem of nanomechanics of composites distinguishing this branch and the old branches (macro-, meso-, micromechanics of composites) consist in an adequate formulation of the above mentioned boundary conditions. The next important distinction of nanomechanics of composites consists in novel for mechanics of materials with very high values of main mechanical properties of nanofillers (for example, extremely high values of Young modulus, which was discussed in previous section). As the most important similarity of all four branches of structural mechanics of composite materials can be considered the fact of applicability of common for all branches continuum models. Mechanics of composites is concentrated on specially designed materials. As a rule, the internal structure of composite materials assumes the jumping (stepwise) change of properties of components (phases) on interfaces and the presence of the soft and stiff components. The stiff component is considered as the arming or reinforcing one and is usually called the filler whereas the soft component is conditionally called the matrix (the binder). A difference in some mechanical properties (for example, Young modulus) of composite components can reach 100 through 1000 and more times. In the case when some areas of free space (voids) between components exist, these areas are treated as pores representing one more component, and such a composite is called the porous composite. The most commonly known and used composites are granular (with granules as reinforcing fillers), fibrous (with fibers as reinforcing fillers), and layered (with thin layers as reinforcing fillers) composites. Complexities in analytical description of mechanical phenomena in composite materials have resulted in the creation of approximate

170


continuum models which, on the one hand, save the main physical properties of the system and, on the other hand, these models are quite simple and assume the analytical solutions for boundary problems. At present, many different approximate models are proposed and well developed. They take into account the internal structure of materials, determine the necessary mechanical parameters and solve practically all important problems. These models can be divided on the structural models of different orders. The basic model (structural model of the first order) is based on assuming the material as a homogeneous continuum, mechanical properties of which should be determined on the base of standard tests. The internal structure of a composite is displayed here in the same way as it is done for engineering and building materials (steel, iron, wood or plastics). Which are found using the procedure of averaging properties and depend on the basic parameters of internal structure. As it can be seen later, they are offered mainly in the form of algebraic relationships. This circumstance permits to foresee on the stage of design the averaged properties of composite material. These abilities of the model together with technological possibilities for designing the engineering composites formed one of the main directions in the development of mechanics of composites. It must be noted that in most cases when the matter concerns the averaging properties, it is understood as working within the framework of classical continuum model of elasticity. Let us consider here some notions and formulas from the classical theory of constitutive equations needed for mathematical modeling of composites by the classical elastic continuum. The rigorous procedure of obtaining such equations will be described later. Here, we will note only that in the classical theory of elasticity the constitutive equations describe only properties of elastic deformation, they couple strains and stresses, and are obtained as a result of observations in experiments or some theoretical considerations about the internal energy (energy of deformation) representation.


171

In the linear theory of elasticity, energy is postulated as the quadratic function of strains. Respectively, constitutive equations in this theory are linear and in conventional symbols have the form

σ ik = Ciklmε lm .

(3.12)

Here, the strain tensor ε lm and stress tensor σ ik are symmetric tensors of the second rank; tensor of the fourth rank Ciklm has elastic constants as its components. Strains are assumed to be small (or infinitesimal), that means quite definite linear relationships between components of the displacement vector u = {um } and the strain tensor ε = {ε lm } (classical Cauchy relationships)

1  ∂um ∂ul  + . 2  ∂xl ∂xm 

ε lm = 

(3.13)

Nonlinear theory of elasticity use mainly finite strains, which will be specially studied in Chapter 4. The linear relationships (3.12) are the generalized Hooke law for elastic materials. Owing to symmetry of strain and stress tensors, matrix Ciklm is symmetric. That means it involves 21 independent constants. Usually, real materials (and composite ones especially) have additional attributes of symmetry. Crystalline materials permit many types of symmetries. Ten of them are the most widespread. For those traditionally used in engineering, building materials etc., the classification, which highlights three above mentioned basic groups of composite materials, is more often used. This classification is linked more likely with the potentiality of a theoretical analysis, than with realistic properties of materials. For these three groups namely, analytical methods were developed. − Orthotropic materials are symmetric by elastic properties relative to three reciprocally perpendicular axes. The number of independent constants is equal to 9. Matrix Ciklm has the form

172


C1111 C1122  C2222       

C1133 C2233 C3333

0 0 0

0 0 0

C2323

0 C3131

     0  0   C1212  0 0 0

The technology progress of composite materials manufacturing caused a new interest concerning the anisotropic theory of elasticity. In most cases, the constructors’ requirements were of favouring the creation of modern layered (laminate) and fibrous composites of a complicated microstructure, which in the average sense can be treated as orthotropic or monotropic.

−

Transversal isotropic (monotropic) materials have the following symmetry properties: so-called main axis exists (as a rule, the axis Ox is chosen), and all perpendicular to this axis planes are isotropic from the point of view of elastic properties (that is, in an arbitrary point of this axis the properties are the same). The number of independent constants is equal to 5. The matrix Ciklm has the form

C1111 C1122 C1133  C1111 C1133   C3333       

0 0 0

0 0 0

C4444

0 C4444

     0   0   1 (C1111 − C1122 )  2  0 0 0

173


−

Isotropic materials are characterized by elastic constants, which don’t depend on the coordinate system choice. In particular, tensor Ciklm is invariant relative to transforms of: rotation, inversion relative to a point, mapping in a plane. Only scalar or unit tensor δ ik has such a property. Tensor Ciklm is written as Ciklm = λδ ik δ lm + µ (δ il δ km + δ imδ kl ) ,

(3.14)

where

C1111 = C2222 = C3333 = λ + 2 µ ; C1212 = C2323 = C1313 = λ ;

1 C4444 = C5555 = C6666 = (C1111 − C1212 ) = µ . 2 The number of independent elastic constants is equal two. The elastic constants λ , µ are usually called the Lamé constants; the constant µ is also called the shear modulus; Lamé constants are linked with Young modulus E and Poisson ratio ν by the relationships

λ=

Eν E , µ= , 2 (1 + ν ) (1 + ν )(1 − 2ν )

E=

µ ( 3λ + 2 µ ) λ , ν= . λ+µ 2(λ + µ )

The matrix Ciklm has the form

174


C1111 C1212 C1212  C1111 C1212   C1111         

0 0 0

0 0 0

1 (C1111 − C1212 ) 2

0

      0    0   1 (C1111 − C1212 )   2 0 0 0

1 (C1111 − C1212 ) 2

or λ + 2µ        

λ λ + 2µ

λ λ λ + 2µ

0 0 0

0 0 0

µ

0

µ

0 0  0   0 0  µ 

.

3.5 Classical Model of Macroscopic (Effective) Moduli The conventional practice in mechanics of materials consists in that physical constants are determined using special tests. The procedure of tests is standardized. The problem of evaluation of physical constants arose in the composites theory as the special problem of optimization of properties of materials and is associated with an artificial origin of these materials. This situation can be commented on in this way, that it isn’t necessary to evaluate elastic constants for materials, internal structure of which can’t be essentially changed for obtaining some important changes in

175


their properties. The material is manufactured, and later the set of its elastic constants is determined from standard tests. But for composite materials, the procedure becomes expedient. In this way, the theory of macroscopic moduli has come into being. Thus, this theory proposes formulas for the evaluation of elastic constants of heterogeneous materials (as a rule, two-phase). For this effect, it needs knowledge of physical properties of phases (elastic constants, density) and geometrical parameters of the internal structure (sizes and form of granule-grain or fibre etc). As it was already mentioned above, the granular, fibrous, and laminate (layered) composite materials are traditionally distinguished. May be it must be added to the prior information that used as the filler granules in granular materials and used as the filler fibres in fibrous materials can be diversed by their properties and forms. For example, granules can have the form of microspheres, but can also be elongated, and not be similar to the usual granules. Fibres can be curvilinear. As a rule, for granular composites the model of an isotropic elastic material is adopted, for fibrous composites - the model of a transverse isotropic elastic material, for laminate composites - the model of an orthotropic elastic material. Let us consider the main formulas for the mentioned types of composites. The simplest formulas are obtained for the case, when the matrix and spheroidal inclusions are isotropic, and a material is also isotropic as a whole. First of all, Voigt and Reuss have proposed the simple but crude (inexact) formulas kVeff = cM kM + cI k I ,

µVeff = cM µ M + cI µ I ,

c c 1 = M + I , k Reff kM k I

µ Reff

1 =

cM

µM

+

cI

µI

.

(3.15) (3.16)

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Used symbols are traditional: k = λ + 2 µ is the bulk modulus, indices 3 V ( R ) mean the correspondence to Voigt (Reuss) approach, indices M ( I ) correspond to a matrix (an inclusion - a filler), cM ( I ) is a volume fraction of a matrix (an inclusion). All other formulas are linked with the theoretical scheme used by one or another authors. Let us show some of them. The most complicated formulas, probably, were proposed by Hill

3k

eff

3cM 3cI 3 = + eff eff + 4µ 3k M + 4 µ 3k I + 4µ eff

 cµ 3cM k M 3cI k I c µ + + 5  effI M + effM I eff eff 3k M + 4µ 3k I + 4 µ  µ − µM µ − µ I

,

(3.17)

 +2=0 . 

(3.18)

The complexity consists in that (3.17) is an equation of fourth order with respect to µ eff . More convenient formulas are given by Dewey

 k eff = k M +  cI ( k I − kM )  

µ eff

 kI − kM  1 +  k M + ( 4 3) µ M

  µ  15(1 −ν M )  1 − I  cI   µM  = µ M 1 −  µI  7 − 5ν M + 2(4 − 5ν M ) µ M 

   , 

   .   

(3.19)

(3.20)

Here, the condition of smallness of the volume fraction of spherical inclusions cI = (a / b)3 +3k ′

(3.21)

6cM cI ( µ M − µ I ) 2 < k + 2 µ > , 5 < µ >< 3k + 4 µ > +6 < k + 2 µ > µ ′

(3.22)

k eff =< k > −

µ eff =< µ > −

where notations < m >= cM mM + cI mI , m′ = (cI − cM )(mM − mI ) are used. Some more simple and differing from prior formulas were proposed by Sendeckyi k eff =

( k c ( 3k ( c ( 3k 1 1

1

+ 4 µ1 ) ) + ( k2 c2 ( 3k2 + 4 µ2 ) )

1

1

+ 4 µ1 ) ) + ( c2 ( 3k2 + 4 µ 2 ) )

µ eff = where α =

,

(3.23)

µ2 (αµ 2 c2 + c1 ) , αµ1c1 + c2

15 (1 − ν 1 )

( 7 − 5ν 1 ) µ1 + (8 − 10ν 2 ) µ2

(3.24)

.

For the case, when solid spherical inclusions are embedded into an isotropic infinite matrix, Vanin proposed expressions 4 µ 2 ( k1 − k2 ) ( 7 − 5ν 2 )( µ1 − µ2 ) 2 + c2 3k2 ( 3k1 + 4k2 ) 3 ( 7 − 5ν 2 ) µ 2 + ( 8 − 10ν 2 ) µ1 (3.25) = E2 E2 ( k1 − k2 ) ( 4 − 5ν 2 )( µ1 − µ2 ) 2E 1 − c1 − c2 2 3k2 ( 3k1 + 4k2 ) 3µ2 ( 7 − 5ν 2 ) µ2 + ( 8 − 10ν 2 ) µ1 1 + c1

E eff

k eff =

3k1k2 + 4k1µ1c1 + 4k2 µ 2 c2 3k1c2 + 3k2 c1 + 4 µ2

.

(3.26)

For composites with hollow inclusions, Vanin also proposed E eff = E2

9k2 + 4c1 L µ2 − 6c1k2 ( 7 − 5ν 2 ) H 9k2 − c1 LE2 + 12c1k2 (1 + ν 2 )( 4 − 5ν 2 ) H

,

(3.27)

178

k eff =


12k1k2 (1 − q 3 ) µ1 − c2 q 3 µ 2  + 16 µ1 µ2 (1 − q 3 ) c1k1 − c2 q 3 k2  9c1q 3 k1k2 + 12q 3 µ2 k1 + 12c1µ1k2 + 12c2 (1 − q 3 ) µ1k1 + 16µ1µ 2

(3.28)

where q = ( r1 r0 ) ( r1 is the internal radius of inclusion, r0 is the external radius of inclusion) 3 1 ( 4µ 2 + 3k2 )  2 (1 − 2ν 1 ) + (1 + ν 1 ) q  , L =1− 4 (1 + ν 1 ) (1 − q )3 µ1 + q 3 µ2  + 2 (1 − 2ν 1 ) µ 2  

( 7 − 5ν 1 )  µ2 + (1 − q ) µ1  + (8 − 10ν 1 ) q3 µ2 H= 3 ( 8 − 10ν 1 )( 7 − 5ν 1 )(1 − q ) µ1 + ( 7 − 5ν 1 ) 7 − 5ν 1 + ( 8 − 10ν 1 ) q3  3

.

More simple formulas for solid inclusions were obtained by Christensen 

µ eff = µ 2 1 − c1 

15 (1 −ν 2 ) 1 − ( µ1 µ2 )   , 7 − 5ν 2 + 2 ( 4 − 5ν 2 )( µ1 µ2 ) 

k eff = k2 + c1 ( k1 − k2 ) 1 + ( k1 − k2 ) ( k2 + ( 4 3) µ2 )

(

( (

(3.29)

))) .

(3.30)

Based on the virial expansion method, we have

  3 (1 − ν m ) ( k f − km ) , k eff = km 1 + c1 2km (1 − 2ν m ) + k f (1 + ν m )    

µ eff = µ m 1 + c1 

 . µm ( 7 − 5ν m ) + µ f ( 4 − 5ν m )  15 (1 − ν m ) ( µ f − µm )

Shermergor used the self-consistent method and found

(3.31)

(3.32)

179

Materials with Micro- or Nanostructure 2

2

ci = 1, ∑ ∗  ki  i =1 1+α  eff − 1   k 

∑ i =1

ci

µ  1 + β  i eff − 1  µ 

=1 .

(3.33)

∗

Let us now consider the fibrous composites. They are usually described by the model of transversally isotropic deformation. The linear law of elastic stresses and strains coupling for transverse isotropic materials is ν 1 ε11 = (σ 11 − ν 1σ 22 ) − 2 σ 33 ; E11 E22

ε 22 =

ν 1 (σ 22 − ν 1σ 11 ) − 2 σ 33 ; E11 E22

ε 33 =

1 ν* σ 33 − 2 (σ 11 + σ 22 ); E22 E11

ε13 =

(3.34)

σ 13 , σ σ ε 23 = 23 , ε12 = 12 . G2

G2

G1

Here, the aggregate number of independent elastic constants is five: E11 , E22 , G2 ,ν 1 ,ν 2 (two constants G1 , ν 2* are expressed by other constants

ν2 E22

=

ν 2* , G1 = E11

E11 ). 2(1 + ν 2 )

For fibrous unilateral materials the axis Ox3 is usually chosen in the direction along fibres, and the index L (longitudinal) is used for notation of properties along this axis, the index T (transverse) is used for notation of properties in the plane x1Ox2 . The simplest formulas are again credited to Voigt and Reuss ELeff = cF EF + cM EM , ν TT = cFν F + cMν M , c c 1 = F + M , eff ET E F EM

1

µ

eff LT

=

cF

µF

+

cM

µM

.

(3.35)

180


Some complicated and exact formulas are proposed by Vanin ELeff = cF EF + cM EM +

eff =ν M − ν LT

2cF cM (ν F − ν M )2 EM EF , cF EM LF + (cM LM + cF ) EF

(3.36)

2cF (ν M − ν F )(1 − ν M2 ) EF , cM EM LF + [cF LM + (1 + ν M )]EF

(3.37) k eff =

eff µ LT =

( k F + µ M ) k M − cF µ M ( k F − k M ) , k F + µ M − cF ( k F − k M )

eff (cM µ F + cF µ M ) µ M , eff 2k eff (1 − ν TT ) ELeff , ET = eff eff 2 EL + 4k eff (ν LT ) cM ( µ F − µ M )

(3.38)

(3.39)

LM = 1 − ν M − 2ν M2 , LF = 1 − ν F − 2ν F2 . The more general formulas for the elastic constants of layered composites will not be considered here. Before considering the next models, let us recall a few statements from the homogeneous elastic materials theory. It is known that in such materials only five families of universal deformations can be realized. The notion of universality of deformations is linked with two demands: − the possibility to realize in any material; − the realization with the aim of surface loading only. In spite of their simplicity, universal deformations are the select ones, their peculiar importance consists in the possibilities of using them for determination of material properties from experiments. Such types of deformations as the simple shear, the simple tension, the uniform volume compression were studied in the theory of incremental deformations. They were found applicable to the effective moduli theory, in which materials of complicated microstructure with internal constraints are treated as the homogeneous elastic materials. This possibility was used in


181

the determination of the particular types of effective constants in different ways and by different authors.

3.6 Other Microstructural Models All microstructural theories exceeding the framework of macroscopic constants theory, can be related to other direction of microstructural analysis. Here, we will dwell on the first, principally important studies. It should be noted that the prior model of effective constants and all of the next models are based on knowledge of components properties and the microstructure as a whole. Let us cite in this connection: “It should be understood, of course, that the mechanical behaviour of the components is not always known with an accuracy needed by theoreticians. Owing to manufacturing processes, the properties of components may differ to a certain degree, when the components are part of a composite as compared with the isolated state. In view of this, it appears to the author that it is hardly necessary to require agreement between accuracy to three significant figures and “precise results”. The agreement represented here with experimental data, which is within the limit of 5-10%, is completely adequate.” This opinion is a general opinion in nature. In real composites, the properties of an individual component and a component of specimens formed by individually taking material are actually different. First, due to variation in the properties of components during the production process of manufacturing (sintering, heating, irradiation, etc.). Second, due to the difference between the mechanical properties of granules, fibres, foils, etc. and the properties inherent to standard specimens of the same material. This reduces the value of models based on calculation of constants on the base of mechanical and geometrical parameters of components of the microstructure, and offers certain advantages over approaches based on experimental determination of the constants.

182


3.6.1 Bolotin model of energy continualization In Bolotin’s studies, a variational approach was proposed for derivation of basic equations and boundary conditions, and conversion from microstructure to macrodescription was accomplished on the basis of the concept of simplifying the problem, which was reached by spreading by change of a layered plate on one-layered anisotropic plate with characteristics of a stress-strain state, which continuously change with the thickness. The author later defined more precisely the term energy spreading initially introduced, and this method of constructing the continuum theory is now called the principle of energy continualization. The constructing of basic equations uses two assumptions: − arming elements are essentially more rigid than binding layers; is small − the characteristic dimension of microstructure compared to characteristic dimensions of a body and to characteristic lengths of a change of functions describing a stress-strain state of arming elements. As a result, the continuum equations of equilibrium for layered composite are obtaining as follows Lα k uk* + X α* = 0 ; L3k uk* + h 2 M 3k uk* + X 3* = 0 .

(3.40)

Here, Lα k , M 3k are linear differential operators of the second and fourth orders, respectively (the last introduce into the third equation the notion of effective stiffness); displacements uk* are so chosen that in the middle plane of each layer they are equal approximately to displacements uk( m ) of this layer; potential energy of the layered body is equal to potential energy of the equivalent quasi-homogeneous body (the fictitious body) ; analogically, components of volume forces X k are also spread; h is the characteristic thickness of reinforcing layers.


183

Let us stop on the last characteristic h ; for a two-layered composite with alternating layers – soft and stiff – h corresponds to the stiff one thickness. Thus, the equations corresponding to the Bolotin model of effective stiffness include the dimensional quantity equal to the thickness of a stiff layer. In this way, the model becomes sensitive to transition, for example, from microlayers in 10 microns to nanolayers in 10 nanometers. It turned out afterward that other microstructural models possess this property too. Also, equations (3.40) have the peculiar characteristic for many of subsequent microstructural theories - they are analogous to equations of the moment theory of elasticity for anisotropic media. Besides them, the solutions will have two parts: slowly varying inside of the layer, and quickly attenuated of the edge effect type. The principle of energy continualization has been found to be fruitful, and has subsequently been used to formulate various continual theories.

3.6.2 Achenbach-Hermann model of effective stiffness The Achenbach and Herrmann model of effective stiffness is the most advanced model of the above-mentioned models. The modern description of this model is different from the first model proposed. The method of conversion from a discrete system of alternating layers or cells with an internal fiber to a continuum body is the principal method in the model, as in any microstructural model though. It includes several stages. The first stage consists in representation of displacements and describing the local effects excitations. In the case of a fibrous composite of cellular checkerboard structure, the displacements for the cell (k , l ) in the polar coordinate system with the origin at the center of the fiber are represented in the form uif ( k ,l ) = ui( k ,l ) + r cos Θ ψ 2fi ( k ,l ) + r sin Θψ 3fi ( k ,l ) (r < a ); uim ( k ,l ) = ui( k ,l ) + a cos Θ ψ 2mi ( k ,l ) + a sin Θψ 3mi ( k ,l )

(3.41)

184


+ (r − a ) cos Θψ 2mi ( k ,l ) + (r − a) sin Θ ψ 3mi ( k ,l ) (r > a ). The next step consists in the representation of internal and kinetic energies for cells as functions of displacements. The spreading operation is then carried out, and energy is determined at the fibre centre. Averaged global displacements are then equated to the displacements at the centre of the fibre in the initial discrete cell system. These averaged displacements define a new homogeneous continuum. The equality between energies in the discrete and spread cells complete the scheme of energy continualization. The density of strain energy of the new continuum depends on the effective moduli, but include additional constants depending on parameters of the microstructure and have sense of effective stiffness. The model (theory) has therefore named the model (theory) of effective stiffness. At the final stage of the theory's construction Hamilton’s principle is used. On the whole, continuum theories of different order were obtained as a function of the number of terms retained in the representations of the energy densities.

3.6.3 Models of effective stiffness of high orders Drumheller and Bedford provide a bibliography on theories of effective stiffness of orders greater than the first one. They proposed a new approach, which makes it possible to construct a model of any order and can model stresses in layered composites. The theory of effective stiffness permitted to investigate the problems of harmonic wave propagation in infinite layered and fibrous composites using different variants of model. Like other approximate theories, the theory of effective stiffness poorly describes the properties of wave filtration by a layered medium during the propagation of waves normal to the layers. At the same time, the exact theory catches these characteristic features of waves well, and


185

they are observed experimentally. A variant of the theory, which its authors call the theory of equalized effective stiffness, has been developed in order to improve the description of the frequency spectrum to bring the latter as closely as possible to the exact description. This theory approximates well zones of wave cutting-off and transmission in layered media. However, let us pay our attention to deficiency of the theory of effective stiffness, which are also characteristic of the theory of mixtures and other micro-structural theories. It is a question that such models describe only the first (lowest) modes. A series of experiments have confirmed for some time now, however, that the contribution of the first modes is dominant in the majority of cases. This is also required of approximate models.

3.6.4 Asymptotic models of high orders In the 1970’s the theory of directionally reinforced (layered) composites has been proposed, which, like the theory of effective stiffness, is based on the expansion of a displacement vector. It is complementary assumed that in this theory the wavelength is essentially larger than the typical size of microstructure. This assumption ensures the continualization. Depending on the numbers of terms, which are retained in asymptotic expansions on the obtained small parameter, the theories of various orders are built. The fact of asymptotic expansion assumed here to be the main, and, in the connection of this, such a model is related to models, which are based on asymptotic expansions. This model differs essentially from the model of effective stiffness, since here the displacements contain the gradients of high orders and constitutive equations have the form of equations of materials of N-th order. Also continuum approach to the layered continua based on asymptotic expansions was developed. Within the framework of these models plane waves in layered composites were studied. On the whole, such continuum approaches give the better exactness in description of dispersion curves than many other theories.

186


3.6.5 Drumheller-Bedford lattice microstructural models Theories based on discrete models of lattices form the isolate direction in the microstructural analysis of composites. A feature of these theories consists in that here the discrete structure of composites is modeled by the discrete lattice model, too. For the first time the approach, which used the lattice model, was applied to layered plates. An introduction of lattices was based on Bloch and Floquet theorems. This model was developed to continuum-discrete models and others. One should remark that the authors of the noncontinuum lattice models have been participated in the constructing of many other microstructural continuum models. For example, the one-dimensional lattice can be used for the description of waves in fibrous materials. As a ground for this model, the fact is taken that in the case of wave propagation in the direction normal to fibres the last ones work as rigid obstacles similar to the particles in lattice nodes. This lattice model is proposed for description of phenomenon of geometrical dispersion of waves. It is also assumed that a form of fibres has the small effect on dispersion and a periodicity of fibres in space is essential. Lattice parameters are proposed to determine from a system of the correspondence to basic physical parameters of the composite as the medium, through which waves propagate. For example, − the stopping (cut) frequencies of a composite and a lattice, − the wave numbers which correspond to these frequencies, − the phase velocities of longitudinal waves in the lattice and the composite, have to coincide and mean densities of the lattice and the composite must be equal. It is widely assumed that continuum theories of composite materials


187

have some advantages over the discrete ones. First of all, for the reason that they are related to wider classes of problems and they are used more than the applying in mechanics mathematical methods. Practically, all microstructural theories are sufficiently complicated in their constructions. May-be, that highlights the complexity of macroscopic processes during the composite materials deformation. Or, probably, that highlights the level of development of continuum mechanics. The medieval philosopher Grigory Skovorod, once said: “We should be obliged to God, that He founded the world in such a way that all that is simple is the truth, and all that is complicated, is not the truth”.

3.6.6 Mindlin microstructural theory This theory has a few features: A. It is based on the notion of a vector-director; B. It has been one of the first microstructural theories; C. It was not further developed in theory and had not been used in practice. Mindlin was one of the first to construct microstructural theory for materials with the repeated unit cell. Applicability to the determined cell structure constitutes an essence feature of this theory. The proposed, by Mindlin, microstructural theory, is based on analysis of the cell as a linear variant of a deformable director. If the cell is considered as absolutely rigid during rotations, the microstructural model is reduced to the Cosserat model of moment continuum. In the general case, having its own microstructure cell is described on the microlevel by an elastic continuum, the linear deformation of which is characterized by three independent tensors: − the tensor of macro strains 1 (3.42) ε ij = (∂ i u j + ∂ j ui ); 2 − the tensor of relative strains

188


γ ij = ∂ i u j −ψ ij ; −

the tensor of micro strain gradient κ ijk = ∂ iψ jk .

(3.43) (3.44)

Here, ui are components of the macro displacements vector, and

ψ ij = ∂ i/ u /j are the components of the tip of deformable director. The continuum theory is constructed on the basis of formalism of the classical theory of elasticity on the bases of assigned in the microstructural model kinematic parameters ε ij , ψ ij , κ ijk . Overwhelming difficulties arise in the determination of the physical constants of the theory. In fact, there are 903 constants in the general case, and 18 independent constants for the isotropic case (2 independent constants in the theory of elasticity, for example).

3.6.7 Eringen microstructural model. Eringen-Maugin model Approximately at the same time with the above mentioned Mindlin micro-structural model, Eringen and Suhubi proposed a model that they called the micromorphic theory. In this theory, the macrocontinuum is described by an elastic medium, each point xk of which is additionally rigged with three deformable directors X K (or one deformable vector director X ). Kinematics of the micromorphic continuum is described by three tensors: −

−

Green classical tensor of deformation CKL = xk , K xk , L ;

(3.45)

the two microstrain tensors S KL = xk , K X k , L , Γ KLM = xk , K X k , LM .

(3.46)

Thereafter, the micromorphic model is constructed in the same way as the previous Mindlin model. In general, they are very similar. Their fates


189

have been completely different, however, even despite their obvious abstractness. The micromorphic theory has found unusual application in the theory of waves in piezoelectric materials. The so-called piezopowders with a small compacting become the granular composite materials. They testify the three-stage evolution of piezomaterials. Piezoelectric phenomena were initially associated with crystals; since the structure of a single crystal satisfactorily characterized the material, however, a question of the material microstructure didn’t arise here. It might seem, the discovery of piezoceramics and the study of waves in these materials could drawn attention to the behaviour of the components of the ceramics, i.e., to their internal structure. But assumptions concerning the mechanical uniformity of a characteristic volume of ceramics and the domain structure of polarization were found, however, to be sufficient. Again, microstructure was found not to be necessary. Only the study and use of the third generation of piezoelectric materials the granulated piezoelectric powders - has finally forced us to consider the microstructure of the material in models of its piezoelectric behavior. This was done in the framework of the micromorphic theory, and later was continued for the mixture theory. It is rather natural to convert the theory of piezoelasticity to a micromorphic theory. With respect to piezoelectric powders consisting of a mixture of piezoceramic granules and usually granules of naphthalene or lead, it is expedient to use the microstructural model. In these powders, the electrical polarization, which is normally modeled by a vector field in physics, is the determining physical parameter. Moreover, it is easy to represent the domains in the form of such a microstructural characteristic as a granule. On the whole, the micromorphic theory elegantly explains a number of physical effects. Where the subject of interest is wave propagation in granular powders such that the wavelengths are sufficiently close to the characteristic size of the microstructure (for example, the mayor diameter of the granule), a good approximation yields another microstructural model, the model of a

190


mixture. This model gives a good approach for the linear and nonlinear wave investigation and will be discussed later.

3.6.8 Pobedrya microstructural theory The classical micropolar approach arises, when the modern procedure of averaging is applied to the composite with the regularly repeated elementary cell. Such approach was proposed by Pobedrya. The initial inhomogeneous problem of the linear theory of elasticity (written in terms of displacements) ∂ 2u Cijkl ( x)uk ,l  + X i = ρ 2i ; ,j ∂t Σ Σ aij C jkl n ul , n nk + bij u j = SiΣ ;

ui ( x,0) = U i ( x);

(3.47)

∂ui ( x,0) = Vi ( x) ∂t

is reduced to the recurrent sequence of problems of the linear theory of elasticity for an anisotropic homogeneous medium with some effective elastic moduli

hijmn wm( k, )nj + X i( k ) = ρ

∂ 2 wi( k ) ; ∂t 2

aijΣ h jlmn wm( k, n) ni + bijΣ w(jk ) = SiΣ ( k ) ; wi( k ) ( x,0) = U i( k ) ( x);

(3.48)

∂wi( k ) ( x,0) = Vi ( k ) ( x) . ∂t

In addition, the solution ui ( x, t ) of the initial problem (3.47) is associated in a sufficiently complicated form with the solution wi( k ) ( x, t ) of the new problem

191


∞

ui ( x, t ) =

∑

α p + q N ijk( p )...k (ξ ) 1

∂ p v j , k1 ...kq ( x, t )

q

p + q =0

∂τ p

∞

vi = ∑ α k wi( k ) .

;

(3.49) (3.50)

α =0

Here, the small parameter α is equal to the ratio of the characteristic size of microstructure and the construction, and ξ = x α . The feature of this model is that in the null approximation the local displacement within the cell can be evaluated as ui = vi + α N ijk (ξ )v j , k ( x) .

(3.51)

In the notation of equations (3.46) and (3.47) the constitutive equations are absent. Therefore, the system of equations is not complete. In other notation, when the problem is written in stresses, this problem is reduced to the problem of moment homogeneous theory of elasticity. In one's turn, the moment problem is reduced to recurrent sequence of problems of the theory of elasticity in stresses for anisotropic homogeneous medium with effective moduli of mechanical compliances. One considers that the procedure of evaluation of effective moduli of elasticity and compliance is well developed. Scheme of the microstructural theory permits, in theory, to use successfully the modern numerical methods.

3.7 Structural Model of Elastic Mixtures This continuum microstructural model is, among all microstructural models, the most developed. Its basic advantage consists in the fact that it can be constructed similar to the classical theory of elasticity, and in point of fact it can be meant as the direct generalization of the onecontinuum model on the multi-continuum model with the same set of thermodynamical parameters. Also, in this book most of the problems

192


on wave propagation in materials with microstructure, are studied within the framework of the mixture theory. Let us start with the remark that the studies of composite materials had one collateral consequence: they have been demonstrated as very expressive by the well-known physical principle: the one and the same physical object can be investigated with the aim of different physical models depending on which phenomenon in the motion of this object is the main goal of study. As applied to composite materials, this principle was transformed in such a way: one and the same composite material can be described by a few tens of the various and, sometimes, very exotic models; all these models will be reasonable and expedient within the framework of some restrictions, which should be thoroughly fixed, and the procedure of such fixing is usually called the fixing of the model practicability boundaries. One important consequence from observing plenty of models is worthy of a special attention: one and the same material in its description by different models will be described within the framework of each model by its own set of physical constants. Very often, sets for two distinct models have no coinciding constant. The determination of full sets of constants is the lamest point for microstructural theories. There exists the subconscious persuasion that material constants are the thing in itself and should not depend on choosing the model of this material. For example, the traditionals understood density in a number of theories is changed on the set of densities. Therefore, the understanding of relativity of used for description of materials set of physical constants must be considered the one of unexpected value in structural mechanics of materials. Let us return now to the model of mixtures so as to confirm the mentioned above facts. First of all, it is necessary to note that multi-phase mixtures as an object of studies begun with the Fick (1855) and Stefan (1871) publications. The concept of interpenetrating and interacting continua forms a theoretical basis. In the consequence of that, all further


193

development within the framework of such a conception theories are continuum ones. The essence of the concept consists in that each geometrical point of a filled by the mixture domain (body) is simultaneously occupied by two (if a mixture is two-phase) or three (if a mixture is three-phase) particles (phases, components), between which the relative motion occurs. Unfortunately, the term “mixture” is used both for real physical objects, and for theoretical models of these objects. The most number of publications is observed in that part of mechanics of multi-phase mixtures, which studies the soils saturated by gas or fluid. Transnational gas and oil companies nourish interest of scientific community to such mixtures. The concept of mixtures was applied to composite materials already at that time, when for other types of mixtures, suspensions, emulsions and others, accumulated a lot of theoretical and experimental facts. Therefore, the first variant of the mixture model of elastic materials proposed by Green and Steel had not even a proper matter of elastic deformation name the diffusional model. The point is that the hypothesis on diffusional character of the transmission of force pulse from one phase to other phase (so called the force interphase interaction) is natural for the fluid saturated porous materials and is adopted for solid mixtures from there. Unfortunately, authors of the diffusional model have not paid attention on some contradiction between the hypothesis of diffusivity for a force pulse and more global hypothesis of elasticity for deformation of mixture. The elasticity can’t have as a consequence the energy attenuation, whereas the diffusional force pulse is directly proportional to a relative velocity of phases and, therefore, it introduces into the full energy some term testifying the energy scattering in an elastic material. This circumstance was the implicit cause of inadequateness of proposed diffusional model with observed effects in conventional composite materials. The diffusional model has gone the full cycle of evolution, and is very seldom remembered at present time.

194


A construction of the adequate to composite materials mixture theory was begun by Lempriere, Bedford, Stern publications, in which the shear model of the force interaction between phases was proposed. For layered composites an intuitive structural one-dimensional model was constructed, where it was observed that owing to the distinction of shear properties of layers, the macro interaction force arises between contacting layers, and it is directly proportional to difference of macrodisplacements in layers. In terms of the phenomenological mixture model, that means the force interaction between phases is directly proportional to the difference of phase displacements. This mechanism was called the shear mechanism by virtue of its physical nature. As it turned out, the shear mechanism introduces not dissipation into the mixture full energy and doesn’t contradict the basic concept. It describes well the geometrical dispersion phenomenon, everywhere observed in wave processes in composite materials. The shear model demonstrates a high practicability of the mixture concept to composite materials and show the close interlink between the microstructural mixture theory and other various approaches in mechanics of composites. A slightly more general concept of the linear model of elastic twophase mixtures was proposed practically simultaneously in different works. First of all, it was shown that basic hypothesis of a linear elastic deformation admit only two mechanisms of the force interaction - shear and inertial. Methods of constructing the linear theory by different authors are distinct. The inertial interaction mechanism is caused by the distinction in inertial properties of phases. It can be introduced formally by taking into account the phase cross-interaction in mixture kinetic energy as a whole. This mechanism is essential in many type mixtures and was taken into account in many models. Now, let us consider the two-phase mixtures. Firstly, the basic hypothesis of any mixture theory should be adopted: the microstructure of a two-phase composite can be described by two continua, the material particles of which are placed simultaneously at each geometric point of a domain and interact with each other. Each continuum is characterized by


195

its own set of field characteristics of the partial density ραα , the partial vector of displacements u (α ) , and the partial tensors of the stresses σ ik(α ) , strains ε ik(α ) , and rotations ωik(α ) . Here and later, the Greek superscripts and Latin subscripts are equal to 1 and 2, and 1,2, and 3, respectively. In accordance with the traditions of mechanics of heterogeneous media, the parameter is named partial if it characterizes one phase only. For the well-posed description of a mixture, three hypotheses are also formulated: 1. All properties of a mixture should be the mathematical consequence of mixture phases properties. 2. For description of the motion of a separately taken component, this component should be mentally isolated from the rest of a mixture with the condition that an interaction between phases is taken into account. 3. The motion of a mixture is described by the same balance equations as the motion of separately taken phase. The basic equations of the theory of mixtures are derived, assuming that the laws of mass, momentum, angular momentum, and energy conservation are valid for the mixture. It is necessary to say that: − The first three laws are always written for each component separately. − The theories of mixtures are divided into two parts, depending on the law of energy conservation written either for each component separately or for the mixture as a whole. This division is essential in view of its effects. In the case “separately” the physical constants of each phase are used, whereas in the case “as a whole” they are not. In the first case the negative sign characterize the theoretical constants which may differ from the real one, while the positive sign represent these constants known from experiments. In the second case, the negative sign testifies

196


that the complete set of physical constants has to be determined from special experiments. Positive sign corresponds to these values for concrete composite material. Writing the basic equations needs an introduction of some classical notions and symbols. Let the mixture fill the body with a volume V . The motion of each phase of the mixture will be described relatively to the fixed orthogonal Cartesian coordinates. It should be noted that the equation of energy balance is a consequence of previous balance equations (mass, momentum, angular momentum), if the energy scattering isn’t taken into account in the mixture. Since it is usually assumed that the mixture is elastic deformed only, hence it is not taken into account. The assumption about the elastic character of deformation led to the theory of elastic mixtures. The assumption about linearity of deformation and all other processes simplifies the basic system of equations. Let us restrict, for the time being, the linear model and consider somewhat more in detail the linear elastic model, and later the viscoelastic and piezoelastic linear models. The mixture as a thermodynamical system is described by two kinematical parameters: partial strain tensors ε ik(α ) and relative displacements vectors ( uk(1) − uk(2) ) . Therefore, the internal energy of mixture as a whole is represented as a function of these parameters U = U ( ε ik(1) , ε ik(2) , ωik(1) , ωik(2) , uk(1) − uk(2) ) .

(3.52)

For internal energy, a few important formulas can be obtained, which are consequences of some thermodynamic hypotheses dU =

∂U ∂U ∂U d ε ik(α ) + (α ) d κ ik(α ) + d ( uk(1) − uk( 2) ) ; (α ) (1) ∂ε ik ∂κ ik ∂ ( uk − uk(2) )

σ ((ikα )) =

in (1) ∂U ∂U ∂U (α ) ; σ = ; R ; k = [ ik ] (α ) (α ) (1) ∂ε ik ∂κ ik ∂ ( uk − uk(2) )

(3.53)

(3.54)

197


U=

1 2  ∂ 2U (0) (α ) ( β ) ∂ 2U (0) (β ) ε ε + κ (α )κ lm ∑ lm ( β ) ik ( β ) ik 2 α ,β =1  ∂ε ik(α ) ∂ε lm ∂κ ik(α ) ∂κ lm  +

∂ ( uk(1)

+2

+2

∂ 2U (0) ( uk(1) − uk(2) )( um(1) − um(2) ) − uk(2) ) ∂ ( um(1) − um(2) )

∂ 2U (0) (α ) ( β ) ∂ 2U (0) ε κ + 2 ε ik(α ) ( um(1) − um( 2) ) lm (α ) ( β ) ik (α ) (1) ( 2) ∂ε ik ∂κ lm ∂ε ik ∂ ( um − um ) ∂ 2U (0) ∂κ ik(α ) ∂ um(1) − um( 2)

(

)

κ ik(α ) ( um(1) − um(2) ) +

 1 3 d U + ... 3! 

(3.55)

After some simplifications, the constitutive equations for a linear anisotropic mixture can be written as (α ) (α ) (3) (δ ) σ ik(α ) ( x, t ) = ciklm ε lm ( x, t ) + ciklm ε lm ( x, t )

(α + δ = 3) .

(3.56)

In the linear theory, an interaction between mixture phases is described by three mechanisms. They are introduced into the linear theory by means of phenomenological considerations. The interaction force is represented as a sum of two forces, which characterize a change of kinetic and internal energies owing to the phase interaction, respectively. From the kinetic energy consideration follows that the phases interaction is displayed by the presence of a new additive term in the kinetic energy, which has the form of added mass energy. This mechanism is called the inertial mechanism. The second mechanism consists in the cross influence of one phase strains on stresses of another phase. It is displayed in constitutive equations (3.56). The third mechanism was initially offered for the one-dimensional model. Here, the interaction force is directly proportional to the relative displacement of phases at each point. This force in the layer composite, along layers of which the shear wave propagates, is the shear force over the boundary between layers. This mechanism is therefore called the shear mechanism.

198


Owing to Bedford works the theory with such a type of interaction mechanism is called the Bedford theory. The basic system of equations of linear theory of elastic mixtures is constituted from the coupled system of six equations of the motion

  ∂ 2uk(1) ∂ 2 uk(1)   ∂σ ik(α ) ∂ 2 uk(α ) + Fk(α ) + (−1)α  β k ( uk(1) − uk(2) ) − ρ12  − = ρ   αα 2 ∂xi ∂t 2   ∂t 2  ∂t  and six linear constitutive equations (3.56), thus giving six coupled hyperbolic differential equations of the second order (α ) (α ) (3) (δ ) ciklm ε lm,i + ciklm ε lm,i + Fk(α )

(3.57)

  ∂ 2uk(1) ∂ 2 uk(1)   ∂ 2 uk(α ) . + (−1)α  β k ( uk(1) − uk(2) ) − ρ12  − = ρ  αα 2 ∂t 2   ∂t 2  ∂t  The next important fragment of the theory is the physical constants. There exist a few different approaches to the determination and clearing (α ) up of a physical sense of constants β n , ciklm , ρ12 . On the method of

determination of partial densities ραα there exists a common position. Let us comment on this position. At the beginning, to focus the attention on stresses and to remember that theoretically each phase of a mixture occupies fully with other phases of the elementary (unit) volume, whereas all phases together occupy the same elementary volume and each phase occupies only its own part of the elementary volume. All partial quantities are therefore not quantities acting over a given phase. They are spread over all unit volumes. For example, the partial stress tensor σ ik(α ) should be considered as only the characteristic from a mixture model. To go to real stresses acting over the real phase, it is required to know the volume fraction of a given phase cα . Then real stresses σ ik(α ) real acting in a given phase of a mixture are equal to

σ ik(α ) real = σ ik(α ) cα .

199


Analogously, each partial density ραα is not equal to the density of the separate phase ρα . It is the density of a separate phase spread over all unit volume. Therefore, the partial density is calculated by the formula

ραα = ρα cα .

(3.58)

Let us now start with the simplest formulas for the evaluation of physical constants, based on the simplest considerations and deterministic concept. Case 1. The layered composite with layers assumed to be plane. Layer properties are as follows: Lamé constants λ m , µ m , density ρ m , layer thickness 2h (the first layer); Lamé constants λ f , µ f , density ρ f , layer thickness 2δ (the second layer). It is also assumed in this model that the cross interaction stress-strain and inertial interaction are absent (3) ciklm = 0, ρ12 = 0 .

