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 |  موضوع: كتاب Computational Techniques for Structural Health Monitoring الأربعاء 25 أغسطس 2021, 4:48 pm | |
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Computational Techniques for Structural Health Monitoring
Srinivasan Gopalakrishnan, Massimo Ruzzene, Sathyanarayana Hanagud
و المحتوى كما يلي :
Contents
Part I Introductory Concepts in Structural Health Monitoring
1 Introduction 3
1.1 Overview on Structural Health Monitoring 3
1.1.1 Why Do We Need Structural Health Monitoring? . 3
1.1.2 Basic Elements of SHM Systems . 5
1.1.3 Levels of Structural Health Monitoring . 9
1.1.4 State-of-Art and Technological Needs 10
1.2 Dynamics-Based Structural Health Monitoring 14
1.2.1 Passive SHM . 15
1.2.2 Classification of Inspection Techniques Based
on Frequency Range of Analysis . 16
1.2.3 Vibration-Based Techniques 17
1.2.4 Guided Waves Inspection 19
1.2.5 Ultrasonics and Nonlinear Ultrasound 20
1.3 Sensing and Actuation Strategies . 21
1.3.1 Piezoelectric Actuators and Sensors . 21
1.3.2 Fiber Optics Sensors 25
1.3.3 Laser Vibrometer 29
1.4 Modeling and Simulation Techniques for SHM . 31
1.4.1 The Importance of Modeling in Structural Health
Monitoring 31
1.4.2 Finite Difference Techniques 32
1.4.3 Finite Element Method 33
1.4.4 Boundary Element Method . 34
1.4.5 Spectral Finite Element Method 34
1.4.6 Perturbation Techniques . 36
1.5 Organization of the Book 37
References . 38
ix2 Fundamentals Concepts in Elasticity, Mechanics
and Wave Propagation 41
2.1 Introduction 41
2.2 Basic Concepts in Elasticity . 41
2.2.1 Description of Motion . 41
2.2.2 Strain 44
2.2.3 Strain–Displacement Relations . 46
2.2.4 Stress 48
2.2.5 Constitutive Relations . 50
2.2.6 Elastic Symmetry 52
2.3 Governing Equations of Motion and the Solution Methods . 54
2.3.1 Solution Procedures in Linear Theory of Elasticity . 56
2.3.2 Plane Problems in Elasticity 59
2.4 Introduction to Theory of Composites 60
2.4.1 Theory of Laminated Composites . 60
2.4.2 Stress–Strain Relation for a Lamina with
Arbitrary Orientation of Fibers . 66
2.5 Introduction to Wave Propagation in Structures . 69
2.5.1 Spectral Analysis 70
2.6 Characteristics of Waves in Anisotropic Media . 74
2.7 Governing Equations for Beams and Plates 75
2.7.1 Governing Equation for an Elementary Beam 76
2.7.2 Governing Differential Equation
for a Higher Order Beam 77
2.7.3 Governing Equations for a Composite Plate 79
2.8 Spectrum and Dispersion Relations 81
2.8.1 Efficient Computation of the Wavenumber
and Wave Amplitude . 81
2.8.2 Spectrum and Dispersion Relation for an
Elementary Beam 84
2.8.3 Spectrum and Dispersion Relation for a
Higher Order Beam 86
2.8.4 Spectrum and Dispersion Relation for an
Anisotropic Plate 89
References . 94
3 Signal Processing Techniques . 97
3.1 Integral Transforms 97
3.1.1 Fourier Transforms . 97
3.1.2 Fourier Series 99
3.1.3 Discrete Fourier Transform . 101
3.1.4 Wavelet Transforms 103
3.1.5 Wavelet-Based Numerical Solutions
of Wave Equations . 109
x Contents3.1.6 Comparative Advantages and Disadvantages
of Different Transforms 109
3.2 Signal Processing Issues . 110
3.2.1 Wraparound Problems . 110
3.2.2 Signal Processing of Sampled Waveforms . 116
3.2.3 Artificial Dispersion in Wavelet Transform 117
3.2.4 Excitation Signals and Wave Dispersion 126
3.3 Frequency/Wavenumber Analysis . 128
3.3.1 Analysis of a One-Dimensional Propagating Wave . 130
3.3.2 Analysis of 2D Wave Propagation 134
3.3.3 Numerical Examples: Wave Propagation
in a Damaged Rod . 138
3.3.4 Numerical Examples: Wave Propagation
in a Homogeneous Medium . 143
