This seminal book unites three different areas of modern science: the micromechanics and nanomechanics of composite materials; wavelet analysis as applied to physical problems; and the propagation of a new type of solitary wave in composite materials, nonlinear waves. Each of the three areas is described in a simple and understandable form, focusing on the many perspectives of the links among the three. All of the techniques and procedures are described here in the clearest and most open form, enabling the reader to quickly learn and use them when faced with the new and more advanced problems that are proposed in this book. By combining these new scientific concepts into a unitary model and enlightening readers on this pioneering field of research, readers will hopefully be inspired to explore the more advanced aspects of this promising scientific direction. The application of wavelet analysis to nanomaterials and waves in nanocomposites can be very appealing to both specialists working on theoretical developments in wavelets as well as specialists applying these methods and experiments in the mechanics of materials.
Author(s): Carlo Cattani, Jeremiah Rushchitsky
Series: Series on Advances in Mathematics for Applied Sciences
Edition: illustrated edition
Publisher: World Scientific Pub Co (
Year: 2007
Language: English
Pages: 473
Tags: Специальные дисциплины;Наноматериалы и нанотехнологии;Методы исследования наноматериалов;
Contents......Page 12
Preface......Page 10
1. Introduction......Page 16
2.1 Wavelet and Wavelet Analysis. Preliminary Notion......Page 28
2.1.1 The space L 2 (R)......Page 30
2.1.2 The spaces L p (R) (p = 1)......Page 31
2.1.3 The Hardy spaces H p ( R) (p = 1)......Page 32
2.1.4 The sketch scheme of wavelet analysis......Page 33
2.2.1 System of Rademacher functions......Page 41
2.2.2 System of Walsh functions......Page 43
2.2.3 System of Haar functions......Page 47
2.3 Integral Fourier Transform. Heisenberg Uncertainty Principle......Page 59
2.4 Window Transform. Resolution......Page 67
2.4.1 Examples of window functions......Page 69
2.4.2 Properties of the window Fourier transform......Page 72
2.4.3 Discretization and discrete window Fourier transform......Page 74
2.5 Bases. Orthogonal Bases. Biorthogonal Bases......Page 78
2.6 Frames. Conditional and Unconditional Bases......Page 86
2.6.1 Wojtaszczyk’s definition of unconditional basis (1997)......Page 96
2.6.4 Definition of conditional basis......Page 97
2.7 Multiresolution Analysis......Page 98
2.8 Decomposition of the Space L 2 (R)......Page 110
2.9 Discrete Wavelet Transform. Analysis and Synthesis......Page 124
2.9.1 Analysis: transition from the fine scale to the coarse scale......Page 126
2.9.2 Synthesis: transition from the coarse scale to the fine scale......Page 128
2.10 Wavelet Families......Page 131
2.10.1 Haar wavelet......Page 132
2.10.2 Strömberg wavelet......Page 135
2.10.4 Daubechies-Jaffard-Journé wavelet......Page 138
2.10.5 Gabor-Malvar wavelet......Page 139
2.10.6 Daubechies wavelet......Page 140
2.10.7 Grossmann-Morlet wavelet......Page 141
2.10.8 Mexican hat wavelet......Page 142
2.10.9 Coifman wavelet – coiflet......Page 143
2.10.11 Shannon wavelet or sinc-wavelet......Page 145
2.10.12 Cohen-Daubechies-Feauveau wavelet......Page 146
2.10.13 Geronimo-Hardin-Massopust wavelet......Page 147
2.10.14 Battle-Lemarié wavelet......Page 148
2.11.1 Definition of the wavelet transform......Page 152
2.11.2 Fourier transform of the wavelet......Page 153
2.11.3 The property of resolution......Page 154
2.11.5 The main properties of wavelet transform......Page 156