On the base of the approach described above for an effective moduli calculation, we have

 λ f (λ f − λ m )  f f m , c = c λ + 2µ − c D   f 11

f

 λ f (λ f − λm )  f m , c = c λ − c D   f 12

c13f = c f

λ f ( λ m + 2µ m )

c44f = c f

where

f

D

, c33f = c f

µ f µm , c f µ m + cm µ f

(λ

f

+ 2µ f

)( λ

(3.59) m

+ 2µ m )

D

1 f ( c11 − c12f ) = c f µ f , 2

,

200


cf =

δ h+δ

,

cm =

h m f f f m m , D = c ( λ + 2 µ ) + c ( λ + 2µ ) h+δ

(formulas for cikm can be obtained from cikf by change f → m, m → f ). Constants β k are calculated by

β1 = β 2 =

3 ( λ f + 2µ f

)( λ

m

D(h + δ )

+ 2µ m )

2

, β3 =

3µ f µ m

(c

m

µ f + c f µ m )(h + δ )

2

.(3.60)

(α ) From these formulas follows that constants ciklm are determined through

mechanical properties of layers and their volume fractions, whereas the interaction between phases and the transmission of the force pulse from one component to another component coefficients β n are inversely proportional to the square of microstructure characteristic size. Case 2. The fibrous composite with the small volume fractions of uniformly distributed fillers which are assumed to be strong in tension and weak by shear and bending. The formula for calculating β3 is 2

β3 = −

µm (1 − ε 2 ) (1 h 2 ) 2 1 1 1 3 1 ln ε + ε 2 − ε 4 − + (1 − ε 2 ) ( µ m µ f 2 2 8 8 8

,

(3.61)

)

where µm , µ f are shear moduli of the matrix and fibre, δ is the fibre radius, s is the distance between fibres, ε = ( δ h ) , h 2 =

(

3 2π s 2 .

)

Case 3. The fibrous unidirectional composite with ideal contact with the matrix and placed in knots of hexagonal lattice fibres. Waves are propagating along fibres. Properties of the matrix (subindex 1) and fibres (subindex 2): Lamé constants µα , λα , density ρα , fibre radius r1 , distance between centres of fibres 2r2 .

201


Constitutive equations are written in the form

σ xx(α ) = cαα u x(α, x) + c12u x(δ, x)

(α , δ = 1, 2;α + δ = 3)

(3.62)

where cαα = nα ( λα + 2 µα ) − ( λα2 E ) , c12 = ( λ1λ2 E ) , E=

1  n2 ( λ1 + 2 µ1 ) + n1 ( λ2 + 2µ 2 ) + µ 2 n1  , n1n2 

(3.63)

n1 = ( r12 r22 ) , n2 = ( r22 − r12 ) r22  . For the shear interaction coefficient β we have: 8n1µ1µ2 , β= r1  r1µ 2 + 4 µ1Q ( r2 − r1 ) 

3

 2 r2 1 −  .  r2 + r1  3 r2 − r1 4 

(r − r ) Q= 2 1

(3.64)

Case 4. The layered composite with plane layers. Constitutive equations are assumed in the form (3.56) with allowance for transversal isotropy of the mixture. For three constants the formulas are written as follows cαα = nα Eα − ( λα2 E ) , c12 = ( λ1λ2 E ) ,

E = ( E1 n1 ) + ( E2 n2 ) , Eα = λα + 2 µα , nα =  hα

(3.65)

( h1 + h2 )

.

The force interaction coefficient is proposed to calculate according to the formula 3µ1 µ2 , (3.66) β= 2 n + n h + h µ µ ( 2 1 1 2 )( 1 2 ) which coincides with formula (3.60). It should be noted that the above formulas are very simple. But this simplicity is caused by the very simple theoretical schemes used and by strong restrictions on the mixture model.

202


Case 5. Stochastic approach. This model assumes that the geometrical structure of a mixture has a stochastic and slightly differing from the regular structure character. The approach developed before for effective moduli calculation has been used. It is supposed that the small difference of mixture components properties, also fluctuations of the elastic constants tensor of initial heterogeneous elastic medium are neglected. For granular microstructures, the physical constants are determined as follows  K ∗2 2 µα∗2  µα∗2 (α ) =  α∗ − δ δ + ciklm (δ ilδ km + δ kmδ il ) ,  ik lm 3 µ∗  µ∗  K  K ∗ K ∗ 2 µ1∗ µ2∗  µ1∗ µ 2∗ (3) ciklm = 1 ∗2 − δ δ + (δ ilδ km + δ kmδ il ) ,  ik lm 3 µ∗  µ∗  K Kα∗ = (−1)δ

Kα ( K ∗ − Kδ ) K1 − K 2

, µα∗ = (−1)δ ∗2

β = 2 5c1c2 a 2

( λ + 2µ )

µα ( µ ∗ − µδ )

(3.67)

,

µ1 − µ2

+ 2 µ ∗2 .

(3.68)

For the force interaction coefficients we have, for the layered composites

βα = 2 3c1c2 a 2 µ ∗ , β3 = 2 3c1c2 a 2 ( λ + 2µ ) , ∗

(3.69)

for the fibrous composites

βα = 4c1c2 a 2

∗2

( λ + 2µ )

+ 2 µ ∗2 , β3 = 4 3c1c2 a 2 µ ∗ ,

(3.70)

for another variant of fibrous composites

βα =

1 + ν  ( 2 + c1 )( 3 − 4ν )  c1c2 a 2 E  12 (1 − ν )

(3.71)


4 +

c2π

∞

2

  , π π m c n c sin sin 1 1  ( m2 + n 2 )

ξ m2 + 2n 2

∑ mn

m , n =1

203

−1

where E ,ν are effective elastic constants, a is expressed through ( c1c2 ) ,

ξ = (1 − 2ν ) (1 + ν ) . Some more complicated and closed to the above described stochastic approach was realized for granular composites with an ellipsoidal form of granules. Since the considered granular composites are assumed to be deformed nonlinearly, then these analytical and numerical results will be shown in Chapter 5, devoted to quadratically nonlinear waves in composite materials. Case 6. Three-layered composites. Layers are assumed to be plane, elastic homogeneous, and isotropic. Two layers are the basic; they alternate. The third layer is inserted between these layers, it models the thin interface layer cemented of two basic layers. The properties of the third layer are taken into account in formulas for constants of a two-component mixture (α ) c1111 = nα ( λα + 2 µα ) − ( λα2 E ) + no ( λo + 2 µo ) kδ2 , (3) c1111 = ( λ1λ2 E ) + no ( λo + 2 µo ) k1k2 ,

(3.72)

2

ραα = nα ρα + no ρ o kδ , ρ12 = no ρ o k1k2 . Here, the subindices 0,1,2 belong to the third, first, and second layers respectively; no = 2ho h , nα = hα h , h = h1 + h2 + 2ho , kα = κ o κ κα = hα 3µα ,

κ = κ1 + κ 2 , E = ( ( λ1 + 2 µ1 ) n1 ) + ( ( λ2 + 2 µ2 ) n2 ) , 2hk is the layer thickness. By assuming ho = 0 we obtain the known formulas for two-layered composite.

204


In this model, the presence of the inertial mechanism of interaction between mixture phases is displayed. This fact can be commented on in the way that neglecting the third thin interface cemented layer in the basic scheme causes the new modeling mechanism of interaction between mixture phases. Now the more general statement can be formulated: the inertial mechanism arises in cases, when the condition of an ideal contact between mixture components are violated. One such violation is the presence of the thin interface layer, that can be called debonding or delamination. Case 7. McNiven – Mengi model approach for layered composites. This is the variant taking into account all three interaction linear mechanisms in the mixture theory. The mixture is assumed transversal isotropic one. It contains 19 physical constants: 15 independent constants of the cross (α ) , two constants of the shear interaction β k , and two interaction ciklm

constants of the inertial interaction ρ12( k ) . Formulas for evaluations of the four last constants are as follows 3r1r2 3E E , β 2 = 21 2 , 2 ∆ r ∆ E 2 2 ρr +ρ r ρ E2 + ρ E2 = 1 2 2 2 1 , ρ12(2) = 1 2 2 2 1 . 5r 5E

β1 = β 3 = ρ12(1) = ρ12(3)

(3.73) (3.74)

Here the notations rα = µα nα , r = r1 + r2 , ∆ = h1 + h2 ( hα is the layer thickness), nα = hα ∆ , Eα = ( λα + 2µα ) nα , E = E1 + E2 are adopted. These formulas have been obtained in the approach valid only for large time. For the other constants we have: (α ) c1111 = nα Eα − ( λα2 E ) ,

(3) (3) c1111 = ( λ1λ2 E ) = c1133 ,

(3) (α ) c1122 = nα λα ( Eδ E ) , c1122 = n2 λ1 ( E2 E ) , (α ) 2 (3) c2211 = n1λ2 ( E1 E ) , c1133 = nα λα − ( λα E ) ,

(3.75)

205


(1) (2) (3) (1) (2) (3) (3) c1313 + c1313 + 2c1313 = ( r1r2 r ) , c1313 c1313 − c1313 c1313 = 0 , (1) (2) (3) (1) (2) (3) (3) c3333 + c3333 + 2c3333 = ( E1 E2 E ) , c3333 c3333 − c3333 c3333 =0, 2

(1) (2) ρ 2 c1313 + ρ1c1313 =  ρ1 ρ 2 − ρ12(1) ( ρ1 + ρ 2 )  ( c∞r ) + ρ12(1) ( r1r2 r ) , 2

(1) (2) ρ 2 c3333 + ρ1c3333 =  ρ1 ρ 2 − ρ12(1) ( ρ1 + ρ 2 )  ( c∞E ) + ρ12(1) ( E1 E2 E ) ,

( ( r r ρ ρ ) )  ( r ρ ) + ( r ρ )  , = ( (E E ρ ρ ))  (E ρ ) + (E ρ )  

c∞r = c∞E

1 2

1

1

2

2

1

1

2

1

1

2

1

2

2

2

.

Case 8. Two based on experiment approaches. It is assumed in the first approach that the real mixture isn’t necessary two-phase and the two phases are not necessary in the ideal contact, but two dominant phases always exist. An essence of the approach consists in that it proposes to determine the physical constants using the parameters of dispersion curves obtained in a few experiments on samples made of real materials. It is necessary in the approach to determine experimentally the dependences of phase velocity of plane longitudinal, transverse horizontal and transverse vertical waves on frequency. To calculate four unknown constants, it is necessary to observe experimentally four characteristics of the dispersion curve: 1. The value v1 (0) = v1o of the phase velocity for ω = 0 (that is, the phase velocity of a basic mode for very small frequency); 2. Two values v1∞ , v2∞ of the phase velocity of both modes (basic and side waves) for ω → ∞ (that is, for very large frequency); 3. The value ωcut of the cutoff frequency of the side wave. The question arises on what are the zero and infinite frequencies in experiments, that is, for very small and very large frequencies. In this case, the value of a characteristic dimension of a microstructure lCDM of

206


the tested material should be used. Usually, when lCDM is significantly (two or more orders) smaller than a wavelength and therefore this wave can be referred to as the long waves for this material, then corresponding to this wavelength frequency can be treated as very small. When lCDM is close (one or less orders) to the wavelength and this wave is the so-called short wave, then corresponding to this wavelength frequency can be treated as very large. Note that, in this case we are approaching the wavelengths when the continual approach is no longer valid. The approach proposes to determine the physical constants from relations 2

a1 + a2 + 2a3 = ( v1o ) ( ρ11 + ρ 22 ) ,

( ρ 22 − ρ12 ) a1 + ( ρ11 − ρ12 ) a2 − 2 ρ12 a3 2 2 = ( v1∞ ) + ( v2∞ )   ρ11 ρ 22 − ρ12 ( ρ11 + ρ 22 )  ,   2

2

a1a2 − a32 = ( v1∞ ) ( v2∞ )  ρ11 ρ 22 − ρ12 ( ρ11 + ρ 22 )  ,

ωcut =

(3.76)

β ( ρ11 + ρ 22 ) . ρ11 ρ 22 − ρ12 ( ρ11 + ρ 22 )

Let us now consider three classical cases of a material symmetry. Isotropic mixture It is necessary to determine eight constants: a1 = λ1 + 2µ1 ; µ1 ; a2 = λ2 + 2µ 2 ; µ2 ; a3 = λ3 + 2 µ3 ; µ3 ; ρ12 , β through two sets in fours of the above described characteristics (one for the P-wave (longitudinal wave) and one for either the SH-wave (transverse horizontal wave) or the SV-wave (transverse vertical wave) ).


207

Transversal-isotropic mixture It is necessary to determine 18 constants: (n) (n) c11( nkk) , c3333 , c1313 , β1 = β 2 , β 3 , ρ12

through five sets in fours of the above described characteristics (in each set, one for a longitudinal wave (P-wave) and one for either a transverse horizontal wave (SH-wave) or a transverse vertical wave (SV-wave)). Here and later the redundancy of the number of equations relative to constants is seeming, since the fourth equation from (3.76) for different types of waves will be the same owing to the same mechanism of phase interaction. The cutoff frequency for any type of waves is determined by parameters ραβ of kinetic energy

and

coefficients of shear interaction β k . The last

depends only on symmetry of mechanical properties of the mixture and doesn’t depend on a wave type. For example, for longitudinal and transverse waves identical relation between ωcut and ραβ , β k will be obtained. Now, the number of equations already becomes seeming. Orthotropic mixture It is necessary to determine 31 independent constants: (n) (n) (n) (n) cmmkk , c1212 , c1313 , c2323 , β n , ρ12

through nine sets in fours of the above described characteristics ( Pwave, SH-wave and SV-wave in three coordinate directions). The second approach is close to the first approach. The main distinction consists in that the second assumes the constant ρ12 is no (1) (2) (3) more the scalar but the vector ρ12 = ( ρ12 , ρ12 , ρ12 ) and matrices (3) (4) ciklm , ciklm are no longer equal, but can be obtained from one another by a

transposition.

208


The essence of this approach consists in that for one part of constants the formulas are proposed, which are based (as all previous formulas) on an analysis of the microstructure. For the second part of constants, the experimental dispersion curves for waves propagating in directions along the layers and normal to layers should be used. The standard set of three polarized plane waves is assumed - the P-wave, the SH-wave, and the SV-wave. We propose to use two basic characteristics of dispersion curves: 2 1. the value c of the phase velocity for ω = 0 2

c 2 = S ( ρ11 + ρ 22 ) , S =

∑ Sαβ α β

,

(3.77)

, =1

2. the value ωα of cutoff frequency of one mode 2

(ωα )

=

βα ( ρ11 + ρ 22 ) . ρ11 ρ 22 + ρ12(α ) ( ρ11 + ρ 22 )

(3.78)

The quantity S , in (3.77), will be: 4

−

(n) for the P-wave in the direction of abscissa axis S = ∑ c1111 ; n =1

4

−

(n) for the SV-wave in the direction of abscissa axis S = ∑ c1313 ; n =1

−

for the SH-wave in the direction of abscissa axis 1 4 (n) (n) S = ∑ ( c1111 − c1212 ); 2 n =1 4

−

(n) for the P-wave in the direction of applicate axis S = ∑ c3333 ; n =1

−

for the SV- and SH-waves in the direction of applicate axis 4

(n) S = ∑ c1313 . n =1

From the mentioned approaches there follows that although the proposed formulas are applicable to some limited classes of microstructures they can indeed be used for many real materials analysis.

209


The approach, based on experiments, has no visible restrictions on a microstructure, but such experiments are very seldom used. Some microcomposite materials, to which the mixture model has been applied and for which physical constants of this theory has been calculated are, among others: aluminum – tungsten, boron - phenolcarbon rosin, thornel - phenol -carbon rosin, quartz - phenol rosin, nylon - glass, stainless steel - Epon 828 rosin, Solitan113 - glass, Solitan113 nylon etc. Some useful relations for mixture theory constants Now, we will write two relations linking physical constants of the same granular material, but calculated within the framework of two different models - the isotropic model of effective elastic moduli (the structural model of the first order) and the model of a two-phase isotropic elastic mixture (the structural theory of the second order). These relations have already been written, but they have not been specially commented together

( λ1 + 2µ1 ) + ( λ2 + 2µ2 ) + 2 ( λ3 + 2µ3 ) = ( λ eff

+ 2 µ eff

µ1 + µ2 + 2µ3 = µ eff .

),

(3.79) (3.80)

The effective constants are linked with the macroscopic phase velocities of longitudinal and transverse plane polarized waves by formulas 2

(v ) = (λ long

eff

+ 2µ eff

2

)

ρ eff , ( vtran ) = µ eff ρ eff ,

(3.81)

ρ eff = ξ1 ρ1 + ξ 2 ρ 2 = ρ11 + ρ 22 . Also, it is appropriate to note that relations (3.79), (3.80) appear not only in formulas for calculation of physical constants, but also in the process of solving a quite different problem in the mixture theory - the problem of beam torsion, what testifies the general character of mentioned relations. Of course, classical constants λ eff , µ eff , ρ eff are more available - for a real material they can be either taken from the handbook, or determined

210


from standard experiments, or calculated from effective moduli theory formulas. That defines such a feature of relations (3.79), (3.80) : they are convenient for testing the exactness and validity of mixture theory constants λk , µk obtained in some other way.

3.7.1 Viscoelastic mixtures Let us apply the general principle of constructing the basic system of equations of the theory of viscoelasticity. We should only change, in the basic system of classical theory of elasticity, all elastic constants by viscoelastic operators (according to Volterra correspondence principle). Let us assume that the basic system of equations for elastic solid mixtures (3.56), (3.57) is given. The problem of the theory of constructing for viscoelastic mixtures presents itself now as the problem of transition from the set of independent elastic constants to the set of independent integral viscoelastic operators. It is necessary to fix two features of the present approach. The first consists in that just the waves will be studied in the mixtures. The second is concerned with that this theory is applied to composite materials. In structural mechanics of composite materials we have the rule: viscoelasticity in deformation of components of composite materials is displayed in the viscoelastic dispersion of waves. This dispersion, in one's turn, has as a consequence two basic effects: no constant phase velocities and wave amplitude attenuation. One must also say that the usual way for stationary processes (including the waves) consists in introducing the complex moduli and loss tangents and in the following consideration of wave attenuation and dispersion phenomena. The second of the above mentioned features shows that both components in two-component composite materials are usually very seldom viscoelastic together. For example, real composites manufactured


211

on the base of an epoxide or other rosins (as a rule, these rosins form the matrix of a composite) are armed by such fillers, which are deformed purely elastic. Therefore, only one component is deformed viscoelastic. That has a consequence in the features of modeling viscoelastic composites. Let us consider, for example, two-phase composite, phases of which are isotropic. The first component (filler) is assumed to be elastic, and is characterized by the volume concentration c f , two Lamé moduli λ f , µ f , and the density ρ f . The second component (matrix) is assumed to be viscoelastic, and is characterized by two viscoelastic operators

λm ≡ λmo (1 − λm* ) , µm ≡ µmo (1 − µ m* ) ,

(3.82)

where t

λm* ⋅ u ( x, t ) ≡ ∫ L(t − τ )u ( x,τ ) dτ , 0 t

µm* ⋅ u ( x, t ) ≡ ∫ M (t − τ )u ( x,τ )dτ , 0

the difference kernels of integral operators L ( z ) , M ( z ) have the mechanical sense of relaxation rate; λmo , µmo are the instantaneous Lamé moduli. The density ρ m and the volume concentration cm (c f + cm = 1) are assumed to be constant. Let us suppose additionally that the matrix displays a creep property only by shear deformations and its volume deformations are elastic. Such behaviour is characteristic for many polymer and no polymer materials, which are used as the matrix in composite manufacturing. This assumption permits us to represent two operators λm , µm with the aim of one operator. Let us demonstrate the afore mentioned assumptions as applied to the analysis of the viscoelastic plane wave in a two-phase mixture. We go

212


from characterizing the matrix Lamé moduli to the Young modulus Em and bulk modulus km . Also, let us state that a specific representation of the operator analogue of Young modulus is determined from creep experiments Em = Emo (1 + γ Em* ) . The creep of many structure materials is described with a sufficient accuracy by the so-called Rabotnov fraction-exponential operators, which are used commonly in the creep theory. Therefore, we will use such an operator representation Em* by the fraction-exponential operator in its traditional form Em* = Э α* (−κ ) .

(3.83)

The assumption about elasticity of the volume deformation is transformed into the statement that bulk modulus is purely elastic (let us denote it later by kmo ) and that all other types of deformations are expressed through kmo and operator Em . This operator involves four parameters: instantaneous Young modulus o m

E , rate relaxation modulus γ , and two parameters from the fractionexponential operator (3.83) α , κ . Analogous operators λm , µm of Lamé moduli can now be written through operator (3.83)

where

o  γ (1 − 2ν mo ) Э *  −κ − γ (1 − 2ν m )   , λm = λmo 1 − o   α o 2(1 + ν mo )     2ν m (1 + ν m )

(3.84)

γ (1 − 2ν mo )    3γ (1 − 2ν mo ) *  Э α  −κ − µm = µmo 1 −  , 2(1 + ν mo )   2(1 + ν mo )  

(3.85)


µmo =

Emo 2 (1 + ν mo )

; λm0 =

213

ν mo Emo Emo  . 1 o ; ν = − 1   m 2  3kmo  2 (1 + ν mo )(1 − 2ν mo )

It is expedient to use these formulas in modeling the composite materials by mixtures. The viscoelastic dispersion of waves is insufficiently investigated. Especially, when the viscoelastic dispersion is displayed simultaneously with the so-called geometrical dispersion.

3.7.2 Piezoelastic mixtures The model of piezoelastic solid mixtures is constructed on the classical principles. Such assumptions should be adopted: 1. The representative volume contains the particles, granules of both phases, both piezoelectrics with different mechanical and piezoelectric properties. 2. The separate phase (α ) (α = 1;2) is characterized by its set of (α ) physical parameters: the vector of displacements u , the linear tensor of mechanical strains ε ik(α ) , the tensor of mechanical (α ) stresses σ ik(α ) , the vector of electric induction D , the vector of (α ) electric intensity E . 3. Each phase corresponds to two ordered pairs of thermodynamical parameters (σ ik(α ) , ε ik(α ) ) , E (α ) , D (α ) , for which the relationships

(

)

in the form of linear constitutive equations are stated. 4. The interaction between mixture phases is linear and reflects the interaction between mechanical and electrical fields of both phases of a mixture. Particularly, the shear mechanism of force interaction of particles of different phases and couple influence

214


of electrical fields of different phases on polarization of separate phase P (α ) ( p ) (α ) ( p ) (δ ) P (α ) = χαα E + χαδ E

(3.86)

( p) ( p) where χ11( p ) , χ 22 , χ12( p ) = χ 21 are coefficients of the dielectric

permeability. 5. The internal energy U ( ε ik(1) , ε ik(2) , uk(1) − uk(2) , Dk(1) , Dk(2) ) is defined as the function of all given parameters, is written for a mixture as the whole, and supposed to be the quadratic function of variables. Within these assumptions the full system of equations for two-phase piezoelectrics can be constructed similar to the case of one-phase piezoelectrics and has the form: −

equations of motion

σ ik(α, k) + (−1)δ β ( ui(1) − ui(2) ) = ραα −

∂ 2u ∂t 2

(3.87)

equations of induced electrostatics in dielectrics div D (α ) = 0 ,

E (α ) = − grad ϕ (α )

(3.88)

(α ) (α ) ( purely electromagnetic waves are absent, and vectors D , E may

depend on time but in acoustical range of frequencies, only; ϕ (α ) are the standard introduced electric scalar potentials ); −

classical linear Cauchy relations

ε ik(α ) =

1 (α ) ( ui,k + uk(α,i ) ) , 2

(3.89)


−

215

linear constitutive equations (α ) E (α ) (3) E (δ ) σ ik(α ) = Ciklm ε lm + Ciklm ε lm − eikl(α ) El(α ) − eikl(3) El(δ ) ,

(α ) (α ) Dk(α ) =∈(ikα ) Ei(α ) + ∈ik(3) Ei(δ ) + eikl ε il + eikl(3)ε il(δ ) ,

(3.90) (3.91)

(n)E ( Ciklm are tensors of elastic coefficients under constant electric fields; (n) eikl are tensors of piezoelectric coefficients; ∈ik( n ) are tensors of

dielectrical permeability ). Altogether, system (3.87) - (3.91) involves 32 equations, and respectively the same number of unknown functions. Similarly to classical theory, for study of running waves the system (3.86)-(3.90) can be written more compactly in the form of three groups of equations: − equations over the volume of a piezoelectric body (α ) E (α ) (3) E (δ ) Ciklm ul ,km + Ciklm ul ,km + ( −1)δ βi ( ui(1) − ui(2) ) (α ) ikl

+e ϕ

(α ) ,lk

(3) ikl

+e ϕ

(δ ) , lk

= ραα

∂ 2ui(α ) , ∂t 2

(α ) (α ) (3) (δ ) ∈ik(α ) ϕ,(kiα ) + ∈ik(3) ϕ,(kiδ ) − eikl uk ,li − eikl uk ,li = 0 ,

−

(3.92)

(3.93)

equations over the body surface (α ) E (α ) iklm l ,m

(C (∈

(α ) ik

u

(3) E (δ ) (α ) (α ) + Ciklm ul ,m − eikl ϕ,l − eikl(3)ϕ,(lδ ) ) nk = Si(α ) o ,

ϕ,(iα ) + ∈ik(3) ϕ,(iδ ) + eikl(α ) ui(,αl ) + eikl(3) ui(,δl ) ) nk = ω (α ) o ,

(3.94) (3.95)

( Si(α )o , ω (α )o are respectively the components of vectors of partial forces given over the body surface and the partial surface currents), −

equations for potentials ϕ (α ) outside a body

216


∆ϕ (α ) = 0 ,

ϕ (α ) → 0 .

(3.96)

r →∞

Eight equations (3.92), (3.93) relative to eight unknown functions (two partial displacement vectors and two partial electric scalar potentials) form together the system, which describes the linear dynamical processes in two-phase dielectrics. This system corresponds to system (3.57) of the linear theory of elastic mixtures.

3.8 Computer Modelling Data on Micro- and Nanocomposites All structural theories have one and the same weak point: the problem on systems of basic experiments is not adequately elaborated. These experiments are understood as those which give for the material and accepted for its description model the full set of physical constant. This problem isn’t a trivial one because while the transition from isotropic media to more complicated by symmetry ones is being done, the problems of the absence of necessary number of independent experiments occur. Of course, each model has another way, i.e. to evaluate theoretically such sets on the base of knowledge of internal structure. But for some models this way is also of little use, because the number of constants can reach hundreds (for example, the micromorphic Eringen model includes 903 constants). Nevertheless in the cases when the model has a lot of constants they were studied together with computer techniques. One such technique was utilized below for the evaluation of constants for granular microcomposites. The essence of the technique consists in that: the granular composite is assumed as a random inhomogeneous system involving two phases with constant volume fractions. Further the formalism of statistical averaging is used. It coincides with averaging over the representative volume owing the hypothesis that mechanical fields are ergodic.


217

The problem is solved approximately. Each separate phase is assumed to be made of homogeneous material with density ρ (α ) , elastic moduli of the 2nd order λ (α ) , µ (α ) and elastic moduli of the 3rd order A(α ) , B (α ) , C (α ) . This corresponds to the nonlinear Murnaghan model with five elastic constants. The adopted approach permits us to calculate macroscopic moduli both for the effective moduli model (total number of constants six – elastic constants λ , µ , A, B, C and macroscopic density ρ ) and for the model of mixtures (total number of constants 16 – two partial densities ραα , six elastic constants of the second order (Lamé constants)

λk , µk (k = 1; 2;3) , six elastic constants of the third order (Murnaghan constants) Aα , Bα , Cα ; two constants of the interaction between mixture phases β , β / ). Physical constants of six types of granular microcomposite materials were evaluated. For each material, three modifications are considered. They differ by volume fractions: K1 - c1 = 0.2; c2 = 0.8 ; K2 - c1 = 0.4; c2 = 0.6 ; K3 - c1 = 0.6; c2 = 0.4 . K means the material number. Materials No.KM ( K=1; 2; 3; 4; 5; 6; M=1; 2; 3 ) are next granular composites: No.1M No.2M No.3M No.4M No.5M No.6M

→ granules - steel, matrix - polystyrene; → granules - copper, matrix - polystyrene; → granules - copper, matrix - tungsten; → granules - copper, matrix - molybdenum; → granules - tungsten, matrix - aluminium; → granules - tungsten, matrix - molybdenum.

Values of physical constants of separate phases are presented in Table 3.1. The SI system for physical units is used, where the conventional dimension for elastic constants is Pascal Pa. The negative values of Murnaghan constants means that the nonlinearity of all these materials is soft (the heavy nonlinearity very seldom occurs).

218


Table 3.1. material

ρ ⋅ 10 −4

λ ⋅ 10−10

µ ⋅ 10−10

A ⋅ 10−11

B ⋅ 10−11

C ⋅ 10 −11

aluminium tungsten copper molybdenum polystyrene steel

0.27 1.89 0.893 1.02 0.105 0.78

5.2 7.5 10.7 15.7 0.369 9.4

2.7 7.3 4.8 1.1 0.114 7.9

-0.65 -1.08 -2.8 -0.26 -0.108 -3.25

-2.05 -1.43 -1.72 -2.83 -0.0785 -3.1

-3.7 -9.08 -2.4 3.72 -0.0981 -8.0

Full sets of effective physical constants of microcomposite materials No. KM are presented in Table 3.2. Table 3.2 material

ρ ⋅ 10

11 12 13 21 22 23 31 32 33 41 42 43 51 52 53 61 62 63

0.24 0.375 0.51 0.263 0.42 0.578 1.69 1.49 1.29 0.995 0.969 0.944 0.594 0.918 1.24 11.9 13.7 15.4

−4

λ ⋅ 10

−10

0.47 1.1 2.05 0.471 1.11 2.06 8.18 17.0 26.5 14.5 27.9 40.3 5.59 11.6 18.1 14.4 27.5 39.3

µ ⋅ 10−10

A ⋅ 10−11

B ⋅ 10−11

C ⋅ 10 −11

0.175 0.449 0.913 0.173 0.442 0.89 6.71 12.9 18.6 9.41 17.4 24.2 3.26 0.721 1.2 1.53 3.66 6.68

-0.163 -0.439 -0.913 -0.161 -0.442 -1.01 -1.37 -3.07 -5.12 -1.24 -3.15 -5.49 -0.658 -1.33 -2.03 -0.483 -1.13 -1.98

-0.115 -0.302 -0.672 -0.114 -0.297 -0.649 -1.56 -3.21 -4.9 -2.52 -4.79 -6.81 -2.18 -4.45 -6.72 -2.62 -4.95 -6.83

-0.15 -0.1 -0.888 -0.148 -0.389 -0.875 -7.2 -12.9 -17.2 -3.5 -6.73 -9.68 -4.35 -9.5 -15.7 -4.78 -11.1 -19.5

Full sets of physical constants of a mixture theory for the same six microcomposite materials are presented in Tables 3.3 - 3.5.

219

Materials with Micro- or Nanostructure Table 3.3 constants

No.11

No.12

No.13

No.21

ρ11 ⋅ 10−4

0.156 0.084

0.312 0.063

0.468 0.042

0. 179 0.084

0.357 0.063

0.535 0.042

µ1 ⋅ 10−10

0.043

0.117

0.277

0.0424

0.114

0.263

µ2 ⋅ 10−10

0.146

0.0556

0.026

0.146

0.055

0.025

0.0254

0.100

0.112

0.0911

0.097

0.108

λ1 ⋅ 10−10

0.022

0.0657

0.170

0.0222

0.667

0.174

λ2 ⋅ 10−10

0.0282

0.115

0.0570

0.282

0.115

0.0576

λ3 ⋅ 10 −10

0.079

0.085

0.0944

0.0793

0.085

0.0953

k1 ⋅ 10−10

0.109

0.314

0.788

0.109

0.313

0.784

−10

0.993

0.401

0.199

0.992

0.400

0.198

−10

0.329

0.355

0.396

0.329

0.355

0.394

A1 ⋅ 10−10

-3.970

-1.930

-1.693

-5.638

-3.392

A2 ⋅ 10−10

-0.818

-0.106

-0.0331

-0.106

-0.033

B1 ⋅ 10−10

-15.24

-5.570

-4.168

-3.842

-1.435

-0.093

-0.0398

-0.093

-0.0397

-7.926

-7.055

ρ 22 ⋅ 10

µ3 ⋅ 10

k2 ⋅ 10 k3 ⋅ 10

−10

B2 ⋅ 10 C1 ⋅ 10

−4

−10

−10

C2 ⋅ 10−10

-0.452 -70.10 -0.492

-25.21

-18.48

-19.58 -0.816 -17.04 -0.454 -15.34

No.22

No.23

-0.1124

-0.0475

-0.496

-0.113

-0.0171

β1 ⋅ 10−15

0.0613

0.225

0.439

0.061

0.224

0.433

β 2 ⋅ 10−7

0.211

0.580

0.792

0.211

0.576

0.782

It should be noted here that the partial densities have the usual density dimension. The constants of the interaction between mixture phases

220


β1 , β 2 have the dimension Pa m 2 . In the next three tables the values of bulk moduli are introduced additionally. Table 3.4 constants

No.31

No.32

No.33

No.41

No.42

No.43

ρ11 ⋅ 10−4

0.179 1.51

0.357 1.13

0.536 0.756

0.179 0.816

0.357 0.612

0.536 0.408

0.315

0.595

0.850

3.79

1.07

ρ 22 ⋅ 10

−4

0.363

0.700

1.02

−10

9.38

2.55

0.730

µ3 ⋅ 10−10

1.75

1.26

0.813

2.03

1.41

0.878

λ1 ⋅ 10−10

4.26

8.77

1.35

0.466

0.870

1.22

µ1 ⋅ 10−10 µ2 ⋅ 10

14.3

λ2 ⋅ 10−10

48.1

14.2

0.438

9.69

2.53

0.701

−10

15.0

11.7

0.802

2.20

1.54

0.970

3.33

5.08

1.72

3.21

4.52

6.80

2.04

λ3 ⋅ 10

k1 ⋅ 10−10 k2 ⋅ 10 −10

1.64 23.8

43.3

11.4

3.17

k3 ⋅ 10−10

6.25

4.76

3.22

8.62

6.04

3.79

A1 ⋅ 10−10

-73.1

-1.83

-0.818

-3.74

-0.902

-0.389

A2 ⋅ 10−10

-75.1

-0.706

-0.139

-0.700

-0.052

-0.160

−10

-29.1

-1.62

-1.38

-1.80

-1.15

-1.02

−10

-109.0

-1.83

-0.0627

-4.27

-1.42

-3.43

-2.63

-2.49

-2.00

-1.97

B1 ⋅ 10

B2 ⋅ 10

C1 ⋅ 10−10 C2 ⋅ 10−10

β1 ⋅ 10−15 β 2 ⋅ 10

−7

-81.3

-12.8

-4.26

-26.6 -2.38 -33.1

-11.3

1.70

4.99

7.39

2.57

7.24

0.229

0.228

0.141

0.257

0.381

-4.98 10.2 0.300

221

Materials with Micro- or Nanostructure Table 3.5 constants

No.51

No.52

No.53

No.61

No.62

No.63

ρ11 ⋅ 10−4

0.378 1.51

0.756 1.13

1.134 0.756

0.378 0.816

0.756 0.612

1.134 0.408

0.216

0.162

0.108

0.816

0.612

0.408

µ3 ⋅ 10−10

0.043 0.146

0.117 0.0556

0.277 0.026

0.0424 0.146

0.114 0.055

0.263 0.025

λ1 ⋅ 10−10

0.0254

0.100

0.112

0.0911

0.0978

0.108

0.0282

0.115

0.0570

0.282

0.115

0.0576

λ3 ⋅ 10 −10

0.079

0.085

0.0944

0.0793

0.0856

0.0953

k1 ⋅ 10−10

0.109

0.314

0.788

0.109

0.313

0.784

k2 ⋅ 10 −10

0.993

0.401

0.199

0.992

0.400

0.198

k3 ⋅ 10−10

0.329

0.355

0.396

0.329

0.355

0.394

A1 ⋅ 10−10

-3.970

-1.930

-1.693

-5.638

-3.392

−10

-0.818

-0.106

-0.0331

-0.106

-0.033

-5.570

-4.168

-3.842

-1.435

-0.093

-0.0398

-0.0934

-0.0397

-7.926

-7.055

-0.496

-0.113

-0.0171

ρ 22 ⋅ 10

−4

µ1 ⋅ 10−10 µ2 ⋅ 10−10

λ2 ⋅ 10

−10

A2 ⋅ 10

B1 ⋅ 10−10 B2 ⋅ 10−10 C1 ⋅ 10−10

-15.24 -0.452 -70.10

-25.21

-18.48

-19.58 -0.816 -17.04 -0.454 -15.34

-0.492

-0.1124

-0.0475

β1 ⋅ 10−15

0.0613

0.225

0.439

0.061

0.224

0.433

β 2 ⋅ 10−7

0.211

0.580

0.792

0.211

0.576

0.782

C2 ⋅ 10

−10

Four types of fibrous unidirectional composite materials will be further considered. We assume that all types of fibers as being made of carbon and the matrix material is made both epoxy rosin EPON-828 with

222


properties: density ρ = 1, 21 ⋅ 103 kg/m3, Young modulus E = 2.68 GPа, shear modulus µ = 0.96 GPа, Poisson ratio ν = 0.40 and consists of a mixture of the rosin EPON-828 and the polystyrene. The last one is the product of polymerization of styrene -CH 2 - CH ( C6 H 5 ) -  n with properties within the framework of nonlinear Murnaghan model: density

ρ = 1.05 ⋅ 103 kg/m3; elastic moduli of the second order: Young modulus E = 2.56 GPа, shear modulus µ = 1.14 GPа, Poisson ratio ν = 0.30 ; elastic moduli of the third order: A = −10.8 GPа, B = −7.85 GPа, C = −9.81 GPа (Murnaghan constants). Because the organic epoxy rosin EPON-838 is assumed as the material of the matrix and this rosin from technological reasons (for avoiding the crystallization) always contains the addition of high-molecular polymers, then the presence in computer modeling of some hypothetic material consisting of chaotic mixture of high-molecular polystyrene and epoxy rosin seems not to be contradicting the nature of the rosin and quite possible. It is necessary to note that the matrix made of the mixture of epoxy rosin with polystyrene is the material with the soft characteristics of nonlinearity. The material with the heavy characteristics of nonlinearity is found rarely enough, it is studied here in the form of the matrix consisting of the mixture of the epoxy rosin with the Pyrex – glass. For

ρ = 9.95 ⋅ 103 kg/m3; λ = 1.45 GPа; µ = 0.941 GPа; A = 124 GPа; B = – 252 GPа;

this glass the following physical properties are assumed:

C = 350 GPа. Additions to epoxy rosin are assumed to be the tenth portion of the mixture as a whole. Fillers have the following characteristics: Filler N1 – industrial carbon microfiber Thornel-300 with properties: mean diameter 8 µm, density ρ = 1.75 ⋅ 103 kg/m3, Young modulus E = 228 GPа, shear modulus µ = 88 GPа, Poisson ratio ν = 0.30 .

223


Filler N2 – graphite whiskers with properties: mean diameter 1 µm, density ρ = 2.25 ⋅103 kg/m3, Young modulus E = 1.0 ТPа, shear modulus µ = 385 GPа, Poisson ratio ν = 0.30 .

Filler N3 – zig-zag carbon nanotubes with properties: mean diameter 10 nm, density ρ = 1.33 ⋅103 kg/m3, Young modulus E = 0.648 ТPа, shear modulus µ = 221 GPа, Poisson ratio ν = 0.33 . Filler N4 – chiral carbon nanotubes with properties: mean diameter 10 nm, density ρ = 1.40 ⋅103 kg/m3, Young modulus E = 1.24 ТPа, shear modulus µ = 477 GPа, Poisson ratio ν = 0.30 . The shown data about the matrix and fibers (further microT, microW, nanoZZ, nanoCH) are utilized for computer modeling the physical constants in basic models. The modeling is described in the following pages, where a change in the value of basic constants in dependence with the degree of fiber volume concentration as well as the distinction and similarities between micro- and nanocases are studied. Some of the results are predictable. For example, certain constants are very sensitive to the fiber properties (Young and shear moduli) big distinctions and some of constants are almost identical for all types of fibers.

1.45 1.4 1.35

0.05

0.1

0.15

0.2

0.25

0.3

1.25

Fig. 3.2 Dependence of density ρ eff on c f for 4 types of composites (from top – microW, microT, nanoZZ, nanoCH).

224


Within the framework of the linear model of effective constants the eff eff constants ρ eff , E eff , G eff ,ν eff , E ′eff , G ′eff ,ν ′eff , C1111 , C3333 are considered. The results of modeling are shown on the next plots. Everywhere the abscissa axis corresponds to the volume fraction of fibers c f (it is assumed to be small – up to 0.3). The restriction to small

1.6

1.4

1.2

0.05

0.1

0.15

0.2

0.25

0.3

Fig. 3.3 Dependence of longitudinal shear modulus G eff on c f for 4 types of composites.

filling of fibers is caused by that in this case the utilized formulas, on one hand, are exact sufficiently, at present level of technology and, on the other hand, the denser filling of matrix with nanoparticles or nanofibers seems to be problematic. 1.7 1.6 1.5 1.4 1.3 1.2 1.1 0.05

0.1

0.15

0.2

0.25 eff

0.3

Fig. 3.4 Dependence of transverse shear modulus G ′ on c f for 4 types of composites (upper plot - microT).

225


0.05

0.1

0.15

0.2

0.25

0.3

0.395 0.39 0.385 0.38 0.375 0.37 0.365

Fig. 3.5 Dependence of Poisson ratio ν eff for 4 types of composites (upper plot - nanoZZ).

It can be noted that Figs. 3.2–3.5 show the weak nonlinearity in dependence of both shear moduli on the volume fraction of fibers. This type of nonlinearity is a characteristic for the majority of parameters in the case of small filling by fibers which can be seen from the next plots. Also Figs. 3.3–3.5 show that the corresponding parameters are not sensitive to the difference between micro- and nanofiber properties. The strong nonlinear dependence of Poisson ratio ν ′eff and the shift of its values to the range (0.5; 0.7) can be noted as an interesting results.

0.66

0.64

0.62

0.05

0.1

0.15

0.2

0.25

0.3

0.58

Fig. 3.6 Dependence of Poisson ratio ν ′eff on c f for 4 types of composites (from top – microW, nanoZZ, nanoCH, microT).

226

Wavelet and Wave Analysis 140 120 100 80 60 40 20 0.05

0.1

0.15

0.2

0.25

0.3

Fig. 3.7 Dependence of longitudinal Young modulus E eff on c f for 4 types of composites (from top – nanoCH, microW, nanoZZ, microT).

5.5 5 4.5 4 3.5

0.05

0.1

0.15

0.2

0.25

0.3

Fig. 3.8 Dependence of transversal Young modulus E ′eff on c f for 4 types of composites (microW, nanoCH, nanoZZ are practically identical).

Figure 3.8 seems to be very similar to Fig. 3.4 and testifies the almost full insensitiveness of transversal Young modulus E ′eff and transverse shear modulus G ′eff on fiber properties. At the same time both parameters change significantly by changing the filling degree.

227

Materials with Micro- or Nanostructure 140 120 100 80 60 40 20 0.05

0.1

0.15

0.2

0.25

0.3

eff Fig. 3.9 Dependence of elastic modulus C1111 on c f for 4 types of composites

(from top – nanoCH, microW, nanoZZ, microT). 600 500 400 300 200 100

0.05

0.1

0.15

0.2

0.25

0.3

eff Fig. 3.10 Dependence of elastic modulus C3333 on c f for 4 types of composites

(from top – nanoCH, microW, nanoZZ, microT).

Thus, if we assume that two groups of parameters essentially differ microcomposites from nano ones – Young and shear moduli as the mechanical group and fiber diameter as the geometrical group – then the pictures above show a differing influence of the first group on mechanical constants of composites. The most significant difference can be seen in the cases of eff longitudinal Young modulus E eff and elastic modulus C3333 .

228


-0.6 -0.7 -0.8 -0.9

0.05

0.1

0.15

0.2

0.25

0.3

Fig. 3.11 Dependence of constants A (lower plot), B (upper plot), C (middle plot) on c f for the matrix rosin-styrene.

30 20 10

0.05

0.1

0.15

0.2

0.25

0.3

-10 -20

Fig. 3.12 Dependence of constants A (middle plot), B (lower plot), C (upper plot) on c f for the matrix rosin-Pyrex.

The two plots Figs. 3.11 and 3.12 show the variation of elastic constants of the third order (Murnaghan constants, in GPa) against volume fraction of fibers. As it can be seen they are not changed significantly within the chosen range of filling with fibers.

