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

Preface     V
Introduction     1
Wavelet Analysis     13
Wavelet and Wavelet Analysis. Preliminary Notion     13
The space L[superscript 2](R)     15
The spaces L[superscript p](R)(p[greater than or equal]1)     16
The Hardy spaces H[superscript p](R)(p[greater than or equal]1)     17
The sketch scheme of wavelet analysis     18
Rademacher, Walsh and Haar Functions     26
System of Rademacher functions     26
System of Walsh functions     28
System of Haar functions     32
Integral Fourier Transform. Heisenberg Uncertainty Principle     44
Window Transform. Resolution     52
Examples of window functions     54
Properties of the window Fourier transform     57
Discretization and discrete window Fourier transform     59
Bases. Orthogonal Bases. Biorthogonal Bases     63
Frames. Conditional and Unconditional Bases     71
Wojtaszczyk's definition of unconditional basis (1997)     81
Meyer's definition of unconditional basis (1997)     82
Donoho's definition of unconditional basis (1993)     82
Definition of conditional basis     82
MultiresolutionAnalysis     83
Decomposition of the Space L[superscript 2](R)     95
Discrete Wavelet Transform. Analysis and Synthesis     109
Analysis: transition from the fine scale to the coarse scale     111
Synthesis: transition from the coarse scale to the fine scale     113
Wavelet Families     116
Haar wavelet     117
Stromberg wavelet     120
Gabor wavelet     123
Daubechies-Jaffard-Journe wavelet     123
Gabor-Malvar wavelet     124
Daubechies wavelet     125
Grossmann-Morlet wavelet     126
Mexican hat wavelet     127
Coifman wavelet - coiflet     128
Malvar-Meyer-Coifman wavelet     130
Shannon wavelet or sinc-wavelet     130
Cohen-Daubechies-Feauveau wavelet     131
Geronimo-Hardin-Massopust wavelet     132
Battle-Lemarie wavelet     133
Integral Wavelet Transform     137
Definition of the wavelet transform     137
Fourier transform of the wavelet     138
The property of resolution     139
Complex-value wavelets and their properties     141
The main properties of wavelet transform      141
Discretization of the wavelet transform     142
Orthogonal wavelets     143
Dyadic wavelets and dyadic wavelet transform     144
Equation of the function (signal) energy balance     144
Materials with Micro- or Nanostructure     147
Macro-, Meso-, Micro-, and Nanomechanics     147
Main Physical Properties of Materials     156
Thermodynamical Theory of Material Continua     160
Composite Materials     168
Classical Model of Macroscopic (Effective) Moduli     174
Other Microstructural Models     181
Bolotin model of energy continualization     182
Achenbach-Hermann model of effective stiffness     183
Models of effective stiffness of high orders     184
Asymptotic models of high orders     185
Drumheller-Bedford lattice microstructural models     186
Mindlin microstructural theory     187
Eringen microstructural model. Eringen-Maugin model     188
Pobedrya microstructural theory     190
Structural Model of Elastic Mixtures     191
Viscoelastic mixtures     210
Piezoelastic mixtures     213
Computer Modelling Data on Micro- and Nanocomposites     216
Waves in Materials     229
Waves Around the World     229
Analysis of Waves in Linearly Deformed Elastic Materials     232
Volume and shear elastic waves in the classical approach     232
Plane elastic harmonic waves in the classical approach     237
Cylindrical elastic waves in the classical approach     241
Volume and shear elastic waves in the nonclassical approach     244
Plane elastic harmonic waves in the nonclassical approach     247
Analysis of Waves in Nonlinearly Deformed Elastic Materials     253
Basic notions of the nonlinear theory of elasticity. Strains     253
Forces and stresses     260
Balance equations     262
Nonlinear elastic isotropic materials. Elastic Potentials     267
Nonlinear Wave Equations     276
Nonlinear wave equations for plane waves. Methods of solving     276
Method of successive approximations     281
Method of slowly varying amplitudes     283
Nonlinear wave equations for cylindrical waves     285
Comparison of Murnaghan and Signorini Approaches     308
Comparison of some results for plane waves     308
Comparison of cylindrical and plane wave in the Murnaghan model     322
Simple and Solitary Waves in Materials     337
Simple (Riemann) Waves     337
Simple waves in nonlinear acoustics     337
Simple waves in fluids     340
Simple waves in the general theory of waves     344
Simple waves in mechanics of electromagnetic continua     345
Solitary Elastic Waves in Composite Materials     346
Simple solitary waves in materials     346
Chebyshev-Hermite functions     347
Whittaker functions     349
Mathieu functions     352
Interaction of simple waves. Self-generation     353
The solitary wave analysis     359
New Hierarchy of Elastic Waves in Materials     373
Classical harmonic waves (periodic, nondispersive)     374
Classical arbitrary elastic waves (D'Alembert waves)     374
Classical harmonic elastic waves (periodic, dispersive)     375
Nonperiodic elastic solitary waves (with the phase velocity depending on the phase)     377
Simple elastic waves (with the phase velocity depending on the amplitude)     379
Solitary Waves and Elastic Wavelets     381
Elastic Wavelets     381
The Link between the Trough Length and the Characteristic Length     391
Initial Profiles as Chebyshev-Hermite and Whittaker Functions     396
Some Features of the Elastic Wavelets     410
Solitary Waves in Mechanical Experiments     422
Ability of Wavelets in Detecting the Profile Features     435
Bibliography     443
Index     455