Nomenclature ................................................. xiii
Preface ....................................................... xix
Acknowledgments ............................................... xxi
1 Introduction ................................................. 1
1.1 Background .............................................. 1
1.2 A Comparison of Bulk versus Guided Waves ................ 3
1.3 What Is an Ultrasonic Guided Wave? ...................... 5
1.4 The Difference between Structural Health Monitoring
(SHM) and Nondestructive Testing (NDT) .................. 7
1.5 Text Preview ............................................ 7
1.6 Concluding Remarks ..................................... 12
1.7 References ............................................. 14
2 Dispersion Principles ....................................... 16
2.1 Introduction ........................................... 16
2.2 Waves in a Taut String ................................. 16
2.2.1 Governing Wave Equation ......................... 16
2.2.2 Solution by Separation of Variables ............. 17
2.2.3 D'Alembert's Solution ........................... 19
2.2.4 Initial Value Considerations .................... 20
2.3 String on an Elastic Base .............................. 21
2.4 A Dispersive Wave Propagation Sample Problem ........... 24
2.5 String on a Viscous Foundation ......................... 25
2.6 String on a Viscoelastic Foundation .................... 26
2.7 Graphical Representations of a Dispersive System ....... 26
2.8 Group Velocity Concepts ................................ 28
2.9 Exercises .............................................. 32
2.10 References ............................................. 35
3 Unbounded Isotropic and Anisotropic Media ................... 36
3.1 Introduction ........................................... 36
3.2 Isotropic Media ........................................ 36
3.1 Equations of Motion .................................... 36
3.2.2 Dilatational and Distortional Waves ............. 38
3.3 The Christoffel Equation for Anisotropic Media ......... 39
3.3.1 Sample Problem .................................. 42
3.4 On Velocity, Wave, and Slowness Surf aces .............. 46
3.5 Exercises .............................................. 50
3.6 References ............................................. 52
4 Reflection and Refraction ................................... 53
4.1 Introduction ........................................... 53
4.2 Normal Beam Incidence Reflection Factor ................ 53
4.3 SnelPs Law for Angle Beam Analysis ..................... 58
4.4 Critical Angles and Mode Conversion .................... 60
4.5 Slowness Profiles for Refraction and Critical Angle
Analysis ............................................... 63
4.6 Exercises .............................................. 64
4.7 References ............................................. 66
5 Oblique Incidence ........................................... 67
5.1 Background ............................................. 67
5.2 Reflection and Refraction Factors ...................... 68
5.2.1 Solid-Solid Boundary Conditions ................. 68
5.2.2 Solid-Liquid Boundary Conditions ................ 71
5.2.3 Liquid-Solid Boundary Conditions ................ 72
5.3 Moving Forward ......................................... 74
5.4 Exercises .............................................. 74
5.5 References ............................................. 75
6 Waves in Plates ............................................. 76
6.1 Introduction ........................................... 76
6.2 The Free Plate Problem ................................. 78
6.2.1 Solution by the Method of Potentials ............ 79
6.2.2 The Partial Wave Technique ...................... 82
6.3 Numerical Solution of the Rayleigh-Lamb Frequency
Equations .............................................. 84
6.4 Group Velocity ......................................... 87
6.5 Wave Structure Analysis ................................ 88
6.6 Compressional and Flexural Waves ....................... 91
6.7 Miscellaneous Topics ................................... 92
6.7.1 Lamb Waves with Dominant Longitudinal
Displacements ................................... 93
6.7.2 Zeros and Poles for a Fluid-Coupled Elastic
Layer .......................................... 102
6.7.3 Mode Cutoff Frequency .......................... 103
6.8 Exercises ............................................. 104
6.9 References ............................................ 106
7 Surface and Subsurface Waves ............................... 107
7.1 Background ............................................ 107
7.2 Surface Waves ......................................... 107
7.3 Generation and Reception of Surface Waves ............. 114
7.4 Subsurface Longitudinal Waves ......................... 116
7.5 Exercises ............................................. 117
7.6 References ............................................ 118
8 Finite Element Method for Guided Wave Mechanics ............ 120
8.1 Introduction .......................................... 120
8.2 Overview of the Finite Element Method ................. 120
8.2.1 Using the Finite Element Method to Solve
a Problem ...................................... 120
8.2.2 Quadratic Elements ............................. 125
8.2.3 Dynamic Problem ................................ 126
8.2.4 Error Control .................................. 128
8.3 FEM Applications for Guided Wave Analysis ............. 129
8.3.1 2-D Surface Wave Generation in a Plate ......... 129
8.3.2 Guided Wave Defect Detection in a Two-Inch
Steel Tube ..................................... 130
8.4 Summary ............................................... 132
8.5 Exercises ............................................. 133
8.6 References ............................................ 134
9 The Semi-Analytical Finite Element Method .................. 135
9.1 Introduction .......................................... 135
9.2 SAFE Formulation for Plate Structures ................. 136
9.3 Orthogonality-Based Mode Sorting ...................... 140
9.4 Group Velocity Dispersion Curves ...................... 141
9.5 Guided Wave Energy .................................... 142
9.5.1 Poynting Vector ................................ 142
9.5.2 Energy Velocity ................................ 142
9.5.3 Skew Effects in Anisotropic Plates ............. 142
9.6 Solution Convergence of the SAFE Method ............... 143
9.7 Free Guided Waves in an Eight-Layer Quasi-Isotropic
Plate ................................................. 143