Chapter 4

Waves in Materials

4.1 Waves Around the World The wave motion as a subclass of the motion in general is observed very frequently. As a result of observations, a description of the phenomenon is becoming, as a rule, well known. It is sometimes considered that the form of description doesn’t need a theoretical concept. Though the last statement has always given rise to doubt. The fact is that in every description some criterion of the distinction of wave motions from other motions is presented deliberately or not. Practically everyone has seen waves on water, sand or somewhere else. And it seems that it is not very difficult to determine purely by the description that we are observing the waves. However, it should be noted that waves vary in their manifestations: besides the well-known waves in water or in air we may have optic, electromagnetic, magnetoactive, interferential, shock, explosion, radio waves, waves in glaciers, high-flood waves and rolling waves in rivers, waves in transportation streams in tunnels, chemical waves of a metabolism, waves in the process of river and sea sediments, epidemic and population waves etc. For all of these waves, some common attributes may be formulated: the disturbance which is observed in a certain place in space must propagate with a finite velocity to some other place in this space; as a rule, the process must be close to oscillatory, if it is observed in time.

229

230


We would also remember that motion is assumed as oscillatory, when it takes place in the neighbourhood of some fixed state, is restricted in its deviation from this state, and is repeated in most cases. It is universally recognized that any wave observation, which extends beyond the limits of daily description, must be associated with certain theoretical scheme. First of all, this scheme provides with some properties of the medium of waves propagation. The traditional physical schemes are based on the continuum concept (see chapter 3), which uses the physical fields. When the fields are selected, the physical medium (acoustic, elastic, electromagnetic, etc.) is fixed. In contrast to the descriptive approach to wave phenomena which needs the knowledge of the wave attributes only, in the scientificcognitive approach some initial theoretical scheme is always presented and utilized. Every theoretical scheme of wave description has to contain at least two independent parameters - time and space coordinates. The continuum schemes establish the relationships between fields depending on these parameters. As a result, diverse differential equations are derived, among solutions of which must also be the solution describing waves. Wave analysis is divided by different attributes. For example, such a characteristic of wave solutions as its smoothness turned out to be critical in theoretical wave analysis. Knowledge of solution smoothness is equivalent to knowledge of its continuity or discontinuity, and also their quantitative estimates (types of discontinuities, order of continuity, etc). The situation when waves corresponding to discontinuous and continuous solutions are studied separately was formed long ago. The delimitations occurred as a result of the difference in the interpretation of mechanisms of an excitation of waves and process of wave motion. So, two branches of studying the one and the same physical phenomenon exist. The branch of study associated with discontinuous solutions treats the wave as a surface motion, which is a singular solution relative to the given smooth physical field. That is to say, any wave motion is understood as a spatial motion of the field jump on a given surface.

Waves in Materials

231

The second branch is associated with continuous solutions describing continuous motion. Two classes of waves are isolated here. Hyperbolic waves are obtained as solutions of differential equations of hyperbolic or ultra-hyperbolic types and, consequently, are clearly defined by the type of equation. It is also possible to speak of another type – dispersive waves. This type is defined by the form of the solution. It is claimed that a wave propagating medium is dispersive and the waves themselves are dispersive, if the wave is mathematically represented in the form of the known function F of the phase ϕ = kx − ω t ( x is the spatial coordinate, k is the wave number, ω is the frequency, and t is time, and if the phase velocity v = ω k of the wave depends nonlinearly on frequency. Occasionally, it is more convenient to fix the dispersivity in the form of nonlinear function ω = W (k ) . Solutions of the type u = F (kx − ω t ) are admitted not only to hyperbolic differential equations, a parabolic one, but also some integral equations. The criteria of hyperbolic and dispersive waves are not mutually exclusive; hyperbolic and dispersive waves are therefore encountered simultaneously. Among other things, the majority of waves in materials with the microstructure discussed in this book are precisely these kinds of waves. This book deals with continuous waves in materials. As it will be shown later, the structural approach in the wave analysis displays some new types of waves. Therefore, some new classification of elastic waves in solids will be proposed. Alternatively to hyperbolic-dispersive waves, the standard classification in physics consists of four types: 1. Solitary waves or pulses, sufficiently short in time irregular given in a locally space disturbances. 2. Periodic (most often called harmonic or monochromatic) waves, characterized by disturbances in all the spaces. 3. Wave packets - locally given in a space regular disturbances. 4. Trains of waves - harmonic wave packets.

232


4.2 Analysis of Waves in Linearly Deformed Elastic Materials

4.2.1 Volume and shear elastic waves in the classical approach Let us start with the classical wave equation. It is a hyperbolic equation and it has, in Cartesian coordinates Ox1 x2 x3 , the form

 ∂ 2 u ∂ 2u ∂ 2u  ∂ 2u ∂ 2u (4.1) a 2  2 + 2 + 2  − 2 ≡ a 2 ∆u − 2 = f ( x1 , x2 , x3 , t ) . ∂t  ∂x1 ∂x2 ∂x3  ∂t Here u ( x1 , x2 , x3 , t ) is an unknown function describing a wave motion in an isotropic medium, f ( x1 , x2 , x3 , t ) is a known function. Besides the isotropy property, a number of assumptions about the character of the physical process are needed for the procedure of obtaining the wave equation (4.1). Let us explain it on the simplest example of sound waves. In this case the function u = p ( x, y, z , t ) − po describes the deviation of the gas pressure from the static pressure po , a = c = const is a constant sound velocity in the given gas medium. First, the assumption of this medium homogeneity is accepted. It is also assumed that the value of an acoustic deviation (that is, the function u ) is considerably less than the static pressure (this means not less than two orders). And one more assumption - an average free run of gas molecules must be considerably less than the characteristic linear scale of a deviation. It is also presented in the theoretical description of a sound propagation process that infinitesimal changes of the density are formed by the same displacements of gas molecules. So, the sound wave is the longitudinal oscillations of gas molecules, which propagate in a space. That is, they propagate in one direction, which is called the direction of the wave propagation.

Waves in Materials

233

It should be noted that the frequency of oscillation of gas molecules defines the name of a wave. If it is lower than 16 Hz , then it is called the infrasound wave; in the frequency range 16 Hz ÷ 20 KHz the sound wave is propagated; in the frequency range 20 KHz ÷ 1GHz the supersound (ultrasound) wave is propagated; if the frequency is more than

1GHz (up to 1013 Hz ), then it is the hyper-sound wave.

Wave, is the propagation in space disturbance of certain state of a physical or other system. The geometrical form (the profile) of these disturbances can be diverse. Waves are distinguished by their form or, as it is said, by the kinematical attribute. This division highlights only the often used terms. However there are some waves out from this classification. Besides the form or the profile, the wave is also characterized by the velocity of propagation and by the phase. These notions are usually explained by means of the classical D'Alembert solution of wave equation (4.1). When some data are zero, this solution may be found of the form u ( x, t ) = f1 (ct − x) + f 2 (ct + x) . (4.2) It should be noted that each of the two summands in (4.2) is called the running (traveling) wave. The first one propagates in the positive direction, the second one in the opposite direction. Functions f1 and f 2 give the form of a wave. The quantity σ = ct ± x is usually called the phase, the quantity c is the phase velocity of a wave. If running waves are periodical, then it is convenient to transform the solution (4.2) into the form

c  c  u ( x, t ) = f1  (ω t − kx )  + f 2  (ω t + kx )  . ω  ω 

(4.3)

Here the wave number k is introduced as k = ( ω c ) = ( 2π f c ) = ( 2π λ ) ,

the wavelength λ and the circular frequency f are used.

234


It should be noticed that running waves are some abstraction, since they are understood as waves propagating in the infinite space. Sometimes such a medium, which doesn’t have a boundary, is called the open medium. The terms “running”, “traveling” are often used, sometimes the term “progressive” is used, which is equivalent to the term “die fortschreitende Welle” initially used in German classic physics context. Besides the solution in the form of traveling waves (4.3), the classical wave equation permits a solution in the form of standing waves. If the solution (4.3) is constructed by the variables separation method, then this solution will consist of sums of summands with the same structure

ui ( x, t ) = X i ( x) ⋅ Ti (t ) = Ci cos ( ki x + ϕ x ) cos (ωi t + ϕt ) .

(4.4)

Quantities Ci , ϕ x , ϕt are constant, the first one is the amplitude, the second and third terms are the phases. Expression (4.4) describes an oscillation, which doesn’t propagate, it stands. For this reason it is called a standing wave. From the existence theorem it follows that there exists a correlation between the solutions of type (4.3) and (4.4). In fact, the standing wave may be obtained by the procedure of adding two traveling waves, and the traveling wave may be obtained as a result of two standing waves’ beating. Therefore the solutions in the two forms, of standing waves and traveling waves are equivalent. Let us dwell now on traveling harmonic waves, divided into dispersive and nondispersive ones. Usually for harmonic waves the dispersion relation ω = W (k ) is given. In this case the phase velocity is defined as

v ph (k ) = W (k ) k .

(4.5)

If the wave is formed by a few dispersive waves with closely spaced frequencies, then each of these waves will propagate with its own velocity. Such waves diverge in space (disperse), therefore the initial profile of the wave (composed of such waves) will change in time. The diffusion of an

235

Waves in Materials

initially given profile of the wave packet is a characteristic attribute of the dispersivity (or dispersity) of waves. Structured materials are dispersive for many types of waves including the solitary ones, which will be of special interest owing to the possibility of efficient wavelets application. Let us focus on the basic properties of traveling waves, the onedimensional harmonic waves. First of all, such a wave is represented mathematically with the simple formula

 x u ( x, t ) = A cos ω  t −  v ph 

  . 

(4.6)

The more habitual representation of the traveling harmonic wave has the form u ( x, t ) = A cos(ω t − kx) (4.7) where k = (ω v ph ) is

the wave number, which is linked with the

wavelength λ = ( 2π k ) . If we define the period T = (1 f ) and the circular frequency f = (ω 2π ) then the phase velocity can be written as v ph = (ω k ) ; v ph = λ f ; v ph = ( λ T ) .

(4.8)

The phase velocity defines the velocity of a motion of the single crest, or the single valley, or the node on a wave profile. Expression ϕ ( x, t ) = ω t − kx is called the phase function or the phase. It linearly depends on the independent variables x and t . When the phase value is fixed, we focus on certain points on the wave profile. In ∂ϕ ∂ϕ dt + dx = ω dt − kdx = 0 for a fixed fact, from the differential dϕ = ∂t ∂x phase, there follows that for the points of the fixed phase it is

( dx

dt ) = (ω k ) . These points move with the velocity v ph , or, as it is

often written, the observer, which moves with the velocity v ph , sees all

236


the time one and the same point of the wave profile. This explains why v ph is called the phase velocity. Let us consider now the waves in materials. The basic system of equations for waves in the classical structural linear approach of effective constants (that means, within the framework of linear theory of elasticity) has the form of three coupled partial differential equations of the second order relative to three components of the displacement vector uk ( x1 , x2 , x3 , t ) Ciklm

∂ 2 um ∂ 2u + Xi = ρ 2 . ∂xk ∂xl ∂t

(4.9)

This system corresponds to the general case of anisotropy of elastic materials. Classical analysis of volume and shear waves is conventionally carried out for the case of isotropic materials. Because the tensor of elastic properties can be written as follows (see (3.14)) Ciklm = λδ ik δ lm + µ (δ il δ km + δ imδ kl ) ,

(4.10)

the initial system (4.9) becomes simpler and takes the form of classical Lamé equations (4.11) ( λ + µ ) graddivu + µ ∆u = ρ ∂ 2u ∂t 2 . It is known that an arbitrary vector u can be represented as the sum of a potential and a solenoidal vector u =v +w

( rot v = 0, div w = 0 ) .

If the vector u represents the displacements (in elasticity), then the vector v characterizes only the deformations accompanying the volume change because of rot v = 0 whereas the vector w characterizes only the deformations, accompanying the form change because of div u = 0 . The first vector v is usually written in terms of a scalar potential, which is defined with exactness to an arbitrary function of time

Waves in Materials

237

v = grad ϕ . The second vector w is written through the vector potential, which is defined with exactness to an arbitrary vector field w = rotψ .

Therefore by applying the operator div to (4.11) gives the wave equation relative to v

 ∆ − ( ρ ( λ + 2µ ) ) ( ∂ 2 ∂t 2 )  v = 0 .  

(4.12)

as well as the wave equation relative to w  ∆ − ( ρ µ ) ( ∂ 2 ∂t 2 )  w   =0.

(4.13)

Both equations (4.12), (4.13) are the classical wave equations. The first one describes the volume waves of deformations with the constant phase velocity v ph = ρ ( λ + 2 µ ) , whereas the second one describes the shear waves of deformation with the constant phase velocity

v ph = ρ µ . The corresponding potentials ϕ ,ψ

fulfill also the same wave

equations. It should be noted that the waves in hand are the free waves, that is, the waves propagating in the infinite elastic space. In practice, these waves are the waves propagating in the earth’s crust for the cases of earthquake and are recorded by seismic stations. From comparison of phase velocities follows that the volume waves are arriving at the station before the shear ones.

4.2.2 Plane elastic harmonic waves in the classical approach Among all types of waves, the plane waves are the most studied. Let us choose some direction defined by the unit vector n o , the plane wave is defined as the D'Alembert wave propagating in the direction n o with constant phase velocity v ph and amplitude u o

238


u ( x, t ) = u o f t − (ξ v ph )  ,

(4.14)

where

ξ = n o ⋅ r = n1o x1 + n2o x2 + n3o x3 , n o = {nko ≡ cos α k0 } , r = { x1 , x2 , x3 } .

At any time the function u ( x, t ) will be constant in the plane defined by the equation ξ = n o ⋅ r = n1o x1 + n2o x2 + n3o x3 = const . Let the wave move during the time interval ∆t , then the vector r will change on the value ∆r . If at that time the argument of function u ( x, t ) will be invariable, then the function will also be invariable. But this means that the direction k is just the direction in which the wave is moving. On the other hand, ∆t and ∆r is: the relationship between quantities kr − ωt = k ( r + ∆r ) − ω ( t + ∆t ) = const. or k ∆r = ω∆t . The last formula can be considered as the equation of the above plane, but shifted congruently in space on distance ∆r . That is, this plane moves in the direction of the wave vector k . In terms of wave analysis, the harmonic wave (4.14) has the front in the form of some plane and this front moves in direction k . This is why the wave is called the plane wave. Therefore, the plane wave is characterized by four parameters: 1. The arbitrary form of the profile described by twice continuously differentiable function f . 2. The arbitrary amplitude u o . 3. The fixed polarization (direction of the vector u o ) of this amplitude. 4. The fixed phase velocity v ph . In order to find the unknown phase velocity and amplitude polarization, we must substitute (4.14) into the equation of motion. Let us take the linear equation of the theory of elasticity (written for the arbitrary case of anisotropy of elastic properties and in terms of displacements)

Waves in Materials

239

Ciklmum ,lk − ρ uɺɺi = 0 .

(4.15)

Then by substituting (4.14) into (4.15) and introducing the Christoffel tensor Γik = Cijlk k j kl we get the Christoffel equation for plane waves in linear elastic medium 2

Γik uko = ρ ( v ph ) uio .

(4.16)

From these equations follows that the components of the vector u o , which define the wave polarization, are the eigenvectors of the tensor Γik

and the corresponding eigenvalues are ρ ( v ph )

2

and they define the

phase velocity of the polarized wave. Thus, the Christoffel procedure on the problem of determining plane wave parameters reduces to the mathematical eigenvalue problem for tensor Γik . As the tensor of elastic properties Cijkl is symmetric, the Christoffel tensor Γik is also symmetric. Hence its eigenvalues are real quantities and eigenvectors are orthogonal. The density ρ is the positive quantity and the eigenvalues are always positive quantities. There follows that the phase velocities of plane waves are real quantities and the waves really propagate. In an anisotropic medium for each direction corresponding to the eigenvector, there exist three plane waves with orthogonal polarizations and different phase velocities. The Christoffel tensor for an isotropic medium is Γik = ( λ + µ ) kio kko + µδ ik

(4.17)

and the Christoffel equations for an isotropic medium are

( λ + µ ) ( k o ⋅ u o ) k o = ( ρ v 2ph − µ ) u o .

(4.18)

240


Equation (4.18) admits two variants of polarization. The first one agrees with collinearity of wave and displacement vectors. This corresponds to longitudinal wave with the constant phase velocity v ph =

( λ + 2µ ) / ρ .

The second variant defines an infinite number of directions, that follows from orthogonality of wave and displacement vectors. This corresponds to the transverse wave with the constant phase velocity v ph = µ / ρ . Thus the choice of certain direction of plane wave propagation doesn’t have any influence on the wave characteristics: number of waves (two only), the phase velocity, and the mutual polarization. We consider also the case of transversal isotropic properties of the medium propagation, as many structured materials with such properties will be considered in the next chapters. Such a medium has one axis of symmetry (later the applicate axis) and is perpendicular to this axis plane of isotropy. The matrix Cijkl has five independent constants

C1111 , C3333 , C4444 , C1313 , C2211 C1111 C1122 C1133  C1111 C1133   C3333       

0 0 0

0 0 0

C4444

0 C4444

    . 0   0   1 (C1111 − C1122 )  2  0 0 0

When the wave is being propagated along the symmetry axis, two variants of polarization and wave types respectively are possible: the longitudinal wave with the phase velocity v ph =

( C3333 ρ )

and the

transverse wave in the plane of isotropy with polarization along the abscissa axis and the phase velocity v ph =

(C1313 ρ ) . When the waves

being propagated are perpendicular to the symmetry axis, then the three types of polarization are possible: the longitudinal wave with the phase

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Waves in Materials

velocity v ph =

( C1111 ρ ) ,

the transverse wave with polarization in the

direction of the symmetry axis and the phase velocity v ph =

( C4444 ρ ) ;

the transverse wave with polarization in the direction of ordinate axis and the phase velocity v ph =

( (1 2) ( C

1111

− C2211 ) ρ ) .

The next usual step is to assume that the direction of propagation coincides with Ox1 - the abscissa axis so that u = {uk ( x1 , t )} .

(4.19)

Substitution of (4.19) into (4.18) gives one longitudinal plane wave ( P wave) and two transverse plane waves, horizontal polarized wave ( SH wave) and vertical polarized wave ( SV -wave) and the corresponding linear equations ρ uɺɺ1 − (λ + 2 µ ) u1,11 = 0,

ρ uɺɺ2 − µ u2,11 = 0,

(4.20)

ρ uɺɺ3 − µ u3,11 = 0. The corresponding solutions in the form of harmonic waves are i kL x1 −ωt )

u1 ( x1 , t ) = u1o e (

,

k L = ( ω v L ) , vL =

( λ + 2µ ) ρ ,

(4.21)

u2 ( x1 , t ) = u2o ei ( kT x1 −ωt ) , u3 ( x1 , t ) = u3o ei ( kT x1 −ωt ) , kT = (ω vT ) , vT = µ ρ . Thus the harmonic plane waves propagate in the linearly elastic isotropic medium with constant velocity and are the simplest nondispersive waves.

4.2.3 Cylindrical elastic waves in the classical approach The classical cylindrical waves are the waves arising in an infinite linear elastic material with a cylindrical cavity to boundary surface r = ro and

242


with a spatially uniform and harmonic (in time) pressure p(t ) = po eiωt

( po = const ) . If the cylindrical system of coordinates Orϕ z is chosen in the way that the axis Oz coincides with the cavity axis, then the problem on motion of the medium will be axisymmetric one and all field functions describing the motion will depend on one spatial coordinate only – the radius r , and on time t . In this case the displacements and stresses in the material can be represented through a potential Φ ( r , t ) . Only one

of

three

displacements: the radial displacement,

is

different from zero ur ( r , t ) ≠ 0, uϕ ( r , t ) = u z ( r , t ) = 0 , as well as the three stresses σ rr ( r , t ) , σ ϕϕ ( r , t ) , σ zz ( r , t ) . The components of the strain and stress tensor are ∂ur , ε ϕϕ = ur r , ε zz = ε rϕ = ε ϕ z = ε rz = 0, ∂r ∂u u  ∂u u   ∂u u  1 σ rr = λ  r + r  + 2 µ r , σ ϕϕ = λ  r + r  2 + 2 µ 3r , r  ∂r r r r  ∂r  ∂r

ε rr =

 ∂ur ur  + . r   ∂r

σ zz = λ 

(4.22)

(4.23)

They can be expressed through the potential Φ ( r , t ) by the formulas ur =

∂Φ ∂ 2Φ λ ∂ 2Φ ρ 2 , , σ rr = 2µ 2 + λ + 2 µ ∂t ∂r ∂r

σ ϕϕ =

2 µ ∂Φ λ ∂2Φ + , σ zz = λ∆ r Φ. ρ ∂r λ + 2 µ ∂t 2

The Laplace operator ∆ r is ∆r =

∂2 1 ∂ + ∂r 2 r ∂r

(4.24)

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Waves in Materials

and the potential Φ ( r , t ) satisfies the wave equation vL2 ∆ r Φ −

∂ 2Φ λ + 2µ . = 0 vL = 2 ∂t ρ

(4.25)

This equation has a well-known solution through the Hankel functions (cylindrical functions) in the form of the so-called cylindrical wave. The wave propagates along the radial coordinate perpendicularly to the symmetry axis from the cavity to infinity and satisfies at infinity the conditions of finiteness of the radiation. The solution is written using the Hankel function of the first kind and the zero index Φ ( r , t ) = Φ o H o(1) ( k L r ) eiωt ,

(4.26)

where Φ o = const. As it can be seen from (4.25), this solution includes the phase velocity and this last is the same as for the longitudinal plane wave. However the Hankel function in the solution of (4.25) differs essentially from the exponential function in the solution of (4.21) for the longitudinal plane wave. The wave is no longer a harmonic wave (it is asymptotically harmonic) and its intensity decreases with the distance of propagation (or with the time of propagation). The unknown amplitude Φ o can be determined from the boundary condition on the cavity surface

σ rr ( ro , t ) = − po eiωt

(4.27)

and is equal to

po

Φo = kL ( 2µ + λ k L ) H

(1) o

2µ ( k L ro ) − H1(1) ( kL ro ) ro

.

The cylindrical wave of radial displacement is described by the formula

244


ur ( r , t ) =

po k L 2µ (1) k L ( 2 µ + λ k L ) H o(1) ( k L ro ) − H1 ( k L ro ) ro

H1(1) ( k L r ) eiωt . (4.28)

Thus for each fixed time the profile of this wave will repeat the graph of the Hankel function H1(1) ( k L r ) . For a large distance of propagation the cylindrical wave will be very close (through parameters) to the longitudinal plane wave.

4.2.4 Volume and shear elastic waves in the nonclassical approach Let us consider here an isotropic mixture. Its equations are obtained by simplifying the equations of motion (3.57) and constitutive equations (3.56), of the general anisotropy case. A similar simplification in the classical theory of elasticity gives the Lamé equations. In the elastic mixture theory, we obtain two coupled vector equations, which are some kind of generalization of the classical Lamé equations µα ∆u (α ) + ( λα + µα ) grad div u (α ) + µ3 ∆u (δ ) + ( λ3 + µ3 ) grad div u (δ ) + β ( u (δ ) − u (α ) ) (4.29) = ( ραα − ρ12 ) uɺɺ(α ) + ρ12 uɺɺ(α ) (α , δ = 1, 2;α + δ = 3) . The constitutive equations for isotropic elastic mixtures are (α ) (δ ) σ ik(α ) = λα ε mm δ ik + 2 µα ε ik(α ) + λ3ε mm δ ik + 2 µ3ε ik(δ ) .

(4.30)

These equations contain six elastic constants λk , µk . Moreover, in the equations (4.29) four mechanical constants are involved, the densities ρ11 , ρ 22 , ρ12 and one constant β of interaction between phases of the mixture. Thus, the isotropic mixture is characterized by ten constants altogether.

Waves in Materials

245

Let us apply to (4.29) the classical procedure of separation of the motion into two independent motions, one linked with the volume change, and the other with the form change. For this purpose, let us represent vectors u (α ) in the form u (α ) = v (α ) + w(α ) ( rot v (α ) = 0, div w(α ) = 0 ) . As a result we obtain two uncoupled systems of equations ( λα + 2µα ) ∆v (α ) + ( λ3 + 2µ3 ) ∆v (δ ) ∂ 2 v (α ) ∂ 2 v (δ ) + β ( v (α ) − v (δ ) ) = ( ραα − ρ12 ) + ρ , 12 ∂t 2 ∂t 2

(4.31)

(4.32)

µα ∆w(α ) + µ3 ∆w(δ )

∂ 2 w (α ) ∂ 2 w(δ ) (α ) ( δ ) + β ( w − w ) = ( ραα − ρ12 ) + ρ12 . ∂t 2 ∂t 2

(4.33)

Thus (4.29) is divided into two similar and simple systems, which describe separately the propagation of volume waves and shear waves. Consider the system

aα ∆ϕ (α ) + a3 ∆ϕ (δ ) + ∂ 2ϕ (α ) ∂ 2ϕ (δ ) + β (ϕ (α ) − ϕ (δ ) ) = ( ραα − ρ12 ) + , ρ 12 ∂t 2 ∂t 2

(4.34)

which can take the form of system (4.32) or system (4.33) depending on the coefficients ak . Assume that the processes change harmonically with time,

ϕ (α ) ( x, t ) = ϕɶ (α ) ( x) eiωt so that system (4.34) becomes

,

( x ≡ ( x1 , x2 , x3 ) )

(4.35)

246


 aα ∆ − ( ραα − ρ12 ) ω 2 − β  ϕɶ (α ) +  a3 ∆ − ρ12ω 2 + β  ϕɶ (δ ) = 0.

(4.36)

The solution of (4.36) is obtained as a sum of two functions, each of which describes a separate wave motion. That is, the motion is a superposition of two waves. There will be two volume waves for system (4.32) and two shear waves for (4.33). To prove this separation, transform (4.36) into ∆ϕɶ (1) = nɶ11ϕɶ (1) + nɶ12ϕɶ (2) , ∆ϕɶ (2) = nɶ21ϕɶ (1) + nɶ22ϕɶ (2) , nɶαα = mα aɶδ − m3 aɶ3 , nαδ = m3 aɶδ − mα aɶ3 , aɶk =

(4.37)

ak , a1a2 − a32

a1a2 − a32 ≠ 0 , mα = ( ραα − ρ12 ) ω 2 + β , m3 = ρ12ω 2 − β . The solution of system (4.37) can be written as

ϕɶ (α ) = f (α ) + rδ (ω ) f (δ ) ,

(4.38)

(α ) where f is the solution of Helmholtz equation

∆ f (α ) + lα2 f (α ) = 0,

(4.39)

and lα are the roots of the biquadratic equation

(a a

1 2

− a32 ) l 4 − ( a1 ρ 22 + a2 ρ11 ) ω 2 − ( a1 + a2 + 2a3 ) ( ρ12ω 2 + β )  l 2 +  ρ11 ρ 22ω 2 − ( ρ11 + ρ 22 ) ( ρ12ω 2 + β )  ω 2 = 0,

  a l 2 − β − ρ12ω 2 rα = −  2 3 α 2    a1lα + β − ( ραα − ρ12 ) ω 

( −1)α

.

(4.40)

Since the basic system (4.29) describes all waves in linearly elastic twophase mixtures, we expect that the number of any type waves in the mixtures will double in comparison with a classical elastic medium.

Waves in Materials

247

It should be noted that the most essential feature of representation (4.38) isn’t concerned with the number of waves, but with new wave properties. These waves will always be dispersive, because the coefficients in Helmholtz equations (4.39) depend essentially on frequency.

4.2.5 Plane elastic harmonic waves in the nonclassical approach The definition of plane waves in the nonclassical approach (for the model of elastic mixtures) is similar to the classical one. But in the mixture, the motion is already described by two displacement vectors, each phase of a mixture has its own vector. In the plane wave, these vectors are not necessary collinear. However, the plane front of a wave should be the same for both vectors. The representation of the plane wave is therefore as follows u (α ) ( x, t ) = u o (α ) e i ( ξ − ω t )

(4.41)

where u o (α ) are arbitrary constant vectors; ξ = k ⋅ r ; r is the radiusvector of the point x ≡ ( x1 , x2 , x3 ) . By using the general definition of a plane motion in basic equations of the motion of anisotropic mixtures (3.56), (3.57), we obtain 2 (δ )  (α ) ∂ 2um(α ) (3) ∂ um k 2  ciklm + c iklm ∂ξ 2 ∂ξ 2 

  nk nl + 

∂ 2ui(α ) ∂ 2 ui(δ ) (4.42) + ρ 12 ∂t 2 ∂t 2 which describes the plane motion, including the propagation of plane waves. By substituting (4.41) into (4.42), we get the system of equations for phase velocities, + β i ( ui(α ) − ui(δ ) ) = ( ραα − ρ12 )

248


 (α ) β i   o (α ) k2  ciklm nk nl 2 − ( ραα − ρ12 ) + 2  δ im  um ω  ω   

(4.43)

 (3) k2  β   + ciklm nk nl 2 +  i2 − ρ12  δ im  umo (δ ) = 0 ω ω    which is the direct generalization of the case of two-phase mixtures of the classical Christoffel equation. Some special important cases of symmetry in mixtures follow. Let us define the isotropic, transversal-isotropic, and orthotropic mixtures as such media, for which three matrixes of elastic properties (n) ciklm have the same necessary orders of symmetry (isotropic, transversal-

isotropic, and orthotropic). Then from the particular representation of system (4.43) we can draw for each type of symmetry, the corresponding types and numbers of waves. In the isotropic mixture, there exist two modes of longitudinal waves ( P -waves), horizontally polarized transverse waves ( SH -waves), and vertically polarized transverse waves ( SV -waves). In the transversal-isotropic mixture, each wave will also have two modes: three waves in the direction of symmetry axis, the longitudinal wave and two differently polarized and identical by velocities transverse waves, and three waves in the symmetry plane, P -wave, SH -wave, and SV -wave. In the orthotropic mixture, two modes of nine types of waves will occur: threes waves ( P, SH , SV ) in the direction of each of the three axes of symmetry. Let us focus on the characteristic example of the plane waves in an isotropic mixture. Plane waves are conventionally introduced with the notion of plane motion. Let us assume that the wave propagates in the direction of the coordinate axis Ox1 . In this case, the partial displacement vectors u (α ) depend only on two variables u (α ) ≡ {uk(α ) ( x1 , t )} .

(4.44)

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Waves in Materials

By the substitution of (4.44) into (4.29) we get three uncoupled systems

∂ 2u1(α ) ∂ 2u1(δ ) ρ + 12 ∂t 2 ∂t 2 2 (α ) ∂ u1 ∂ 2 u1(δ ) −(λα + 2µα ) − ( + 2 ) − β ( u1(α ) − u1(δ ) ) = 0, λ µ 3 3 2 2 ∂x1 ∂x1

( ραα − ρ12 )

∂ 2um(α ) ∂ 2 um(δ ) ρ + 12 ∂t 2 ∂t 2 ∂ 2um(α ) ∂ 2um(δ ) − µα − µ − β ( um(α ) − um(δ ) ) = 0 (m = 2,3). 3 ∂x12 ∂x12

(4.45)

( ραα − ρ12 )

(4.46)

Each of the obtained systems is coupled separately and, as expected, each of them describes the independent propagation of three types of waves – longitudinal, transverse horizontal, and transverse vertical ones. Systems (4.45), (4.46) can be written in a form which is invariant with respect to the wave type (like systems (4.34))

∂ 2um(α ) ∂ 2um(δ ) + ρ 12 ∂t 2 ∂t 2 2 (δ ) ∂ 2uk(α ) ( k ) ∂ uk − aα( k ) − − β ( uk(α ) − uk(δ ) ) = 0 a 3 2 2 ∂x1 ∂x1

( ραα − ρ12 )

(1) m

(a

(4.47)

= λm + 2 µ m , am(2) = am(3) = µm ) .

The last system has a solution in the form of harmonic waves (α ) um(α ) ( x1 , t ) = Aom e

(m) − i kα x −ω t

(

) + l (k ( m) ) A(δ ) e− i(kδ( m) x −ωt ) δ

om

.

(4.48)

The wave numbers kα( m ) are determined from the dispersion equation M 1( m ) k 4 − 2 M 2( m ) k 2ω 2 + M 3( m )ω 4 = 0, 2

M 1( m ) = a1( m ) a2( m ) − ( a3( m ) ) ; 2 M 2( m ) = a1( m ) ρ11 + a2( m ) ρ 22

250


 β  − ( a1( m ) + a2( m ) + 2a3( m ) )  2 + ρ12  , ω 

(4.49)

 β  M 3( m ) = ρ11 ρ 22 − ( ρ11 + ρ 22 )  2 + ρ12  , ω  and the coefficients of the matrix of amplitude distribution are determined by a simple algebraic formula ( −1)α

 a ( m ) ( k ( m ) )2 + β − ρ ω 2  α α αα  (4.50) . l (kα( m ) ) =  − 2 m m ( ) ( )   a3 ( kα ) − β   The main features of these waves are: 1. There exist two simultaneous modes which differ by the wave numbers

kα( m ) ( α fixes the number of a mode,

characterizes the type of a wave). 2. Both modes are essentially dispersive waves. 3. Mixture filters one of the modes, it is cutoff frequencies, leading off the frequency

while m

for low

β ( ρ11 + ρ 22 ) * ωcut = ρ11 ρ 22 , which is called the cutoff frequency. 4. Two modes propagate in each phase and with their amplitudes, the matrices of amplitudes distribution are given by (4.50) and depend essentially nonlinearly on frequency, as a consequence the energy of modes is pumped from a mode to a mode with a change of frequency. The characteristic view of coefficients l ( k1(1) , ω ) and l ( k1(2) , ω ) depending on frequency can be seen in Fig. 4.1 for fibrous materials (the composite with volume fraction 0.2 of microfibers Thornel-300 and nanofibers in the form of zigzag nanotubes). It must be noted that all the above described effects (especially, the wave dispersion effect) have a structural character and the waves are

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Waves in Materials

linear. If we assume that dispersion in materials can be constructional, geometrical and viscoelastic, then the dispersion observed in mixtures is geometrical. We should also notice that if we choose the first longitudinal mode, it propagates in both phases of the mixture.

1.1 5

1.075 4

1.05 3

1.025 2

2 1010 4 1010 6 1010 8 1010 1 1011 0.975

1

0.95 1 108 2 108 3 108 4 108

5 108 6 108

1 1

0.5

0.8 0.6

1 108 2 108 3 108 4 108 5 108 6 108 0.4

-0.5 0.2

-1 11

2 10

11

4 10

11

6 10

11

8 10

12

1 10

Fig. 4.1 Amplitudes via frequencies for plane waves in composites.

In the first phase of a mixture the longitudinal wave propagates as

u1(1) ( x1 , t ) = Ao(1)1 e

(1) − i k1 x −ω t

(

),

(4.51)

and in the second phase of a mixture we have the same longitudinal wave, but with a special amplitude coefficient

252


u1(2) ( x1 , t ) = l (k1(2) ) u1(1) ( x1 , t ) .

(4.52)

If, instead, the second mode is chosen, in the first phase of a mixture, the wave propagates as

u2(1) ( x1 , t ) = l (k1(1) ) u2(2) ( x1 , t ) ,

(4.53)

while in the second phase of a mixture, we have

u2(2) ( x1 , t ) = Ao(2) 1 e

(2) − i k1 x −ω t

(

)

(4.54)

which differs for an amplitude wave. It is necessary to pay our attention on two features of plane wave phase velocities in the mixtures. The first consists of the phase velocities of both modes tend for high frequencies to some finite values. The second testifies the link of the mixture theory as the structural theory of the second order with the effective moduli theory as the structural theory of the first order. The question is that at the small frequencies (consequently, at the large wavelengths) the phase velocity of the first mode tends to the value of the phase velocity given by formulas of the effective moduli theory. The last question to be discussed in this subsection is associated with experiments with plane waves in materials. The only materials for which experiments were thorough and extensive are the composite ones. Therefore the wave dispersion was one of the most studied phenomena. Let us formulate a few notes. The first one deals with the observation of wave phenomena in tests with real materials. The prevailing point of view is that any observation should be concentrated on some chosen phenomena and the description or the setting of this observation is based on some theoretical scheme. The theory defines what should be observed. The second note is that when being wished to observe certain wave phenomenon, it turned out to finding (very difficult and very often) the real material in which this phenomenon will be shown isolated. The real wave picture is always more complicated as it is the theoretical scheme.

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Waves in Materials

This becomes apparent when we study the wave dispersion in composite materials. In experiments with plane and volume waves, the dispersion of different nature – geometrical, viscoelastic, and constructional – seems to be much difficult for separation. It should be noted that the geometrical dispersion in materials is called the dispersion which arises owing to the presence in materials both of the internal structure and of the interaction of waves with elements of the structure. The viscoelastic dispersion in composite materials is usually the effect of the viscoelastic properties of the matrix of composites. The constructional dispersion arises when waves interact with the boundary surfaces of the constructions. The third note focuses on the special role of the wavelength as a parameter, which determines the bounds of applicability of all theoretical schemes of wave propagation. As a consequence the frequency ranges, with which the experiments were realized, are essential in experiments.

4.3 Analysis of Waves in Nonlinearly Deformed Elastic Materials 4.3.1 Basic notions of the nonlinear theory of elasticity. Strains Motion is a primary notion in mechanics. When studying the elastic bodies (elastic materials), motion is related to the change of the form and dimensions of the body. The motion is studied with respect to a reference system. The notion of material continua permits to identify the material with the geometrical domain of a three-dimensional space, occupied by the material (body). Thus, the geometrical domain transforms into some physical abstraction, which is called the body. Suppose the body B in Euclidean space ℝ 3 is given. The motion is then defined as the mapping of a set B on the domain χ ( B, t ) of the space ℝ 3 at a given time t

x = χ ( X , t ) , X ∈ B, t ∈ ℝ1 .

(4.55)

254


The motion is differentiable by time (as a rule, not more than twice), so that we define the velocity xɺ = χɺ ( X , t ) , and the acceleration

ɺxɺ = χɺɺ( X , t ) . The image χ ( B, t ) of the mapping χ at the moment t is the

configuration. That is, the configuration might be considered as a fixed (at time t ) picture of the motion. The configuration of a body at time t is called actual. The configuration of a body at any arbitrarily chosen initial moment is called reference. The description of the body motion with respect to the reference configuration is called the reference description. This description is mainly used in mechanics of materials. The reference and actual configurations are linked with the concept of Lagrangian and Eulerian reference systems.

−

The Lagrangian system is when the material particles of a body are individualized, i.e. each particle is associated with Cartesian coordinates

xk

(or the curvilinear coordinates

x k ). The

individualization is carried out in the reference configuration. Moreover it is assumed that during the motion (transition from the reference configuration to the actual one) the coordinates xk don’t vary, i.e., the particle and its coordinates are always linked.

−

The Eulerian system is when the particle occupies the point in the actual configuration characterized by coordinates X α (or X α ). The coordinates of the particle are not linked now with a motion, because the last has already been taken place (a body is already in the actual configuration).

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k α Let us choose in ℝ 3 the Lagrangian {x } and the Eulerian { X }

reference systems, and

assume x k = x k ( X α ) , X α = X α ( x k ) for the

transitions from a system to another. The motion in the reference description is denoted by

χκ ( X α , t ) or x m = χκm ( X α , t )

(4.56)

where functions χκm are continuously differentiable up to a necessary order, real, and one-value. For a given Lagrangian coordinate system the coordinate transform

dθ i = ( ∂θ i ∂ϑ k ) dϑ k = aiiki dϑ k is the correspondence between two systems (θ 1 ,θ 2 ,θ 3 ) , (ϑ 1 ,ϑ 2 ,ϑ 3 ) , which is assumed to be continuously one-to-one, that is, homeomorphic. It should be noted that on this stage of identification, the notion of metrics in the space (θ 1 ,θ 2 ,θ 3 ) is not needed, the space can also be a non-metric one. The next step consists in using the covariant vectors of basis ( e1 , e2 , e3 ) , ( e1′, e2′ , e3′ ) (these vectors are by definition directed along the tangents to the corresponding coordinate lines θ k ) and representation of an arbitrary infinitesimal vector r = M 1M 2 ( M 1 , M 2 are infinitely near points of continuum) in the form d r = dθ i ei , d r = dϑ i ei′ . The contravariant basis ( e 1 , e 2 , e 3 ) is reciprocal to the covariant basis ( e1 , e2 , e3 ) in the sense that vectors of new basis are directed

normally to the surface θ k = const . Associated with the bases we have the corresponding covariant and contravariant quantities. The metrics is defined by the length of the vector d r

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2 d r = ds 2 = dθ i dθ k ei ⋅ ek = dθ i dθ k gik .

(4.57)

This length should be invariant relatively to the coordinate system 2 d r = dθ p dθ q e′p ⋅ eq′ = dθ p dθ q g ′pq → g ′pq = aii ip aikqi gik .

The expression (4.57) is called the fundamental quadratic form. The fundamental metric tensor is defined as the tensor g = gik e i e k with covariant components gik . It should be noted that contravariant and mixed components are analogously defined. When we go from curvilinear coordinate system to the Cartesian one, all three types of metric tensors are transformed into the tensor with components in the form of Kronecker’s symbols. Based on a given metrics, the nonlinear theory of deformation is constructed as a part of continuum mechanics. The initial configuration (at the initial time) and actual (at the present time) one are fixed. In different configurations, we will have different basis vectors {eko } for the initial configuration and {ek∗ } for the actual one. If two different time moments t ′, t ′′ are considered, then the corresponding configurations will be characterized by different bases {ek′∗ } ,{ek′′∗ } . The fundamental metric tensors will also be different g ′ = gik′ ei′∗ek′∗ ,

( ds′ )

2

= gik′ dθ i dθ k ,

g ′′ = g ik′′ ei′′∗ek′′∗ ,

( ds′′ )

2

= gik′′ dθ i dθ k .

(4.58)

By using (4.58) we obtain

( ds′ )

2

2

− ( ds′′ ) = 2ε ik dθ i dθ k , ε ik =

1 ( gik′ − gik′′ ), 2

(4.59)

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(where ε ik can be treated as the covariant components of a certain tensor) one of the most important elements of kinematics of deformations. From the covariant components ε ik , two kinds of contravariant components can be constructed depending on the chosen basis ( {ek′∗ } or {ek′′∗ } ) ε ′ik or ε ′′ik . Strain tensors are defined conventionally in both bases in terms of covariant components E ′ = ε ik ei′∗ei′∗ , E ′′ = ε ik ei′′∗ei′′∗ . If the displacement vector u = {ui } exists for all points of continuum, then

the strain tensor can be written in the simple form 1  ∂u ε ik =  i 2  ∂θ 1  ∂u ε ik =  i 2  ∂θ

∗ ∂u ⋅ ek + k ∂θ ∂u ⋅ eko + k ∂θ

∗ ∂u ⋅ ei − i ∂θ ∂u ⋅ eio + i ∂θ

∂u ∂θ k ∂u ∂θ k

 , 

(4.60)

 . 

(4.61)

Tensor (4.60) is usually called the Almansi strain tensor, whereas tensor (4.61) is called Green or Cauchy-Green strain tensor. Let us remind that the deformation (strain) of body is understood as the change of the body form or dimensions. The gradient of deformation is defined as F ≡ Fκ ( X α , t ) ≡ ∇χκ ( X α , t ) ,

Fαm = x,mα =

∂ χκm ( X 1 , X 2 , X 3 , t ) . ∂X α

(4.62)

For the description of deformations in materials, the displacement vector of the particle X ∈ B , under its transition from the reference configuration BR to the actual configuration B , is defined as

258


u = {u m } = {u1 , u 2 , u 3 } ,

u m ( X α , t ) = xm ( X α , t ) − X m

(4.63)

so that the deformation gradient (4.62) can be understood as the linear approximation of deformations (4.63). The conventional representation of Cauchy-Green strain tensor in the reference configuration, which defines deformations of body for the body metrics in reference configuration (here, the metrics corresponds to the no deformed state of body) is 1  ∂u n ∂u m ∂u n ∂u i + + 2  ∂x m ∂x n ∂xi ∂x m

ε nm ( χ,kα ) ≡ ε nm ( x k , t ) = 

 . 