3.3.5 Frequency/Wavenumber Filtering
for Mode Separation 144
References . 153
Part II Computational Simulation Techniques
for Structural Health Monitoring
4 Application of the Finite Element Method in SHM . 157
4.1 Overview and Basic Principles . 157
4.2 Modeling Issues in FEM . 159
4.3 Damage Modeling Using FEM . 164
4.3.1 Stiffness Reduction Method . 164
4.3.2 Duplicate Node Method . 165
4.3.3 Kinematics Based Method 166
4.4 Numerical Examples 168
4.4.1 Static and Free Vibration Analysis of a Damaged
Cantilever Beam Using DNM . 168
4.4.2 Response Analysis of a Cantilever Composite
Beam with Different Damage Types . 170
4.5 Finite Element Modeling Suggestions 172
4.6 Modeling Pitfalls in FEM for SHM and Their Remedies . 173
References . 174
5 Spectral Finite Element Method . 177
5.1 The Need for Spectral FEM in SHM . 177
5.1.1 General Formulation Procedure: Fourier Transform
Based SFEM. . 178
5.1.2 General Formulation Procedure: Wavelet Transform
Based SFEM . 180
Contents xi5.2 Spectral Elements for Rods and Beams . 182
5.2.1 Non-dispersive Isotropic Rod: FFT Based
Spectral Element Formulation . 182
5.2.2 Non-dispersive Isotropic Rod: Wavelet Transform
Based Spectral Element Formulation 184
5.2.3 Dispersive Isotropic Timoshenko Beams-FFT
Based Spectral Element Formulation 184
5.2.4 Composite Beams-FFT Based Spectral Element
Formulation 186
5.2.5 Higher Order Composite Beam-FFT Based Spectral
Element Formulation . 188
5.3 Spectral Elements for 2D Composite Layers-FFT Based
Spectral Element Formulation . 190
5.3.1 Finite Layer Element (FLE) . 195
5.3.2 Infinite Layer (Throw-Off) Element (ILE) . 196
5.3.3 Expressions for Stresses and Strains . 197
5.3.4 Prescription of Force Boundary Conditions 197
5.3.5 Determination of Lamb Wave Modes 198
5.4 Anisotropic Plate-FFT Based Spectral Element
Formulation 199
5.4.1 Finite Plate Element 200
5.4.2 Semi-infinite or Throw-Off Plate Element . 201
5.5 Numerical Examples 202
5.5.1 Wave Transmission and Scattering Through
an Angle-Joint 202
5.5.2 Wave Propagation in 2D Portal Frame . 205
5.5.3 Propagation of Surface and Interfacial Waves
in a Composite Layer . 207
5.5.4 Propagation of Lamb Wave . 211
5.5.5 Wave Propagation in a Composite Plate
with Ply-Drop 214
5.6 Conclusions 216
References . 216
6 Simplified Spectral Models for Damaged Waveguides . 219
6.1 Need for Spectral Element Damage Models in Structural
Health Monitoring . 219
6.2 Review of Simplified Models for Structural Defects 220
6.3 Modeling of Single Delamination or Horizontal Cracks . 221
6.3.1 Wave Scattering in a Delaminated Beam Using
Wavelet Spectral Elements . 226
6.3.2 Effect of Wave Scattering Due to Delamination
at Ply-drops 229
xii Contents6.4 Modeling of Fiber Breakage and Vertical Cracks 230
6.4.1 Interface Equilibrium of Forces 232
6.4.2 Assembly of the Element Internal Waveguides . 233
6.4.3 Modeling Dynamic Contact Between
Crack Surfaces 234
6.4.4 Modeling of Surface Breaking Cracks 236
6.4.5 Distributed Constraints at the Interfaces Between
Sub-Laminates and Hanging Laminates . 237
6.4.6 Wave Scattering Due to Transverse Cracks 239
6.4.7 Sensitivity of the Fiber Breakage Location
and Configuration 240
6.5 Modeling of Structures with Multiple Horizontal
Cracks or Delaminations . 241
6.5.1 Wave Scattering from Delamination: Comparison
with 2D FEM . 246
6.5.2 Computational Efficiency of FSFEM Compared
to FEM . 247
6.6 Modeling of Corrosion Pits . 248
6.6.1 Wave Propagation Response Due to Corrosion Pits . 250
6.7 Modeling of Material Degradations . 253
6.7.1 Experimental Degraded Model (EDM) . 254
