Wavelet and wave analysis as applied to materials with micro or nanostructure / Carlo Cattani, Jeremiah Rushchitsky.

Format:

Book

Author/Creator:

Cattani, Carlo, 1954-

Contributor:

Rushchit︠s︡kiĭ, I︠A︡. I︠A︡. (I︠A︡rema I︠A︡roslavovich)

Series:

Series on advances in mathematics for applied sciences ; v. 74.

Series on advances in mathematics for applied sciences ; v. 74

Language:

English

Subjects (All):

Wavelets (Mathematics).

Nanostructures--Mathematics.

Nanostructures.

Mathematics.

Physical Description:

x, 458 pages : illustrations ; 24 cm.

Place of Publication:

Hackensack, NJ : World Scientific Pub. Co., [2007]

Contents:

2.1 Wavelet and Wavelet Analysis. Preliminary Notion 13

2.1.1 The space L[superscript 2](R) 15

2.1.2 The spaces L[superscript p](R)(p[greater than or equal]1) 16

2.1.3 The Hardy spaces H[superscript p](R)(p[greater than or equal]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

2.7 Multiresolution Analysis 83

2.8 Decomposition of the Space L[superscript 2](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 Stromberg wavelet 120

2.10.3 Gabor wavelet 123

2.10.4 Daubechies-Jaffard-Journe 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-Lemarie 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

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

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.

Notes:

Includes bibliographical references (pages 443-454) and index.

ISBN:

9789812707840

9812707840

OCLC:

123485620