(4.64)

The Almansi strain tensor is given in the actual configuration, i.e., in the deformed state of a body and uses the metrics in the deformed state 1  ∂U β ∂U γ ∂U β ∂Uδ  . + −  2∂ Xγ ∂ Xβ ∂ Xδ ∂ Xγ 

εɶβγ ( X α , t ) = 

(4.65)

Both tensors, Cauchy-Green and Almansi strain tensors, are symmetric, positively defined and their main values are positive. It is very important to know the character of dependence of strain tensors on a chosen configuration, that is, on that relative to which configuration these tensors are defined. The usual assumption in the choice of the initial state is the natural one, i.e. the absence of external forces and internal stresses and strains. However the metrics corresponding to this state isn’t necessary Euclidean one. The initial state can even be a virtual state, whereas the real deformation of continuum always takes place in real (Euclidean) space. Kinematics of deformations also use the notions of Christoffel symbols and Riemann-Christoffel tensor, which are fundamental in differential geometry. In general, the Christoffel symbols are replaced by

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the coefficients of affine connection and the Riemann-Christoffel tensor by the curvature tensor of the affine connection. Christoffel symbols appear in nonlinear mechanics of materials when the coordinate system is curvilinear and the basis vectors are changing from point to point. Then the variability of the basis is characterized by the formula

∂ek ∂ei = = Γ kim em = Γ kim e m . i k ∂θ ∂θ

(4.66)

In general, the Christoffel symbols change from point to point. In Euclidean and Riemann spaces, the Christoffel symbols can be expressed through the components of the metric tensor

Γ mki =

1 mn  ∂g kn ∂g in ∂g ki g  i + k − n ∂θ ∂θ 2  ∂θ

 . 

(4.67)

In the first case of formula (4.66), i.e. the expansion over covariant basis {em } ) the quantities Γmki are called the Christoffel symbols of the first kind. In the case of expansion over the contravariant basis {e m } the quantities Γ kim are the Christoffel symbols of the second kind. They don’t form a tensor, but the operation of lowering indexes Γ mki = g mn Γ nki and Γ mki = g mn Γ nki holds true. From (4.67) there follows that if the components of metric tensor gik are constant over all space, then Christoffel symbols will be zero everywhere. Such a coordinate transform (from {θ k } to {θ k∗ } ) can be found in Euclidean spaces, when the equality

 ∂θ k ∂θ i ∂ 2θ m  ∂θ ∗ s Γ∗pqs =  Γ kim ∗ p + =0.  ∗q ∂θ ∗ p ∂θ ∗q  ∂θ m  ∂θ ∂θ

(4.68)

260


is always fulfilled. This is not true in Riemann spaces.

4.3.2 Forces and stresses The strain tensor highlight the geometry of the deformation process, whereas the stress tensor represents the other fundamental aspect of the process, which is associated with the primary notions of the force and the moment. Both these notions can be considered as some physical abstractions, which is intended for the description of the action either of a body on another body or particles of the body on other particles of the same body. The forces are classified into external and internal ones. According to an intuitive knowledge, any force applied to a body is characterized by the point of application, the direction and the intensity. Therefore, it is convenient to represent external forces by a vector field and by a distribution of this field. If such a distribution is the mass distribution in a body, then forces are called the mass forces. If the distribution is over a surface, then forces are called the surface forces. In the same way, we define the linear forces and the point forces.

The internal forces act within the body and are described by the internal stresses. Stresses are treated in mechanics as a primary notion. They are defined through the stress tensor, which can be introduced in many different ways. Following the Euler-Cauchy principle, let us choose the arbitrary volume V and assume that the action of the rest of the body on V can be substituted by the action of some vector field of forces, which is given on the external surface S of the body V .

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The infinitesimal element dS ( x) of the surface S is one piece of the tangent plane to S at the point x . The force acting on the infinitesimal element of the surface dS , is denoted by dP and assumed to be equal

dP ( x) = tN dS ( x) .

(4.69)

Here, t N is the stress vector, which is applied to the surface dS with the normal vector N . The vector t N depends not only on the position x and eventually on time, but on the orientation N of the surface. It is also assumed that 1. Internal forces are contact forces which describe local interactions. 2. The vector field is specified at each point of the surface and it depends both on the point and on the normal to this surface at this point.

If we decompose the vector t N into three components along orthogonal directions linked with dS (i.e., the tangent, normal, and binormal), we obtain three stress vectors along three independent directions. The values of the stresses along the three directions are obtained by the corresponding vectors divided by the area of the surface element dS . In order to introduce the internal stress tensor, let us now consider the infinitesimal coordinate tetrahedron which is in equilibrium under the action of the forces (4.69) applied to the tetrahedron faces. Here we can have two possibilities: the elementary tetrahedron is without or with a deformed state. In any case it can be written a simple formula which links the element area of the tetrahedron face in absence of deformation with the same element in a deformed state. As a result of the equilibrium there are nine

262


stresses on the three coordinate faces which form the (stress) tensor of the second order. When the tetrahedron is fixed, the stress tensor is invariant with respect to the choice of the reference system. For this reason the nine quantities tensor.

t nm ( x k , t ) , which are called

the stresses, form the stress

It describes the stresses at moment t on the surfaces of elementary volume in the deformed state, which are measured on the unit area in the reference configuration (i.e., in the no deformed state). This tensor is called the Piola-Kirchhoff stress tensor, it is asymmetric, and it does not immediately determine the stress state in a body. The Lagrange-Cauchy stress tensor σ ik ( X α , t ) is introduced in the same way. It involves the stresses, which are measured on the unit area in the actual configuration (i.e., in the deformed state). This tensor is symmetric. The above tensors are the often used. But also other stress tensors might be expedient such as the Piola tensor, Hamel tensor, the second Piola-Kirchhoff tensor, tensor of true stresses, tensor of generalized stresses, and others.

4.3.3 Balance equations Let us consider some extensive tensor quantity A ( x, t ) , which characterizes the material continuum. It could be the mass, temperature, pulse (momentum), moment of momentum, energy. Together with A we define the quantities: 1. Ξ( x, t ) is the volume density of the quantity A

A=

∫

Ξ dV

V (t )

( V (t ) is a closed connected domain).

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2. A( x, t ) is an increment of the volume density

Ξ ( x, t )

induced by the influx, for example, owing to the sources placed inside the domain V (t ) . 3. α ( x, t , n ) is the rate of a flux density of the quantity A through the boundary of the domain V (t ) ( n is normal to the surface

S (t )) . It should be noted that all three quantities can be any arbitrary tensor fields. But Ξ( x, t ) and A( x, t ) are fields of the same dimension, whereas α ( x, t , n ) is a field with one more dimension. The conservation law expresses such a balance (equilibrium) law, the change of the quantity A on unit of time is the result of: 1. presence of this quantity flux through the boundary surface; and 2. work of sources (discharges) within the body, So that it is

d dt

∫ V (t )

Ξ dV =

∫ V (t )

AdV −

∫ α dS .

(4.70)

S (t )

It must be reminded that in mechanics, two different operations of the differentiation of tensors are used. The first characterizes the rate of change of the tensor field at the fixed geometrical point (place), for it the symbol ∂ is used and it is called the local derivative. The second ∂t characterizes the same rate at the fixed material point (particle), for it the symbol d is used and it is called the material (substantial) derivative. dt If A is the vector quantity and quantities Ξ , A, α are smooth functions, then balance equations can be written in the differential form

∂ Ξ k +( Ξ k vm + α km ),m = Ak , ∂t

(4.71)

264


where v = {vm } is the vector of particle velocity (velocity at the material point). The conservation laws is the fundamental law for defining most of equations (and theories) of mechanics and in particular of mechanics of materials. The first equations is the continuity equation, which follows from the law of mass conservation. So, let A be the mass, and ρ ( x, t ) the mass volume density (which replace the symbol Ξ ( x, t ) ). Suppose now that the interchange of the material between parts of a body is absent, i.e., the sources and

( A( x, t ) = 0 ) and the flux of absent S (α ( x, t , n ) = 0 ) . Then

discharges of mass are absent in a body mass through the boundary surface is

from equation (4.71), we obtain the classic equation of mass balance

d ρ ( x, t )dV = 0 dt V∫

(4.72)

or

∂ρ + ( ρ vm ), m = 0 ∂t

 ∂ρ  + div ( ρ v) = 0  .   ∂t 

If the quantities: − ρ ( x, t ) , the density at time t ,

−

ρo ( x, t ) , the density at the initial time;

− J ( x, t ) , the Jacobian of transformation are sufficiently smooth functions, then from equation (4.72) we get the classic relationship ρ o = ρ J . The next important balance relation highlights the second Newton law. We consider the balance of momentum where the external forces are

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265

given. The quantity A is the kinetic torque of a body V (the torque of momentum of a body V )

A ( x, t ) = ∫ ρ v( x, t )dV ,

(4.73)

V

that is,

Ξ i ( x, t ) = ρ vi ( x, t ) .

(4.74)

Two other quantities A( x, t ) , α ( x, t , n ) express the external forces and the stress tensor

Ai ( x, t ) = Fi ( x, t )

,

α i ( x, t , nk ) = −σ ik ( x, t ) nk .

(4.75)

The law of conservation of momentum can be written as

d ρ (ξ )vi (ξ , t )dV (ξ ) − ∫ σ ik (ξ , t )nk dS (ξ ) = ∫ Fi (ξ , t )dV (ξ ) . dt V∫ S V

(4.76)

All integrand functions are supposed to be continuously differentiable, so that by applying the Gauss-Ostrogradski theorem to the surface integral the equation (4.76) can be written, in the local (differential) form and with the Euler coordinates,

∂ ( ρ vi ) + ( ρ vi vk ),k = σ ik , k + Fi , ∂t or, after some transformations by taking into account the equation of mass balance we have the usual form of three motion equations

ρ

dvi = σ ik ,k + Fi . dt

(4.77)

The third balance equation concerns with the moment and, according to (4.76), has the form

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d ρ (ξ )ε ilmξl vi (ξ , t )dV (ξ ) dt V∫

− ∫ σ ik (ξ , t )ε ilm nk dS (ξ ) = ∫ ε ilmξ l Fi (ξ , t )dV (ξ ) . S

(4.78)

V

Here, the external moments are supposed to be formed only by the action of external forces. ε ilm is the Levi-Civita tensor. As a consequence of equations (4.78) we have that the stress tensor is symmetric. The fourth group of balance relations consists of equations of the energy balance. The energy of a body is defined as the sum of the kinetic energy of the body with the internal energy of the body. However this equation does not give any further information on the process of deformations. The balance equation of energy is replaced by the first principle of thermodynamics, which can be stated as follows: the full derivative of the energy of a body at arbitrary time is equal to the sum of the power of external forces acting on a body and the heat quantity acquired of a body during unit time. Therefore, the definition of the (energy) power is very important. It is assumed, usually, that the power of external forces (volume forces

F = {Fi } and surface forces S = {Si } ) is defined by Wi = ∫ Fi (ξ , t )vi (ξ , t )dV (ξ ) + ∫ Si (ξ , t )vi (ξ , t )d Σ(ξ ) . V

Σ

If the surface forces are absent (this hypothesis was adopted in the prior three groups of balance equations) and the heat is neglected (this assumption is adopted in the classic theory of elastic deformations), then the representation of the energy balance (first principle of thermodynamics) is as follows

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267

d 1  ρ (ξ , t )  vi (ξ , t )vi (ξ , t ) + e  dV (ξ ) ∫ dt V 2 

= ∫ σ ik (ξ , t )nk (ξ )vi (ξ , t ) d Σ(ξ ) + ∫ Fi (ξ , t )vi (ξ , t ) dV (ξ ) . Σ

(4.79)

V

Here, the specific internal energy is denoted by e . Equation (4.79) is the balance energy equation of an elastic body and corresponds adequately to the general structure of balance equations since

1  Ξ = ρ (ξ , t )  vi (ξ , t )vi (ξ , t ) + e  , A = Fi (ξ , t )vi (ξ , t ) , 2  α = σ ik (ξ , t )nk (ξ )vi (ξ , t ) . Thus we need to write more concretely the internal energy for an elastic body.

4.3.4 Nonlinear elastic isotropic materials. Elastic potentials As it was mentioned in chapter 3, in mechanics the basic property of elastic deformations is the reversibility of deformations after removing the causes of these deformations. This property is shown in a full recoverability of the initial body shape and in the full restoring of the energy stored by the body during a deformation process. The elasticity of deforming materials has the strong classical definitions according to which all materials under elastic deformations are divided into hypoelastic, elastic, and hyperelastic materials. Each of the mentioned groups has the exact definition based on notions of the fields of strains, strain rates, and stresses. As a rule, the elasticity of deformation isn't reduced to the linear description of process only. Therefore, all the above three cited fields are introduced for the general case of nonlinear deformations. Here, the distinction between the reference (initial) and actual states of a body seem to be essential. More concretely, the situation is essential to which metrics (in the natural, the initial no disturbed or disturbed states) are referred to all these quantities.

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For different ways of the deformation description, it is convenient to use different strain tensors (and respectively, different strain rate tensors) and different stress tensors. In the following, two strain tensors will be used in slightly distinguishing form. The Cauchy-Green strain tensor is given by the known displacements vector u ( xk , t ) in the reference configuration and Lagrangian coordinate system { xk } as

ε = ε nm g n g m = ε nm g n g m = ε nm g n g m , ε nm =

1 ( u n , m + um , n + u k , n uk , m ) . 2

(4.80)

Here, the metrics is defined by the basic vectors (covariant g n or contra variant g n ) and metric tensor ( g nm or g nm ). The Almansi strain tensor is given through the known displacements ⌢ vector u ( X α , t ) in the actual configuration and the Eulerian coordinate system { X α } as

⌢

⌢ ⌢ ⌢

⌢ ⌢ ⌢

⌢ ⌢ ⌢

ε = ε nm g n g m = ε nm g n g m = ε nm g n g m , ⌢

ε nm =

1 ⌢ ⌢ ⌢ ⌢ un , m + um , n + u k , n uk , m ) . ( 2

(4.81)

It is adopted that the pair, vector u ( xk , t ) and tensor ε nm ( xk , t ) , or the ⌢ ⌢ pair, vector u ( X α , t ) and tensor ε nm ( X α , t ) , describe fully the kinematic picture of deformations in a body. The Piola-Kirchhoff stress tensor t nm ( xk , t ) includes stresses at moment

t on elements of the elementary volume in deformed state, which are measured on the unit area in the reference state. The Lagrange-Cauchy stress tensor σ ik ( X α , t ) includes stresses at time t on elements of the elementary volume in deformed state, which are measured on the unit area in the deformed state.

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For description of hypoelastic materials, we need to introduce one of the stress change velocities, as in the variant proposed by Jaumann. For symmetric Lagrange tensor, it can be written as

σ ik∇ =

Dσ ik − σ in v[ k , n ] − σ kn v[ i ,n ] . Dt

(4.82)

∂u  ∂u  Here v = = {vk } =  k  is the particle velocity, quadratic brackets ∂t  ∂t  in v[ k ,n ] denote the asymmetric (or skewsymmetric) part of vk , n . In the definition (4.82) we have the asymmetric rotations velocity tensor with components Wkn = = v[ k ,n ] and the symmetric strain velocity tensor Vkn = v( k ,n ) which is the characteristic for hypoelastic materials. These tensors are not used practically in the description of elastic and hyperelastic materials. The rotation tensor appears in (4.82) because it used velocities change of stresses and strains relative to the stationary coordinate system. The constitutive equations which define the hypoelastic material are given in the form

σ ik∇ = Ciklm (σ rs )Vlm .

(4.83)

It should be mentioned that the prefix hypo- means the decreasing of something against the norm. Since the prefix hyper- means the strong increasing of something against the norm, then hypoelastic materials should have the property of elasticity as if in the less grade than elastic ones and in the lesser grade than hyperelastic ones. The hypoelastic materials definition admits the presence of initial stresses. Also, the infinitesimal strains of hypoelastic materials are reversible relatively to the initial stresses. This fact parallel with the impossibility of hypoelastic materials to be deformed viscously (an internal dissipation is absent) justifies, according to Prager, their name.

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The elastic material is defined strongly as a material which can be in the natural (free of stresses) state and in a neighbourhood of this state the stresses in present time can be defined one-to-one by values either of deformation gradient or of strain tensor at present time

σ ik = Fik ( ε lm )

(4.84)

σ ij = Aijkl ε kl + Aijklmn ε kl ε mn + Aijklmnpq ε lm ε mn ε pq +⋯ .

(4.85)

or for rectilinear symmetry

Let us pay attention to the fourth rank tensor Aijkl . It defines the linear properties of elastic materials, when tensors of higher ranks are absent in (4.85). The symmetry of stress and strain tensors decreases the number of independent elastic constants from 81 to 36. For the next decreasing number of constants, the additional symmetry of a material is needed. The hyperelastic material is defined as the elastic material for which the specific internal energy e is an analytical function of the strain tensor components referred to the natural state

e = e ( ε lm ) .

(4.86)

The stresses in hyperelastic materials can be calculated by the formula 1 ∂ ∂ + 2  ∂ε ij ∂ε ji

σ ij =  

  e ( ε lk ) . 

(4.87)

The Noll’s theorem shows that each hyperelastic material is a particular case of the elastic material and hence the hypoelastic material. It isn't true for the case of anisotropy. Formula (4.87) testifies that hyperelasticity decreases the anisotropy level in materials, since in addition it increases the symmetry owing to equalities Aijkl = A jikl , Aijkl = Aijlk , Aijkl = Aklij .

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The number of independent constants decreases from 36 to 21. Usually this is fixed in many books on the theory of elasticity, and attention is not paid to the fact that the case of hyperelastic materials (and not the case of the general elasticity) is considered. So, we have briefly described three basic types of elastic materials. Nonlinear mechanics of materials have a rich history both in experimental and theoretical studies. The most difficult in constructing the nonlinear models have been displayed in the transition from the Hooke linear law to more complicated nonlinear dependences. Some simplification can be reached for isotropic materials. First of all because the simplicity of the Hooke law and comparatively simple experiments on which a validity of the linear model can be tested. The next consideration will be concentrated on hyperelastic materials. It is necessary to note that for arbitrary hyperelastic body the potential can be written as the analytical function of three basic invariants of chosen strain tensor (Green, Almansi, Hencky or any other). Let us remember the main nonlinear models. The simplest one is the Seth model. For this model, the law of stressstrain dependence conserves the classical form of Hooke law, in which the small strains tensor should be changed on the finite strains tensor

tik = λεɶkk δ ik + 2 µεɶik .

(4.88)

In formula (4.88), the nonlinear stress tensor is presented on the left-hand side and the nonlinear Almansi strain tensor is presented on the righthand side, elastic moduli should be evaluated from experiments. The model takes into account a necessity to apply additionally normal forces in the simple shear experiment. It also takes into account a finiteness of the force ruptured by the sample. But Seth model does not have the main property of hyperelasticity because the potential can’t be written. This deficiency of the Seth model was corrected in the Signorini model. The Signorini model gives the link between the stress tensor and

272


Almansi strain tensor, too. But just as internal energy constructed for the model is such that the medium is hyperelastic. For the case, when the reference configuration is natural, the Signorini potential is written as

W ( εɶik ) =

G g

1 c 2  cI 2 (εɶ ) +  λ + µ −  ( I1 (εɶ ) ) 2 2   

c   c  +  µ +  ( I − I1 (εɶ ) )  −  µ +  . 2 2   

(4.89)

Here for the Almansi strain tensor invariants, the symbols I k (εɶ ) are used, the quantities λ , µ , c are the physical constants of the model and are chosen in a such way that the constitutive equation should differ as little as possible from the Hooke law. In nonlinear mechanics of materials, the representation by means of invariants are often used. Let us remind here that the three first algebraic invariants Ak of tensor ε ik are evaluated by the formulas utilizing the trace operator “ tr ” 2 3 A1 = tr ( ε ik ) , A2 = tr ( ε ik )  , A3 = tr ( ε ik )      or more commonly used formulas

I1 ( ε ik ) = ε ik g ik , I 2 ( ε ik ) = ε imε nk g ik g nm , I 3 ( ε ik ) = ε pmε inε kq g im g pq g kn . The constitutive equation has the form  1 c 2 tik =  λ I1 (εɶ ) + cI 2 (εɶ ) +  λ + µ −  ( I1 (εɶ ) )  δ ik 2 2    c   2 +2  µ −  λ + µ +  ( I1 (εɶ ) )  εɶik + 2c ( εɶik ) . 2    

(4.90)

Sometimes the third constant c is neglected. Then the quasi-linear model with two moduli λ , µ is obtained

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1  2 tik =  λ I1 (εɶ ) + ( λ + µ )( I1 (εɶ ) )  δ ik + 2 ( µ − ( λ + µ )( I1 (εɶ ) ) ) εɶik . 2  

(4.91)

Since we will use extensively the Signorini potential in the next chapters, it seems appropriate to insert at this point some historical comments associated with the potential named after him. Signorini potential was proposed in the 1940’s for incompressible and compressible materials, the main part of last Signorini publications was devoted to the incompressible case and comparison this potential with other contemporary potentials developed by others. The main area of application of this potential are rubberlike materials. The starting point is the formula proposed by Signorini notation) 2

2W ( I1 , I 2 ) = h2 ( I1 − 3) + h1 ( I 2 − 3) + h3 ( I 2 − 3) ,

(in his

(4.92)

where hk ( k = 1, 2,3) are elastic constants. Signorini noted the simplest potential from the set of potentials for incompressible materials as the one proposed in 1943 by Treloar 2W ( I1 , I 2 ) = h2 ( I1 − 3) .

(4.93)

The next potential among the potentials being compared by Signorini with (4.92) was proposed in 1940 by Mooney 2W ( I1 , I 2 ) = h2 ( I1 − 3) + h1 ( I 2 − 3) .

(4.94)

Signorini observed that in his publications of 1949 Rivlin used the Mooney potential. But in 1951 Rivlin and Saunders proposed the most general potential for incompressible materials 2W ( I1 , I 2 ) = h2 ( I1 − 3) + ψ ( I 2 − 3) ,

(4.95)

where ψ is a function of the second invariant of strain tensor ( I 2 − 3) .

274


There follows that the Signorini potential corresponds to a particular case of the function ψ 2

ψ = h1 ( I 2 − 3) + h3 ( I 2 − 3) . It is in place to note that the simplest variant of representing the potential through invariants (the potential for a linear elastic body with elastic Lamé modulus λ , µ ) is quadratic relative to components of the linear Green strain tensor W (ε ) =

1 2 λ ( I1 (ε ) ) + µ I 2 (ε ). 2

(4.96)

John proposed to take into account in equation (4.96) the invariants of the nonlinear Green tensor. Then the potential becomes suitable for description of nonlinear strains. This potential has been used by John for various problems of mechanics. When the John potential is used for plane problems of elasticity, there appear some problems of the harmonic functions theory. Therefore, this potential is also called the harmonic John potential. Next, after the quadratic potential (4.96) comes the cubic potential. One variant of this potential was firstly proposed by Murnaghan for the Green strain tensor ε ik 1 W (ε ik ) = λ (ε mm )2 + µ (ε ik ) 2 2 1 1 + Aε ik ε imε km + B (ε ik ) 2 ε mm + C (ε mm )3 3 3

(4.97)

or through the first algebraic invariants I k of the tensor ε ik W ( I1 , I 2 , I3 ) =

1 2 1 1 λ I1 + µ I 2 + AI 3 + BI1 I 2 + CI13 . 2 3 3

(4.98)

Here λ , µ are the Lamé elastic constants (constants of the second order), A, B, C are the Murnaghan elastic constants (constants of the third order).

Waves in Materials

275

From (4.98) we have three representations of the potential ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ ⌢ l + 2m ⌢ 3 1 W ( I1 , I 2 , I 3 ) = ( λ + 2 µ ) I12 − 2 µ I 2 + nI 3 − 2mI1 I 2 + I1 , 2 3 1 4 1 W ( I1 , I 2 , I 3 ) = λ I12 + µ I 2 + ν 3 I 3 + ν 2 I1 I 2 + ν 1 I13 , 2 3 6 1 c a W ( I1 , I 2 , I 3 ) = λ I12 + µ I 2 + I 3 + bI1 I 2 + I13 . 2 3 3 ⌢ The main invariants I k are linked with the algebraic ones I k by the formula

⌢ ⌢ 1 ⌢ 1 I1 = I1 , I 2 = ( I12 − I 2 ) , I 3 = ( I13 − 3I1 I 2 + 2 I 3 ) . 2 6

(4.99)

The Murnaghan potential which describes a large class of industrial materials, is widely used and is thoroughly commented on in the fundamental books on nonlinear solid mechanics. A modification of the Murnaghan potential has been proposed by Guz. This model, of deformation process, can be considered as a model from the classical elasticity. Guz’s potential has the form

W ( I1 , I 2 , I 3 ) =

1 c a K iklmε ik ε lm + I 3 + bI1 I 2 + I13 . 2 3 3

(4.100)

The Green strain tensor Kiklm is the fourth rank tensor of the second order elastic constants. The potential consists of two different non-linear parts - quadratic and cubic. According to the author, the quadratic one characterizes the anisotropy properties of material in the unloaded state of a body and corresponds to the potential of a linearly elastic anisotropic body. The cubic one corresponds to an isotropic body. This potential has been used for studying the regularities of wave propagation in polycrystallic bodies having weak anisotropy of properties in the natural state (so called quasi-isotropic bodies).

276


4.4 Nonlinear Wave Equations 4.4.1 Nonlinear wave equations for plane waves. Methods of solving In this subsection we consider the solutions of wave problems, based on classical Murnaghan potential. The pioneer publications on plane nonlinear elastic waves were made as studies in nonlinear acoustics. These results have been based on some simplifications of the Murnaghan potential 2 2 1 1 W = λ ( u m , m ) + µ ( ui , k + u k , i ) 2 4 2 1  1  +  µ + A  ui , k u m , i u m , k + ( λ + B ) u m , m ( ui , k ) 4  2 

+

3 1 1 1 Aui ,k uk ,m um,i + Bui ,k uk ,i um,m + C ( um ,m ) . 12 2 3

(4.101)

This potential is still nonlinear and it keeps the basic system nonlinearity order. Now the nonlinearity is defined relative to the deformation gradient and not relative to the strain tensor as it is assumed in (4.98). To obtain the nonlinear wave equations from (4.101), two steps must be taken.

Step 1. First, write the formula for the Kirchhoff stresses representation in terms of strains (in order to obtain the so-called constitutive equations) according to the relationship tik = ( ∂W ∂ui ,k ) tik = µ ( ui ,k + uk ,i ) + λ uk ,k δ ik

+  µ + (1 4 ) A ( ul ,i ul ,k + ui ,l uk ,l + 2ul ,k ui ,l ) 2 + (1 2 ) ( B − λ ) ( um ,l ) δ ik + 2ui ,k ul ,l  + (1 4 ) Auk ,l ul ,i   2

+ B ( ul ,mum,l δ ik + 2uk ,i ul ,l ) + C ( ul ,l ) δ ik .

(4.102)

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Step 2. Second, substitute (4.102) into the motion equations tik ,i + X k = ρuɺɺk to obtain some nonlinear equations analogously to the classical Lamé equations

ρ uɺɺm − µ um,kk − (λ + µ ) un ,mn = Fm .

(4.103)

The right-hand side collects all the nonlinear terms Fi =  µ + (1 4 ) A ( ul ,kk ul ,i + ul ,kk ui ,l + 2ui ,lk ul ,k ) +  λ + µ + (1 4 ) A + B  ( ul ,ik ul ,k + uk ,lk ui ,l )

(4.104)

+ ( λ + B ) ui ,kk ul ,l + ( B + 2C ) uk ,ik ul ,l + (1 4 ) A + B  ( uk ,lk ul ,i + ul ,ik uk ,l ) . Let us show first on the plane polarized waves, and corresponding wave equations. We start from (4.101) , with (4.19). The waves propagate only along Ox1 , so that u = {uk ( x1 , t )} and 2 2 2 2 W (2,3) = (1 2 ) λ ( u1,1 ) + µ ( u1,1 ) + (1 2 ) ( u2,1 ) + (1 2 ) ( u3,1 )    2 2 2 +  µ + (1 4 ) A u1,1 ( u1,1 ) + ( u2,1 ) + ( u3,1 )    2 2 2 + (1 2 )( λ + B ) u1,1 ( u1,1 ) + ( u2,1 ) + ( u3,1 )    3

3

+ (1 12 ) A ( u1,1 ) + (1 2 ) B ( u1,1 ) + (1 3) C ( u1,1 )

3

2 2 2 = (1 2 ) ( λ + 2 µ ) ( u1,1 ) + µ ( u2,1 ) + ( u3,1 )      

+  µ + (1 2 ) λ + (1 3) A + B + (1 3) C  ( u1,1 ) 2 2 + (1 2 )( λ + B ) u1,1 ( u2,1 ) + ( u3,1 )  .   The constitutive equations are

3

(4.105)

278


t11(2,3) = ( λ + 2 µ ) u1,1 + ( 3 2 ) λ + 2 µ + 2 ( A + 3B + C )  ( u1,1 )

2

2 2 + (1 2 )  λ + 2 µ + (1 2 ) A + B  ( u2,1 ) + ( u3,1 )  ,  

t12(2,3) = µ u2,1 + (1 2 )  λ + 2 µ + (1 2 ) A + B  u1,1u2,1 ,

(4.106)

t13(2,3) = µ u3,1 + (1 2 )  λ + 2 µ + (1 2 ) A + B  u1,1u3,1 . Substitution (4.106) into the motion equations gives the quadratically nonlinear wave equations for three polarized plane elastic P, SH , SV waves

ρ u1,tt − ( λ + 2 µ ) u1,11 = N1 u1,11u1,1 + N 2 ( u2,11u2,1 + u3,11u3,1 ) ,

(4.107)

ρ u2,tt − µ u2,11 = N 2 ( u2,11u1,1 + u1,11u2,1 ) ,

(4.108)

ρ u3,tt − µ u3,11 = N 2 ( u3,11u1,1 + u1,11u3,1 ) ,

(4.109)

N1 = 3 ( λ + 2 µ ) + 2 ( A + 3B + C )  , N 2 = λ + 2µ +

1 A + B. 2

Let us now compare the linear wave equations (4.20) with the nonlinear ones (4.107)-(4.108)-(4.109). The last include, separately, two parts: the linear one and the quadratically nonlinear one (on the righthand side). This structure turned out to be very convenient in future studies. The second novelty consists in that in contrast to the linear wave equations the nonlinear ones are coupling equations and this coupling isn’t symmetric, as we will comment on later. Let us return now to potentials (4.98) and (4.101). Owing to the nonlinearity of Green strain tensor, the representation of potential (4.98) through deformations gradient will involve terms, not only the second and third orders of nonlinearity, but also the fourth, fifth, and sixth orders. As a result, potential (4.98) can describe not only quadratic nonlinearity which is realized using potential (4.101), but also the cubic nonlinearity and nonlinearities of the fourth and fifth orders. The full representation of Murnaghan potential (4.98) is

Waves in Materials

279

2 2 1 1 1   W = λ ( um , m ) + µ ( ui , k + uk ,i ) +  µ + A  ui , k um , k um ,i 2 4 4   2 3 1 1 1 1 + ( λ + B ) um,m ( ui ,k ) + Aui ,k uk ,mum,i + Bui ,k uk ,i um,m + C ( um ,m ) 2 12 2 3 4 2 1 1 1 + λ ( un,m ) + µ ( un,i un , k ) + A ( ui , k + uk ,i )( ui , m + um ,i ) us , k us , m 4 4 8

+ ( ui ,k + uk ,i )( uk , m + um, k ) us ,i us , m + ( ui , m + um ,i )( uk , m + um , k ) us ,i us , k  1  1  + B ( ui , k + uk ,i ) un ,i un , k um , m + ( ui , k uk ,i + ui , k ui , k ) un , m un , m  2  2  2 2 3 + C ( um , m ) ( un , m ) 2 1 + A ( ui ,k + uk ,i )( us ,i us ,m )( ul , k ul , m ) + ( ui , m + um,i )( un ,i un, k )( ul , k ul , m ) 24

+ ( uk , m + um, k )( un ,i un , k )( us ,i us , k )  2 2 1 + B ( un,i un ,k ) um , m + ( ui , k + uk ,i ) un ,i un , k ( us , m )    4 4 1 1 + Cum , m ( un , m ) + A ( un ,i un,k )( us ,i us ,m )( ul ,k ul ,m ) 12 24 2 2 6 1 1 + B ( un , i un , k ) ( u s , m ) + C ( un , m ) . 8 24

If we restrict only to the cubic nonlinearity, the Murnaghan potential becomes 2 2 1 1 1   W = λ ( um , m ) + µ ( ui , k + uk ,i ) +  µ + A  ui , k um , k um ,i 2 4 4   2 3 1 1 1 1 + ( λ + B ) um,m ( ui ,k ) + Aui ,k uk ,mum,i + Bui ,k uk ,i um,m + C ( um ,m ) 2 12 2 3 4 2 1 1 1 + λ ( un,m ) + µ ( un,i un , k ) + A ( ui , k + uk ,i )( ui , m + um ,i ) us , k us , m 4 4 8

+ ( ui , k + uk ,i )( uk , m + um , k ) us ,i us , m + ( ui , m + um,i )( uk , m + um , k ) us ,i us , k 

280


1  1  + B ( ui , k + uk ,i ) un ,i un , k um , m + ( ui , k uk ,i + ui , k ui , k ) un , m un , m  2  2  2 2 3 + C ( um , m ) ( un , m ) . (4.110) 2 If we simplify, according to (4.19) the potential (4.110) becomes 2 2 2 W ( 2,3,4) = W (2,3) + (1 8 )( λ + 2 µ + A + 2 B ) ( u1,1 ) + ( u2,1 ) + ( u3,1 )   

2

2 2 2 2 + (1 8 )( 3 A + 10 B + 4C ) ( u1,1 ) ( u1,1 ) + ( u2,1 ) + ( u3,1 )  . (4.111)  

Constitutive equations have the form

t11(2,3,4) = t11(2,3) + (1 2 )  λ + 2 µ + 4 ( A + 3B + C )  ( u1,1 )

3

2 2 + (1 4 )  2 ( λ + 2 µ ) + 5 A + 14 B + 4C  ( u1,1 ) ( u2,1 ) + ( u3,1 )  ,  

t12(2,3,4) = t12(2,3) + (1 4 ) [ λ + 2µ + A + 2 B ] ( u2,1 )

3

+ (1 4 )  2 ( λ + 2 µ ) + 5 A + 14 B + 4C  u2,1 ( u1,1 )

2

2

(4.112)

+ (1 2 )( 3 A + 10 B + 4C ) u2,1 ( u3,1 ) , 3

t13(2,3,4) = t13(2,3) + (1 2 ) [ λ + 2 µ + A + 2 B ] ( u3,1 )

+ (1 4 )  2 ( λ + 2 µ ) + 5 A + 14 B + 4C  u3,1 ( u1,1 )

2

2

+ (1 4 )( 3 A + 10 B + 4C ) u3,1 ( u2,1 ) . The plane polarized wave equations, when we take into account both quadratic and cubic nonlinearities can be written as follows

ρ u1,tt − ( λ + 2 µ ) u1,11 = N1 u1,11u1,1 + N 2 ( u2,11u2,1 + u3,11u3,1 ) 2

+ N 3 u1,11 ( u1,1 ) + N 4 ( u2,11u2,1u1,1 + u3,11u3,1u1,1 ) ,

ρ u2,tt − µ u2,11 = N 2 ( u2,11u1,1 + u1,11u2,1 )

(4.113)

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Waves in Materials 2

2

2

2

2

+ N 4 u2,11 ( u2,1 ) + N 5 u2,11 ( u1,1 ) + N 6 u2,11 ( u3,1 ) , (4.114)

ρ u3,tt − µ u3,11 = N 2 ( u3,11u1,1 + u1,11u3,1 ) 2

+ N 4 u3,11 ( u3,1 ) + N 5 u3,11 ( u1,1 ) + N 6 u3,11 ( u2,1 ) , (4.115) N 3 = ( 3 2 )( λ + 2µ ) + 6 ( A + 3B + C ) , N 4 = (1 2 )  2 ( λ + 2 µ ) + 5 A + 14 B + 4C  , N 5 = ( 3 2 )( λ + 2 µ + A + 2 B ) ,

N 6 = 3 A + 10 B + 4C .

These equations keep two basic features of quadratically nonlinear wave equations (4.107)-(4.109): the linear and nonlinear parts are separated and the equations are coupled. But they include some new peculiarity in coupling what will be commented on later. Let us consider now two basic methods of analysis of nonlinear elastic wave equations. First of all, we must note that all problems of nonlinear wave analysis to be discussed here are considered with the essential restriction that the nonlinearity of materials is small. This condition for materials as the medium of wave propagation means that wave amplitudes are restricted by some finite upper values. For different waves and for different materials, this value will be different. When weak nonlinearity is assumed, all elastic wave equations from section 3 can be analyzed using two basic methods – method of successive approximations and method of slowly varying amplitudes. 4.4.1.1 Method of successive approximations The main feature of this method consists in the introduction of a small parameter ε . Since the displacements vector u ( x , t) (or something similar to it) is assumed sufficiently smooth, then it can be expanded to the form u ( x, t ) = u (0) ( x, t ) + ε u (1) ( x, t ) + ε 2u (2 ( x, t ) + ⋯ . (4.116)

282


The nonlinear vector F , which collects on the right-hand side all nonlinearities of nonlinear wave equations (for example, the simplest ones - (4.107)-(4.109) or (4.113)-(4.115)) , should also be expanded

F = ε F (1) + ε 2 F (2) + ⋯ .

(4.117)

Representations (4.116), (4.117) should be substituted into one of non-linear wave equations, for example (4.107). Thus we obtain the quadratically nonlinear wave equation for P -waves 2

2

2

uɺɺ1 − ( cP ) u1,11 = α1 ( cP ) u1,11u1,1 + α 2 ( cP ) ( u2,11u2,1 + u3,11u3,1 ) ,

( cP )

2

= ( λ + 2µ ) ρ ,

(4.118)

α k =  N k ( λ + 2 µ )  .

Then for the first three approximations it is 2

(0) uɺɺ1(0) − ( cP ) u1,11 =0, 2

2

2

2

2

2

(4.119)

(1) (0) (0) (0) (0) (0) (0) uɺɺ1(1) − ( cP ) u1,11 = α1 ( cP ) u1,11 u1,1 + α 2 ( cP ) ( u2,11 u2,1 + u3,11 u3,1 ) , (4.120) (2) (1) (1) (1) (1) (1) (1) uɺɺ1(2) − ( cP ) u1,11 = α1 ( cP ) u1,11 u1,1 + α 2 ( cP ) ( u2,11 u2,1 + u3,11 u3,1 ) ,

(4.121)

so that the solution in zero approximation is linear. In the harmonic wave class of solutions, it has the form

u1(0) ( x, t ) = u1(0) cos ( kx − ωt )

( (ω k ) = c ) . P

(4.122)

Let us narrow the problem and consider the first standard problem of nonlinear acoustic waves, when only the longitudinal wave is given at the entrance to the medium. Then in the first approximation, we have the equation 2

(1) ρ uɺɺ1(1) − ( λ + 2µ ) u1,11 = (1 2 ) N1 ( u1(0) ) k 3 sin 2 ( kx − ωt ) .

The solution can be written as

(4.123)

283


u1(1) ( x, t ) =  N1 8 ( λ + 2 µ )  ( u1(0) ) k 2 x cos 2 ( kx − ωt ) .

(4.124)

Thus, the main advantage of this method is that in each approximation the non-homogeneous linear wave equation should be solved. But this method is only good when the initial amplitudes don’t increase essentially (in some cases, half as much again). However, it doesn't permit to study many nonlinear effects - self-influence, big energy pumping, self-switching and so on. In this case, the method of slowly varying amplitudes is used instead. 4.4.1.2 Method of slowly varying amplitudes This method of approximation was proposed by Baltazar van der Pol for problems of nonlinear oscillations in radio physics. Sometimes it is shortly called the van der Pol method. It was first used in mechanics then by developing the nonlinear optics and radio physics, the method was applied to wave problems as well. Now it is returned to mechanics and found to be a good tool for waves in materials. The basic hypothesis is that the solution of the weak nonlinear system is close to the linear solution. “Slowly varying amplitude” means that the wave amplitude will vary slightly (more exactly, will not vary) over a distance equal to one wavelength. Let us consider as an example the quadratically nonlinear wave equation (4.107). The base for the analysis is formed by the linear solution. Therefore, the nonlinear solution is searched in the form u1 ( x , t) = A1 ( x )e

or

i k x −ωt )

u1 ( x, t ) = Re A1 ( x)e ( 1

{

i(k1 x −ω t )

(4.125)

} = a ( x) cos [ k x − ωt + ϕ ( x)] . 1

1

1

(4.126)

The second step in the method consists of considering the interaction of some finite number (mainly, two, three, four) of waves, so that the solution of (4.125) is assumed as

284


M

u1 ( x, t ) = ∑ A1m ( x)eiσ m ,

σ m = k1m x − ωm t.

(4.127)

m =1

The next few steps are: 1. Substitution of (4.127) into (4.107). 2. Taking into account that (4.127) is the solution of a linear equation. 3. Neglecting the second derivatives because of the energy external flux is absent. 4. Saving only products of amplitudes for taking into account an interaction and self-interaction of waves-participants. As a result, we obtain the shorten equation M

∑ k1m ( A1m ),1 e m =1

iϕm

=−

M M i (ϕ +ϕ ) N1 k1n k12p A1n A1 p e n p . (4.128) ∑∑ 2(λ + 2 µ ) n =1 p =1

Another way to the shorten equation can also be used. The amplitude in (4.127) can be assumed as the function of the slow coordinate as A( xɶ ) = A(ε x) , where ε is the small parameter. Then the four steps above must be changed: first two steps will rest, next two must be changed on the step-saving on both sides of equation (4.107) only terms with the first order ε and neglecting the terms of higher orders. Further, we must consider some additional assumptions relative to the relationships between wave numbers k1m and frequencies ω m . The first assumption is the condition of frequency synchronism (matching) (4.129)

ω 1 ± ω 2 = ω3 .

In this case, the shorten equation splits up into three evolution equations

i( k13 − k12 − k11 ) x

,

i( k13 − k12 − k11 ) x

,

i( k13 − k12 − k11 ) x

,

( A11 ),1 = σ 1 A12 A13e

( A12 ),1 = σ 2 A11 A13e ( A13 ),1 = σ 3 A11 A12 e

(4.130)

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σα = −

N1 k1δ k13 (k1δ + k13 ) Nk k ; σ α = − 1 11 12 ; 2(λ + 2 µ )k1α 2(λ + 2 µ )

(α + δ = 3).

Usually evolution equations (4.130) are analyzed under conditions, when the second matching condition, wave numbers matching, holds (4.131)

k11 ± k12 = k13 .

Studying the evolution equations is a separate problem accompanying the slowly varying amplitude method use. This method has been used for analyzing the problems of nonlinear wave effects - self-influence, big energy pumping, self-switching and so on.

4.4.2 Nonlinear wave equations for cylindrical waves Let us first consider the procedure of derivation of nonlinear wave equations for cylindrical waves. Introduce the cylindrical orthogonal system of θ 1 = r , θ 2 = ϑ , θ 3 = z . In this system, the vector length is given by

( ds )

2

2

2

2

= g ik dθ i dθ k = ( dr ) + r 2 ( dϑ ) + ( dz ) ,

(4.132)

the metric tensors have the components 1

0

gik = 0 r 0

2

0

0

1

0,

g

ik

0

= 0 1r

1

0

0

0 2

0 ,

(4.133)

1

the basis vectors ( e1 , e2 , e3 ) , ( e 1 , e 2 , e 3 ) have the lengths e1 = 1, e2 = r , e3 = 1 ,

e 1 = 1, e 2 = (1 r ) , e 3 = 1 ,

δ nk = e k ⋅ en .

(4.134) (4.135)

Only three Christoffel symbols of the first kind Γ mki differ from zero

286

Wavelet and Wave Analysis 2 2 Γ122 = − r , Γ12 = Γ 21 = (1 r ) .