2.11.6 Discretization of the wavelet transform......Page 157
2.11.7 Orthogonal wavelets......Page 158
2.11.9 Equation of the function (signal) energy balance......Page 159
3.1 Macro-, Meso-, Micro-, and Nanomechanics......Page 162
3.2 Main Physical Properties of Materials......Page 171
3.3 Thermodynamical Theory of Material Continua......Page 175
3.4 Composite Materials......Page 183
3.5 Classical Model of Macroscopic (Effective) Moduli......Page 189
3.6 Other Microstructural Models......Page 196
3.6.1 Bolotin model of energy continualization......Page 197
3.6.2 Achenbach-Hermann model of effective stiffness......Page 198
3.6.3 Models of effective stiffness of high orders......Page 199
3.6.4 Asymptotic models of high orders......Page 200
3.6.5 Drumheller-Bedford lattice microstructural models......Page 201
3.6.6 Mindlin microstructural theory......Page 202
3.6.7 Eringen microstructural model. Eringen-Maugin model......Page 203
3.6.8 Pobedrya microstructural theory......Page 205
3.7 Structural Model of Elastic Mixtures......Page 206
3.7.1 Viscoelastic mixtures......Page 225
3.7.2 Piezoelastic mixtures......Page 228
3.8 Computer Modelling Data on Micro- and Nanocomposites......Page 231
4.1 Waves Around the World......Page 244
4.2.1 Volume and shear elastic waves in the classical approach......Page 247
4.2.2 Plane elastic harmonic waves in the classical approach......Page 252
4.2.3 Cylindrical elastic waves in the classical approach......Page 256
4.2.4 Volume and shear elastic waves in the nonclassical approach......Page 259
4.2.5 Plane elastic harmonic waves in the nonclassical approach......Page 262
4.3.1 Basic notions of the nonlinear theory of elasticity. Strains......Page 268
4.3.2 Forces and stresses......Page 275
4.3.3 Balance equations......Page 277
4.3.4 Nonlinear elastic isotropic materials. Elastic Potentials......Page 282
4.4.1 Nonlinear wave equations for plane waves. Methods of solving......Page 291
4.4.1.1 Method of successive approximations......Page 296
4.4.1.2 Method of slowly varying amplitudes......Page 298
4.4.2 Nonlinear wave equations for cylindrical waves......Page 300
4.5.1 Comparison of some results for plane waves......Page 323
4.5.2 Comparison of cylindrical and plane wave in the Murnaghan model......Page 337
5.1.1 Simple waves in nonlinear acoustics......Page 352
5.1.2 Simple waves in fluids......Page 355
5.1.3 Simple waves in the general theory of waves......Page 359
5.1.4 Simple waves in mechanics of electromagnetic continua......Page 360
5.2.1 Simple solitary waves in materials......Page 361
5.2.2 Chebyshev-Hermite functions......Page 362
5.2.3 Whittaker functions......Page 364
5.2.4 Mathieu functions......Page 367
5.2.5 Interaction of simple waves. Self-generation......Page 368
5.2.6 The solitary wave analysis......Page 374
5.3 New Hierarchy of Elastic Waves in Materials......Page 388
5.3.2 Classical arbitrary elastic waves (D’Alembert waves)......Page 389
5.3.3 Classical harmonic elastic waves (periodic, dispersive)......Page 390
5.3.4 Nonperiodic elastic solitary waves (with the phase velocity depending on the phase)......Page 392
5.3.5 Simple elastic waves (with the phase velocity depending on the amplitude)......Page 394
6.1 Elastic Wavelets......Page 396
6.2 The Link between the Trough Length and the Characteristic Length......Page 406
6.3 Initial Profiles as Chebyshev-Hermite and Whittaker Functions......Page 411
6.4 Some Features of the Elastic Wavelets......Page 425
6.5 Solitary Waves in Mechanical Experiments......Page 437
6.6 Ability of Wavelets in Detecting the Profile Features......Page 450
Bibliography......Page 458
Index......Page 470