6.7.2 Average Degraded Model 259
6.7.3 Wave Scattering in a Degraded Composite Beam
Using ADM 262
6.8 Modeling of Vertical Cracks in 2D Waveguides 263
6.8.1 Flexibility Along the Crack . 266
6.8.2 Scattering Due to a Transverse Crack 267
6.9 Conclusions 269
References . 270
7 Perturbation Methods for Damaged Structures 273
7.1 Perturbation Methods for Notched Structures . 273
7.2 Modal Analysis of Damaged Plates . 274
7.2.1 Governing Equations 274
7.2.2 Perturbation Solution . 277
7.2.3 Fourier Series Solution of e1 Equations . 278
7.2.4 Strain Energy Ratio for Damage Localization 281
7.2.5 Effect of Notch Damage on the Plate
Modal Properties 283
7.2.6 Notch Damage Localization Through the Strain
Energy Ratio . 285
7.2.7 Effect of Line Damage on the Plate
Modal Properties 287
7.3 Analysis of Wave Propagation in Notched Beams Through
Spectral FE Solution 289
Contents xiii7.4 Governing Equations 290
7.4.1 Spectral Finite Element Discretization 300
7.5 Wave Propagation in Notched Beams: Numerical Examples . 302
7.5.1 Technique Validation: FSFEM Versus
FE Predictions 302
7.5.2 FSFEM and Modal Superposition Results . 304
7.5.3 Time Domain Results . 305
7.5.4 Frequency Domain Results . 310
References . 311
8 Bridging Scale Analysis of Wave Propagation in Heterogeneous
Structures with Imperfections . 313
8.1 Overview . 313
8.2 Theoretical Background . 315
8.2.1 Coarse and Fine Scale Discretization
and Bridging Matrices 315
8.2.2 Multiscale Lagrangian 316
8.2.3 Reduction of the Degrees of Freedom 317
8.2.4 Time Domain Formulation . 318
8.2.5 Frequency Domain Formulation . 319
8.3 Results for Time-Domain Bridging 321
8.3.1 Application to a One-Dimensional Rod 321
8.3.2 Homogenized Bi-material Rod with Imperfections 325
8.3.3 Energy-Based Time Integration Scheme 331
8.3.4 Propagation of In-plane Waves in a 2D
Elastic Domain 332
8.4 Results for Frequency-Domain Bridging . 339
8.4.1 Time Domain Spectral Element Discretization 339
8.4.2 Rod 340
8.4.3 Damaged Timoshenko Beam 340
8.4.4 Two Dimensional Waveguides . 344
8.5 Conclusions 347
References . 348
9 Modeling of Actuators and Sensors for SHM . 349
9.1 Introduction 349
9.2 Modeling of Lamb Wave Generation 350
9.2.1 Governing Equations 351
9.2.2 Harmonic Far Field Response . 353
9.2.3 Actuator Directivity 355
9.2.4 Example: Circular Actuator . 355
9.2.5 Experimental Validation . 358
9.2.6 Finite Element Evaluation of the Interface Stresses . 359
xiv Contents9.2.7 Example: Circular Patch . 362
9.2.8 Rectangular Isotropic Piezo Patch 364
9.3 Beamforming Through One-Dimensional Phased Arrays:
A Quick Overview . 364
9.3.1 Response Due to a Single Component 366
9.3.2 Array Response . 367
9.3.3 Beam Steering Strategies 368
9.4 Two Dimensional Arrays for Frequency Based
Beam Steering . 372
9.4.1 Application to SV Waves in a Membrane . 374
9.4.2 Application to Guided Waves in Thin Plates . 383
9.5 Modeling of Lamb Wave Sensors 391
9.5.1 Plate Configuration and Piezoelectric
Constitutive Relations 392
9.5.2 Voltage Generated by Piezo Sensors
of Arbitrary Shape . 394
9.5.3 Examples of Directivities for Simple Geometries . 397
9.5.4 Frequency Steerable Acoustic Transducer
Periodic Array . 399
References . 403
Part III Computational Methodologies for Damage Detection
and Quantification
10 Computational Techniques for Damage Detection, Classification
and Quantification . 407
10.1 Overview . 407
10.2 A General Introduction to Vibration-Based Techniques . 408
10.2.1 Early Techniques Based on Natural Frequency Shifts 408
10.2.2 Mode Shape Analysis . 410
10.2.3 Mode Shape Curvature Changes 410