(4.136)

Let us consider four configurations (states) of elastic medium:

−

−

−

−

State I. The configuration depends on coordinates r ,ϑ and doesn’t depend on coordinate z . This state of plane strain is characteristic, for example, for some types of waves or for the rotational Volterra distortions in hollow cylinder. State II. The axisymmetry depends on coordinates r , z and doesn’t depend on coordinate ϑ . The configuration admits Oz as the symmetry axis. This state is characteristic, for example, for the longitudinal torsional wave propagating along cylinder. State III. It depends on angular coordinate ϑ , with the symmetry axis Oz . This state is characteristic, for example, for the transverse torsional wave propagating along cylinder. State IV. The axisymmetric depends on the radial coordinate with the symmetry axis Oz . This state is characteristic, for example, for the classical cylindrical wave or for the translational Volterra distortions in the hollow cylinder.

Let us now derive the nonlinear equations of motion and show the principal scheme of evaluation of nonlinear strain and stress tensors.

State I. Because the particular case of configuration, then displacement vector u (θ 1 ,θ 2 ,θ 3 ) is

u = uk e k = u k ek → u1e 1e1 = u1e1e1 , u2 e 2 e2 = u 2 e2 e2 → u1 = u1 , u2 = u 2 r 2 , u (θ 1 ,θ 2 ,θ 3 ) = u ( r ,ϑ , z ) (4.137) = {u1 = ur ( r ,ϑ ) , u2 = r ⋅ uϑ ( r ,ϑ ) , u3 = u z = 0}. The components of nonlinear Cauchy-Green strain tensor are evaluated using the covariant derivatives of co- and contravariant components of displacement vector

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1 ( ∇ i u j + ∇ j ui + ∇ i u k ∇ j u k ) , 2 ∂u j ∂u k ∇i u k = i + u j Γ kji , ∇ i u j = i − uk Γ kji , ∂θ ∂θ

ε ik =

2 1 ∇1u1 = u1,1 − u1 Γ111 − u2 Γ11 = ur ,r , ∇1u1 = u,11 + u1 Γ11 + u 2 Γ121 = ur ,r ,

1 = ruϑ ,r , r u 1 u u  2 ∇1u 2 = u,12 + u1 Γ11 + u2 Γ 221 =  ϑ  + ϑ ⋅ = ϑ ,r , r  r , r r r

∇1u2 = u2,1 − u1 Γ121 − u2 Γ 221 = ( ruϑ ),r − ruϑ

∇ 2u2 = u2,2 − u1Γ122 − u2 Γ 222 = ( ruϑ ),ϑ − ur ( − r ) = r ( uϑ ,ϑ + ur ) ,

u 1 u  2 ∇ 2u 2 = u,22 + u1Γ12 + u 2 Γ 222 =  ϑ  + r = ( uϑ ,ϑ + ur ) ,  r ,ϑ r r

1 1 2 ∇ 2u1 = u1,2 − u1 Γ12 − u2 Γ12 = ur ,ϑ − ruϑ   = ur ,ϑ − uϑ , r u  1 ∇ 2u1 = u,21 + u1 Γ12 + u 2 Γ122 = ur ,ϑ +  ϑ  ( − r ) = ur ,ϑ − uϑ ,  r  1 1 ε11 = ε rr = ∇1u1 + ∇1u1∇1u1 + ∇1u2∇1u 2 = 2 2 2 2 1 1 = ur ,r + ( ur ,r ) + ( uϑ ,r ) , 2 2 1 1 ε 22 = r 2ε ϑϑ = ∇ 2 u2 + ( ∇ 2u2∇ 2u 2 ) + ( ∇ 2u1∇ 2 u1 ) 2 2 2 2 1 1 = r ( uϑ ,ϑ + ur ) + ( ur ,ϑ − uϑ ) + ( uϑ ,ϑ + ur ) , 2 2 1 ε12 = rε rϑ = ( ∇1u2 + ∇ 2 u1 + ∇1u1∇ 2u1 + ∇1u2∇ 2u 2 ) (4.138) 2 1 =  ruϑ ,r − uϑ + ur ,ϑ + ur ,r ( ur ,ϑ − uϑ ) + uϑ ,r ( uϑ ,ϑ + ur )  , 2 1 1 1 ε 33 = ε zz = ∇3u3 + ( ∇3u3∇3u 3 ) + ( ∇3u2∇3u 2 ) + ( ∇3u1∇3u1 ) = 0 , 2 2 2

(

)

(

)

288


ε13 = ε rz = ε 23 = rε ϑ z

1 ( ∇1u3 + ∇3u1 + ∇1u1∇3u1 + ∇1u2∇3u 2 + ∇1u3∇3u 3 ) = 0 , 2 1 = ( ∇ 2u3 + ∇ 3u2 + ∇ 2u1∇ 3u1 + ∇ 2u2∇ 3u 2 + ∇ 2u3∇ 3u 3 ) = 0 . 2

State II. The displacement vector and components of strain tensor have the form u (θ 1 ,θ 2 ,θ 3 ) = u ( r ,ϑ , z ) = {u1 = ur ( r , z ) , u2 = r ⋅ uϑ = 0, u3 = u z ( r , z )} .

(4.139)

2 1 ∇1u1 = u1,1 − u1 Γ111 − u2 Γ11 = ur ,r , ∇1u1 = u,11 + u1 Γ11 + u 2 Γ121 = ur ,r ,

1 = ruϑ ,r , r u 1 u u  2 2 ∇1u 2 = u,12 + u1 Γ11 + u2 Γ 21 =  ϑ  + ϑ ⋅ = ϑ ,r , r  r , r r r

∇1u2 = u2,1 − u1 Γ121 − u2 Γ 221 = ( ruϑ ),r − ruϑ

∇ 2u2 = u2,2 − u1Γ122 − u2 Γ 222 = ( ruϑ ),ϑ − ur ( − r ) = r ( uϑ ,ϑ + ur ) ,

u 1 u  2 2 ∇ 2u 2 = u,22 + u1Γ12 + u 2 Γ 22 =  ϑ  + r = ( uϑ ,ϑ + ur ) ,  r ,ϑ r r

1 1 2 ∇ 2u1 = u1,2 − u1 Γ12 − u2 Γ12 = ur ,ϑ − ruϑ   = ur ,ϑ − uϑ , r u  1 ∇ 2u1 = u,21 + u1 Γ12 + u 2 Γ122 = ur ,ϑ +  ϑ  ( − r ) = ur ,ϑ − uϑ ,  r 

ε11 = ε rr = ∇1u1 +

2 2 1 1 1 1 ∇1u1∇1u1 ) + ( ∇1u3∇1u 3 ) = ur ,r + ( ur , r ) + ( u z ,r ) ( 2 2 2 2

ε 22 = r 2ε ϑϑ = ∇ 2u2 +

∂u2 1 1 ∇ 2u2∇ 2u 2 ) + ( ∇ 2 u1∇ 2u1 ) = − um Γ m22 ( 2 2 ∂θ 2

  ∂u1  ∂u1   ∂u 2 1  ∂u m 2 m  +  22 − um Γ m22   + u Γ + − u Γ + u m Γ1m 2  m2  m 12   2 2 2    2  ∂θ   ∂θ   ∂θ   ∂θ

   

289

Waves in Materials

1 (4.140) ( −umΓ m22 )( u mΓm2 2 ) = ur r + 12 ( ur )2 , 2 1 1 1 ε 33 = ε zz = ∇3u3 + ( ∇3u3∇3u 3 ) + ∇3u2∇3u 2 + ( ∇3u1∇3u1 ) 2 2 2 3 ∂u 1  ∂u m   ∂u m 3  m = 33 − um Γ33 +  33 − um Γ33   3 + u Γm3  ∂θ 2  ∂θ   ∂θ  = −um Γ m22 +

(

)

2

1 2 2 1  ∂u 1 1 m   ∂u m 3  +  13 − um Γ33   3 + u Γ m 3  = u z , z + ( u z , z ) + ( ur , z ) , 2  ∂θ 2 2   ∂θ 

ε12 = rε rϑ =

1 ( ∇1u2 + ∇ 2u1 + ∇1u1∇2u1 + ∇1u2∇ 2u 2 ) = 0 , 2

1 ( ∇2u3 + ∇3u2 + ∇2u2∇3u 2 + ∇ 2u3∇3u3 + ∇ 2u1∇3u1 ) = 0 , 2 1 ε13 = ε rz = ∇1u3 + ∇3u1 + ∇1u1∇3u1 + ∇1u2∇3u 2 + ∇1u3∇3u 3 2

ε 23 = rεϑ z =

(

=

)

1  1  ∂u3 ∂u1  ∂u1 m m   ∂u m + − u Γ − u Γ + − u Γ + u m Γ1m3   m 31 m 13 m 11   3 1 3 1  ∂θ 2  ∂θ  ∂θ   ∂θ 

3  1  ∂u m   ∂u m 3  u z , r + ur , z + ur , r u r , z + u z , r u z , z . +  33 − um Γ33  1 + u Γ m1   =   ∂θ   ∂θ  2

(

)

State III. The displacement vector and components of strain tensor have the form

u (θ 1 ,θ 2 ,θ 3 ) = u ( r ,ϑ , z ) = {u1 = ur = 0, u 2 = r ⋅ uϑ ( r , z ) , u 3 = u z = 0} .

(4.141)

ε11 = ε rr = ∇1u1 +

2 1 1 1 1 ∇1u1∇1u1 ) + ( ∇1u2∇1u 2 ) + ( ∇1u3∇1u 3 ) = ( uϑ ,r ) , ( 2 2 2 2 2 ε 22 = r ε ϑϑ

290


= ∇ 2 u2 +

1 1 1 1 2 ∇ 2 u2 ∇ 2 u 2 ) + ( ∇ 2 u1∇ 2 u1 ) + ( ∇ 2 u3∇ 2 u 3 ) = ( uϑ ) , ( 2 2 2 2

ε 33 = ε zz = ∇ 3u3 +

2 1 1 1 1 ∇ 3u3∇ 3u 3 ) + ( ∇ 3u2∇ 3u 2 ) + ( ∇ 3u1∇ 3u1 ) = ( uϑ , z ) , (4.142) ( 2 2 2 2 ε12 = rε rϑ

1 ∇1u2 + ∇ 2u1 + ∇1u1∇ 2u1 + ∇1u2∇ 2 u 2 + ∇1u3∇ 2u 3 ) = ruϑ ,r − uϑ , ( 2 ε 23 = rεϑ z

=

=

=

1 ∇ 2u3 + ∇ 3u2 + ∇ 2u3∇ 3u 3 + ∇ 2 u2∇ 3u 2 + ∇ 2u1∇ 3u1 ) = ruϑ , z , ( 2 ε13 = ε rz

1 1 ∇1u3 + ∇ 3u1 + ∇1u1∇ 3u1 + ∇1u2 ∇ 3u 2 + ∇1u3∇ 3u 3 ) = uϑ ,r uϑ , z . ( 2 2

State IV. The displacement vector and components of strain tensor have the form u (θ 1 ,θ 2 ,θ 3 ) = u ( r ,ϑ , z ) = {u1 = ur ( r ) , u 2 = r ⋅ uϑ = 0, u 3 = u z = 0} .

ε11 = ε rr =

∂u1 1  ∂u m m − um Γ11 +  11 − um Γ11 1 ∂θ 2  ∂θ = ur , r +

ε 22 = r 2εϑϑ =

(4.143)

1    ∂u + u m Γ1m1    1   ∂θ 

2 1 ur , r ) , ( 2

 ∂u 2  ∂u2 1  ∂u2 m m  − u Γ + u + u m Γ 2m 2  − Γ  m 22 m 22   2 2 2   ∂θ 2  ∂θ   ∂θ 

m = −um Γ 22 +

1 1 2 m −um Γ 22 u m Γ 2m 2 ) = ur r + ( ur ) , ( )( 2 2 2

ε 33 = ε zz =

 ∂u 3 ∂u 3 m 3 m 3  + u Γ +  3 + u Γm3  = 0 , m3 3 ∂θ  ∂θ 

(4.144)

291

Waves in Materials

1  ∂u 2

∂u1



1  ∂u 3

∂u1



1  ∂u 2

∂u 3



ε12 = ε rϑ =  1 + 2 − u m Γ 2m1 − u m Γ1m 2  = 0, 2  ∂x ∂x  ε13 = ε rz =  1 + 3 − u m Γ3m1 − u m Γ1m 3  = 0, 2  ∂x ∂x  ε 23 = εϑ z =  3 + 2 − u m Γ 2m3 − u m Γ3m 2  = 0. 2  ∂x ∂x  Let us now return to the general relationships and write the nonlinear equations of motion ∇ k σ ki (δ in + ∇ i u n )  = ρ uɺɺi .

(4.145)

It should be noted that the system of equations more convenient for analysis can be obtained from (4.145) by means of formulas ∇ n t nm =

∂t nm km n ∂u m nk m m + t Γ + t Γ , ∇ u = + u n Γ nim , ∇ k g nm = 0 . kn kn i ∂θ n ∂θ i

Equations (4.145) can be analyzed as equations relative to the components of displacement vector only. The components of the stress tensor

1 ϑϑ σ , σ 33 = σ zz , r2 1 1 = σ rϑ , σ 13 = σ rz , σ 23 = σ ϑ z r r

σ 11 = σ rr , σ 22 = σ 12

should be evaluated using the formula σ ik = ( ∂W ∂ε ik ) in the form of nonlinear functions of components of deformation. We assume that the quadratic nonlinearity of these functions is kept while the higher order nonlinearities are neglected, as we did with Cartesian coordinates.

292


Let us assume that the deformation of hyperelastic medium is described by the Murnaghan elastic potential (4.98) 1 1 1 W ( I1 , I 2 , I 3 ) = λ I12 + µ I 2 + A I 3 + B I1 I 2 + C I13 . 2 3 3 In this case the three first invariants have the form I1 ( ε ik ) = ε ik g ik = ε 11 ⋅1 + ε 22 ⋅

1 + ε 33 ⋅1 , r2

(4.146)

I 2 ( ε ik ) = ε imε nk g ik g nm 2

(4.147) 2

2

1 1  1 2 2 2   = ( ε11 ⋅ 1) +  ε 22 ⋅ 2  + ( ε 33 ⋅ 1) +  ε12 ⋅  +  ε 23 ⋅  + ( ε 13 ⋅ 1) , r r r       I 3 ( ε ik ) = ε pmε inε kq g im g pq g kn

(4.148)

3

1 1 3 3    = ( ε11 ) +  ε 22 2  + ( ε 33 ) + ( ε 13 ⋅ 1)  ε13ε11 + ε 23ε12 2 + ε13ε 33  r  r    1  1   +  ε 12 ⋅ 2  ε 12ε11 + ε12ε 22 2 + ε13ε 23  r  r   1  1   +  ε 23 ⋅ 2  ε 12ε 13 + ε 23ε 22 2 + ε 23ε 33  . r r    The necessary components of stress tensor are

σ 11 = λ I1

 ∂I ∂I1 ∂I ∂I  ∂I 1 ∂I + µ 2 + A 3 + B  I1 2 + I 2 1  + CI12 1 ∂ε 11 ∂ε11 3 ∂ε 11 ∂ε11  ∂ε11  ∂ε 11

1 1 2 2 2  = λ I1 + 2µε11 + A ( ε11 ) + 2 ( ε12 ) + ( ε 13 )  + B ( 2ε 11 I1 + I 2 ) + CI12 , 3r 3  

σ 22 =

1 r2

 ∂I1 ∂I 1 ∂I +µ 2 + A 3  λ I1 ∂ε 22 ∂ε 22 3 ∂ε 22 

 ∂I ∂I  1 ∂I  + B  I1 2 + I 2 1  + CI12 1  ∂ε 22  3 ∂ε kk   ∂ε 22

293

Waves in Materials

 1 1 1 2 2  λ I1 + 2µε 22 + A r 2  r 2 ( ε 22 ) + 3 ( ε12 )  

=

1 r2

+

1 2  2 ( ε 23 )  + B  2 ε 22 I1 + I 2  + CI12  , 3r 2 r    

 ∂I ∂I1 ∂I ∂I  ∂I 1 ∂I + µ 2 + A 3 + B  I1 2 + I 2 1  + CI12 1 ∂ε 33 ∂ε 33 3 ∂ε 33 ∂ ε ∂ ε ∂ ε 33 33 33  

σ 33 = λ I1

1 1 2 2 2 2  = λ I1 + 2µε 33 + A ( ε 33 ) + 2 ( ε 23 ) + ( ε13 )  ( ε 33 ) 3 r 3  

(4.149)

+ B ( 2ε 33 I1 + I 2 ) + CI12  ∂I ∂I1 1 ∂I ∂I ∂I ∂I  + µ 2 + A 3 + B  I1 2 + I 2 1  + CI12 1   ∂ε 12 ∂ε 12 ∂ε12 3 ∂ε12 ∂ε 12   ∂ε 12 (crossed summands are zero, because the first invariant doesn’t depend on shear components of stress tensor)

σ 12 = λ I1

=

σ 13 = λ I1

1 r2

 2  2   2 µε 12 +  3 A + 2 B  ε12 I1 + 3r 2 Aε 13ε 23  .    

 ∂I ∂I1 1 ∂I ∂I ∂I + µ 2 + A 3 + B  I1 2 + I 2 1  ∂ε 13 ε13 ∂ε13 ∂ε13 3 ∂ε 13 ∂ 

 ∂I  + CI12 1  ∂ ε13 

 2  2  =  2 µε 13 +  A + 2 B  ε13 I1 + 2 Aε12ε 23  . 3r 3   

σ 23 = λ I1

 ∂I ∂I1 ∂I 1 ∂I ∂I + µ 2 + A 3 + B  I1 2 + I 2 1  ∂ε 23 3 ∂ε 23 ∂ε 23 ∂ε 23  ∂ε 23

=

1 r2

 ∂I  + CI12 1  ∂ε 23 

 2  2   2 µε 23 +  3 A + 2 B  ε 23 I1 + 3r 2 Aε 12ε 13  .    

For the derivatives we have ∂I1 ∂I1 ∂I1 ∂I ∂I 1 ∂I1 = 1, = 2, = 1, = 1 = 1 = 0, ∂ε 11 ∂ε 22 r ∂ε 33 ∂ε12 ∂ε13 ∂ε 23

294


∂I 2 ∂I 2 ∂I 2 2 = 2ε 11 , = ε 22 , = 2ε 33 , ∂ε 11 ∂ε 22 r 2 ∂ε 33 ∂I 2 ∂I ∂I 2 2 2 = 2 ε12 , 2 = 2 2ε 23 , = 2ε13 , ∂ε 12 r ∂ε 23 r ∂ε 13 2

∂I 3 ∂I 3 ∂I 3 3 1 2 2 = 3 ( ε 11 ) , = 2  ε 22 2  , = 3 ( ε 33 ) , ∂ε 11 ∂ε 22 r  r  ∂ε 33 ∂I 3 2  1  ∂I 1    = 2 ε12  ε11 + 2 ε 22  + ε 13ε 23  , 3 = 2 ε 13 ( ε11 + ε 33 ) + 2 ε12ε 23  , ε ∂ε12 r   r ∂ r     13 ∂I 3 2 1   = 2 ε 23  2 ε 22 + ε 33  + ε12ε13  . ∂ε 23 r   r   From equations (4.149) it follows that the procedure of writing the components of the stress tensor in terms of the displacement vector starts with the representation of the first two algebraic invariants I1 , I 2 as well 2

2

as the quantities ( I1 ) , ( ε kk ) , ( ε ik ε lm ) . Let us write now the equations of motion for the four cases under consideration. State I. The third equation degenerates into the identity and two first ones become 1 r

1 (σ rr − σ ϑϑ ) − ρ uɺɺr r 1 + rur ,rr ) σ rr − ( ur ,ϑr + ur ,rr − 2uϑ ,r ) σ rϑ r

σ rr ,r + σ rϑ ,ϑ +

1 ( ur , r r 1 − 2 ( ur ,ϑϑ − 2uϑ ,ϑ + ur ) σ ϑϑ − ur , rσ rr ,r r 1 1 1 − ( ur ,ϑ − uϑ ) σ rϑ ,r − ur ,rσ rϑ ,ϑ − 2 ( ur ,ϑ − uϑ ) σ ϑϑ ,ϑ , r r r =−

(4.150)

Waves in Materials

295

1 (4.151) (σ rϑ ,r + σ ϑϑ ,ϑ + 2σ rϑ ) − ρ uɺɺϑ r 1 2 1 = − ( uϑ ,r + ruϑ ,rr ) σ rr − ( uϑ ,ϑ r + ur ,r )σ rϑ − 2 ( uϑ ,ϑϑ − 2ur ,ϑ + uϑ ) σ ϑϑ r r r 1 1 1 −uϑ , rσ rr ,r − ( uϑ ,ϑ − ur ) σ rϑ ,r − uϑ ,rσ rϑ ,ϑ − 2 ( uϑ ,ϑ − ur ) σ ϑϑ ,ϑ . r r r The main feature of equations (4.150), (4.151) is that in both the linear parts corresponding to the classical linear wave theory are selected on the left-hand side 2 r

σ rϑ ,r + σ ϑ z , z + σ rϑ − ρ uɺɺϑ

(4.152)

1 1  2  1   = 2 µ  uϑ , rr − uϑ , r + 2 uϑ  + 2µ  uϑ , r − uϑ  r r r    r  1 1 ρ +2 µ uϑ , zz − ρ uɺɺϑ → uϑ ,rr + uϑ , r − 2 uϑ + uϑ , zz − uɺɺϑ = 0 , r r 2µ and on right-hand side two types of nonlinear components are collected: they contain the components of stress tensor and the components of derivatives of stress tensor respectively. The form of these summands follows that the motion equations will be nonlinear even in the case of linear relationships between stresses and strains that correspond to the physically nonlinear theory. Nevertheless the geometrical nonlinearity rests and generates the nonlinear motion equation. The advantage of these representations consists in that such nonlinear equations arise when Cartesian coordinates are used, they are studied thoroughly, and the method of successive approximations can be applied with success. State II. The second equation degenerates into the identity whereas the first and third ones have the form

296


σ rr ,r + σ rz , z + =−

1 (σ rr − σ ϑϑ ) − ρ uɺɺr r

u 1 1 ur ,r + rur ,rr )σ rr − ( 2ur , zr + ur , z ) σ rz − ur , zzσ zz − 2r σ ϑϑ ( r r r −ur ,rσ rr ,r − ur , zσ rz ,r − ur ,rσ rz , z − ur , zσ zz , z . =−

(4.153)

1 1 u z ,r + ru z , rr ) σ rr − ( 2u z , zr + u z , z ) σ rz − u z , zzσ zz ( r r −u z ,rσ rr ,r − u z , zσ rz ,r − u z , rσ rz , z − u z , zσ zz , z .

(4.154)

State III. The third equation degenerates into the identity whereas the first and third ones have the form − ( uϑσ rϑ ),r + uϑ ,rσ rϑ − ( uϑσ ϑ z ), z + uϑ , zσ ϑ z = −uϑ σ rϑ ,r + σ ϑ z , z = 0 ,

(



2

)

1

(4.155)



σ rϑ , r + σ ϑ z , z + σ rϑ − ρ uɺɺϑ = −  uϑ , rr + uϑ ,r σ rr − uϑ ,rzσ rz − uϑ , zzσ zz r r   −

1 uϑσ ϑϑ − uϑ , rσ rr , r − uϑ , rσ rz , z − uϑ , zσ zz , z . r2

(4.156)

It should be observed that the right side of (4.156) corresponds to the linear theory. But allowance for equation (4.155) results in that the obtained linear part (in the absence of crossed expression) will not correspond to classical theory. This situation is uncommon and should be commented on. State IV. The last two equations of the three motion equations will be satisfied identically, the first one has the form

σ rr ,r +

1 (σ rr − σ ϑϑ ) − ρ uɺɺr r

(4.157)

297

Waves in Materials

1 1   = −  ur , rr + ur , r  σ rr − 2 urσ ϑϑ − ur , rσ rr , r . r r   The nonlinear equation (4.157) is simpler of three prior cases however in this equation there is a conditional separation of physical and geometrical nonlinearities. In fact, the right side shows the geometrical nonlinearity only before the substitution expressions for stress tensor through vector of displacements (constitutive relations). The physical nonlinearity can be introduced using the constitutive relations. From the experience with analogous equations in Cartesian coordinates it follows that four variants are possible: 1. Using the linear Cauchy-Green strain tensor in the nonlinear constitutive relations and neglect the nonlinear right side (as a result, the nonlinearity will be pure physical). 2. Using the nonlinear Cauchy-Green strain tensor in the nonlinear constitutive relations and neglect the nonlinear right side parts (as a result, the nonlinearity will not be pure physical). 3. Taking into account in the nonlinear constitutive relations both the nonlinear Cauchy-Green strain tensor and the nonlinear right side (as a result, all nonlinearities will be involved). 4. Using the linear constitutive relations, the nonlinear CauchyGreen strain tensor and the nonlinear right side parts (as a result, the nonlinearity will be pure geometrical). The variants 1 and 2 were used first in nonlinear acoustics and then in nonlinear wave analysis in structured materials. Further the distinctions between variants will be shown within the framework of state 4. Let us evaluate for each of the states the necessary components of the stress tensor as functions of displacements. State I. The algebraic invariants are simply 2

I1 = ε 11 +

ε 22 r2

, I 2 = ( ε11 )

2

ε  ε  +  222  +  12  r   r 

2

so that the representations, through the deformation gradient, is

298


2 2 1 1 ur ,r ) + ( uϑ ,r ) ( 2 2 2 2 1 1 1 + ( uϑ ,ϑ + ur ) + ( ur ,ϑ − uϑ ) + ( uϑ ,ϑ + ur ) r 2 2 2 2 2 2 1 1 + ( uϑ ,ϑ + ur ) + ( ur ,r ) + ( uϑ ,r ) + ( ur ,ϑ − uϑ ) + ( uϑ ,ϑ + ur )  .   r 2

I1 ( ε ik ) = ur ,r +

= ur ,r

It must be noted here that the first two terms are the part of the invariant corresponding to the linear theory. The other terms represent six types of quadratically nonlinear summands. Because the strain tensor is quadratically nonlinear, then the first invariant can’t include higher nonlinearities. In the next expressions, the summands of strain components with more than the second order are neglected 2 2 1 1  I 2 ( ε ik ) = ur , r + ( ur ,r ) + ( uϑ ,r )  2 2  

2

2 2 1 1 1 +  ( uϑ ,ϑ + ur ) + 2 ( ur ,ϑ − uϑ ) + 2 ( uϑ ,ϑ + ur )  2r 2r r 

2

 1  +   ruϑ ,r − uϑ + ur ,ϑ + ur , r ( ur ,ϑ − uϑ ) + uϑ ,r ( uϑ ,ϑ + ur )    2r  2 2 2 1 1 → I 2 ( ε ik ) = ( ur , r ) + 2 ( uϑ ,ϑ + ur ) + 2 ( ruϑ ,r − uϑ + ur ,ϑ ) . r 4r 2 2 1 2 2 ( I1 ) = ( ur ,r ) + 2 ( uϑ ,ϑ + ur ) + ur ,r ( uϑ ,ϑ + ur ) , r r 2

(ε11 )

2

2

2

= ( ur , r ) , ( ε 22 ) = r 2 ( uϑ ,ϑ + ur ) ,

2 1   2ε 11 I1 = 2 ( ur , r ) + ur , r ( uϑ ,ϑ + ur )  , r  

2 2ε 22 I1 = 2 ( uϑ ,ϑ + ur ) + rur ,r ( uϑ ,ϑ + ur )  ,  

2

299

Waves in Materials

σ 11 = σ rr = ( λ + 2µ ) ur ,r + λ ( uϑ ,ϑ + ur )

1 r

(4.158)

2 2 1  1  +  ( λ + 2µ ) + A + 3B + C  ( ur , r ) +  λ + B + C  ( uϑ ,ϑ + ur ) 2  2  2 2 B + ( B + C ) ur ,r ( uϑ ,ϑ + ur ) + 2 ( ruϑ ,r + ur ,ϑ − uϑ ) r 4r ( λ + 2µ ) u 2 + λ u − u 2 , + ( ϑ ,r ) 2 ( r ,ϑ ϑ ) 2 1 1 σ 22 = 2 σ ϑϑ = ( λ + 2 µ ) ( uϑ ,ϑ + ur ) + λ ur ,r (4.159) r r 2 2 1  1 1  +  ( λ + 2µ ) + A + 3B + C  2 ( uϑ ,ϑ + ur ) +  λ + B + C  ( ur , r ) 2 r 2 

+

2 2 B ( B + C ) ur ,r ( uϑ ,ϑ + ur ) + 2 ( ruϑ ,r + ur ,ϑ − uϑ ) r 4r ( λ + 2µ ) u − u 2 + λ u 2 , + ( r ,ϑ ϑ ) 2 ( ϑ ,r ) 2

σ 12 =

1 rϑ 1 σ = 2 r2 r

 1 1   2 µ r  uϑ ,r + ur ,ϑ − uϑ  r r   

(4.160)

2 2     +  2 µ + A + 2 B  ur , r ( uϑ ,ϑ + ur ) +  2 µ + A + 2 B  uϑ , r ( uϑ ,ϑ + ur ) 3 3     1 2   +  A + 2 B   rur ,r uϑ , r + ( ur ,ϑ − uϑ )( uϑ ,ϑ + ur )   . r 3   The nonzero stress tensor components involve nine types of quadratically 2

nonlinear summands:

( ruϑ

r ,r

2

,r

2

2

(u ) , (uϑ ϑ + u ) , u ( uϑ ϑ + u ) , (u ϑ − uϑ ) , + u ϑ − uϑ ) , ( uϑ ) , u uϑ , ( u ϑ − uϑ )( uϑ ϑ + u ) , uϑ ( uϑ ϑ + u ) r,

,

r ,r

r

,

r,

r

2

r ,r

,r

,r

r,

,

r

,r

,

r

which can be expanded into 19 simplest types: 2

2

1. ( ur ) , 2. ( uϑ ) , 3. ( ur , r ) 8. ( uϑ ,ϑ ur ) ,

2

2

4. ( uϑ ur ) , 5. ( uϑ ,ϑ ) , 6. ( uϑ ,ϑ ur , r ) , 7. ( uϑ ,ϑ uϑ ) ,

9. ( ur ,r ur ) ,

2

10. ( ur ,ϑ ) ,

11. ( ur ,ϑ ur ) ,

300


12. ( ur ,ϑ uϑ ) ,13. ( ur ,ϑ uϑ , r ) ,14. ( uϑ , r ) ,

15. ( uϑ ,r uϑ ) ,

16. ( ur ,r uϑ ,r ) ,

17. ( uϑ ,ϑ ur ,ϑ ) , 18. ( uϑ ,ϑ uϑ ,r ) , 19. ( uϑ ,r ur ) . State II. The algebraic invariants will be as follows I1 = ε 11 + = ur , r +

ε 22 r2

+ ε 33

2 2 2 2 ur 1 1 1 1 1 2 + u z , z + ( ur , r ) + 2 ( ur ) + ( u z , z ) + ( ur , z ) + ( u z , r ) . r 2 2r 2 2 2

The first three terms represent the part of the invariant corresponding to the linear theory. The last terms include five types of quadratically nonlinear summands. 2

2

2

I 2 = ( ε11 ) + ( ε 22 ) + ( ε 33 ) + ( ε13 ) 2

= ( ur , r ) +

( I1 )

2

2

2

2

2 2 1 1 2 u + ( u z , z ) + ( ur , z + u z , r ) , 2 ( r) r 4 2

2

2

2

= ( ur , r ) + ( u z , z ) + 2u z , z ur , r , ( ε 11 ) = ( ur , r ) , ( ε 33 ) = ( u z , z ) , 2

(ε13 )

=

2 1 ur , z + u z , r ) , 2ε 11 I1 = 2ur ,r ( ur ,r + u z , z ) , ( 4

2ε 22 I1 = 2u z , z ( ur , r + u z , z ) , 2ε 13 I1 = ( ur , z + u z , r )( ur ,r + u z , z ) u



σ 11 = σ rr = ( λ + 2µ ) ur ,r + λ  r + u z , z   r  2 1  +  ( λ + 2 µ ) + A + 3B + C  ( ur , r ) 2 

2 2 1 1   (u ) +  λ + B + C   r2 + ( u z , z )  + ( A + 3B ) ur , z u z ,r 2   r  6

+

2 2 1 1 6 ( λ + 2 µ ) + A + 3B  ( ur , z ) + [ 6λ + A + 3B ] ( u z ,r ) 12 12

1 1   +2C  ur , r u z , z + ur ur , r + ur u z , z  , r r  

(4.161)

301

Waves in Materials

σ 22 =

u 1 σ ϑϑ = ( λ + 2 µ ) r + λ ( ur ,r + u z , z ) r2 r 2 1  1 +  ( λ + 2 µ ) + A + 3B + C  2 ( ur ) r 2   2 2 1 1  +  λ + B + C  ( ur , r ) + ( u z , z )  + Bur , z u z , r  2 2  2 2 1 + ( 2λ + B ) ( ur , z ) + ( u z ,r )  + 2Cur ,r u z , z ,   4



(4.162)

u 

σ 33 = σ zz = ( λ + 2 µ ) u z , z + λ  ur ,r + r  r   2 1  +  ( λ + 2 µ ) + A + 3B + C  ( u z , z ) 2  2 1 1 2 1  +  λ + B + C   ( ur , r ) + 2 ( ur )  + ( A + 3 B ) ur , z u z , r r 2   6 2 2 1 1 + 6 ( λ + 2 µ ) + A + 3B  ( u z ,r ) + [ 6λ + A + 3B ] ( ur , z ) 12 12 1 1   +2C  ur , r u z , z + ur ur , r + ur u z , z  , (4.163) r r  

σ 13 = σ rz = µ ( ur , z + u z ,r ) 1   1  +  µ + A + B  ( ur , r ur , z + u z , r u z , z ) +  A + B  ( ur , r u z , r + ur , z u z , z ) . (4.164) 3   3  It should be noted that 12 types of nonlinearities are present in these expressions: 2

2

2

2

1. ( ur , r ) , 2. ( u z , z ) , 3. ( ur , z ) , 4. ( u z , r ) , 5. ( ur ,r ur , z ) , 6. ( ur ,r u z ,r ) , 7. ( ur , z u z , z ) , 8. ( u z ,r u z , z ) , 9. ( ur ,r u z , z ) , 10. 1  1  11.  2 ur ur , r  , 12.  2 ur u z , z  . r  r 

1 2 u , 2 ( r) r

302


State III. The algebraic invariants and some other computations give 2

I1 =

2 2 2 u  1 1 2 2  uϑ , r ) + 2 ( uϑ ) + ( uϑ , z )  , I 2 =  uϑ ,r − ϑ  + ( uϑ , z ) , ( I1 ) = 0, (  2 r r   

2

ε kk I1 = 0,

2

ε11 = (1 2 ) ( uϑ ,r ) → ( ε11 ) = 0, 2

2

2

2

ε 22 = (1 2 )( uϑ ) → ( ε 22 ) = 0, ε 33 = (1 2 ) ( uϑ , z ) → ( ε 33 ) = 0, 2

2

ε12 = ( ruϑ ,r − uϑ ) → ( ε12 ) = ( ruϑ ,r − uϑ ) , 2

2

2

ε 23 = ruϑ , z → ( ε 23 ) = r 2 ( uϑ , z ) , ε13 = (1 2 ) uϑ , r uϑ , z → ( ε13 ) = 0 . 1

σ 11 = σ rr = 3 ( λ + 2 µ ) + 2 ( A + 3B )  ( uϑ ,r ) 6

2

(4.165)

2 1 1 2 1 2 λ  + 3λ + 2 ( A + 3B )  2 ( uϑ ) + ( A + 3B ) uϑ uϑ , r +  + B  ( uϑ , z ) , r r 6 3 2  1 1 1 2 σ 22 = 2 σ ϑϑ = 3 ( λ + 2µ ) + 2 ( A + 3B )  2 ( uϑ ) r 6 r 2 1 + 3λ + 2 ( A + 3B )  ( uϑ ,r ) 6 2 1 2 1 + 3λ + 2 ( A + 3B )  ( uϑ ,r ) + ( A + 3B ) uϑ uϑ ,r , (4.166) 6 3 r 2 1 σ 33 = σ zz = 3 ( λ + 2 µ ) + 2 ( A + 3B )  ( uϑ , z ) 6 1 1 2 λ  +  + B  ( uϑ , r ) + 2 ( uϑ )  + 2 B uϑ uϑ , r , (4.167) r r 2  

 2  1   2µ ( ruϑ ,r − uϑ ) + 3 A  uϑ ,r uϑ , z − r uϑ , z uϑ   ,   

1 r

1 r2

1 r

2 2  2 1  µ uϑ , z , σ 13 = σ rz =  µ + A  uϑ ,r uϑ , z − A uϑ , z uϑ . 2 r 3  3 r 

σ 12 = σ rϑ = σ 23 = σ ϑ z =

This state is characterized by six types of nonlinearities:

(4.168)

303


2

2

1. ( uϑ ) , 2. ( uϑ , r ) , 3. ( uϑ , z ) , 4. ( uϑ ,r uϑ , z ) , 5. ( uϑ uϑ , r ) , 6. ( uϑ uϑ , z ) . State IV. Starting from the representation of components of stress tensor 2

σ 11 = σ rr = λ I1 + 2 µε11 + A ( ε11 ) + B ( 2ε11 I1 + I 2 ) + CI12 , σ 22 =

2  1 1  1   2  2 σ = λ I + 2 µε + A ε ϑϑ 1 22  22 2  + B  2 ε 22 I1 + I 2  + CI1  2 2  r r  r   r  

with further computation we get 2

2 2 (u ) u 1 1 2 I1 = ur , r + r + ( ur , r ) + ( ur ) , I 2 = ( ur ,r ) + r2 , r 2 2 r 2 uu 2 2 u 2 2 ( I1 ) = ( ur , r ) + r2 + 2 r r ,r , (ε11 ) = ( ur ,r ) , r r 2 1 2   2ε 11 I1 = 2 ( ur , r ) + ur , r ur  , 2ε 22 I1 = 2 ( ur ) + rur ,r ur  .   r  

The stresses are  

σ 11 = σ rr = λ  ur ,r +

ur  2  + 2 µ ur , r + ( B + C ) u r , r ur r  r

(4.169) 2

2 1  1  (u ) +  ( λ + 2µ ) + A + 3B + C  ( ur ,r ) +  λ + B + C  r2 , 2  2  r



u 

u

2

σ 22 r 2 = σ ϑϑ = λ  ur ,r + r  + 2µ r + ( B + C ) ur ,r ur r  r r 

(4.170) 2

2 1  1  (u ) +  λ + 3B + C  ( ur ,r ) +  ( λ + 2µ ) + A + B + C  r2 . 2  2  r

Since this state is the simplest of the four considered, the number of presented nonlinearities is the smallest: 2

2

1. ( ur , r ) , 2. ( ur ) , 3. ( ur ,r ur ) .

304


The above wave equations were written through the stress tensor and separately the representations are evaluated for the stress tensor through the displacement vector. However these equations can be written through the displacement vector only. Let us show such equations. State I. Consider the full variant 2 when the physical and partially geometrical nonlinearities are taken into account  

µ  ur , rr +

1 1 2 1  u + ur , r − 2 uϑ , r − 2 ur  2 r ,ϑϑ r r r r  1 1 1   + ( λ + µ )  ur , rr + uϑ ,ϑ r + ur , r − 2 ur  − ρ uɺɺr r r r  

(4.171)

2 2   ≡ µ  ∆ur − 2 uϑ ,r − 2 ur  + ( λ + µ ) I1,r − ρ uɺɺr r r   = −  λ + 2 µ + 2 ( A + 3B + C )  ur ,r ur ,rr −  λ + 2µ + 2 ( B + C )  ( uϑ ,ϑ r + ur ,r )( uϑ ,ϑ + ur ) −

B ( ruϑ ,rr + ur ,ϑ r )( ruϑ ,r + ur ,ϑ − uϑ ) 2r 2

2 B ruϑ , r + ur ,ϑ − uϑ ) − ( λ + 2µ ) uϑ ,rr uϑ ,r − λ ( ur ,ϑϑ − uϑ ,ϑ )( ur ,ϑ − uϑ ) 3 ( 2r 2 1 − ( A + 3B ) {ur ,rϑ uϑ ,r + uϑ ,rϑ ur ,r + 2 ( ur ,ϑϑ − uϑ ,ϑ )( uϑ ,ϑ + ur ) 3 r

−

−

2 1 ( B + C ) ur ,rr ( uϑ ,ϑ + ur ) + ur ,r ( uϑ ,ϑ r + ur ,r ) − ur ,r ( uϑ ,ϑ + ur )  r r  

1 3µ + 2 ( A + 3B )  ( ur ,rϑ + uϑ ,rϑ )( uϑ ,ϑ + ur ) + ( uϑ ,ϑϑ + ur ,ϑϑ )( ur ,r + uϑ ,r )  3r  + ( uϑ ,ϑϑ + ur ,ϑ )( ur ,ϑ − uϑ ) 

}

−  

2 2 1 ( λ + µ + A + 4 B + 2C ) ( ur , r ) − ( uϑ ,ϑ + ur )  r

µ  uϑ ,rr +

1 1 2 1  u + uϑ , r + 2 ur ,ϑ − 2 uϑ  2 ϑ ,ϑϑ r r r r 

,

305

Waves in Materials

1 1   + ( λ + µ )  ur ,rϑ + uϑ ,ϑϑ + ur ,ϑ  − ρ uɺɺϑ r r  

(4.172)

2 1   ≡ µ  ∆uϑ + 2 ur ,ϑ − 2 uϑ  + ( λ + µ ) I1,ϑ − ρ uɺɺϑ r r   = −  λ + 2µ + 2 ( B + C )  ur ,rϑ ur ,r −  λ + 2µ + 2 ( A + 3B + C )  ( uϑ ,ϑϑ + ur ,ϑ )( uϑ ,ϑ + ur ) −λ ( ur ,ϑϑ − uϑ ,ϑ )( ur ,ϑ − uϑ ) B ( ruϑ ,rϑ + ur ,ϑϑ )( ruϑ ,r + ur ,ϑ − uϑ ) − ( λ + 2µ ) uϑ ,rϑ uϑ ,r 2r 2 1 − 3µ + 2 ( A + 3B )  uϑ ,rϑ ( uϑ ,ϑ − ur ) + ur ,r ( uϑ ,ϑϑ − ur ,ϑ )  3r −

−

1 3µ + 2 ( A + 3B )  ( ur ,rr + uϑ ,rr )( uϑ ,ϑ + ur ) + ( uϑ ,ϑ r + uϑ ,rr )( ur ,r + uϑ ,r )  3r  2 1 − ( A + 3B )  ur , r uϑ , r + ( ur , rr uϑ , r + uϑ , rr ur , r ) 3 r +

1 ( ur ,ϑϑ − uϑ ,ϑ )( uϑ ,ϑ + ur ) + ( uϑ ,ϑϑ + ur ,ϑ )( ur ,ϑ − uϑ )   .  r2 

State II. Let us consider the variant 4, where we take into account only the geometrical nonlinearity.  

1 r

µ  ur ,rr + ur , zz + ur ,r −

1  ur  r2 

1 1   + ( λ + µ )  ur , rr + u z , zr + ur , r − 2 ur  − ρ uɺɺr r r   ≡ µ ∆ur + ( λ + µ ) I1,r − ρ uɺɺr = −2 ( λ + 2 µ ) ur ,r ur , rr − 2 ( λ + 2 µ )

2 1 ur , r ) ( r

1 2 u − ( λ + 2 µ ) ur , z u z , zz − ( λ + 2 µ ) u z , z ur , zz − ( λ + µ ) ur ,r u z ,rz 3 ( r) r 2 1 − ( λ + 2 µ ) ur , r ur , zr − ( λ + µ ) ( ur , z ) − ( λ + µ ) ur ,r ur , zz − 3µ ur ,r ur , zr r − ( λ + 2µ )

306


1 1 −2 µ u z , r ur , zr − µ ur , z u z ,rr − µ ur , z u z ,r − λ ur ,r u z , z − λ ur ur , zz r r 1 1 1 −λ ur ur ,rr − λ 2 ur ur ,r − λ 2 ur u z , z − λu z , z ur ,rr − λ ur , z ur ,rz , (4.173) r r r 

1





1



µ  u z ,rr + u z , zz + u z ,r  + ( λ + µ )  ur , rz + u z , zz + ur , z  − ρ uɺɺz r r     1 ≡ µ ∆u z + ( λ + µ ) I1, z − ρ uɺɺz = −2 ( λ + 2µ ) u z , z u z , zz − ( λ + 2 µ ) ur ,r u z ,r r − ( λ + 2 µ ) ur , r u z ,rr − ( λ + 2 µ ) u z ,r ur ,rr − 3µ u z ,r u z , zr − 2µ ur , z u z , zr 1 1 − ( λ + µ ) u z , z ur ,rz − ( λ + µ ) u z , z u z ,r − ( λ + µ ) u z ,r u z , z r r 1 1 − µ u z ,r ur , zz − µ u z , z ur , z − λ ur ,r u z , zz − λ ur u z ,rr r r 1 1 1 −λ u z , r ur ,r − λ u z , z ur , z − λ ur u z , zz − λur , r u z , zz − λu z , r u z , zr . r r r

(4.174)

State III. Let us consider the variant 4, when the geometrical nonlinearity is taken into account. Since all components of the stress tensor, which are on the right of the wave equation do not have linear terms, then the realization of the variant 4 gives zero at the right side. Therefore the nonlinear equation will only be cubically nonlinear. Hence in the quadratic approximation the variant 4 coincides with the variant 3. State III. Consider the variant 2, when the physical nonlinearity and partially the geometrical nonlinearity are taken into account. 1 1   2 µ  uϑ , rr + uϑ , r − 2 uϑ + uϑ , zz  − ρ uɺɺϑ = r r   2 1 − A 2  2 ( uϑ ,r uϑ , z ) + ( uϑ ,rz uϑ ) + ( uϑ ,rr uϑ , z ) + ( uϑ ,rz uϑ , z )  . 3 r

(4.175)

Waves in Materials

307

State IV. Consider the variant 3, when the physical nonlinearity and the geometrical nonlinearity are taken into account.

( λ + 2µ )  ur ,r + 

ur   − ρ uɺɺr r , r

(4.176)

1 = − 3 ( λ + 2µ ) + 2 ( A + 3B + C )  ur ,rr ur ,r − ( λ + 2 B + 2C ) ur ,rr ur r 2 λ 1 − 2 ur , r ur − [ 2λ + 3µ + A + 2 B + 2C ] ( ur ,r ) r r 1 2 − [ 2λ + 3µ + A + 2 B + C ] 3 ( u r ) . r The first row in (4.176) represents the linear part of equation and can be transformed as  

λ  ur , r +

ur   + 2 µ ( ur ,r ),r r , r

1  +  λ  ur , r r  u  λ  ur , r + r r 

ur  ur     + 2 µ ur ,r − λ  ur ,r +  − 2  − ρ uɺɺr = r r    u r , r ur   u    − 2  − ρ uɺɺr = ( λ + 2 µ )  ur ,r + r  − ρ uɺɺr .  + 2 µ  ur ,rr + r r  r , r , r  

+

ur r

State IV. Consider the variant 2, when the physical nonlinearity and partially geometrical nonlinearity are taken into account.

( λ + 2µ )  ur ,r + 


(4.177)

1 = −  λ + 2 µ + 2 ( A + 3B + C )  ur ,rr ur ,r − 2 ( B + C ) ur ,rr ur r 2 λ 1 1 2 − 2 ur , r ur − [ µ + A + 2 B + 2C ] ( ur ,r ) − [ λ + µ + A + 2 B + C ] 3 ( ur ) . r r r

308


State IV. From the variant 4, when only the geometrical nonlinearity is taken into account we have. 

( λ + 2 µ )  ur , r + 

ur  − ρ uɺɺr r , r

(4.178)

1 λ = −2 ( λ + 2 µ ) ur ,rr ur ,r − λ ur , rr ur − 2 ur ,r ur r r 2 1 1 2 −2 ( λ + µ ) ( ur , r ) − ( λ + 2µ ) 3 ( ur ) . r r Thus in this subsection we have considered nonlinear wave equations in cylindrical (orthogonal) coordinates. These nonlinear equations are very novel and not widely known. Their construction is based on some notions of modern mechanics of nonlinear continua, very rarely shown.

4.5 Comparison of Murnaghan and Signorini Approaches

4.5.1

Comparison of some results for plane waves

Different variants of theoretical approaches were previously offered as a description of deforming the hyperelastic materials. The main part of them is based on postulating the explicit nonlinear dependence of an internal energy on the finite strain tensor e = e (ε lm ) . All these variants have the phenomenological character, that is, according to principles of phenomenological mechanics, the explicit dependence of energy must be postulated and further the physical constants should be determined for each material by phenomenological considerations (for example, by experimental methods of mechanics or molecular physics). In exploring the elastic waves, the Murnaghan potential turned out to be the most utilized one. At least, in studies of acoustic plane waves and waves in materials with initial stresses the presence of the Murnaghan potential is prevailing. This can be partially explained by the fact that this

Waves in Materials

309

potential contains the third algebraic potential, which permits us to take into account a row of essential wave effects. Both Murnaghan and his followers assumed the experimental researches on determination of physical constants as very important, which have been included in the analytical representation of the potential (two Lamé constants and three Murnaghan constants). Actually these constants are known for many tens of different materials used in engineering applications. But not all of the initiators of new models have realized how important the knowledge of the physical constants presented in the model is. For example, Signorini doesn’t show the visible trails of analysis of the new constant from his model. Despite a certain advantage of his potential (it contains one new constant of the third order, whereas the Murnaghan potential contains three such constants), Signorini was neglected by the following researchers. It is well-known in physics that for equal possibilities to describe some physical phenomena it is more advantageous to use that approach which uses fewer numbers of physical constants, as in Signorini model. This model was proposed by Signorini about 60 years ago. In constructing the elastic potential he uses the nonlinear Almansi finite strain tensor ε A ≡ {ε ikA } . The natural pair with the Almansi strain tensor is formed by the Euler-Cauchy stress tensor T ≡ {Tik } , i.e. the tensor of true stresses. The equations of motion, written by using this tensor, include the operator of substantial derivative relative to time ∇iT ik − ρ

D2 k u =0, Dt 2

(4.179)

2

D2  ∂ ∂u n  m n ≡ + v ∇ , ∇ u ≡ + Γ nsmu s . m m  Dt 2  ∂t ∂θ m  The energy is given in the form of the quadratically nonlinear relationships (4.89)

310


W = − ( µ + (c 2)) + 1

(

2 AA3 cAA2 + (1 2 ) ( λ + µ − ( c 2 ) ) ( AA1 ) + ( λ + ( c 2 ) ) (1 − AA1 )  ,  

)

The constitutive equations for Signorini model can be written in the form (4.90)  1 c 2 T =  λ AA1 + cAA2 +  λ + µ −  ( AA1 )  g 2 2   2  c   +2  µ −  λ + µ +  AA1  ε A + 2c ( ε A ) 2   

where g ≡ { gik } is the metric tensor. The last equation testifies that the constitutive equations are quadratically nonlinear and besides two Lamé elastic constants λ , µ includes the new constant с. Consider now the classical approach to plane wave analysis based on the Murnaghan model. The primary step is a representation of the vector of displacement as a function of phase variable z = x1 − v pht ( v ph is the phase velocity) u = {uk ( x1 , t )} , uk ( x1 , t ) ≡ uk ( x1 − v ph t )

and assuming the properties of propagation medium as described by the hyperelastic Murnaghan potential. At the final step, two systems of coupled nonlinear wave equations should be analyzed for two different approximations: 1. quadratically nonlinear approximation (4.107)-(4.109)

ρ u1,tt − ( λ + 2µ ) u1,11 = N1 u1,11u1,1 + N 2 ( u2,11u2,1 + u3,11u3,1 ) , ρ u2,tt − µ u2,11 = N 2 ( u2,11u1,1 + u1,11u2,1 ) ,

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ρ u3,tt − µ u3,11 = N 2 ( u3,11u1,1 + u1,11u3,1 ) , N1 = 3 ( λ + 2 µ ) + 2 ( A + 3B + C )  , N 2 = λ + 2µ +

1 A + B. 2

2. cubically and quadratically nonlinear approximation (4.113)(4.115)

ρ u1,tt − ( λ + 2 µ ) u1,11 = N1 u1,11u1,1 + N 2 ( u2,11u2,1 + u3,11u3,1 ) 2

+ N 3 u1,11 ( u1,1 ) + N 4 ( u2,11u2,1u1,1 + u3,11u3,1u1,1 ) ,

ρ u2,tt − µ u2,11 = N 2 ( u2,11u1,1 + u1,11u2,1 ) 2

2

2

2

2

+ N 4 u2,11 ( u2,1 ) + N 5 u2,11 ( u1,1 ) + N 6 u2,11 ( u3,1 ) ,

ρ u3,tt − µ u3,11 = N 2 ( u3,11u1,1 + u1,11u3,1 ) 2

+ N 4 u3,11 ( u3,1 ) + N 5 u3,11 ( u1,1 ) + N 6 u3,11 ( u2,1 ) , N 3 = ( 3 2 )( λ + 2 µ ) + 6 ( A + 3B + C ) , N 4 = (1 2 )  2 ( λ + 2 µ ) + 5 A + 14 B + 4C  , N 5 = ( 3 2 )( λ + 2 µ + A + 2 B ) ,

N 6 = 3 A + 10 B + 4C .

The coupled system of nonlinear wave equations (4.107) - (4.109) permits us to consider different problems of propagation of plane polarized waves. There arise three so-called standard problems enabling the analysis of basic wave effects, which correspond to different propagations. Analogously to classical acoustics, these problems can be formulated as follows: − −

Problem 1. At the entrance into medium only a longitudinal wave is given. Problem 2. At the entrance into medium only a transverse wave is given.

312


−

Problem 3. At the entrance into medium both longitudinal and transverse waves are given.

We obtain these three problems by the method of successive approximations and their solutions are analyzed up to the second approximation. Problem 1. It is necessary to solve the equation

ρ u1,tt − ( λ + 2 µ ) u1, xx = N1u1, xx u1, x

(4.180)

with boundary conditions u1 (0, t ) = u1o cos ωt ; u2 ( 0, t ) = u3 ( 0, t ) = 0 and the following parameters of the plane longitudinal wave: u1o is the initial amplitude, k1 is the wave number, ω is the frequency, v1 ph = ( k1 ω ) =

( λ + 2µ ) ρ

is the phase velocity.

Problem 2. We have to solve two coupled equations

ρ u1,tt − ( λ + 2 µ ) u1, xx = N 2u3, xx u3, x , ρ u3,tt − µ u3, xx = 0

(4.181)

with boundary conditions u3 ( 0, t ) = u3o cos ωt; u1 ( 0, t ) = u2 ( 0, t ) = 0 and the following parameters of the plane longitudinal wave: u3o is the initial amplitude, k3 is the wave number, ω is the frequency, v3 ph = ( k3 ω ) = µ ρ is the phase velocity. Problem 3. There are two coupled equations

Waves in Materials

ρ u1,tt − ( λ + 2µ ) u1, xx = N1u1, xx u1, x + N 2u3, xx u3, x ; ρ u3,tt − µ u3, xx = N 2 ( u3, xx u1, x + u1, xx u3, x )

313

(4.182) (4.183)

with boundary conditions u1 (0, t ) = u1o cos ωt , u2 ( 0, t ) = 0, u3 ( 0, t ) = u3o cos ωt . The corresponding solutions of Problems 1-3 up to the second order approximation are: Problem 1. u1 ( x, t ) = u1∗ ( x, t ) + u1∗∗ ( x, t ) 2 1  N1  o 2 = u1o cos ( kx − ωt ) − x   ( u1 ) ( k1 ) cos 2 ( k1 x − ω t ) . (4.184) 8  λ + 2µ 

Problem 2. u3 ( x, t ) = u3∗ ( x, t ) + u3∗∗ ( x, t ) = u3o cos ( k3 x − ωt ) ;

(4.185)

u1 ( x, t ) = u1∗ ( x, t ) + u1∗∗ ( x, t ) = u1o cos ( k1 x − ωt )

(4.186)

2

+

N 2 ( u3o ) k3 2 2 4 ρ ( v1 ph ) − ( v3 ph )   

sin ( k3 − k1 ) x  cos ( k1 + k3 ) x − 2ω t  .

Problem 3. u1 ( x, t ) = u1∗ ( x, t ) + u1∗∗ ( x, t )

(4.187)

2 1  N1  o 2 = u1o cos ( k1 x − ωt ) − x   ( u1 ) ( k1 ) cos 2 ( k1 x − ω t ) 8  λ + 2µ  2

+

N 2 ( u3o ) k3 2 2 4 ρ ( v1 ph ) − ( v3 ph )   

sin ( k3 − k1 ) x  cos ( k1 + k3 ) x − 2ω t  ;

314


u3 ( x, t ) = u3∗ ( x, t ) + u3∗∗ ( x, t ) = u3o cos ( k3 x − ωt ) +

N 2u1o u3oω

ρ ( v3 )

3

(4.188)

2 2 ( v1 ) + ( v3 ) 1   sin  ( k3 − k1 ) x   4 2 2 2   3 ( v1 ) − 2 ( v1 ) + ( v3 )

2 v3 ) ( 1  1   × cos  ( 3k3 + k1 ) x − 2ωt  − cos k − k x ( ) 2 2  2 3 1  . 2  ( v1 ) − ( v3 )   

The main nonlinear waves are respectively: Problem 1. The longitudinal harmonic wave self-generates and generates its second harmonic. Problem 2. The transverse harmonic wave propagates without changes, self-generating in its linear approximation, at the same time it generates the transverse wave which is modulated in space and having a double frequency relatively to the frequency of transverse wave. Problem 3. The longitudinal harmonic wave self-generates and generates its second harmonic (as in Problem 1) and the transverse wave, also the transverse wave self-generates and generates a longitudinal wave. The wave equations (4.113)-(4.115) contain not only the quadratically nonlinear summands, but also the cubically nonlinear ones with coefficients N m ( m = 3, 4,5,6 ) . This creates new possibilities in the description of the wave picture. The main ones can be displayed when the standard problems is being solved. Possibility 1. In the first standard problem, owing to the presence in the first equation (4.113) of the cubic summand with coefficient N 3 , we can describe the generation by longitudinal wave of its third harmonic. This theoretical result suggests to organize an experiment for the observation of the third harmonic of longitudinal wave.

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315

Possibility 2. To analyze, with allowance for the Possibility 1, the influence of both nonlinearities (quadratic and cubic) on the evolution of the initial harmonic profile of longitudinal wave. Possibility 3. Owing to the presence in the third equation (4.115) of the term with coefficient N 4 , the generation by the transverse wave of its third harmonics can be described. Possibility 4. A new fourth standard problem can be formulated. It consists in the setting at the entrance into the medium of one transverse wave and the observation of generating another transverse wave on the same frequency. This is provided by the summands with coefficient N 6 . Possibility 5. This and the next possibilities are associated with the analysis of evolution equations. The presence of cubic nonlinearities enables the analysis of the quadruplets problem (the problem on interaction of four waves). Possibility 6. Each of the three equations (4.113)-(4.115) can be transformed into the evolution equation describing the phenomenon of self-switching from the first harmonic to the third one and vice versa. Let us choose now the Signorini potential as the starting point in constructing the nonlinear wave equations. We have two different ways. The first way follows from the general Signorini scheme, when the Almansi nonlinear strain tensor is used. This way seems to be logical, but in our case it is inconvenient because of the results. In fact, the propagation of harmonic waves will be difficult to compare with the propagation of similar waves which utilize the Lagrangian coordinates (with respect to the initial body configuration) and the Murnaghan potential (written through the invariants of the Cauchy-Green finite strain tensor). The second way is based on classical works and consists in the transition from Eulerian coordinates to Lagrangian ones. The elastic

316


potential is not written through the invariants of Almansi tensor but through the invariants of Cauchy-Green tensor. Thus the procedure of derivation of the motion equation in Lagrangian coordinates follows. In order to compare some results of plane wave analysis obtained for Murnaghan with the corresponding for Signorini potentials, the second way is preferable. Let us show the necessary steps for this comparison. It is possible to start with the traditional representation of the Signorini potential using the Almansi strain tensor. But the next step will be nontraditional. Here the dependence between three first algebraic invariants of Almansi AAk and Cauchy-Green Ak finite strain tensors will be utilized 2

3

A1 + 2 ( A1 ) − 2 A2 + 2 ( A1 ) − 6 A1 A2 + 4 A3 ; 4 8 2 3 1 + 2 A1 + 2 ( A1 ) − 2 A2 + ( A1 ) − 4 A1 A2 + A3 3 3 1 1 2 3 ( A1 ) − A2 + ( A1 ) − 3 A1 A2 + 2 A3 1 2 2 2 AA2 = ( A1 ) − ; (4.189) 4 8 2 3 2 1 + 2 A1 + 2 ( A1 ) − 2 A2 + ( A1 ) − 4 A1 A2 + A3 3 3 AA1 =

3

( A1 ) − A1 A2 + 2 A3 2 1 3 AA3 = A1 A2 − . ( A1 ) 4 8 2 3 3 4 3 1 + 2 A1 + 2 ( A1 ) − 2 A2 + ( A1 ) − 4 A1 A2 + A3 3 3 This link between the invariants has no general character. Formulas (4.180) are true for a group of three directions, which are orthogonal both in the initial configuration and the actual configuration and are the main directions for both tensors – Cauchy-Green and Almansi. This means that these wave equations describe the propagation of such waves, which generate such a type of deformation. The kinematical picture of plane waves satisfies this condition.

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The next step consists in the substitution of (4.189) into Signorini potential (4.89) and in writing the potential and constitutive equations through invariants of the Cauchy-Green strain tensor. This way leads us away from the scheme proposed by Signorini: Eulerian system of coordinates, Almansi finite strain tensor, nonsymmetrical Euler-Cauchy tensor of true stresses etc., and push us towards the characteristic scheme for nonlinear Murnaghan model: concomitant system of coordinates, Cauchy-Green finite strain tensor, Lagrange symmetric stress tensor or Kirchhoff nonsymmetric stress tensor. So that we have some advantages in using concomitant system but we loose the simplicity in the representation with constitutive equations. Let us focus on the constitutive equations taking into account that the plane waves are studied and the next considerations can be done without the expression for potential. Let us use the constitutive equations for the Signorini model in their classical form (4.90). Writing the equations (4.90) in a concomitant system of coordinates and using the corresponding strain and stress tensors we have not only the transition to corresponding system of algebraic invariants (4.189), but also the transition from the Euler-Cauchy true stress tensor T nm to the Kirchhoff stress tensor t nm or (linked with it) true stress tensor τ αβ t nm =

A3τ nk ( g kn + ∇ k u m ) .

(4.190)

∂X n ∂X m , ∂θ α ∂θ β

(4.191)

We can use the formula T nm = τ αβ

where X n ,θ α are the curvilinear coordinates in the actual and initial configurations, respectively. Owing to the particular chosen case of the deformation process (the plane wave), these coordinates will coincide. Hence, tensors T nm and τ αβ coincide too.

318


Actually, the tensor T nm on the left side of constitutive equations (4.90) should be changed into the tensor τ αβ and take into account on the right side the relationships (4.189). We substitute into constitutive equations the quadratically nonlinear forms of invariants through the components of displacement vector and its first and second spatial derivatives (relative to coordinate x1 ) instead of the full representations of invariants. To obtain these representations, one must complete a few steps, first of which is to evaluate the representations of Cauchy-Green nonlinear strain tensor components through the displacement vector components. The Cauchy-Green nonlinear strain tensor components are evaluated by means of covariant derivatives of co- and contravariant components of displacement vector. For the plane wave, in Cartesian coordinates, we have 1 ε ij = ( ui , j + u j ,i + ui ,k u j ,k ) , 2 2 1 1 1 ε11 = u1,1 + ( u1,1 ) , ε12 = u2,1 , ε13 = u3,1 , ε 22 = ε 33 = ε 23 = 0 . 2 2 2 The next step consists of the evaluation of the first three algebraic invariants of the Cauchy-Green finite strain tensor. In the case of plane waves in hand these invariants have the form 2 1 u1,1 ) ; ( 2 2 3 2 2 1 1 1 A2 = ε ik ε ki = ( u1,1 ) + ( u1,1 ) + ( u2,1 ) + ( u3,1 ) ; 2 2 2

A1 = ε ii = u1,1 +

(4.192)

3

A3 = ε ij ε jk ε ki = ( u1,1 ) . Further expressions (4.192) should be substituted into (4.189), in which the degrees

2

( A1 ) , ( A1 )

3

and product A1 A2 are presented. These

three quantities should be first evaluated

319


( A1 )

2

2 2 3 1  = u1,1 + ( u1,1 )  = ( u1,1 ) + ( u1,1 ) 2   3

( A1 )

3

2 3 1 5  = u1,1 + ( u1,1 )  = ( u1,1 ) ; 2 2  

(4.193)

2 2 3 2 2 1 1 1 1  A1 A2 = u1,1 + ( u1,1 )  ( u1,1 ) + ( u1,1 ) + ( u2,1 ) + ( u3,1 )  2 2 2 2    2 2 2 1 1  = u1,1 ( u1,1 ) + ( u2,1 ) + ( u3,1 )  . 2 2  

Take into account in (4.189) the new expressions (4.193) and evaluate (approximately, with allowance for smallness of three invariants comparing with unit and with restriction by the quadratic and cubic nonlinearities only) the containing in the constitutive equations two first algebraic invariants I A1 , I A2 2 2 3 2 2 1 − ( u2,1 ) + ( u3,1 )  − 6 ( u1,1 ) − 2u1,1 ( u2,1 ) + ( u3,1 )      AA1 = 2 3 2 2 2 2 1 + 2u1,1 + ( u1,1 ) + 3 ( u1,1 ) − ( u2,1 ) + ( u3,1 )  − 2u1,1 ( u2,1 ) + ( u3,1 )      2 3 2 2 = 1 − 2u1,1 − ( u1,1 ) − 6 ( u1,1 ) + u1,1 ( u2,1 ) + ( u3,1 )  ;  

AA2 =

2 3 1  u + u ( ) ( ) 1,1 1,1  2 

(4.194)

(4.195)

2 2 3 2 2 − ( u2,1 ) + ( u3,1 )  + 7 ( u1,1 ) − 6u1,1 ( u2,1 ) + ( u3,1 )      − 2 3 2 2 2 2 4 1 + 2u1,1 + ( u1,1 ) + 3 ( u1,1 ) − ( u2,1 ) + ( u3,1 )  − 2u1,1 ( u2,1 ) + ( u3,1 )      2 3 2 2 2 2 1 = 2 ( u1,1 ) − 5 ( u1,1 ) − ( u2,1 ) + ( u3,1 )  + 6u1,1 ( u2,1 ) + ( u3,1 )  .     4

{ {

}

}

320


In the expression for the first invariant of Almansi tensor there appear terms with cubic non-linearity which, in general, are not characteristic for the first invariant. Their presence is due to the fact that we are writing this tensor not in the Eulerian system of coordinates, but in Lagrangian one. It is important to note that the absence, in the initial constitutive equations (4.90), of the third invariant of Almansi tensor is filled up by the presence of the third invariant of Cauchy-Green tensor in transition formulas (4.189). We can repeat now the classical procedure of derivation of wave equations. First evaluate the components of the stress tensor T11 = ( λ + 2 µ ) u1,1 +

2 2 2 1 1 ( −λ + 5c ) ( u1,1 ) + c ( u2,1 ) + ( u3,1 )  ; 2 2

2 c  T21 = µ u2,1 −  λ + µ +  u1,1u2,1 + 2c ( u2,1 ) ; 2 

(4.196)

2 c  T31 = µ u3,1 −  λ + µ +  u1,1u3,1 + 2c ( u3,1 ) . 2 

By substituting (4.196) into motion equation and taking into account (4.194),(4.195) we obtain the nonlinear wave equations 1 1 ( −λ + 5c ) u1,11u1,1 + c ( u2,11u2,1 + u3,11u3,1 ) (4.197) 2 2 c  − µ u2,11 = 2  λ + µ +  ( u2,11u1,1 + u1,11u2,1 ) + 4cu2,11u2,1 ; (4.198) 2 

ρ u1,tt − ( λ + 2 µ ) u1,11 = ρ u2,tt

 

c

ρ u3,tt − µ u3,11 = 2  λ + µ +  ( u3,11u1,1 + u1,11u3,1 ) + 4cu3,11u3,1 . 2 

(4.199)

Finally, the wave equations obtained for Murnaghan and Signorini models, can be compared.

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321

Their comparison can be very useful, because the nonlinear systems (4.107)-(4.109) (quadratic approach in the Murnaghan model) and (4.197)-(4.199) (quadratic approach in the Signorini model) are similar but not identical. Similarity is displayed first of all in the equations for longitudinal waves (4.107) and (4.197) and coincide with exactness to elastic constants in nonlinear terms. Distinction consists of the presence in the equations for transverse waves (4.198) and (4.199) of the new terms 4cu2,11u2,1 and 4cu3,11u3,1 . This comparison should be taken as a preliminary one, because only the two basic wave effects are compared. They follow from two standard problems of the second (nonlinear) approximation: 1. The problem on propagation of longitudinal harmonic waves when initially the longitudinal oscillations are excited. 2. The problem on excitation of transverse harmonic waves when initially the transverse oscillations are excited. The first problem permits to describe, in the framework of Murnaghan model, the self-generation of longitudinal wave and subsequently the transformation of the first harmonic into the second one. The second problem, within the same approach, has a solution in which the selfgeneration of transverse waves is absent. Comparison of equations (4.107) and (4.197) shows that the Signorini model is identical to the Murnaghan model, when the first standard problem is studied, and identically describes both the self-generation of longitudinal wave and the generation of its second harmonic. Comparison of equations (4.198) and (4.199) shows that the Signorini model differs from the Murnaghan model, when the second standard problem is studied, and describes, in contrast with the Murnaghan model, both the self-generation of transverse wave and the generation of its second harmonic. This advantage of the Signorini model should be interpreted in the future by taking into account that within the next (i.e. third) approximation the Murnaghan model reflects an influence of nonlinearity on the evolution of the transverse wave. This

322


influence was observed in the experiments on propagation of transverse waves in materials. The similarity of equations (4.107) and (4.197) permits to suppose that because of the first standard problem these equations have the form

ρ u1,tt − ( λ + 2 µ ) u1,11 = 3 ( λ + 2 µ ) + 2 ( A + 3B + C )  u1,11u1,1 , ρ u1,tt − ( λ + 2 µ ) u1,11 =

1 ( −λ + 5c ) u1,11u1,1 2

(4.200) (4.201)

and because of the good correlation between the theoretical description by (4.200) and experimental observations of the second harmonic generation, the coefficients of nonlinear summands in (4.200) and (4.201) can be equaled. As a result, the relationship 1 c =  7λ + 12 µ + 12 ( A + 3B + C )  5

(4.202)

holds true. Thus, the new Signorini elastic constant can be identified by means of Lamé and Murnaghan elastic constants, which are known for many engineering materials. This new constant has the order of the elastic constants of the third order (Murnaghan constants, which as a rule exceed values of Lamé constants by one order).

4.5.2 Comparison of cylindrical and plane wave in the Murnaghan model Nonlinear plane waves were analyzed starting in the 1960’s mainly on the bases of the Murnaghan model. The nonlinear cylindrical waves for this model have a very short history. Nevertheless some comparison can be done. To this purpose, let us consider the nonlinear wave equation (4.176), where all nonlinearities are taken into account. This equation can be represented in the convenient form for analysis

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ur  ρ  uɺɺr = S ( ur∗ , ur∗, r , ur∗, rr ) ;  ur , r +  − r , r λ + 2 µ 

(4.203)

1 S ( ur∗ , ur∗, r , ur∗, rr ) = − Nɶ 1ur , rr ur , r − Nɶ 2 ur , rr ur r 2 1 1 1 2 − Nɶ 3 2 ur , r ur − Nɶ 4 ( ur , r ) − Nɶ 5 3 ( ur ) ; r r r  2 ( A + 3B + C )  ɶ λ + 2 B + 2C ɶ λ Nɶ 1 = 3 + ; N3 = ;  ; N2 = λ + 2µ λ + 2µ λ + 2µ   2λ + 3µ + A + 2 B + 2C ɶ 2λ + 3µ + A + 2 B + C Nɶ 4 = ; N5 = . λ + 2µ λ + 2µ We study the harmonic (in time) cylindrical waves, described by (4.176), which arise in a medium with a cylindrical cavity of radius ro , when the harmonic (in time) load σ rr ( ro , t ) = po eiωt is applied to the cavity. In the linear case, such waves are given analytically through the Hankel functions of the first kind and zero order ur ∗ ( r , t ) = uro H1(1) ( k L r ) eiωt ,

(4.204)

where uro is the arbitrary amplitude factor, determined by the boundary condition on the cavity surface; k L = (ω vL ) , vL = ( λ + 2µ ) ρ  are the wave number and phase velocity of the longitudinal plane wave, respectively. The nonlinear solution can be constructed by the method of successive approximations ur ( r , t ) = ur ∗ ( r , t ) + ur ∗∗ ( r , t ) 2 =

π

o r

ue

π  i  k L r −ω t −  4 

1  1 i 9 1 − − 2 k L r  8 k L r 128 ( k L r )

(4.205)    

324


+

r ( uro )

π kL

2

( kL )

2

N1e

π  i 2  k L r −ω t −  2 

 2 1 5 i 151 1  + + −  2 288 ( k L r )3   3 k L r 18 ( k L r )

where we have taken for the Hankel functions the well-known asymptotic representation H p(1) ( z ) ∼

2 2 1   π  2 i  z − 2  p + 2   4 p 2 − 1 ( 4 p − 1)( 4 p − 9 ) + − + ⋯ . e 1 i  2 πz 8z 2!( 8 z )  

The solution (4.205) is important because it shows the similarity of the mathematical structure of solutions with the first two approximations for the plane and cylindrical waves. The analysis of similarities and distinctions in the characteristics of propagation of plane and cylindrical nonlinear hyperelastic waves is very special and will be discussed in the following pages. Here, the main attention will be paid to the initial wave profile evolution equations (4.184) for the longitudinal plane wave and (4.205) for the cylindrical wave. The first distinction is obvious: analysis of plane and cylindrical waves is based on distinguishing approaches. This statement can be substantiated by two assertions −

Assertion 1. Analysis of plane waves follows to the first works on nonlinear plane hyperelastic waves, where the gradients of components of deformations u,nm and associated with them by the elastic potential W non-symmetrical Piola-Kirchhoff stress tensor t nm = ∂W ( u,i j , x k ) ∂u,nm are used.

(

−

)

Assertion 2. Analysis of cylindrical waves is carried out according to the classical approach to the analysis of the linear cylindrical waves, where the nonlinear Cauchy-Green strain

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Waves in Materials

tensor and associated with it by the elastic potential symmetrical Lagrange stress tensor σ ij = ∂W ( ε nm ,θ k ) ∂ε ij are used.

(

)

The common factor in both approaches is utilizing the Murnaghan elastic potential and subsequent restriction of analysis to quadratic nonlinearity. Let us consider the solutions (4.184) and (4.205) and compare the plane and cylindrical waves by the basic parameters. First of all, they are similar in that: they are characterized by two identical parameters: the wave number and frequency. Of course, the defining fact is the equality of wave numbers, according to which both waves are referred to as the type of longitudinal waves. − The first distinction consists in that the plane wave is a harmonic one, whereas the cylindrical wave can be considered as the harmonic wave only conditionally. − The second distinction is that the plane wave can be related to the class of free or running waves; it is generated “at infinity” and goes out to “infinity” and therefore its amplitude is not defined, whereas the cylindrical wave is generated on a definite cylindrical surface and therefore its amplitude is defined exactly by the given boundary condition. − The third distinction is associated with the shape of the wave front: the plane wave has the plane front (it is the plane according to the definition of the wave front), the cylindrical wave has the curvilinear front (it is the circular cylindrical surface). The solution (4.205) can be considered in more detail, by analyzing the linear and nonlinear summands. From the form of the linear part of (4.205) follows that the cylindrical wave becomes very similar to the longitudinal plane wave with three more concrete distinctions: 1. The amplitude decreases rapidly with the distance of the wave propagation.

326


2. The phase variable k L r − ωt − ( 3π 4 ) with nonzero shift ( 3π 4 ) . 3. The great inexactness of approximation for small values of radius. Return now to (4.184) and (4.205) anew. Formula (4.184) expresses two basic nonlinear wave effects for the longitudinal plane wave in quadratically nonlinear elastic material: self-generation of the wave and generation of the second harmonic. − The first fact consists in that, when the longitudinal wave being propagated in the nonlinear medium, then this wave generates only the wave like its own. − The second fact is that (4.184) describes quite exactly the second harmonic starting at some time (or after some distance) of the wave propagation. The solution (4.205) shows the same two basic nonlinear effects, but referred to the cylindrical wave. To understand these statements, the computer modeling of evolution of the wave initial profile turned out to be very useful. This modeling was carried out for different micro- and nanocomposite materials described in chapter 3. For plane waves five stages of evolution were observed. −

−

Stage 1. The initial profile is cosinusoidal, it tilts downwards under a constant angle, i.e. maximal positive values decrease and maximal negative values increase. Stage 2. The tops of the profile become lower, and a plateau is gradually formed instead of the peak. Later the plateau lowers even further, and further the middle part of the plateau begins to sag and the profile becomes two-humped instead of one-humped. The frequency of repetition of the same profile is equal to the initial oscillation frequency. Starting with this stage, the amplitude increases.

Waves in Materials

−

−

−

327

Stage 3. Saving the prior frequency the profile becomes more clearly two-humped with an increasing sag up to the point when it touches the abscissa axis. Stage 4. The sag increases and the profile becomes similar to the harmonic one with the second harmonic frequency but with the unequal amplitude swings: upwards, large amplitude, downwards, roughly half the size of the prior one, upwards, slightly bigger than the prior upper one, downwards, roughly twice as big as the prior lower one. Stage 5. The gradual change (progress) of the first harmonic profile transforms into the second harmonic profile and we can observe the transformation of one harmonic into the other one. The amplitude increases significantly (on one order), that is in order to obtain the second harmonic, it is necessary to pump energy into the propagating wave. This testifies the arising inadequateness of the initial model because the Sommerfeld finiteness condition forbids such situation in elastic materials.

The following set of computer plots represents the wave profile evolution for the most characteristic materials - 11, 41, and 62 (see section 3.8). On all plots abscissa axis shows the distance x in m, the ordinate axis shows the displacement - amplitude u1 in mm. Time is fixed. The first group of graphs corresponds to the same initial amplitude o 1

u = 0.1 mm and changing frequencies. Figure 4.1 presents the plot only for material 62, plots for the other two materials are very similar. It demonstrates only the 1st stage of the profile evolution (profile tilting downwards). For different materials we can see the effect, more or less, clearly, its presence could be seen sufficiently well for material 62 with relatively low wave phase velocity:

328


0.1

0

2

4

6

8

10

Fig. 4.1 The initial stage of evolution due to the second harmonics progress for fixed initial amplitude.

v11 = 1.848 ⋅103 m/s, v62 = 1.7 ⋅103 m/s ) and not well enough for material 41

(with relatively high wave phase velocity v41 = 5.78 ⋅103 m/s ). This plot corresponds to the smallest studied frequency of 10 kHz and small initial amplitude of 0.1 mm. The next figure demonstrates first three stages for material 62. This plot corresponds to the frequency of 50 kHz , the amplitude about 10 m and the distance of 2 m.

0.1

0

0.5

1

1.5

2

Fig. 4.2 The developed stage of evolution due to the second harmonic progress for fixed initial amplitude.

Further we look at the plot for the frequency of 100 kHz which is extremely high for materials 11 (here the significant distortion is observed to the right of the beginning of wave propagation). Figure 4.3 shows the plots for 11.

329

Waves in Materials

0.1

0

0.2

0.4

0.6

0.8

1

Fig. 4.3 The highly developed stage of evolution due to the second harmonic progress for fixed initial amplitude.

Finally Fig. 4.4 presents the exact transition from the frequency ω to 0.4

0

0.025 3.058

0.050 3.083

0.075 3.108

0.100 3.133

0.125 3.158

0.150 3.183

Fig. 4.4 The transformation of the initially excited first harmonic into the second one.

frequency 2ω which is the pure effect of the transformation of the, initially given, first harmonic into the second harmonic. Here two different (initial and advanced) parts of one and the same plot built for material 11 with ω = 100 kHz, u1o = 0.1mm are superposed (see two rows of the distance values on horizontal axis). Analogous computer modeling shows quite a different scheme of evolution for the cylindrical wave. It consists of four stages. −

−

Stage 1. The ascending (bottom-up) branch of the plot of periodic damped oscillations is increasingly more tilted to abscissa axis with each new oscillation cycle. The descending (up-bottom) branch is practically invariable. Stage 2. Around the point of intersection with abscissa axis and tilting branch as if the plateau is formed, which is further

330

−

−


transformed into the sinusoid with small and slowly increasing number of cycles with increased amplitude. Simultaneously the basic ascending branch becomes more symmetric to the descending branch, they form together one new cycle of oscillations tending to the cycle with double initial frequency. Stage 3. The originated cycle of oscillations with small amplitudes increases little by little the amplitude up to values commensurable with the amplitude of the basic cycle. At that the frequency of the small cycle of oscillations tends to be the double initial frequency. Stage 4. The basic and originated small cycles of oscillations draw together asymptotically by frequency and amplitude showing a change of the initial first “harmonic” into its second “harmonic” with all reserves mentioned above – both harmonics aren’t harmonics in the classical sense. This stage is characteristic by stopping the amplitude decrease and its gradual increase. The wave from damped is transformed into the wave storing energy. The comment from above relative to increasing the plane wave can be related equally to the case in hand (the cylindrical wave).

The following set of computer plots represents the wave profile evolution for the two types of fibrous composite materials (see section 3.8) microcomposites with industrial fibers Thornel-300 and nanocomposites with fibers in the form of zig-zag carbon nanotubes. The matrix is assumed the mixture of epoxy rosin EPON-828 and polystyrene with volume fraction of last cPS = 0.05; 0.10; 0.20. The volume fraction of fibers is also assumed to be small c f = 0.05; 0.10; 0.20. The composites are the unidirectional ones and the direction of wave propagation is perpendicular to fibers. The phase velocity of plane longitudinal waves in this direction is v ph =

( C1111 ρ ) ,

2

eff 1111

C

=E

1 − (ν ) ( E ′ E )

(1 + ν ′ ) (1 − ν ′) − 2 (ν ) ( E ′ E )  2

.

331

Waves in Materials

In the modeling such values were assumed: the initial amplitude uro = 1 ⋅ 10−4 m, the wave frequency ω = 1 МHz, the wave number k L = 145.14 1/м. On all computer plots the distortion of the initial profile of cylindrical wave is shown for the case of the tenth part of the polystyrene volume fraction and the same part for micro- and nanofibers. The plots correspond to different ranges of changing the quantity x = k L r , but everything else is of the same kind: the abscissa axis shows the distance x = k L r in m corresponding to the passed-by wave distance (for example, x = 30 corresponds about 20 cm and x = 150 - 1m), the ordinate axis shows the displacement – oscillation amplitude in mm, here the unit is equal to uro . Time is fixed. The next three figures (Fig. 4.5 - Fig. 4.7) correspond to the case of macro-composites and show the slow evolution of cylindrical wave. This process of evolution can be accelerated by increasing the initial wave amplitude or decreasing the initial wavelength. Figure 4.8 and Fig. 4.9 correspond to nanocomposites with differing wavelength. 0.75 0.5 0.25

5

10

15

20

25

-0.25 -0.5 -0.75

Fig. 4.5 The initial stage of evolution of the cylindrical wave in microcomposites.

30

332

Wavelet and Wave Analysis 0.1

0.05

215

220

225

230

235

240

-0.05

-0.1

Fig. 4.6 The developed stage of evolution of the cylindrical wave in microcomposites. 0.1

0.05

905

910

915

920

925

930

-0.05

-0.1

Fig. 4.7 The highly developed stage of evolution of the cylindrical wave in microcomposites. 0.75 0.5 0.25

5

10

15

20

25

30

-0.25 -0.5 -0.75

Fig. 4.8 The initial stage of evolution of the cylindrical wave in nanocomposites.

333

Waves in Materials

0.75 0.5 0.25

605

610

615

620

625

630

-0.25 -0.5 -0.75

Fig. 4.9 The highly developed stage of evolution of the cylindrical wave in nanocomposites.

As can be seen from plots, the periodicity of oscillations becomes very quickly two times more. For equaling the amplitudes, the significantly more time of propagation is needed. But with the condition that both harmonics aren’t harmonics in the classical sense, the phenomenon of transition of the first harmonic into the second one is here revealed very sharply. Thus, the comparison of schemes for evolution of the longitudinal plane wave and the cylindrical wave shows the striking distinction of these schemes. It is striking first of all by that in the final step both cases show the transformation of the first harmonic into the second one. Consider now two different cases for an analysis of cylindrical waves: the case of using the Murnaghan model and the case of using the Signorini model. For the first case the nonlinear wave equation (4.203) should be solved ur  ρ  uɺɺr = S ( ur∗ , ur∗, r , ur∗, rr ) ;  ur , r +  − r , r λ + 2 µ  1 S ( ur∗ , ur∗, r , ur∗, rr ) = − Nɶ 1ur , rr ur , r − Nɶ 2 ur , rr ur r 2 1 1 1 2 − Nɶ 3 2 ur , r ur − Nɶ 4 ( ur , r ) − Nɶ 5 3 ( ur ) . r r r

334


For the second case the Signorini nonlinear wave equation (4.200) should be analyzed

( λ + 2µ )  ur ,r + 


2 1 1 1 1 2 = S1 ur , rr ur , r + S 2 ur , rr ur + S3 2 ur , r ur + S4 ( ur , r ) + S5 3 ( ur ) ; r r r r 1 1 1 S1 = ( −6λ + 4 µ + 5c ) ; S2 = ( 4λ − 2 µ − 5c ) ; S3 = ( 2µ − 5c ) ; 2 2 2 1 1 S4 = ( 2λ − µ − 3c ) ; S5 = ( 5µ − 3c ) . 2 2

The comparison of the above written two wave nonlinear equations shows that they are identical with exactness to coefficients (they contain five identical quadratically nonlinear summands which are distinguished by coefficients only). The solutions of these equations will also distinguish the coefficients and will be in this sense identical. Finally, consider in more detail the distinctions between the two solutions obtained within the framework of Signorini and Murnaghan models for the second standard problem for the plane wave. The corresponding wave equations for this problem were discussed above. This will be two coupled nonlinear equations for Signorini model 1 2

ρ u1,tt − ( λ + 2 µ ) u1,11 = cu3,11u3,1 , ρ u3,tt − µ u3,11 = 4cu3,11u3,1

(4.206) (4.207)

with conditions on the boundary

u3 ( 0, t ) = u3o cos ωt; u1 ( 0, t ) = u2 ( 0, t ) = 0 . The solution is

(4.208)

Waves in Materials

335

u3 ( x, t ) = u3∗ ( x, t ) + u3∗∗ ( x, t ) = u3o cos ( k3 x − ωt ) −

1 c o 2 2 x   ( u3 ) ( k3 ) cos 2 ( k3 x − ω t ) ; (4.209) 4 µ

u1 ( x, t ) = u1∗ ( x, t ) + u1∗∗ ( x, t ) = u1o cos ( kx − ωt ) 2

+

c ( u3o ) k3 2 2 2µρ ( v1 ph ) − ( v3 ph )   

sin ( k3 − k1 ) x  cos ( k1 + k3 ) x − 2ω t  . (4.210)

Thus the solution (4.209) for the transverse wave contains the summand with the second harmonic providing two basic nonlinear effects for this wave: the self-generation and the second harmonic generation. Therefore, the Signorini model turns out to be the more adequate to the real wave picture model starting with the second step in approximations. All results discussed in this section testifies the presence of many similarities and many distinctions in description of both the plane wave and the cylindrical waves within the framework of Murnaghan and Signorini models. On the whole, both potentials can be characterised as quite close in the analysis of many wave problems.