10.3 Damage Measure Based on Energy Functional Distributions . 411
10.3.1 Formulation for Beams and Plates 412
10.3.2 Spline Interpolation of Operational Deflection Shapes 414
10.3.3 Numerical Results on Beams . 416
10.3.4 Numerical Results on Plates 418
10.3.5 Experimental Results on Beams . 421
10.3.6 Experimental Results on Plates 426
10.4 Wave Propagation Techniques: Time Domain
Damage Measure 429
10.4.1 Theoretical Background . 430
10.4.2 Numerical Examples: Wave Propagation
in a Homogeneous Medium 433
10.4.3 Experimental Results: Aluminum Plate 435
Contents xv10.5 Phase Gradient and Conversion Coefficients Evaluation
for Damage Localization and Quantification . 436
10.5.1 Simplified Description of a Multi-Modal Wave . 437
10.5.2 Phase Gradient for Damage Localization 437
10.5.3 Reflection, Transmission and Conversion
Coefficients for Damage Quantification 439
10.5.4 Application to Simulated Data 440
10.5.5 Application to Experimental Data 446
10.6 Damage Force Indicator Technique . 450
10.6.1 Identification of Single Delamination Through
Damage Force Indicator . 452
10.6.2 Identification of Multiple Delamination Through
Damage Force Indicator . 453
10.6.3 Sensitivity of Damage Force Indicator Due to
Variation in Delamination Size 454
10.6.4 Sensitivity of Damage Force Indicator Due to
Variation in Delamination Depth . 457
10.7 Summary . 459
References . 459
11 Use of Soft Computing Tools for Damage Detection 463
11.1 Genetic Algorithms 463
11.1.1 A Brief Introduction to Genetic Algorithms . 463
11.1.2 Genetic Algorithm Process for Damage Detection
and Definitions . 466
11.1.3 Objective Functions in GA for
Delamination Identification . 468
11.1.4 Case Studies with a Cantilever Beam 472
11.1.5 Identification of Delamination Location, Size
and Depth 476
11.2 Artificial Neural Networks . 478
11.2.1 Simple Model of Neuron . 478
11.2.2 Types of Activation Function . 480
11.2.3 Multilayer Feedforward Networks 481
11.2.4 Neural Network Integrated with SFEM 482
11.2.5 Numerical Results and Discussion 487
11.3 Summary . 490
References . 493
Index .
Index
A
Acoustic emission, 15
Acoustic tomography, 12
Activation function
bi-polar sigmoidal
function, 486
piecewise linear function, 480
sigmoid function, 481
threshold function, 480
types, 480
Active sensors
piezoelectric, 21
PVDF, 5
PZT, 5
TERFENOL-D, 5
Actuator
circular, 355, 362, 385
directivity, 355
piezoceramic, 126, 426
rectangular, 364
surface bonded, 350
Actuator array
plate response, 388
Aliasing, 112, 117, 178
Artificial Neural Network
(ANN), 478
back propagation error, 487
effect of noise, 489
multiple layer feed forward
networks, 479, 481
single layer feed forward
networks, 479
Artificial dispersion, 117, 125
Averaged degraded model, 260
Axial–Flexural coupling, 85
B
b-splines, 432
Beam forming algorithms, 373
Beam steering
2D phased arrays, 372
frequency based steering, 370
linear phase delay, 369
strategies, 368
Beltrami–Mitchell equations, 58
Bessel function, 322, 356
Boundary element method, 33, 157, 349
fundamental solutions, 34
Bragg wavelength, 28
Bridging scale method, 314
1D rod time domain formulation, 321
2D elastic medium, 332
bi-material rod with imperfections, 325
bridging matrices, 315
energy based time integration
scheme, 331
frequency domain formulation, 319
reduction of degrees of freedom, 317
shape functions, 316
time domain formulation, 318
time domain spectral element, 339
Bulk waves, 431
C
Cauchy’s stress tensor, 49
Christoffel symbol, 75
Circulant matrix, 120, 124
Classical plate theory, 79, 264
Companion matrix, 82, 91
Compatibility equations, 58
C (cont.)
Composites
material property determination, 61
micromechanics, 61
Computational efficiency of FSFEM, 247
Condition based monitoring, 6
Constitutive relations, 50, 69
elastic symmetry, 52
Hooke’s law, 51
isotropic system, 53
monoclinic system, 52
orthotropic system, 53
piezoelectric material, 362
triclinic system, 52
Contractional mode, 87
Corrosion pit, 248, 250
Curvature mode, 281–282, 284, 286–287, 413
Cut-off frequency, 74, 88, 195, 386
D
Damage detection
vibration based, 407
wave propagation based, 407
Damage force indicator, 450, 452, 457
multiple delaminations detection, 453
single delamination detection, 452
variation in delamination depth, 457
variation in delamination size, 454
Damage force vector, 451, 452
Damage localization, 448
phase gradient, 436, 437, 441
Daubechies basis function, 110
Daubechies wavelets, 106, 109, 110
Decimation factor, 416, 432
Deformation gradient, 42, 43
Description of motion, 41
Differential equation
weak form, 32
Dirac delta function, 32, 99, 132, 198, 267,
276, 280, 298, 363, 441
Directional sensing, 391
Dispersion relation, 70, 74, 81
composite plate, 89
elementary beam, 84
higher order beam, 86
Dispersive wave, 70, 81
Distributed dynamic contact, 167
Doppler shift, 29
Duplicate node method, 164–165
Dynamic stiffness matrix, 179, 454
2D composite layer, 196
beams, 186
rods, 183
E
Eigen solvers
Jacobi–Davidson Method, 83
Krylov method, 83
QZ algorithm, 83, 91
Eigenfunctions
orthogonal property, 279
Energy functional, 41
Energy harvesting systems, 12
Energy method, 41
Engineering strain, 44
Equations of motion, 54
Equivalent single layer theory, 220
Eulerian coordinates, 46
Evanescent mode, 71, 74
F
FEM, 8, 31, 33, 157, 349
coarse scale, 317
damage models, 164
direct time integration, 34
fine scale, 317
interface stresses evaluation, 359
mode superposition, 34
modeling issues, 159
modeling pitfalls, 173
modeling suggestions, 172
plane strain model, 451
Fiber optic sensors, 5, 7, 11, 21, 25
bio-medical, 26
chemical, 26
EPFI, 27
extrensic, 26–27
FBG, 27–28
interferometric, 26
intrensic, 26
modulation, 26
physical, 26
Finite difference, 32, 157
central difference, 32
explicit method, 32, 35
implicit method, 33, 35
First order shear deformation theory, 77, 184,
249, 289
Flexibility function, 263, 266
Force boundary condition, 197
Fourier transform, 70, 97, 109–110, 438
2D, 129, 132, 364, 376–377
3D, 19, 129, 354
Continuous, 97, 110, 177, 178
DFT, 70, 97, 99, 101, 111, 178, 319
FFT, 35, 101, 99, 110, 177, 356
Fourier series, 99, 110, 179, 198
496 Indexhigher dimensional, 19
inverse FFT, 321
multidimensional, 144
short time, 19
windowed, 104
Frequency response function, 31, 35, 310
Frequency/wavenumber domain, 179–180
Frequency/wavenumber filtering
1D, 130
2D, 132
mode separation, 144
FSFEM
general procedure, 178
G
GA
alleles, 467
chromosome, 464, 466
crossover, 465
crossover probability, 465
damage location & size identification, 475
damage location identification, 473
damage location, size & depth
identification, 476
damage size identification, 474
genes, 467
genotype, 467
introduction, 463
locus, 467
mutation, 465
objective function, 464
phenotype, 467
population, 464
probability of mutation, 465
process, 466
Galerkin approximation, 273
Gauss–Lobatto–Legendre point, 339
Governing equation
composite plate, 79
electro-mechanical system, 362
elementary beam, 76
higher order beam, 77
isotropic damaged plate, 274
lamb wave modeling, 350
notched beam, 289
SV waves, 374
weak form, 300
Green’s function, 378
Group speed, 70, 72, 89
Guided ultrasonic waves, 429
Guided wave, 11, 16, 18, 21, 24
thin plates, 383
tomography, 21
H
Haar wavelet, 103
Hamilton’s principle, 75, 77–78, 80,
290, 293, 298
Hankel function, 357, 379
Harmonic far field response, 353
Heaviside function, 276–277, 293
Helmhöltz equation, 109
Helmholtz decomposition, 75, 198, 351
Hilbert transform, 137, 143
Hilbert–Huang transform, 19
I
Integral transforms, 97
Interdigitated electrodes, 22
Inverse problem, 463
J
Jacobian, 44
K
Kinematics based method, 164, 166,
220, 254
material degradation, 260
modeling of delaminations, 166
modeling of fiber breakage, 167
Kirchoff plate theory, 418
L
Lagrange equations, 316
Lagrange multiplier, 361–362
Lagrange polynomials, 339
Lamb wave, 220, 241, 429, 437
antisymmetric, 22, 352, 355, 358, 386, 436
composites, 190
directional excitation, 384
dispersion, 211, 443
experimental validation, 358
modeling, 350
modes determination, 198
non-linear optimization, 211
propagation, 211
reflection/conversion coefficients,
440, 441, 452
symmetric, 22, 352, 355, 358, 386,
436–437, 447
Lamb waves, 440
Lambda matrix, 91
Laminated composite, 61
theory, 60
Index 497L (cont.)