Chapter 5

Simple and Solitary Waves in Materials

5.1 Simple (Riemann) Waves The concept of simple waves has arisen in hydrodynamics. As a mathematical notion, it corresponds to the solution of hydromechanics equations, which have some structure. As a physical notion, it deals with waves of some form. Let us review this notion in different fields of physics. 5.1.1 Simple waves in nonlinear acoustics It is widely recognised that the exact solution for plane waves in an incompressible fluid was obtained, for the first time, by Poisson in 1808. The solution had the form of a plane running wave, which was called simple. The theory of simple waves was extended in the works of Stokes, Airy, Earnshow. Riemann gave, in 1860, the general solution of the one-dimensional problem of hydrodynamical equations in the case when the disturbance is plane and the constitutive equation has the form of an arbitrary functional relationship between pressure and density. For the plane case, the nonlinear equations of hydromechanics in conventional symbols and in Eulerian coordinates are as follows

ρt + ρvx + vρ x = 0; ρvt + ρ vvx + px = 0.

337

(5.1)

338

Wavelet and Wave Analysis ρ

The function σ ( ρ ) = ∫ c( ρ ) ρ0

dρ

was introduced by Riemann. Here

ρ

ρ0 is the density of no disturbed medium and c( ρ ) = dp d ρ is the sound velocity. With σ x = c

ρx

ρt ρ ; σ t = c ρ then equation (5.1) can be rewritten as

Pt + (c + v) Px = 0, Qt + (c − v)Qx = 0 ,

(5.2)

where P = v + σ , Q = v − σ . The solution of system (5.2) has the form v=

1  x  x   Pt −  + Qt +  .  2  c+v  c − v 

(5.3)

Each, separately taken, function P or Q , which corresponds to a perturbation of finite amplitude moving in one direction, is called a simple wave. There are other definitions in acoustics, as e.g.: The simple wave is such a wave process, in which all parameters describing this process can be expressed in the form of a function of one parameter: ρ = ρ (u ) , p = p (u ) or u = u ( ρ ) , p = p ( ρ ) . Where, in the last relationships, should not involve integrals and derivatives. Let us consider system (5.1) as ∂u  1 ∂p  ∂u ∂ρ ∂ ∂ρ + u + = 0; + ( ρu ) = 0 . ∂t  ρ ∂u  ∂x ∂t ∂ρ ∂x

(5.4)


339

Through the relationships ∂u = ± c , which enable us to write explicitly ∂ρ ρ c dp u = ±∫ d ρ = ±∫ , and substituting into (5.4) we obtain the system of ρ ρc differential equations with respect to u and ρ ∂u ∂u + [u ± c(u ) ] = 0; ∂t ∂x ∂ρ ∂ρ + [u ( ρ ) ± c( ρ )] = 0. ∂t ∂x

(5.5)

The condition on the boundary x = 0 is u (0, t ) = Φ (t ) . The equations for the characteristics are dt dx du = = 1 u ± c(u ) 0 and they have two integrals u = C1 ;

− x + [u ± c(u )] t = C2 .

(5.6)

From equations (5.5) and (5.6) follows C1 = Φ (t );

C2 = [ Φ ± c(Φ ) ] t ; C2 = [C1 ± c(C1 )] Φ −1 (C1 ) .

Finally, we get

 . x u = Φt ±   c(u ) ± u 

(5.7)

Therefore, the solutions (5.3) and (5.7) are really identical. But they were obtained by rather different methods. The initial definitions of simple waves are distinguishing significantly.

340


5.1.2 Simple waves in fluids The most brilliant description of Riemann waves is given by Lighthill. In his description of the simple Riemann wave he used the word brilliant often: “The brilliant mathematical discovery of Riemann - one of the outstanding mathematicians of the middle 19 century - has become the basis of all subsequent work on nonlinear theory of plane sound waves. This discovery is equivalent to transformation of motion equations to the form, which is wonderfully easy of use for a study of waves, produced, in due time, to a beautiful level of the subject understanding.” Further, the author in order to avoid some hypothesis of Riemann works gave a not less brilliant physical reasoning, which produced the same brilliant result and the clearer physical understanding of Riemann theory. Lighthill formulates such a question: can the evolution of a plane wave with an arbitrary amplitude, in the absence of dissipation, be predicted with the aim of simple physical reasoning assuming the knowledge of the linear acoustics only? It is reasonable to answer this question with the aim of an initial remark that for arbitrary concrete place x = x1 and moment t = t1 some space interval around x = x1 and some time interval around t = t1 exist. They are both so small that for belonging to these intervals x and t the corresponding disturbances of quantities u and p according to values u1 and p1 , which they have at the point ( x1 , t1 ) , rest so necessarily small that the linear theory should be enough to describe their behaviour. Therefore, the small disturbances from a nonzero value u1 are studied and the disturbances in the intervals relative to a co-moving reference system (with constant velocity), are investigated. The place is determined by the new space coordinate x − u1t . The velocity in this system is

341


( u − u1 )

and it remains small everywhere (quite as in the linear theory)

in the chosen intervals. In addition, it is needed to give the sound velocity c some local value c1 , according to the general formula

∂ p . c1 =    ∂ρ  p = p1 ; ρ = ρ1

(5.8)

For this linear problem, the linear theory gives the general solution of D’Alembert type in the form of simple wave in the reference system, in which the space coordinate is x − u1t p − p1 = f [ x − u1 t − c1 t ] + g [ x − u1 t + c1 t ] , u − u1 =

1 1 f [ x − u1 t − c1 t ] − g [ x − u1 t + c1 t ] . ρ1c1 ρ1c1

By defining p − p1 = δ p ; u − u1 = δ u it is easy to see that

δu +

δp is a function of x − ( u1 + c1 ) t only, ρ1c1

whereas δ u −

δp is a function of x − ( u1 − c1 ) t only. ρ1c1

(5. 9) (5.10)

In Riemann method this integral p

∫ p0

dp = P( p) , ρ1c1

(5.11)

is used, where p0 is a given initial pressure. It is obvious, in the case of a very small deviation δ p from p = p1

δ P = P / ( p1 ) δ p =

δp . ρ1c1

(5.12)

342


So that the quantity on the left-hand side of (5.9) can be treated as a very small deviation δ ( u + P ) of u + P in a small neighbourhood of point ( x1 , t1 ) from its value at this point. Since δ ( u + P ) is a function of x − ( u1 + c1 ) t only, there follows

δ ( u + P ) , when δ x − ( u1 + c1 ) δ t ,

(5.13)

where δ x , δ t are small deviations of x, t from their values ( x1 , t1 ) . However, the function δ ( u + P ) is equal to zero at the point ( x1 , t1 ) , and therefore at all points x, t , as testified by (5.13). Equation (5.13) means also that some space-time curve C+ is considered, in all points of which the differential equation dx = ( u + c ) dt along C+ is valid. Obviously C+ is the trajectory of point, which always advances with a local wave velocity c in the reference system, moving together with the fluid with a local velocity u . Equation (5.13) tells us that exactly along the curve C+ , which passes through the point x1 , t1 , the quantity u + P is stationary and is equal to its value at the point x1 , t1 . That is wonderful, according to Lighthill result, since the quantity u + P is defined by (5.12) independently at the point x1 , t1 . And therefore, all previous considerations can be applied to any other point of the curve C+ . Since the function, which is stationary at all points of C+ , has to be constant everywhere along C+ , then this fact is formulated as the first result of Riemann u + P = const along the curve C+ : dx = ( u + c ) dt .

(5.14)

The second result of Riemann can be obtained analogously u − P = const along the curve C− : dx = ( u − c ) dt .

(5.15)

343


It is not quite clear from the Riemann’s classic consideration, whence the integral P (5.11) appears. At the same time, the expression the simple wave signifies the immediate generalization of the notion of the running wave from a linear theory on perturbations with finite amplitudes. A simple wave is defined as the domain, where a constant value of u + P along any curve C+ in each case is equal to zero. p − p0

That has as the outcome u = P ( p − p0 ) =

∫ 0

d ( p − p0 ) . ρc

The corresponding curves C+ are straight lines

along which

the

function u has a constant value and it is, generally speaking, different dx for different lines C+ . Quantities p − p0 , c and u + c = have the dt same behaviour on these curves. It should be noted that simple waves differ from plane running waves of a linear theory by that: 1. Now, the conductivity on the unit of cross-section 1 is the ρc differential conductivity, which depends on the excess pressure p − p0 and means the increase of a fluid velocity u on the unit of increasing of p − p0 ; 2. Different excess pressures propagate with different velocities c , which are proper to each of them, relatively to itself (proper) velocity of the fluid, so that the absolute velocity of their propagation is equal to u + c . Some authors distinguish first of all the linear running waves, which propagate without the form change along the axis Ox with a constant (and same) velocity a0 , for all disturbance, and the waves, which are a solution of the nonlinear system (5.1).

344


This system doesn’t have solutions depending on x ± a0t only, but it has a solution, which is a plane wave and is some generalization of D’Alembert type solutions of the linear theory f ( x ± a0t ) . These solutions are the partial solutions of system (5.1), for which the velocity u is a function of the density ρ only, that is u = u ( ρ ) ρ = ρ ( x, t ) . Such solutions are sometimes called the Riemann solutions and the corresponding motions are called the Riemann waves or the simple waves. Some textbooks on the theory of fluids and gases discuss the simple waves traditionally in the section on the simplest problem of propagation of small perturbation in a gas. The problem is then reduced to a classical wave equation, for which the D'Alembert solution is constructed in the form of two waves with constant phase velocities. Such waves are called here the simple ones.

5.1.3 Simple waves in the general theory of waves Most people working in the general theory of waves treat the problem of simple waves on the base of classical system of nonlinear equations of gas dynamics (5.1). The nonlinear plane waves, which are a solution of system (5. 1), are usually studied with the aim of characteristic equations. Therefore, Riemann invariants (5.14),(5.15), which are written here as

∫

a( ρ )d ρ

∫

a( ρ )d ρ

+ u = const on C+ :

dx =u+a, dt

(5.16)

− u = const on C− :

dx =u−a , dt

(5.17)

ρ ρ

are introduced at once.


345

The simple wave is defined as the solution of system (5.1), for which Riemann invariants remain constant everywhere (on all the characteristics). It is interesting that the consideration explains an equivalence of different simple wave definitions. It is contended that three basic equations of gas dynamics for three quantities p, ρ , u in the particular case of a simple wave have two integrals. It follows that two of three basic equations can be excluded and two quantities p, ρ , u can be written as a function of the third one. So, there remains only one equation. The choice of ρ as the basic variable is more convenient for general statements, whereas the formulas are written in a more simple form by u . In the last case, the evolution of waves can be studied more simply.

5.1.4 Simple waves in mechanics of electromagnetic continua The simple waves are defined here for an electromagnetic media. At the beginning, the notion of uniform (which doesn’t change with time) states (solutions of the conservative system of equations) should be introduced. The solution of this system is a vector, all components of which are constant in some area of space ( x, t ) . The simple waves are defined in such a way: The continuously differentiable solution is the simple wave in the area R of the space ( x, t ) , if all components are constant along lines from one-parametric family covering the entire area R . The importance of solutions in the form of simple waves is explained by the circumstances that the solution with no constant parameters is joined without fail with the solutions with constant parameters by the area, in which a solution is the simple wave. Let

l (σ ) : x = c(σ ) t + a (σ ) , σ ∈ [σ 1 ;σ 2 ]

(5.18)

346


be the lines of a one-parametric family of lines

{l (σ )}

with the

parameter σ . Functions c(σ ) , a(σ ) have to be differentiable. For solving equation (5.18) relative to σ and obtaining σ = σ ( x, t ) , it is necessary that the condition

∆ (σ , t ) = c / (σ )t + a / (σ ) ≠ 0, σ ∈ [σ 1 ;σ 2 ] be fulfilled. This condition ∆ (σ , t ) = 0

means that the family of lines {l (σ )}

admits a geometrical envelope. At the time, when such an envelope appears, it has the form of a cusp. From this moment, the solution in the form of simple waves is no longer valid. Its extension is realized in the form of shock waves. In this section besides the direct goal to describe simple waves there is also one additional goal, i.e. the comparison of the point of view of different authors. The above shown considerations and definitions sometime use differing terms, but they practically contradict one another only on the question: D’Alembert waves can be classified as simple waves or not. On the other hand, they supplement each other well and probably form the frame, which can be used by any reader as an introduction to the simple wave theory.

5.2 Solitary Elastic Waves in Composite Materials 5.2.1

Simple solitary waves in materials

In the first section we have considered simple waves in nonlinear physics. But the simple waves in solids have not yet been analysed. Also, the consideration of simple waves has the goal to dissociate ourselves from harmonic periodical waves and to show features of simple waves.


347

In this section, the aperiodic and no harmonic waves will be considered, that is, mainly waves with such an initial profile form, which allow us to call these waves the solitary waves. A solitary wave is a wave with an initial profile which is equal to zero nearly everywhere besides some finite interval where one or more humps from the wave profile. So, the shape of a solitary wave is very similar to the shape of the wavelet. For instance a typical solitary wave is the classical bell-shaped signal. Two types of solitary waves in materials will be studied here: − waves with initial profiles in the form of one of Chebyshev-Hermite, Mathieu or Whittaker functions − waves with profiles observed in some experiments on wave propagation in materials. It is expediently therefore to write some preliminary information about the mentioned functions.

5.2.2 Chebyshev-Hermite functions The functions named after Chebyshev and Hermite have the form

ψ n ( z) = e

−z

2 2

H n* ( z ) ,

(5.19)

where H n* ( z ) are the Chebyshev-Hermite polynomials n

H n* ( z ) = 2 2 H n ( 2 z ) ;

H n ( z ) = (−1)n e

x2

2

d n  − x2 2  e  . dx n  

(5.20)

These polynomials are determined by the generating function e

tx −

t2 2

∞

= ∑ H n ( x) n=0

tn . n!

The general polynomials are given by the formula

(5.21)

348


H n ( x) = x n −

1 n(n − 1) n − 2 1 n(n − 1)(n − 2)(n − 3) n − 4 x + 2 x − ⋅⋅⋅ 2 1 2 1⋅ 2 j≤

n 2

=∑ j =0

(−1) j (−n, 2 j ) n − 2 j . x 2j (1, j )

Recurrent relationships have the form H n +1 ( z ) = 2 zH n ( z ) − 2nH n −1 ( z );

dH n ( z ) = 2nH n −1 ( z ). dz

Let us show also some useful properties from the theory H 2 n (− x) = H 2 n ( x); H 2 n +1 (− x) = − H 2 n +1 ( x); H 0* ( z ) = 1; H1* ( z ) = z ; H 2* ( z ) = z 2 − 1; H 3* ( z ) = z 3 − 3 z ; H 4* ( z ) = z 4 − 6 z 2 + 3; H 5* ( z ) = z 5 − 10 z 3 + 15 z . For real values of the argument these functions are real, and they are orthogonal with weight 1 on the real axis (−∞, ∞) , i.e. ∞

∫ψ

−∞

m

0 m ≠ n ; ( x)ψ n ( x)dx =   1 m = n.

These functions are also solutions of the Weber’s differential equation

ψ n// ( z ) + (1 + 2n − z 2 )ψ n ( z ) = 0,

(5.22)

whereas the polynomials H n* ( z ) are solutions of the Chebyshev-Hermite differential equation w// − 2 zw/ + 2nw = 0.

(5.23)

From Figure 5.1 it is clear that functions ψ n ( z ) are defined over the infinite interval, but with a finite mass.

349


If these functions are chosen as functions describing the initial profile of simple waves, then such waves will have all attributes of solitary waves. Special attention will be paid to waves with the initial profile in a form of the Chebyshev-Hermite function of the zero index

ψ 0 ( x) = e

−x

2 2

.

(5.24)

This is the well known, from the signal theory, bell-shaped signal, which is also used in the nonlinear wave theory. Later the function (5.24) will be used in computer modeling of the propagation of solitary waves. The results obtained for (5.24) will be extended on waves with profiles, described by functions with more complicate shape. 3 n=4

2

n=0

1

n=1

n=2

0 n=3

−1

0

1

2

3

4

Fig. 5.1 Plots of the first five Chebyshev-Hermite functions.

5.2.3 Whittaker functions The Whittaker functions M k ,m ( z ) and Wk , m ( z ) are generated by the differential equation

350


1  2   1 k 4 −m  w +− + + w = 0. 4 z z2      //

Functions Wk , m ( z )

and W− k , m (− z )

(5.25)

form, for this equation, the

fundamental system of solutions. The infinite-valued analytical function Wk , m ( z ) is defined over all values k , m and for all nonnegative real z (that is, it has a real branch for these values of z ). Whittaker functions can be represented by the degenerate hypergeometric functions Ψ ( a, c, z ) , Φ ( a , c, z ) z

−

M k ,m ( z ) = e 2 z z

m+

1 2

1   Ψ  m − k + , 2m + 1, z  , 2  

1 2

1   Φ  m − k + , 2m + 1, z  , 2   Γ(−2m) Γ(2m) Wk , m ( z ) = M k ,m ( z ) + M k ,− m ( z ) . 1  1  Γ − m − k  Γ + m − k  2  2  −

Wk , m ( z ) = e 2 z

m+

(5.26)

Whittaker functions also have the representation − m+

1

z

z 2e2 d p p+ 2m − z . M ( z ) = (z e ) 1 p+ m+ ,m (2m + 1)⋯ (2m + p ) dz p 2 There exist some important links of the Whittaker function, with given values of indexes, with other well-known functions of mathematical physics. Let us indicate those, which will be useful in our simple wave analysis: 1. Functions of a parabolic cylinder Dn ( z )

Simple and Solitary Waves in Materials 1

Dn ( z ) = 2 2

n+

1 4

1 −

z 2 W1

1 1 n+ ,− 2 4 4

 z2     2

351

(5.27)

remember also the simple link of Chebyshev-Hermite functions ψ n ( z ) and parabolic cylinder functions Dn ( z ) Dn ( z ) = ( 2π ) ∞

−

1 4

1 − ( n!) 2 ψ n ( z )

.

2. Error function Erfc( z ) = ∫ e− t dt 2

z

Erfc( z ) =

1 − 12 − 12 z 2 z e W 1 1 ( z2 ) . − , n 2 4 4

(5.28)

3. Bessel functions of the imaginary argument K m ( z ) , I m ( z )

π z Km   = W0,m ( z ) , z 2 z M 0, m ( z ) = 22 m Γ(m + 1) zI m   . 2

(5.29) (5.30)

Bessel functions of an imaginary argument have no oscillatory character, the first is monotonic increasing, and the second is monotonic decreasing. For the following computer simulation, we choose four functions with one hump: A. W0,0 ( z ) , which is the solution of the equation d 2W  1 1 +  − + 4 2 W = 0 . 2 dz z   4 B. W0,1 ( z ) , which is the solution of the equation

(5.31)

352


d 2W  1 3  +  − − 2 W = 0 . 2 dz  4 4z 

(5.32)

Wm,n ( x) 0.6

0.4 W1

1 , 4 4

(x)

W 1 1 (x) −

, 8 20

0.2

W0,0 ( x) W0,0 ( x) W0,0 ( x) 1

0

2

3

4

x

5

Fig. 5.2 Plots of Whittaker functions.

C. W

1 1 − , 4 4

( z ) , which is the solution of the equation d 2W  1 1 3  + − − + W =0. 2 2  dz  4 4 z 16 z 

D. W

1 1 − , 8 20

(5.33)

( z ) , which is the solution of the equation d 2W  1 1 99  + − − + W =0. 2 2  dz  4 8 z 400 z 

(5.34)

5.2.4 Mathieu functions Mathieu functions are defined as functions of four classes ∞

ce2 n ( x) = ∑ A2(2i n ) cos 2ix , i =0

(5.35)


353

∞

ce2 n +1 ( x) = ∑ A2(2i +n1+1) cos(2i + 1) x ,

(5.36)

i =0 ∞

se2 n ( x) = ∑ B2(2i n ) sin 2ix ,

(5.37)

i =0 ∞

se2 n +1 ( x) = ∑ B2(2i +n1+1) sin(2i + 1) x .

(5.38)

i =0

Representations (5.35)-(5.38) are Fourier series, periodical with period 2π , solutions of the equation w′′ + ( p − 2q cos 2 x) w = 0 .

(5.39)

The coefficients in series are determined by recurrent formulas, for each series some different recurrent formulas exist. Each of the functions (5.35) - (5.38) has n zeros over the interval

( 0;π 2 ) .

The behaviour of Mathieu functions is illustrated by the limits lim ce0 ( x) = q →0

1 , lim cen ( x) = cos nx (n ≠ 0) , lim sen ( x) = sin nx . (5.40) q →0 2 q →0

Equation (5.39) also has four solutions, which are linearly independent of the first four solutions (5.35) - (5.38). They are fe2 n ( x), fe2 n +1 ( x), ge2 n ( x), ge2 n +1 ( x) .

5.2.5

Interaction of simple waves. Self-generation

Our goal is to show that classic quadratic nonlinear theory permits one to analyse the interaction of not only harmonic (periodic) waves with the sinusoidal profile, but also the non-periodic waves with arbitrary smooth initial profiles. Let us start with the first standard problem for elastic plane waves. As it is known from prior sections, this is the problem on radiation into the medium of the longitudinal wave only. In this case, the wave propagation is described by the equation

354


ρ

∂ 2u1 ∂ 2u ∂ 2u ∂u − ( λ + 2µ ) 21 = N1 21 1 . 2 ∂t ∂x1 ∂x1 ∂x1

Rewrite this equation in the form

ρ

 N1 ∂u1  ∂ 2u1 ∂ 2u1 − λ + 2 µ 1 + ( )   2 = 0. ∂t 2  λ + 2 µ ∂x1  ∂x1

(5.41)

Let c1 be the phase velocity of longitudinal wave in the elastic body c1 =

N1 λ + 2µ and introduce the nondimensional quantity α = . ρ λ + 2µ

We can write the nonlinear equation (5.41) in the form where some nonessential indexes are omitted

∂ 2u ∂u  ∂ 2u 2  c − 1 + α =0. 1   ∂t 2 ∂x  ∂x 2 

(5.42)

Equation (5.42) permits the analysis of two simple waves: displacements and deformations. The waves of longitudinal displacements will be considered later. We seek a solution in the form of a simple wave. This will be interpreted as some initial disturbance, which propagates in space and varied on time. The disturbance is the finite longitudinal displacement, and as initial profile u ( x,0) = F1 ( x) is taken a twice continuously differentiable function. The problem of the initially given pulse motion can be treated as the problem of a motion in the semi-infinite rod, when at the initial time on the rod end the longitudinal displacement (disturbance) is given and it moves at later time along the rod in the form of a longitudinal wave. For this wave, the solution of equation (5.42) is sought in the form of D’Alembert wave with some variable unknown phase velocity v ph

355


u ( x, t ) = F1 ( x − v ph t ) .

(5.43)

The phase velocity v ph may be treated as the local velocity at the point x and time t . Then an analysis of solution (5.43) agrees with the procedure for simple waves, previously described and agrees also with the Lighthill’s description of Riemann waves. In this case the formula

∂u holds true. It follows from this formula that the phase ∂x velocity depends nonlinearly on the solution, and this fact reveals that a wave has all the attributes of a simple wave. ∂u Let us pay attention to the quantity α . Since the factor α is a finite ∂x v ph = c1 1 + α

quantity of a few tens and deformation usually has the order 10−3 , we can assume that

α

∂u T > 0 ,

then

formula (2.54) is valid  π  sin ( xT − nπ ) . f ( x) = ∑ f  n   T  xT − nπ n∈Z This theorem, as well as its discrete version, gives some additional possibilities of using the Shannon wavelets. As it states that for some class of functions an application of wavelet analysis based on Shannon functions will need the knowledge of values of the signal in a finite number of points and will be similar to discrete wavelet families cases (for example, Daubechies wavelets), when only a finite number of nonzero wavelet coefficients will present in the wavelet series representations. Newland has sought wavelets that satisfy two conditions: the spectrum is like the box function and they are complex functions. Shannon wavelets fulfill the first condition. A further step towards a complex generalization of Shannon wavelets can be done by assuming as a father wavelet w( x) =

ei 4π x − ei 2π x , i 2π x

(6.1)

consisting of imaginary and real parts. This step can be the step to classical Fourier analysis, when the function is expanded into series through the basis {ei 2π kt / T } . The family of wavelets, corresponding to the father (6.1), is i 4π ( 2 x − k ) i 2π ( 2 x − k ) e −e j

j

j

w(2 x − k ) =

i 2π ( 2 j x − k )

with Fourier transform in the form of trapezoidal box

(6.2)

385

Solitary Waves and Elastic Wavelets

(1 2π ) (1 2 j ) e − iω k / 2 , ω ∈  2π 2 j , 4π 2 j  ,    W (ω ) =  j j 0, ω ∉  2π 2 , 4π 2  .  j

(6.3)

The mother wavelet is defined as some generalization of the usual sincfunction ϕ ( x) =

e i 2π x . i 2π x

ϕ j ,k ( x) = ϕ (2 j x − k ) = ei 2π (2

(

j

x −k )

(6.4) −1

)

( i 2π (2

j

x − k ) ).

(6.5)

Four graphs of both functions (mother and father wavelets in the real and imaginary parts) are shown below.

-4

1

1

0.75

0.75

0.5

0.5

0.25

0.25

-2

2

4

-4

-2

2

-0.25

-0.25

-0.5

-0.5

-0.75

-0.75

-1

-1

Fig. 6.4 Plots of mother harmonic wavelets in the real and imaginary parts.

4

386


-4

1

1

0.75

0.75

0.5

0.5

0.25

0.25

-2

2

4

-4

-2

2

-0.25

-0.25

-0.5

-0.5

-0.75

-0.75

-1

-1

4

Fig. 6.5 Plots of father harmonic wavelets in the real and imaginary parts.

Thus the Newland harmonic wavelets can be referred to as physical family of wavelets, because they are specially proposed for the analysis of physical problems with oscillations. Elastic wavelets, proposed by the authors, are based on three elements: 1. Kaiser’s idea of constructing the physical wavelets on the base of specially chosen (admissible) solutions of wave equations; 2. the theory of solitary waves (with profiles of the ChebyshevHermite functions) propagated in elastic dispersive media; 3. the theory and practice of using the Mexican hat wavelet family, the mother and father wavelets (and their Fourier transforms) of which are analytically represented as the Chebyshev-Hermite functions of different indexes. Only the first element is sufficient for constructing an elastic wavelets set. The other elements are necessary, when we try to obtain certain wavelet family. The elastic wavelet is a solution of the wave equations for the linear elastic dispersive medium. As an example of such a medium, we will consider the two-phase elastic medium. The base system of plane wave equations is presented in Chapter 5 as equations (5.55). They describe the propagation of solitary waves for some conditions in the form (5.74),

387


where profiles of solitary waves are chosen as the Chebyshev-Hermite functions of arbitrary indexes. The Chebyshev-Hermite functions can be associated in turn with the Mexican hat wavelets. This is the main reason why the MH wavelet was appointed as the first candidate for the elastic wavelets. Mexican hat wavelet family is an example of continuous wavelets with infinite support (that is, it is defined over all real axis). The most famous wavelet families represent the wavelets with finite support. The Mexican hat wavelet was first introduced for the exact detection of sharp changes in images and is utilized for the complex signals analysis. It has a narrow energy spectrum and two zero moments (the zeroth and first ones). The mother Mexican hat wavelet and its Fourier transform have the normalized form x2

2

2

 x2  − 2σ 2 2 2σ 5 4 π 2 − σ 2ω ω e ψ ( x) = − 4 , ψˆ (ω ) = . (6.6)  2 − 1 e 3 π 3σ  σ  2

The Fourier transform of the Mexican hat father wavelet for σ = 1 is as follows 2 24 π ϕɵ (ω ) = 3

2

ω +1 e

−

ω2 2

.

(6.7)

The Mexican hat father wavelet for σ = 1 can be analytically represented as ∞

2(−1)k ( 2k − 1)!!

k =1

( 2 k )!

ϕ ( x) = ∑

 1 

1

π x 2 kU  − , k ,  , 2 2 

(6.8)

388


1 b − a −1 e − zt t a −1 (1 + t ) dt is the hypergeometric ∫ Г (a) 0 ∞

where U ( a, b, z ) = function.

All of these four functions are shown in the following figures.

−ψˆ ( x )

−ψ ( x ) 1

1.5

0.8 0.6

1 0.4 0.2 -4

0.5

-2

2

4

-0.2

-4

-2

2

4

-0.4

-0.5

Fig. 6.6 The plots of Mexican hat father wavelet and its Fourier transform.

ϕ ( x)

ϕˆ ( x) 1.5

0.8

1.25

0.6 1 0.75

0.4

0.5

0.2 0.25

-4

-2

2

4

-4

-2

2

4

Fig. 6.7 The plots of Mexican hat mother wavelet and its Fourier transform.

Formulas (4.2), (4.3) can be expressed in terms of the ChebyshevHermite functions


where

389

ψ ( x) =

 x  x  ψ 2  σ  − 2ψ 0  σ   , 2 π 3σ     

(6.9)

ψˆ (ω ) =

σ9 4 π  ω   ω  ψ 2  σ  + 2ψ 0  σ   , 6     

(6.10)

ψ 0 ( x) = e

−

x2 2

1

4

, ψ 2 ( x) = e

−

x2 2

2

( −2 + 4 x )

are the Chebyshev-

Hermite functions of the zeroth and second indices. Let us return to the solitary elastic waves propagating in composite materials and the basic description of these waves in Subsection 5.2.3 using the model of elastic mixtures. The corresponding wave equation system is (5.55). The most important fact in (5.55) is that when the initial profile of solitary wave is represented in the form of the Chebyshev-Hermite function of arbitrary index, then the solution of (5.55) can be written as the solitary waves with the same profiles (5.74). That is, the Chebyshev-Hermite functions of the corresponding wave arguments are the solutions of wave equations (5.55). As Mexican hat wavelets are expressed in terms of the Chebyshev-Hermite functions, this creates the unique situation when the Mexican hat wavelet family satisfies the basic wave equation (5.55). This means that the basic requirement of physical wavelets is fulfilled. Also the translated and scaled instances of the Chebyshev–Hermite functions are still solutions of the basic mixture wave equations. When we introduce the scaling factor k into the argument of the ChebyshevHermite function, the basic formula for phase velocities (5.73) remains valid, but some of its components change 2M1 =

a1

+

ρ11 M2 =

a1 a2

ρ11 ρ 22

−

a2

−

ρ 22 a32

ρ11 ρ 22

ρ11 + ρ 22 K ( z ), ρ11 ρ 22 −

a1 + a2 + 2a3

ρ11 ρ 22

K ( z),

(6.11)

390


K (z) =

βk2 (1 + 2n) − z 2 k 2

.

(6.12)

The initial pulse in a composite material cannot be chosen arbitrarily if we try to take into account the microstructure of the material (i.e., if we abandon so-called long waves). Therefore the factor k may be treated as the base length (bottom) of the initial pulse. Experience on harmonic waves suggests that the wavelength must exceed the basic representative dimension of a microstructure by one order of magnitude (by a factor of 7 to 10). This ensures the applicability of the continuum approach to harmonic waves. Otherwise, the microstructure cannot be considered as homogenized, and a two-phase material must be considered as piecewise continuous. Obviously the size of the base of a solitary wave, that exceeds the representative dimension of microstructure by one order of magnitude, is a threshold value. This threshold value is used in computer modeling of wave-profile evolution as the minimum base length at which evolution is most intensive. By the maximum base length we mean that value where the evolution is still imperceptible. Thus, a relationship among the Mexican hat wavelet, Chebyshev– Hermite functions, and the wave solutions in the theory of elastic twophase mixtures is established. Following Kaiser idea, we: (i) chose a wave equation characteristic of some field in physics, (ii) select a partial solution of this equation (in the form of a radar signal – similar to a “chirp,” yet not “chirp”), and (iii) used this solution to construct a set of physical wavelets. Thus we obtain a relationship between the known Mexican-hat wavelet set, on the one hand, and dispersive physical media, their wave equations, and the solutions of these equations, on the other hand.

391


6.2 The Link between the Trough Length and the Characteristic Length Consider the free plane solitary waves in a material with a microstructure (composite) by combining theoretical analysis and computer simulation. Let us remember that solitary waves are understood as aperiodic smooth waves whose initial profile is practically zero everywhere except for some finite interval. A structural material is described by the structural theory of two-phase elastic mixture. A mixture is always a dispersive medium for classical free harmonic waves. The phase velocities of these waves in a mixture depend non-linearly on the frequency or wavelength. In the case of solitary waves, the dispersity of a mixture is manifested as a phase dependence of their velocity. Therefore, solitary waves in a mixture are a new type of waves in elastic materials. In Chapter 5 we considered the analytical description of solitary waves based only on the Chebyshev–Hermite and Whittaker functions. The graphs of the Chebyshev–Hermite functions and of the Whittaker functions with different indices are shown in Figs. 5.1 and 5.2. (a) 0

2

4

6

4

6

λ = 1 2,µ = 1 2 0.6

λ = 0, µ = 0 0.3

λ = −1 4 , µ = 1 4

0 0

2

392


(b)

Fig. 6.8 Graphs of three Whittaker functions (a), and experimental graph of high-speed impact (b).

Waves with such profiles are encountered in experimental mechanics. In the Flügge’s encyclopedia, Bell presents one of the results of Filbey’s high-speed impact experiments on pure aluminium specimens. The high-peak profile of a stress wave σ , pound/inch ⋅10−4 ↔ t, µsec, shown in Fig. 6.7 fits fairly well the graph of the Whittaker function. The above-mentioned phase dependence of the velocity of a solitary wave results in a variation in the wave profile during its propagation. Therefore, there is a good reason to speak about an evolution of the wave. This fact is quite unusual, since we use the linear approach according to which classical free harmonic waves (sinusoidal waves) do not evolve. It turns out that the profile shape of a wave in structural materials is related, in some fashion, to its evolution. Evolution studies of solitary waves have revealed that there are at least two questions: A. How can we establish correctly the relationship between the parameters of the wave (its trough as the most typical parameter) and of the material (the characteristic microstructural dimension as the most important parameter in any microstructural theory)? and


393

B. At what distance from the starting point of motion or after what time from the beginning of motion are data on the change of the initial wave profile already incorrect? Both questions are interrelated; therefore, the answer to one of them helps to answer the other question. The objective of this subsection is to answer reasonably to the first question. Let us start with modification of the analytical solution (5.74) and (5.78)

u1(α ) ( x, t ) = A(α )ψ n ( zψ(α ) ) + p( zψ(δ ) ) A(δ )ψ n ( zψ(δ ) ) , zψ(α ) = x − v (α ) ( zψ(α ) ) t , u1(α ) ( x, t ) = A(α )Wk ,m ( zW(α ) ) + p ( zW(δ ) ) A(δ )Wk ,m ( zW(δ ) ) , zW(α ) = x − v (α ) ( zW(α ) ) t . They are related to the analytical representation of the initial impulse

u1(α ) ( x, t ) = A(α )ψ n ( z ) or u1(α ) ( x, t ) = A(α )Wk ,m ( z ) . The analysis reveals that solutions (5.74) and (5.78) should be modified for the following reason. Harmonic waves are characterized by the following quantities: sinusoidal profile, amplitude, frequency, and wave number (or wavelength). In dispersive media, the wave frequency and wavelength are related by the dispersion law. The wavelength is a parameter which is compared with the characteristic structural dimension of a material to determine whether the continuum approach can be applied to describe the propagation of a wave in a structural medium. If waves are not so short that the continuum approach is already incorrect, then the wavelength as a special parameter allows specifying of the limits of applicability of this approach. For example, the wavelength separates of frequency ranges where the basic structural model of the first order (the classical model with effective elastic moduli) and

394


microstructural models of the second order (for example, a two-phase elastic mixture with a set of physical constants) are valid. A solitary wave is defined by two parameters: profile and amplitude. Usually, the wave profile has a so-called trough (bottom), i.e., an interval of the abscissa axis within which the corresponding ordinates on the profile are considerably different from zero. The word “considerably” in the definition of trough implies that its length can be calculated differently; nevertheless, the trough does have a definition. The trough of a solitary wave may be regarded in the next analysis as the length of a harmonic wave, i.e., it is possible to try to compare the trough with the characteristic structural dimension of medium (material) and to find out how, if at all, to account for the internal structure in the model of the medium where the wave propagates. The simplest way to account for the trough length in solutions (5.74) and (5.78) is to introduce a coefficient k , which has the dimension of a length, u1(α ) ( x, t ) = A(α )ψ n ( zψ(α ) k ) + p ( zψ(δ ) k ) A(δ )ψ n ( zψ(δ ) k ) ,

(6.13)

u1(α ) ( x, t ) = A(α )Wk ,m ( zW(α ) k ) + p ( zW(δ ) k ) A(δ )Wk ,m ( zW(δ ) k ) .

(6.14)

Because the functions ψ n ( x ) and Wλ , µ ( x ) satisfy the equations 2

ψ n′′ ( z k ) + 1 + 2n − ( z k ) ψ n ( z k ) = 0,

(

)

 1 λ k (1 4 ) − µ 2 2  Wλ′′, µ ( z k ) +  − + − k Wλ , µ ( z k ) = 0, z2  4 z 

then the propagation velocities v (α ) ( zψ(α ) ) and v (α ) ( zW(α ) ) of the initial impulse in the form of two solitary waves (6.13) and (6.14) in a mixture are determined by

395


v ( zψ(α(W) ) ) = M 1ψ (W ) − ( −1)

α

a1

2 M 1ψ (W ) =

ρ11 M 2ψ (W ) =

a1 a2

−

ρ 22

a32

−

ρ11 ρ 22

a2

+

−

ψ (W ) 2 1

ρ11 ρ 22

Kψ ( zψ ) =

)

− M ψ2 (W ) ,

(6.16)

Kψ (W ) ( zψ (W ) ) ,

βk2 (1 + 2n) − zψ2 k 2

,

(6.17)

βk2 1 4 + λ k zW + (1 4 − µ 2 ) k 2  zW2 (α )

(6.15)

ρ11 + ρ 22 Kψ (W ) ( zψ (W ) ) , ρ11 ρ 22

a1 + a2 + 2a3

ρ11 ρ 22

KW ( zW ) =

(M

.

The amplitude factor p zψ (W ) is given by

(

(α )

p zψ (W ) =

(

)

)

aα + v 2 zψ( (W) ) ραα − Kψ (W ) ( zψ (W ) )

(

α

)

a3 + Kψ (W ) ( zψ (W ) )

.

It is obvious that the factor β k 2 did not appeared accidentally. In the composite analysis among the nine physical constants ραα , µ k , λk , β (α = 1, 2; and k = 1, 2,3) of the isotropic theory of elastic two-phase mixtures, only the shear interaction constant β depends strongly on the characteristic structural dimension of material. In all analytical expressions (derived in various ways and by various authors), the constant β depends, in inverse proportion, on the squared characteristic microstructural dimension. Let us assume now that k is the unit trough length of the initial impulse. As experience suggests, for the continuum approach to be applicable to harmonic waves, their wavelength should be by an order of

396


magnitude (by a factor of 7 to 10) greater than the utilized in our analysis quantity - characteristic structural dimension. Otherwise, the internal structure could no longer be treated as homogeneous, and a twophase material should be regarded as a piecewise inhomogeneous medium. For a solitary wave, it is obvious that a trough length that exceeds the characteristic structural dimension of material by an order of magnitude should also be considered a threshold value. This minimum trough length, corresponding, possibly, to the most intensive manifestation of profile evolution, is used in computer simulation of wave profile evolution. The maximum trough length, at which the evolution is still somewhat noticeable, can be found from computer simulation.

6.3 Initial Profiles as Chebyshev-Hermite and Whittaker Functions This section can be understood as an introduction to computer analysis based on elastic wavelets. Here the general scheme of analysis will be shown as well as a few features of this scheme being revealed. Let us first describe the materials involved in the computer simulation. Consider two composite materials consisting of an aluminium matrix and tungsten fibers. The composites are different by the volume fraction of fibers: ξ 21 = 0.022 (for one material M1) and ξ 22 = 0.221 (for the other material M2). The average distances between fibers, s1 , and between fiber layers, s2,

are s11 = 0.894 ⋅10 −3 , s12 = 0.18 ⋅10 −3 ,

s12 = 0.648 ⋅10−3 , and s22 = 0.3175 ⋅10−3 m. Then the characteristic material structural dimension can be evaluated : a1 = 0.648 ⋅10−3 m for Ml

and a2 = 0.3825 ⋅10−3 m

for M2. Since the characteristic

wavelength must be, by an order of magnitude, greater than the characteristic material structural dimension, the minimum admissible

397


coefficient k for the two materials may be assumed k11 = 1.5 ⋅10−3 and

k12 = 0.9 ⋅10−3 . The ultimate goal of the computer simulation is to plot three-dimensional graphs showing how the initial profiles evolve in time. The following waves have been simulated u (α ) ( x, t ) = A(α )Wλ , µ ( z (α ) ) , u (δ ) ( x, t ) = p ( z δ ) A(δ )Wλ ,µ ( z (δ ) ) , u (α ) ( x, t ) = A(α )ψ n ( z (α ) ) , u (δ ) ( x, t ) = p ( z δ ) A(δ )ψ n ( z (α ) ) ,

i.e., two modes of a solitary wave are considered separately. The phase z is not constant for both modes and is determined as z = x − v( z )t , where the phase velocity v ( z ) depends nonlinearly on the phase. The phase and, hence, the wave profile are expressed implicitly, just as the profile of a simple wave is expressed. Thus, the dispersity of an elastic medium for solitary aperiodic waves transforms into a nonlinear dependence of the wave velocity on the phase. This dependence is determined after imposing some additional constraints on the time of propagation and is shown in Figs. 6.9-6.16.

V 6.5 6

k 31

5 .5

k 21

5

z

z 1∗ 0.1

0.2

0.3

k 11

0.4

Fig. 6.9 Phase velocities versus phase for the first mode of waves with a Chebyshev–Hermite initial profile ψ 0 ( z ) in M1.

398


V k 11

k 31

k 21

80

80

40

20

z

z 1∗ 0.1

0.3

0.5

0.7

Fig. 6.10 Phase velocities versus phase for the second mode of waves with a Chebyshev–Hermite initial profile ψ 0 ( z ) in M1.

V 6 5 .5

k 32

5 4 .5

k 22 4

k12

3 .5

z

∗ 2

z 0 .1

0 .2

0 .3

0. 4

Fig. 6.11 Phase velocities versus phase for the first mode of waves with a Chebyshev–Hermite initial profile ψ 0 ( z ) in M2.

399


V 80

k 22

60

40

k 32

40

k 12

z

z 2∗ 0 .1

0.3

0 .5

0.7

0 .9

Fig. 6.12 Phase velocities versus phase for the second mode of waves with a Chebyshev–Hermite initial profile ψ 0 ( z ) in M2.

Figures 6.9 - 6.12 show the phase velocity as a function of the phase 2

(V in (Km/sec), z in 10 (m)) for both materials through which a wave with a Chebyshev–Hermite initial profile ψ 0 ( z ) propagates. Figures 6.9 and 6.11 correspond to the first mode and Figs. 6.10 and 6.12 to the second mode. Each figure presents three curves for three different values of k:

k11 = 1.5 ⋅10−3 , k21 = 2 ⋅10−3 , k31 = 4 ⋅10−3 (Figs. 6.9 and 6.10) and k12 = 0.9 ⋅10−3 , k22 = 2 ⋅10 −3 , k32 = 4 ⋅10 −3 (Figs. 6.11 and 6.12). The material M1 is represented by Figs. 6.9, 6.10 and the material M2 by Figs. 6.11, 6.12. The threshold values for the second mode are z 1∗ = 1, 5 ⋅ 1 0 − 3 m , and z2∗ = 0,9 ⋅10−3 m (for smaller values the second mode doesn’t exist as a free wave). For waves with Chebyshev–Hermite profiles, these values derive from the condition that the term

400


Kψ ( zψ ) =

βk2 2

k (1 + 2n) − zψ2

(the first one in (6.17)) tends to infinity. For Whittaker waves, z1∗ and z2∗ can be obtained from a similar condition (the second one in (6.17)) for the expression K W ( zW ) =

βk2 − (1 4 ) k 2 + ( λ zW ) k 3 +   (1 4 ) − µ 2  k 4  zW2

.