Laser vibrometer, 5, 7, 14, 18–19, 29, 129,
144, 274, 358, 410, 412, 415,
422–423, 429, 448
Latent eigenvector, 82
Layerwise theories, 220
Legendre polynomial, 339
Line damage
modal properties, 287
Linear combiner output, 479
Linearization of PEP, 83
Littlewood–Paley wavelet, 104
Loading
broad band, 127, 162, 226, 239, 246, 263,
269, 443, 455, 473, 479
tone burst, 127, 160–161, 226, 240, 246,
251–252, 258–259, 262, 303, 305,
309, 340, 342, 388, 441, 447
Logarithmic strain, 44
Love–Kirkchoff layered theory, 360
M
Macro fiber composite, 22
Mass spring lattice model, 143, 435
Material degradation, 219
Matrix crack, 254, 260
Matrix cracking, 219
Matrix debonding, 254
Mean squared error, 486
MEMS, 7
Modal analysis
damaged plates, 274
Modal assurance criterion, 412
Mode conversion, 220, 407, 436–437, 448–449
estimation, 443, 448
Mode-II fracture, 242
Molecular dynamics, 316
Multi-layer perceptron, 19, 482
Multiscale lagrangian, 316
N
Navier’s equation, 57
Neuron, 478
activiation function, 479
adder, 479
simple model, 478
synapses, 478
Newmark method, 139, 239
Non destructive evaluation, 6, 12, 14–15, 20,
249, 313, 368, 372
Non-dispersive wave, 70
Nonlinear ultrasound, 20
Normal stress, 49
Notch damage
modal properties, 283
Notch type damage, 274, 408, 411, 436
Nyquist frequency, 103, 110, 117, 182, 184,
247, 452, 469, 483, 487
O
Objective functions, 463, 468, 470
displacement based, 468
power flow based, 470
Operational deflection shape, 408, 414–416,
420, 423, 429, 433
Orthotropic material, 65
P
P wave, 351, 366
Partial wave technique, 190, 198
Penalty parameter, 238
Permittivity matrix, 360
Perturbation analysis, 283, 410
Perturbation solution, 277
Fourier series solution, 278, 284
Perturbation technique, 36, 273
Phase speed, 72
Phased arrays
1D, 364
2D quadrilateral, 379
2D rectangular, 375
Piezocomposite, 362
Piezocomposites, 22
Piezoelectric coupling matrix, 360
Piezoelectric discs array, 383–384
Piezoelectric patches, 349
Piezoelectric transducers, 431
Pitting corrosion, 248
Plane strain, 59, 260, 366, 440
Plane stress, 59, 65–66, 260, 262
Polynomial eigenvalue problem, 82, 87, 185
Preface, v
Principal direction, 69
Propagating mode, 71, 74, 87
Q
Quasi P wave, 75, 195
Quasi S wave, 75, 195
R
Random decrement technique, 15
Rayleigh Lamb wave dispersion, 354
498 IndexRayleigh–Ritz solution, 273
Recurrent networks, 479
Refractive index, 27, 28, 28
Representative volume, 62
Residue theorem, 356
Rigid links, 166, 221, 230
Rule of mixtures, 63, 325, 328, 330
S
S wave, 351, 366
Selection procedure, 465
deterministic selection, 466
enlarged sampling, 466
mixed selection, 466
regular sampling, 466
stochastic selection, 466
Semi analytical finite element, 349
Sensitivity
fiber break configuration, 240
SH waves, 353
Shear stress, 49
SHM
diagnosis, 5, 9
dynamics based, 14, 16, 407
elements, 5, 7
guided wave based, 25, 373
levels, 9
modeling, 31
need, 3
off-line, 10
on-line, 10
overview of SHM, 3
passive and active, 14
prognosis, 5, 9
sensing and actuation strategies, 21
vibration based, 17
Signal leakage, 117
Signal periodicity, 102
Signal processing issues, 110
Simplified damage models