The phase velocity–phase curves for solitary waves with a Whittaker profile W0,0 ( z ) are shown in Figs. 6.13 and 6.15 (for the first mode), in Figs. 6.14 and 6.16 (for the second mode), in Figs. 6.13 and 6.14 for the material M1, and in Figs. 6.15 and 6.16 for the material M2. Each figure presents three curves for three different values of k:

k11 = 1.5 ⋅10−3 , k21 = 2 ⋅10−3 , k31 = 4 ⋅10−3 (Figs. 6.13 and 6.14) and k12 = 0.9 ⋅10−3 , k22 = 2 ⋅10 −3 , k32 = 4 ⋅10 −3 (Figs. 6.15 and 6.16). The curves of the first mode have a section on which the phase velocity changes drastically and after which the phase velocity becomes constant and no longer depends on the phase. In this section, points of the profile with a smaller phase have higher velocities, which results in a distortion of the initial profile. For small troughs (k is by an order of magnitude smaller than the characteristic microstructural dimension), all phases of profile points belong to this section. Such a profile deforms almost instantaneously. As the trough length increases, the effect of the section of strong dispersion on the initial profile weakens. Hence, the effect of the internal structure of the material on the evolution of the wave profile weakens with an increased ratio of the trough length to the characteristic structural dimension in material.

401


It is also possible to establish the difference between the evolution processes for waves with the Whittaker and Chebyshev–Hermite initial profiles. In the latter case, the section of strong variation in the phase velocity is flatter and smoothes down with an increase in the trough length. This confirms the physical fact that the material internal structure affects considerably the behaviour of a solitary wave with small characteristic dimensions (wave bottom).

9

V

8

7

k 31

k 21

6

0.1

k 11

z

z 1∗ 0.2

0.3

0.4

Fig. 6.13 Phase velocities versus phase for the first mode of waves with a Whittaker initial profile W0,0 ( z ) in M1.

V 100

k 31 80 60 40

k 21

20

k 11

z

z 1∗ 0.11

0.2

0.4

0.6

0.8

Fig. 6.14 Phase velocities versus phase for the second mode of waves with a Whittaker initial profile W0,0 ( z ) in M1.

402

Wavelet and Wave Analysis V 12

10

8 6

k 32 z

0.05

z

∗ 2

0.1

k 22 k 12

0.15

Fig. 6.15 Phase velocities versus phase for the first mode of waves with a Whittaker initial profile W0,0 ( z ) in M2. V 100 2

k

3

80

k

60

2 2

40

k 12 20

z

z

∗ 2

0.1

0.2

0.4

Fig. 6.16 Phase velocities versus phase for the second mode of waves with a Whittaker initial profile W0,0 ( z ) in M2.

This fact is well displayed by the wavelet analysis, since it assumes different bottoms for wavelets in different scales. Discovered dependence of solitary wave propagation parameters on the bottom size creates together with assumption on weak nonlinearity of the material the main physical base for next elastic wavelets application. The plots in Figs. 6.9-6.16 give an opportunity for formulation of a row of conclusions: C1. Plots show the break-down of the wave into two modes each propagating with distinguishing phase velocity and amplitude.


403

C2. The stability of the first mode is revealed, namely it was observed that while the initial impulse is broken-up on two modes, then the first one is stable whereas the second one displays the greater signs of evolution. The stability is meant as the form invariance when the wave propagates the distance in a few wave bottoms. The fact of stability follows from that shown in Figs. 6.9-6.16: the phase velocity for the first mode is nearly a straight line, whereas the second mode velocity is changed near the phase value corresponding to the inflection point in the initial profile. C3. Two modes of solitary waves propagate simultaneously in two components of the composite material. C4. The distance on which an evolution is displayed depends essentially on characteristic sizes of the initial profile and the internal structure of material. Further the three-dimensional plots of dependence of displacements on the spatial and time coordinates in both components of the mixture will be constructed. Here the implicit dependence of the phase velocity on phase will be taken into account on the base of special algorithm. These plots are shown in Figs. 6.17-6.20. The first two figures correspond to the Chebyshev-Hermite initial profile, the next two figures correspond to the Whittaker initial profile. Each figure contains three pictures: the first picture corresponds to 10-fold excess of the wave bottom length over the CML, next two correspond to 50-fold and 100fold excess. In the process of evolution of the solitary wave three characteristic stages can be separated. On the first stage the profile moves without significant distortions. The second stage consists in the separation (breaking-up) of the wave on two waves which propagate with different phase velocities. The profile distorts on the leading edge of an impulse and then the separation of the second impulse is started. The third and final stage shows the separate propagation of two waves with distinguishing phase velocities and amplitudes.

404


Fig. 6.17 Evolution of the wave first mode with Chebyshev-Hermite profile for three different sizes of wave bottom.


Fig. 6.18 Evolution of the wave second mode with Chebyshev-Hermite profile for three different sizes of wave bottom.

405

406


Fig. 6.19 Evolution of the wave first mode with Whittaker profile for three different sizes of wave bottom.


Fig. 6.20 Evolution of the wave second mode with Whittaker profile for three different sizes of wave bottom.

407

408


Profiles of propagating in the composite material two waves (modes) distort weakly owing to the weak dispersion – the phase velocity changes essentially only in the small neighbourhood of the phase value. This is the value where the second mode appears and in which the obtained solution is already not valid. All these three stages can be seen for each choice of the wave bottom, but they have different durability. For the large bottom l = 100 the first stage displays on the distance of three bottom lengths. For the middle bottom l = 50 this distance permits to see the second stage also. The first two stages display on the distance of two bottom lengths for small bottom lengths l = 10 . The reason of the next four pictures is to show more in detail the character of evolution for different initial profiles.

x

Fig. 6.21 The cross-sections of the Whittaker wave profile for different time wave propagation. V (z)

1 4 1 2 1 0 8 6 4 2

z 0

1

2

3

Fig. 6.22 Phase velocities versus phase for material I and Whittaker profile.

409


x

Fig. 6.23 The cross-sections of the Chebyshev-Hermite wave profile for different time wave propagation.

V (z ) 5 4 3 2 1

z 0

0.5

1

1.5

2

2.5

Fig. 6.24 Phase velocities versus phase for material I and Chebyshev-Hermite profile.

Figures 6.21 – 6.24 involve the pictures of two different kinds: Figs. 6.21 and 6.23 show the cross-sections of the profiles for different time of propagation of the wave with Whittaker (Fig. 6.21) and Chebyshev-Hermite (Fig. 6.23) initial profiles with the small bottom l = 10 for the composite material MI and the first component of the material; Figs. 6.22 and 6.24 show the corresponding curves of dependence of phase velocities for both modes on the phase.

410


The above pictures permit to observe some distinctions in evolution of solitary waves with Whittaker and Chebyshev-Hermite initial profiles. Even for small values of the bottom length, when on the initial stages (two bottom lengths) of propagation the second mode should be visible, the waves with Whittaker and Chebyshev-Hermite initial profiles show different pictures. The second mode isn’t so visible for the Whittaker case as for the Chebyshev-Hermite one. This fact displays the dependence of the process of wave evolution on the initial profile of the wave.

6.4 Some Features of the Elastic Wavelets The starting point is the intention to use the Mexican hat wavelets and the fact that these wavelets are solutions of the elastic wave equations in structured materials. j The efficiency of using the family of MH-wavelets ψ j , k ( x) = ψ ( a x − uo k )

depends on the choice of the translation step uo and scale base a. It is convenient to choose uo and a so that the system {ψ j ,k ( x)} j ,k∈ℤ forms a tight frame

∑

f ,ψ j ,k

2

2

=A f

,

(6.18)

j , k∈ℤ

where f ( x) is a given function, A ≈ Cψ

( uo ln a )

is a frame bound.

The value Cψ is usually commented as the admissibility condition ∞

2 Cψ = ∫ (ψˆ (ω ) ) ω  d ω < ∞ .   0

In what follows that wavelets are binary with a half-step translation a = 2; uo = 0,5 , and with father wavelet

411

Solitary Waves and Elastic Wavelets 2

3 π −1 4 (1 − x 2 ) e − x 2 .

ψ ( x) = 2

(

(6.19)

)

Then Cψ = ( 4 3) π , A ≈ 6.819.

An approximate (over a finite number of wavelet scales from jo to joo )

representation

of

the

signal-function

in

the

form

2

F ( x ) = (1 − x 2 ) e

−

x 2

by the family (based on (6.19)) of

wavelets

ψ j ,k ( x) = 2 j 2 ψ ( 2 j x − 0,5k ) has the form j = joo

F ( x) ≈ (1 A )

∑∑d

ψ j ,k ( x),

j ,k

(6.20)

j = jo k∈Z

∞

d j ,k =

∫

f ( x)ψ j ,k ( x)dx.

(6.21)

−∞

Note also that the effective support (the interval of the concentration of the function mass) of Mexican hat wavelet (6.19) is assumed as the interval ( −5,5 ) . In computer modeling it is sufficient to integrate over interval k = ±20 rather than over the entire numerical axis. It also turns out that the scale levels from jo = −4 to joo = 4 are sufficient. The representation (6.20) with k ∈ {−20, −19,...,19, 20} , j ∈ {−4, −3,...,3, 4} was evaluated for the known function (6.19).

412

Wavelet and Wave Analysis F−1 F0

1

F1

0.75

F4

0.5

0.25

-4

-2

2

4

-0.25

-0.5

Fig. 6.25 Different approximations for the function under study.

Figure 6.25 shows various approximations corresponding to the transition from the coarse scale joo = −1 to a very fine scale joo = −4 and calculated by F−1 = F0 = F1 =

j − 1 −1 20 2 d 2 ∑ ∑ jk ψ ( 2− j x − 0.5k ) , A j =−4 k =−20 j − 1 0 20 2 d 2 ∑ ∑ jk ψ ( 2− j x − 0.5k ) , A j =−4 k =−20 j − 1 1 20 2 d 2 ψ ( 2− j x − 0.5k ) , ∑ ∑ jk A j =−4 k =−20

j − 1 4 20 2 F4 = ∑ ∑ d jk 2 ψ ( 2− j x − 0.5k ) . A j =−4 k =−20

The difference between the exact plot of the function (dotted line) and the approximate plot for joo = −4 (solid line) does not exceed 1% (see Fig. 6.26).

413

Solitary Waves and Elastic Wavelets 1

0.8

0.6

0.4

0.2 0 -4

-2

2

4

-0.2

-0.4

Fig. 6.26 Exact and approximate plots for the function F ( x) .

Let us return to the propagation of a solitary wave with the initial profile in the form of the Mexican hat father wavelet. The original idea of solving the problem in hand is to represent the initial wave profile in terms of Mexican hat wavelets F ( x) ≈ F4 ( x) =

j − 1 4 20 2 d 2 ψ ( 2− j x − 0.5k ) ∑ ∑ jk A j =−4 k =−20

(6.22)

on the assumption that the initial profile distorts weakly propagating through a weakly dispersive medium. The two-phase nature of the medium is the reason why each of the two original waves Bα F ( z )t = o ( z = x − v ph t ) splits into two similar waves. As a result, two waves with strongly different phase velocities simultaneously propagate in each phase of the mixture. The solution of this problem is similar to the solution for the case of Chebyshev-Hermite initial profile (5.74) 4

u (α ) ( x, t ) = (1 A ) ∑

20

∑ {Bα Fɶ α

( ,l )

j =−4 k =−20

+ p ( z (δ , j ) l ) Bδ Fɶ(δ ,l ) ,

}

Fɶ(α ,l ) = d j ,k 2 j 2ψ 2 j ( z (α , j ) l ) − 0,5k ,

(

)

(6.23)

(6.24)

414


z (α , j ) = x − v (phα , j ) t .

(6.25)

The weak variability of the wave profile means that it slightly distorts after traveling a distance equal to the wave bottom length. This is reflected in (6.23) as we assume the constant wavelet coefficients d jk (6.21). The changes in the profile are then caused by non-linearly varying phase velocities v (phα ) , only. The new parameter l is related to the bottom length of the initial pulse chosen for computer simulation. The characteristic structure length (CML) is known with the choice of specific composite materials. As it was discussed above, the procedure of comparing the length of harmonic waves and the CML helps to find the allowable frequency range or, more generally speaking, to establish the applicability domain of the microstructural model. The wavelength of a solitary wave is assumed as the length of its bottom. One more feature of the wavelet representation is that the different scale levels j in solution (6.23) are associated with different phase velocities v (phα ) and different coefficients of the amplitude matrix p ( z (α , j ) l ) v ph (α , j ) ( z ) = M 1( j ) − ( −1)

α

a1

2 M 1( j ) =

ρ11 M 2( j ) =

a1 a2

−

ρ11 ρ 22

p z

(

(α , j )

l

a2

a32

− M 2( j ) ;

a1 + a2 + 2a3

+

⋅ Z (l , j ,k ) ;

ρ11 ρ 22

aα − v 2 z (α , j ) ραα + Z (l , j ,k ) =

(6.26)

ρ11 + ρ 22 (l , j ,k ) ⋅Z ; ρ11 ρ 22

+

ρ 22

ρ11 ρ22

( −1)α

)

+

( j) 2 1

(M )

(

)

a3 − Z (l , j ,k )

;

(6.27)

415


22 j ⋅ β ⋅ l 2

Z (l , j ,k ) =

.

2

2

( 2− j ( z l ) − k ) −

−j

(2 ( z l ) − k )

2

−1

−5

Figures 6.27 and 6.28 show the plots of the phase velocities v (phα , j ) at scale

j = 0 and at the translation step

k = 0 and k =0 and j =1,

respectively. The dashed and solid lines correspond to the first and second modes, respectively. Figures 6.19 and 6.20 represent two composite materials with aluminium matrix and tungsten filaments. The volume fraction of tungsten fibers for the first material (material I) is ξ1 = 0,022 , for the second one (material II) is ξ 2 = 0.221 . The characteristics of the materials are detailed in Chapter 3. 6

4

2

0

0.002

0.006

0.01

0.014

Fig. 6.27 Phase velocities versus phase for material I.

416


4

2

0

0.001

0.003

0.005

0.007

Fig. 6.28 Phase velocities versus phase for material II.

The characteristic microstructure lengths of the materials (in meters): a = 0.648 ⋅ 10−3 , a II = 0.382 ⋅ 10−3 . The bottom lengths of waves exceed the material characteristic structure length by factors of 100, 50, and 10, i.e., the wave bottom lengths are equal to (in meters) I

0.648 ⋅ 10−1 , 0.324 ⋅ 10−1 , 0.648 ⋅ 10−2 , for material I, and 0.3825 ⋅ 10−1 , 0.191 ⋅ 10−1 , 0.3825 ⋅ 10−2 for material II. The coefficient l, which defines the wave bottom length, has the values l1I = 8.1 ⋅ 10−3 , l2I = 4 ⋅ 10−3 , l3I = 8.1 ⋅ 10−4 and l1II = l1II = 4.7 ⋅ 10−3 , l2II = 2.3 ⋅ 10−3 , l3II = 4.7 ⋅ 10−4 .

The next Figs. 6.29-6.32 show some three-dimensional evolution surfaces, generated by using formula (6.23), for solitary waves with three different bottom lengths in materials I (Figs. 6.29 and 6.30) and II (Figs. 6.31 and 6.32). In the first picture exceeding of the bottom length over the characteristic size of microstructure is equal 100, in the second one is equal 50, in the third one is equal 10.


417

Figures 6.29, 6.31 and 6.30, 6.32 represent the phase velocities in the first and second phases of the composite, respectively. In the figures, the distance x, in centimetres, traveled by the wave is laid off along the abscissa axis; the travel time t, in microseconds, along the ordinate axis; and the wave amplitude (profile) u (α ) ( x, t ) , in tenths of millimetres, along the z-axis. First, the role of the wave bottom length should be emphasized. When the wave bottom length is much greater than the CML, the wave propagates practically without dispersion, like a long harmonic wave in a dispersive medium. This can be observed in the four pictures with 100-fold excess of the wave bottom length over the CML. The influence of dispersion on wave propagation can already be observed in the subsequent four pictures (50-fold excess). The 10-fold excess in the final four pictures is almost critical, and the profile starts distorting immediately. In formulating the problem, it seemed very important to choose the initial pulse in the form of Mexican hat wavelet. The subsequent wavelet-based solution revealed the vital importance of Mexican hat wavelets for the solution procedure used. The fact that the Mexican hat wavelets belong to the family of elastic wavelets defined the solution procedure.

418


Fig. 6.29 Evolution of the wave first mode for material I for three different sizes of wave bottom.


Fig. 6.30 Evolution of the wave second mode for material I for three different sizes of wave bottom.

419

420


Fig. 6.31 Evolution of the wave first mode for material II for three different sizes of wave bottom.


Fig. 6.32 Evolution of the wave second mode for material II for three different sizes of wave bottom.

421

422


We find another advantage in that: the initial pulse can be chosen arbitrarily, except that it must be represented in the wavelet form and, naturally, must be a solitary one. The three-dimensional evolution surfaces are quite informative and indicative. They clearly demonstrate the two basic structural effects: 1. splitting of the wave into two modes propagating with different phase velocities, and 2. simultaneous propagation of both modes in both components of the composite. One more effect is manifested: the distance at which the profile starts distorting depends on the characteristic dimensions of the initial profile and microstructure.

6.5 Solitary Waves in Mechanical Experiments

The present subsection will demonstrate that we can consider the initial profile arbitrary (except it must be single and represented in terms of wavelets) provided that the profile is further approximated by Mexican hat wavelets that has the basic property of elastic wavelets – they are solutions of elastic wave equations. We will address a narrow class of solitary waves – impulses – wellknown in classical experimental solid mechanics. These impulses were observed in the classical impact experiments (to be described below) and may differ in duration (length) and intensity (amplitude). Let us remember that usual excitation of an initial impulse in an elastic two-phase mixture is mathematically interpreted as simultaneous excitation of two identical impulses in the two-phases of the mixture. The specific nature (two-phases) of the medium results in the splitting of each of the two initial waves into two similar waves. This splitting does not occur instantaneously: while propagating, initially one wave gradually transforms into two. As a result, simultaneously two waves

423


propagate in each phase of the mixture. It is this phenomenon that is being studied here. We will consider an initial impulse with a profile well known from classical experiments. Four main types of impulses arising in mechanical experiments are shown by Bell interpretation on Fig. 6.33 (as stress versus time). In mechanics, the durability of loading the body is very often such that reflection of waves from free surfaces of the body creates troubles for interpretation of observed dependences “displacement – time” or “ stress – time”. The cases when the order of impulse length or of the wavelength are commensurable with the sample size require care when making comments.

(а )

(b )

(c )

(d )

Fig. 6.33 Typical profiles of the impact wave in mechanical experiments.

424


Mechanics do not have the wide choice in experimental methods, in which the impulses are generated, the duration of which is essentially less than the duration of the wave traveling along the smallest size of the sample. They are shown in the volume of Flugge encyclopedia written by Bell and will be described below. Let us start with profiles obtained by Fanning and Basset in studying the history of deformation of cylindrical bars suspended from two strings and subjected to symmetric impact. The bars were fairly long (183cm), the end of one bar was flat, and the end of the other had the form of a spherical cap of known radius. Figure 6.33 shows the results of measurement at the specimen end subjected to impact. Hopkinson bars were widely used to determine the dynamic properties of materials under intensive loading. For example, Chiu and Neubert experimentally studied the effect of different impact velocities (from 40 to 80 inches per second).

Fig. 6.34 Typical profile of the impact wave in experiments of Fanning and Basset.

Figure 6.34 shows the pulses detected at the loading and support bars without a specimen.


425

Fig. 6.35 Typical profile of the impact wave in experiments of Chiu and Neubert.

Bell did a similar experiment on a specimen made of aluminium 1100. The pulse detected at the support bar is shown in Fig. 6.36. Similar pulses were detected in studying the growing waves.

Fig. 6.36 Typical profile of the impact wave in experiments of Bell.

Sternglass and Stewart dropped a hammer 4.7 lb in weight from 8-inch height on a shock platform attached to a plastic specimen and detected strain pulses, using electric strain gauges, at three different distances

426


from the platform. Figure 6.37 shows the strains measure data distance of 8

2

37 inches from the impact area for a quasistatic prestress of 2510 N/m . The strains exceeded 0.005.

Fig. 6.37 Typical profile of the impact wave in experiments of Sternglass and Stewart.

After that, a pulse close in shape to that shown in Figs.6.34 – 6.37 was selected as initial and then was approximated by Mexican hat wavelets. We will use Mexican hat wavelets to represent the initial pulse and to describe the variation in its shape during propagation as a solitary wave. As already mentioned, the Mexican hat wavelet is one of the solutions of the elastic wave equation within the basic model of elastic mixtures. A wavelet set ψ j , k ( x ) is known to be created by translating and scaling the mother function ψ ( x ) . Thanks to scaling the wavelets are capable of revealing differences in characteristics at different scales, and owing to translation they allow analyzing the properties of a signal at different points of the interval of interest. It should be noted once again that algorithms for restoration of signals using wavelet coefficients are numerically stable when wavelet sets

{ψ ( x )} j ,k

j , k∈ℤ

form frames. For

427


better convergence, it is necessary to select frames close to so-called tight frames. Mexican hat wavelets may be considered to form a special frame that tends to a tight frame under certain conditions. According to general wavelet theory, the approximation of a function f ( x ) by arbitrary wavelets that form a frame with a bound A (including

Mexican hat wavelets) can be expressed as a series in terms of wavelet functions (6.20) with wavelet coefficients djk (6.21). We will further assume that Mexican hat wavelets form a tight frame with A≈ 3.409. A function is frequently defined on a finite interval (signal of finite length). In wavelet analysis, this simplification is realized by retaining, in the sum over translations, only the terms that together cover the base of the signal. Formally, this means that the sum over k has a finite number of terms. Moreover, we may not use the whole set of wavelet coefficients djk, but only some of them, omitting coefficients that are less than some small quantity ε. This is the method of threshold coefficients. It is based on the fact that if a function is sufficiently smooth in its domain of definition, then many wavelet coefficients are very small. The axis of symmetry of the Mexican hat wavelet is the ordinate axis, and most of its weight is concentrated on the interval [–5,5]. This interval is usually called the effective support. The center of each wavelet ψ j , k ( x ) generated by the mother Mexican j

hat wavelet is the point x= 2 k, and the length of this wavelet support is j

proportional to the effective support with coefficient 2 . j

The general theory is based on the fact that each level of resolution 2 is associated with a set of wavelet coefficients, their position being determined by the centers of the wavelets ψ j , k ( x ) . The absolute value of each coefficient djk depends on the local regularity of the function f ( x ) .

428


How the wavelet coefficients decrease within the neighborhood of the j

point 2 k , the scale depends on the local smoothness of the function. The idea of applying Mexican hat wavelets to study the evolution of solitary elastic waves is to represent the initial wave profile in terms of these wavelets on the assumption that the wave initial profile changes insignificantly in a weakly dispersive elastic medium, i.e., the shape of the solitary wave practically does not change at a distance equal to the wavelength. The wavelet coefficients djk are assumed constant, and the profile changes due to nonlinear changes in the phase velocities appearing in the argument of Mexican hat wavelets. The idea of attributing the evolution of the wave profile with constant functional representation only to the change in the phase variable underlies the theory of simple waves. This fundamental idea is best reflected in the expression of the initial profile n ( x,0 ) = F ( x ) and in the implicit (the phase velocity depends on the solution) representation of a simple wave n ( x, t ) = F  x − v ( n ) t  . In this case, the profile of a solitary wave is also defined implicitly (the phase velocity depends on the phase) x → z = x − v ( z ) t . Naturally, this implicitness complicated the early software programs intended to analyze the evolution of a solitary wave and complicates the Mexican hat wavelet-based program proposed here.

1

-5

-3.5

0

3.5

5

-5

-3.5

0

3.5

5

u (x)

0.5

x Fig. 6.38 Exact and approximate graphs of the initial wave profile.

429


Let us turn to the Mexican hat wavelet approximation of the initial impulse. The shape of the impulse is identical to that shown in Fig. 6.33c, 6.34 – 6.37. It is shown in Fig. 6.38 by a solid line. The abscissa is a distance in meters, though this is conditional since the wavelengths of interest for the composites under consideration vary from 0.0648 to 0.00648 m. Real impulses used in the computer simulation are derived from impulse with a length of 10m by introducing a scaling coefficient l. The dashed line, which practically coincides with the solid line, represents the approximate shape of the pulse fε ( x ) . This approximation is obtained by discarding wavelet coefficients less than 10−4 in absolute value. Figure 6.39 shows the grid of the remaining wavelet coefficients.

2

j x

-15

- 10

-5

5

10

15

-2

-4

-6

Fig. 6.39 Grid of the remaining wavelet coefficients.

The summation over only the coefficients d j , k > 10−4 corresponds to the wavelet approximation of f ( x ) fε ( x ) ≈

1 jmax kmax ∑ ∑ d j ,kψ j ,k ( x ) , A j = jmin k = kmin

jmin = −9, jmax = 2, kmin = −14, kmax = 14.

(6.28)

430


The total number of coefficients is 348, and the number of discarded coefficients is 202. The estimate is then f ( x ) − fε ( x ) < 1.42 ⋅ 10−3 .

(6.29)

For reference, Fig. 6.38 shows rough approximations of function f ( x ) corresponding to sums over j from –9 to 0 and from –9 to 1.

The next step is to represent the solution as two displacements in the two components of the mixture in the form of the initial impulse with the wave phase as the argument u (α ) ( x, t ) =

1 2 14 ∑ ∑ d j ,kψ j ,k ( x − v pht ) . A j =−9 k =−14

(6.30)

This finite sum of wavelets is the solution of the original wave equations in the sense in which Mexican hat wavelets are solutions of these wave equations, i.e., under some constraints. However, each scale level of wavelets is characterized by two individual phase velocities (the phase velocity depends on the index j). It also depends on the wave mode since this is a necessary rule for mixtures. The phase velocity is not essentially dependent on the translation index k. An impulse propagating in a mixture is expressed as u (α ) ( x, t ) =

1 jmax kmax  (α ) ɶ ∑ ∑ B f(α ,l ) + p ( z (δ , j ) l ) B(δ ) fɶ(δ ,l ) , A j = jmin k = kmin 

(6.31)

jmin = −9, jmax = 2, kmin = −14, kmax = 14,

fɶ(α ,l ) = d j ,k 2 j 2ψ  2 j ( z (α , j ) l ) − k  , z (α , j ) = x − v (phα , j ) t.

The phase velocities v (phα , j ) are

(6.32)

431


v ph (

α , j)

(z) =

M 1( j ) − ( −1)

α

j

2

j

− M 2( ) ;

(M ( ) ) 1

2M 1( j ) = ( a1 ρ11 ) + ( a2 ρ 22 ) + Z (l , j ) ( ρ11 + ρ 22 ) ( ρ11 ρ 22 ) ;

M 2( j ) =

a1 a2

ρ11 ρ 22

−

a32

a1 + a2 + 2a3

+

ρ11 ρ 22

⋅ Z (l , j ) ;

2−2 j ⋅ β ⋅ l 2

Z (l , j ) = j

(2 ( z l ) − k )

.

2

2

−

(6.33)

ρ11 ρ 22

j

(2 ( z l ) − k )

2

−1

−5

Consider now a composite with aluminium matrix and tungsten fibers. The volume fraction of the tungsten wire ξ1 = 0.022 (this corresponds to above considered material MI). The typical length scale of the internal composite structure a I = 0.648 ⋅ 10−3 m . Let the initial wave lengths exceed the typical length scale by factors of 100, 50, and 10. The corresponding wave lengths are equal to 0.0648, 0.0324, and 0.00648, respectively, and the coefficient l characterizing the wavelength is the following : l100 = 6.48 ⋅ 10−3 m , l 50 = 3.24 ⋅ 10−3 m and l10 = 6.48 ⋅ 10−4 m . The amplitude coefficients B (α ) = B (δ ) = 10−4 . The result of the computer modeling based on the analytical expression (6.31) is curves demonstrating the evolution of the initial profile. These curves for the first phase of the composite are shown in Fig. 6.40. The profiles have been plotted for five sequential instants of time (in microseconds): t11=25, t12=51, t13=76, t14=100, t15=120; t21=12.5, t22=25, t23=37.5, t24=50, t25=62.5; t31=2.5, t32=5.1, t33=7.6, t34=10, t35=12.

In the case of long waves, the dispersion of the wave is not manifested immediately, which is similar to long harmonic waves. The second mode is blank, and the first one propagates without distortion (see Fig. 6.40).

432

Wavelet and Wave Analysis u ( x)

1

t11

t12

t13

t15

t14

0.5

x 0 0.064

0.129

0.194

0.259

0.32

Fig. 6.40 Evolution of the first mode for the large bottom.

u( x)

1

t21

t22

t23

t24

t25

0.5

x 0

0.0324

0.0648

0.0975

0.128

0.162

Fig. 6.41 Evolution of the second mode for the large bottom. 1

u( x)

t31

t32

t33

t34 t35

0.5

x 0 0.0064

0.0129

0.0194

0.0259

Fig. 6.42 Evolution of the first mode for the small bottom.

0.032

433


As the wavelength decreases, dispersion starts playing a vital part: the second mode affects the initial wave profile, distorting it (Fig. 6.41). For small wavelengths, the influence of the microstructure is manifested immediately. The initial wave profile breaks up into two modes, which then propagate with different phase velocities (Fig. 6.42). Mixture theory bounds the wavelength from below: the wavelength must exceed the typical length scale of the internal structure by a factor often. The evolution of a solitary wave includes three stages. At the first stage, the wave moves without distortion. At the second stage, the profile is distorted on the leading edge, and the wave breaks up into two modes. At the final stage, the two modes independently propagate with different velocities and amplitudes. The duration of each stage depends on the 100

wavelength. The first stage compares to three wavelengths for l , the 50

second stage to the same distance for l , and the first two stages to two 10

wavelengths for l . Thus wavelet analysis is applied to study the evolution of waves of special forms. An advantage of this analysis is that real impulses that do not have analytical representation can be expressed in terms of elastic wavelets, which do have analytical representation and are solutions of the system of wave equations for an elastic microstructural material. This approach considerably extends the class of analyzable initial pulses. The proposed method enables us to study arbitrarily shaped impulses, with no restrictions on the initial profile. Clearly, the requirement that the function describing the initial profile must be a solution of elastic wave equation system drastically reduced the range of resolvable problems. The approximate solution obtained is represented by the sum of solutions of wave equations, and hence, is a solution itself, as well as Mexican hat wavelets in the sum are solutions of the corresponding system. Note that the approximation of this solution is due to the use of wavelets and a finite number of terms in the sum. The number of terms in the series is determined from the condition that the initial pulse is –3

approximated with an accuracy of 10 . Figures 6.43 and 6.44 show the same impulse with different initial approximations at three different

434


instants of time: t1 = 3µ sec , t2 = 7 µ sec , and t3 = 10 µ sec . The solid line corresponds to the exact solution with the initial approximation fε ( x ) dashed lines to solutions with the more coarse approximations f1 ( x ) ≈


jmin = −9, jmax = 0, kmin = −14, kmax = 14. f2 ( x ) ≈


jmin = −9, jmax = 1, kmin = −14, kmax = 14. n=4

n =6

n=8

n=2

Fig. 6.43 Evolution of the first mode profile for different bottom lengths.

n=8 n=6 n=4 n=2

Fig. 6.44 Evolution of the second mode profile for different bottom lengths.


435

The plots indicate that the accuracy of the approximate solution depends on how accurately the initial profile is approximated and does not deteriorate with time.

6.6 Ability of Wavelets in Detecting the Profile Features Many mechanicians have carried out experiments using an explosion and observed very similar pictures for wave profile. For example Kolsky with coworkers used samples made of plastics and glass. The samples made of Perspex have the small phase velocities of shear waves 2.0 km/sec and the length of impulse in 2 µ sec was 4 mm only whereas the stress is 1.0 MPa . Corresponding profile is shown in Fig. 6.45.

Fig. 6.45 The typical profile in mechanical experiments with explosion.

Similar profiles were also observed in experiments with thin aluminium plates. Further, two types of initial impulses (very close to mentioned above experimental profiles for aluminium plates) with identical load 217 kg and duration 100 µ sec but with different duration of loading increase 10 µ sec and 50 µ sec will be analyzed.

436


The problem of evolution of such initial profiles is considered within the same approach used in prior sections of this Chapter. Thus, at the moment t = 0 the special initial profile of displacement in the form of triangle is given. It is shown in Fig. 6.46. The first step consists in the representation of the profile by elastic wavelets. In order to ensure exactness, the practically tight frame is used with intermediate scales u0 = 0.5 and ν = 2 . Then the frame bond can be found approximately A ≈ 13.656 .

F 217

t 10

50

100

Fig. 6.46 Two initial profiles used in the next computer simulation.

The basic wavelet representation is as follows f ( x) ≈

1 ∞ ∞ j 2 j  ∑ ∑ d j ,k 2 ψ ( 2 x − 0.5k ) A  j =−∞ k =−∞  + d ′j ,k 2 2 ψ 2 ( 2 x − 0.5k )  ,  j 2 12

(

12

j

)

∞

d j ,k =

∫2

j 2

f ( x)ψ ( 2 j x − 0.5k ) dx,

−∞ ∞

d ′j ,k =

∫2

−∞

j 2 12

2

f ( x)ψ 21 2 ( 2 j x − 0.5k ) dx ,

(

)

(6.34)

437


where ψ j ,k ( x) = 2 j 2ψ ( 2 j x − 0.5k ) is the family of Mexican hat wavelets. The next step should be done in approximate representation of solution (6.34), when small wavelet coefficients d j , k > ε , ε = 10−3 are neglected and infinite summation is substituted by the finite one. Then u ( x ) = fε ( x ) ≈

1 jmax kmax ∑ ∑ d j ,k 2 j 2ψ ( 2 j x − 0.5k ) A j = jmin k = kmin

+ d ′j ,k 2 j 2 21 2ψ 21 2 ( 2 j x − 0.5k ) .

(

(6.35)

)

jmin = −14, jmax = 7, kmin = −20, kmax = 26. Figure 6.47 shows the lattice of saved wavelet coefficients. Here the voice approximation is utilized in the way that the intermediate voices

ψ 21 2 ( 2 j x − 0.5k )

(

)

have,

as

the

wavelets,

the

centre

points

x = 0.5 ⋅ 2− j k and supports with sizes proportional to the effective support with coefficient 21 2 2− j (the base wavelets have coefficients 2− j ). j 9 6 3

x -1

0.5

-0.5

1

-3

Fig. 6.47 Grid of the saved wavelet coefficients with values greater than 10−3 .

438


B ( )u ( x ) α

4.5 ⋅10−4

2.25⋅10−4

x 4,5

0

Fig. 6.48 Exact (solid line) and approximate (dashed line) graphs of the profile.

In this case the pyramid of coefficients is slightly distinguished from before. It displays the local features of the function and additionally includes the voice coefficients that enables us to focus an attention on the upper angle. jmax=10

0.45

jmax=9 jmax=8

0.5 u(x) 0.446

0.4 0.3

0.442 -0.005

0.2

0.005 0.01

0.1 -1

0.015 0.02

x 1

2

3

4

5

Fig. 6.49. Approximations of the profile near the angle point for different jmax = 8,9,10 .

439


Figure 6.48 shows the exact (solid line) and approximate (dashed line), i.e., renewed by wavelet coefficient on the base of equation (6.35) graphs of the initial profiles. The difference between lines don’t exceed 1.5 ⋅ 10−3 ⋅ B (α ,δ ) . It should be noted that in the computer modeling in hand the material MI was used with the characteristic size of structure a I = 0.648 ⋅ 10−3 m . Then the minimum admissible bottom length should be 4.54mm and the initial

amplitude

should

be

assumed

as

ten

times

less

−4

= B = 4,54 ⋅ 10 m . An approximate solution is looking for the form like prior cases of initial profiles B

(α )

(δ )

u (α ) ( x , t ) =

1 jmax kmax ∑ ∑ d j ,k 2 j 2ψ  2 j x − v(phα , j )t − 0.5k  A j = jmin k = kmin

(

)

(6.36)

+ d ′j ,k 21 2 2 j 2ψ  21 2 2 j x − v(phα , j )t − 0.5k  .  

( (

)

)

jmin = −14, jmax = 7, kmin = −20, kmax = 26. The formulas of dependences of the phase velocities v (phα , j ) on the phase z (

α , j)

α, j

= x − v (ph )t and coefficients of the amplitude matrix are

identical with the prior case, too. The approximate solution is assumed as follows u (α ) ( x , t ) =

1 jmax kmax α δ B ( ) fɶ(α ,l ) + p ( z (δ , j ) l ) B( ) fɶ(δ ,l ) , ∑ ∑ A j = jmin k = kmin

(

)

(6.37)

440


fɶ(α ,l ) = d j ,k 2 j 2ψ 2 j z (α , j ) − 0.5k

(

)

+ d ′j ,k 21 2 2 j 2ψ 21 2 2 j z (α , j ) − 0.5k ,

( (

))

jmin = −14, jmax = 7, kmin = −20, kmax = 26.

Further the results of modeling of evolution for three different shapes of the initial triangle profiles will be shown. The profiles have identical bottom length and differ by different duration of loading increase. The ratio of increasing and decreasing parts in profiles is assumed as xincr xdecr = 0.1;0.2;1.0. Summation in scale level and translation will be different: for the first profile jmin = −14, jmax = 7, kmin = −20, kmax = 26; for the second profile jmin = −7, jmax = 13, kmin = −24, kmax = 25; for the third profile jmin = −6, jmax = 16, kmin = −22, kmax = 22. On all pictures, the abscissa axis corresponds to the passed-by wave distance in cm, the ordinate axis, the time of propagation in µ sec , the applicate axis, the wave amplitude (wave profile) u (α ) ( x, t ) in 10−3 m .


Fig. 6.50 Evolution of the triangle initial profile with xincr xdecr = 0.1;0.2;1.0.

441

442


The next Fig. 6.51 shows three cross-sections of the triangle profile xincr xdecr = 0.2 corresponding t1 = 1, t2 = 5, t3 = 10 µ sec . The main goal here is to show the splitting of one triangle into two.

u (x )

x

Fig. 6.51 Three sequential in time cross-sections of the triangle profile xincr xdecr = 0.2 .

Thus the technique of wavelet representation of solitary elastic waves is shown. It enables us to analyze different experimental impulses, which do not have analytical representation. The profiles with arbitrary shape can be analyzed by using the proposed technique and studied with the evolution of the solitary waves with different singular shapes with high exactness. It should be mentioned that we started with profiles in the analytical form of different special functions of mathematical physics, then the profiles in the analytical form of Gauss functions were analyzed and finally the shock and explosion wave profiles were considered using the technique of elastic wavelets.

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Index

Achenbach, 187 aggregation, 147, 148, 149 gaseous, 147 liquid, 148 plasmic, 147 solid, 148 atoms, 38, 73

D’Alembert solution, 341 deformation, 149, 191, 201, 216, 218 density, 233, 240, 266-268 volume, 266 derivative local, 267 material, 267 Dirac, 52, 61 function, 50 Du Bois-Reymond, 38 Drumheller, 188 dispersion law, 362 dyadic, 42, 43, 119, 144, blocks, 41 functions, 100 intervals, 33, 118 levels, 41 rational numbers, 92 wavelet, 144 wavelet transform, 145 wavelets, 144

basis biorthogonal, 19,65,74 Riesz, 65,80 Schauder, 40 orthogonal, 63 orthonormal, 63 biorthogonal, 74 unconditional, 79,81 conditional, 82 Shannon, 131 Bedford, 188, 189, 198, 203 body, 149, 155, 197, 201, 221 Bolotin, 185, 186 Cesaro, 25 chemical potential, 169 chirp-signal, 381, 382 Christensen, 181 circular frequency, 236 Clausius inequality, 166 coiflet, 129 concentrations, 147, 169 constants Lamé, 176 crystalline, 148 convolution, 47, 48 criterion Dini-Lipschitz, 26 Dirichlet-Jordan, 26

elasticity, 157 elastoplasticity, 158 electroelasticity, 161 equation balance, 262, 265, 269 Christoffel, 240 constitutive, 282 Helmholtz, 248 Klein-Gordon, 361 Lamé, 237 Weber, 370 Eringen, 192, 222

455

456


fibrous, 175, 178, 182, 187, 205, 207, 228 first law of thermodynamics, 164 Fourier, series, 21 transform, 44 trigonometric series, 41 frame, 60, 114 exact, 72 loose, 72 snug, 74 tight, 72 function Bessel, 352 cardinal, 71 Chebyshev - Hermite, 348 energy of, 73 Error, 352 Haar, 26, 33, 34 Hankel, 244 Mathieu, 354 Rademacher, 26 spectral density of, 50 Walsh, 26, 28 Whittaker, 350 Gabor, 57 Gauss, 52, 54, 71 Gauss function, 50 Gramm-Schmidt procedure, 64 Green tensor of deformation, 192 Haar, 33, 42, 92, 103, 117, 121, 125 scaling function, 33 system, 33 wavelets, 35 Hardy, 17, 49, 141, space,17 Heisenberg, 44, 141 Herrmann, 187 homogenization, 156 Hooke law, 174 inequality Bessel, 22 Clausius, 167 Planck, 167 internal energy, 162, 173, 201, 220 Laplace operator, 244

Lebesgue, 15, 20 Littlewood-Paley atoms, 43 function, 43 macro, 149, magnetoelasticity, 161 materials, 147, 149,157, 223 composite, 170 fibrous, 172 granular, 172 isotropic, 176 layered, 172 orthotropic, 174 transversal isotropic, 175 Maugin, 192 McNiven – Mengi, 209 meso, 149, 171 micro, 149, 191, 225, 230, 232 microstrain tensors, 192 Mindlin, 191, 192, 193 mixture, 170, 193, 206 isotropic, 212 orthotropic, 213 transversal isotropic, 212 model, 249, 330, 343 Guz, 279 Murnaghan, 278 Rivlin and Saunders, 277 Seth, 275 Signorini, 275 MRA, Multiresolution Analysis, 83 Murnaghan, 279, 330, 332, 341-343 nano, 150, 234 nanomechanics, 147, 150, 171 Newland, 382, 384, 386 nondispersive, 243 Parseval, 49, 77, 114 equality, 49 identity, 49 phase, 231, 316, 331 piezoelastic, 219 piezoelectric, 193 plasmic, 147 plasticity, 158

Index Pobedrya, 194 Poisson ratio, 176, 228, 229, 232 polynomials Chebyshev - Hermite, 348 power, 270, 334 Rademacher, 26, 33 redundant ratio, 73 Reuss, 178, 179, 183 rigidplasticity, 158 Riemann, 337, invariants, 345 Shermergor, 182 Signorini, 275, 314, 341, space, L2 ( \ ) , 15

Lp ( \ ) ( p ≥ 1) , 17

H p ( \ ) ( p ≥ 1) ,17 symbols, Christoffel, 262 system, 233, 317 Eulerian, 257 Lagrangian, 257 tensor, Almansi strain, 257, 260 Cauchy-Green, 257, 260 Christoffels, 240, 259 Green, 257, 260 Levi-Civita, 269 metric, 256, 258 Piola Kirchoff, 265 stress, 263, 265 theorem, Cauchy, 70 convolution, 48 Daubechies, 125 derivative of convolution, 48 Littlewood-Paley, 42 Meyer-Strömberg ,143 Plancherel, 49 product or convolution, 48 Riesz-Fischer, 22 Shannon, 69 Wojtaszczyk, 81

457

thermoelasticity, 159 thermoplasticity, 159 triangle function, 94, 102 Vanin, 180, vector, displacement, 260 virial, 181 viscoplasticity, 159, 160 viscosity, 158 Voigt, 178, 183 Walsh, 26, 33 wave, cylindrical, 290 D’Alembert, 347, 378 dispersive, 235, 362 elastic, 378 equation, 232 harmonic, 362, 377 length, 234 Love, 379 nondispersive, 235 number, 234 simple, 338, 346 simple elastic, 383 solitary, 347, 381 wavelet, Battle-Lemarié,133 Cohen-Daubechies-Feauveau, 131 Coifman, 128 Daubechies, 125 Daubechies-Jaffard-Journé, 123 dyadic, 144 elastic, 386 Gabor, 123 Gabor-Malvar, 124 Geronimo-Hardin-Massopust, 132 Grossmann-Morlet, 126 Haar, 102, 117 harmonic, 382 Malvar-Meyer-Coifman, 130 Mexican hat, 127, 387 Shannon, 130, 383 Strömberg, 120 window, 53,

458


Blackmann, 56 Gabor, 57 discrete, 59 Hamming, 56 Hanning, 57 properties, 57 rectangular, 55 transform, 53 triangle, 55 Young modulus, 171, 217, 228, 233, 235

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