review, 220
Sinc function, 98, 101
Singular value decomposition, 83
Spectral element, 157, 158
2D layer element, 191
2D layer throw-off element, 196
notched beam, 289
anisotropic plate, 199
anisotropic plate throw-off element, 201
average degraded model, 260, 483
beam throw-off element, 187
corroded region, 248
dynamic contact element, 234
experimental degraded model, 254
fiber breakage, 230
FSFEM for isotropic beams, 184
FSFEM for composite beams, 186
FSFEM for higher order beam, 188
FSFEM for isotropic rods, 182
higher order beam throw-off element, 190
material degradations, 253
multiple delamination, 241
need in SHM, 177, 219
notched beam, 289
plate, 221, 264
plate with vertical cracks, 263
rod throw-off element, 183
single delamination, 221
surface breaking crack, 235
WSFEM isotropic rods, 184
WSFEM procedure, 180
Spectral element method, 351
Spectral finite element, 8, 31, 34
dynamic stiffness matrix, 35
throw-off element, 35
Spectral power flow, 470
Spectrum relation, 70, 74, 81, 87
Spline basis function, 415
Spline function, 423
Spline interpolation, 414, 432
Stiffness reduction method, 164
Strain energy ratio, 281, 288, 413
cumulative, 282
damage index, 411
damage index-beams, 416
damage index-plates, 421
experiments on beam, 421
experiments on plates, 426
notch damage, 287
Strain tensors
Eulerian, 45, 47
Lagrangian, 46–47
Stress intensity factor, 10
Surface-breaking crack, 235
SV waves, 374, 384
membrane, 374
Synaptic weights, 478, 481, 483, 485–487
System identification, 157, 463
T
Theory of elasticity, 41
Time domain spectral element
2D waveguides, 344
analysis of a rod, 340
damaged Timoshenko beam, 340
shape functions, 339, 345
Index 499T (cont.)
Time frequency transforms, 19
Time integration
central difference scheme, 440
Newmark scheme, 323
Time kernel history function, 318
Time signal window
rectangular, 117
Gaussian, 117, 127
Hanning, 117, 127, 132, 140, 149, 151,
161, 303, 358, 441, 447, 448
Tukey, 148
Timoshenko beam theory, 225
Total internal reflection, 27
Transformation matrix, 68
Traveling salesman problem, 471
True strain, 44
U
Ultrasonic inspection, 20
V
Variational principles, 32
Visco-elasticity, 35
W
Wave amplitude, 180, 195
Wave matrix, 180
Wave propagation
2D composite layer medium, 207
2D portal frame, 205
degraded composites using ADM, 262
response due to corrosion pits, 251
angled joint, 202
composite plate with ply-drop, 214
damaged rod, 138
degraded composites using EDM, 258
delaminated beam, 226
delamination at ply-drop, 229
fiber breakage, 238
homogeneous medium, 143, 435
introduction, 69
notched beams, 302
plate with vertical crack, 267
single delamination using multiple
delamination model, 246
spectral analysis, 70
Wave propagation technique
damage index theoretical background, 430
time domain damage index, 430
Wavefield, 16, 25, 30
Wavefield data, 129
Waveguide, 70
Wavelet transform, 19, 70, 103, 177, 412
non-periodic boundary conditions, 121
periodic boundary conditions, 119
Wavenumber, 70, 73
Weighted residual technique, 32
Wigner–Ville distributions, 19
Wireless sensors, 8
Wraparound problem
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