9.8 SAFE Formulation for Cylindrical Structures ........... 145
9.9 Summary ............................................... 153
9.10 Exercises ............................................. 153
9.11 References ............................................ 154
10 Guided Waves in Hollow Cylinders ........................... 155
10.1 Introduction .......................................... 155
10.2 Guided Waves Propagating in an Axial Direction ........ 155
10.2.1 Analytic Calculation Approach .................. 155
10.2.2 Excitation Conditions and Angular Profiles ..... 164
10.2.3 Source Influence ............................... 166
10.3 Exercises ............................................. 171
10.4 References ............................................ 172
11 Circumferential Guided Waves ............................... 174
11.1 Introduction .......................................... 174
11.2 Development of the Governing Wave Equations for
Circumferential Waves ................................. 175
11.2.1 Circumferential Shear Horizontal Waves in
a Single-Layer Annulus ......................... 176
11.2.2 Circumferential Lamb Type Waves in a Single-
Layer Annulus .................................. 180
11.3 Extension to Multilayer Annuli ........................ 184
11.4 Numerical Solution of the Governing Wave Equations
for Circumferential Guided Waves ...................... 187
11.4.1 Numerical Results for CSH-Waves ................ 188
11.4.2 Numerical Results for CLT-Waves ................ 193
11.4.3 Computational Limitations of the Analytical
Formulation .................................... 199
11.5 The Effects of Protective Coating on Circumferential
Wave Propagation in Pipe .............................. 202
11.6 Exercises ............................................. 205
11.7 References ............................................ 206
12 Guided Waves in Layered Structures ......................... 209
12.1 Introduction .......................................... 209
12.2 Interface Waves ....................................... 210
12.2.1 Waves at a Solid-Solid Interface: Stoneley
Wave ........................................... 210
12.2.2 Waves at a Solid-Liquid Interface: Scholte
Wave ........................................... 213
12.3 Waves in a Layer on a Half-Space ...................... 215
12.3.1 Rayleigh-Lamb Type Waves ....................... 215
12.3.2 Love Waves ..................................... 219
12.4 Waves in Multiple Layers .............................. 221
12.4.1 The Global Matrix Method ....................... 222
12.4.2 The Transfer Matrix Method ..................... 227
12.4.3 Examples ....................................... 230
12.5 Fluid-Coupled Elastic Layers .......................... 233
12.5.1 Ultrasonic Wave Reflection and Transmission .... 234
12.5.2 Leaky Guided Wave Modes ........................ 242
12.5.3 Nonspecular Reflection and Transmission ........ 243
12.6 Exercises ............................................. 244
12.7 References ............................................ 245
13 Source Influence on Guided Wave Excitation ................. 246
13.1 Introduction .......................................... 246
13.2 Integral Transform Method ............................. 247
13.2.1 A Shear Loading Example ........................ 247
13.3 Normal Mode Expansion Method .......................... 251
13.3.1 Normal Mode Expansion in Harmonic Loading ...... 253
13.3.2 Transient Loading Source Influence ............. 257
13.4 Exercises ............................................. 267
13.5 References ............................................ 268
14 Horizontal Shear ........................................... 269
14.1 Introduction .......................................... 269
14.2 Dispersion Curves ..................................... 269
14.3 Phase Velocities and Cutoff Frequencies ............... 272
14.4 Group Velocity ........................................ 273
14.5 Summary ............................................... 274
14.6 Exercises ............................................. 275
14.7 References ............................................ 275
15 Guided Waves in Anisotropic Media .......................... 276
15.1 Introduction .......................................... 276
15.2 Phase Velocity Dispersion ............................. 277
15.3 Guided Wave Directional Dependency .................... 281
15.4 Guided Wave Skew Angle ................................ 286
15.5 Guided Waves in Composites with Multiple Layers ....... 287
15.6 Exercises ............................................. 292
15.7 References ............................................ 293
16 Guided Wave Phased Arrays in Piping ........................ 294
16.1 Introduction .......................................... 294
16.2 Guided Wave Phased Array Focus Theory ................. 295
16.3 Numerical Calculations ................................ 303
16.4 Finite Element Simulation of Guided Wave Focusing ..... 307
16.5 Active Focusing Experiment ............................ 310
16.6 Guided Wave Synthetic Focus ........................... 316
16.7 Synthetic Focusing Experiment ......................... 319
16.8 Summary ............................................... 321
16.9 Exercises ............................................. 321
16.10 References ........................................... 322
17 Guided Waves in Viscoelastic Media ......................... 323
17.1 Introduction ......................................... 323
17.2 Viscoelastic Models .................................. 324
17.2.1 Material Viscoelastic Models .................. 324
17.2.2 Kelvin-Voight Model ........................... 324
17.2.3 Maxwell Model ................................. 325
17.2.4 Further Aspects of the Hysteretic and
Kelvin-Voight Models .......................... 326
17.3 Measuring Viscoelastic Parameters .................... 327
17.4 Viscoelastic Isotropic Plate ......................... 328
17.5 Viscoelastic Orthotropic Plate ....................... 329
17.5.1 Problem Formulation and Solution .............. 329
17.5.2 Numerical Results ............................. 330
17.5.3 Summary ....................................... 333
17.6 Lamb Waves in a Viscoelastic Layer ................... 333
17.7 Viscoelastic Composite Plate ......................... 334
17.8 Pipes with Viscoelastic Coatings ..................... 340
17.9 Exercises ............................................ 342
17.10 References ........................................... 343
18 Ultrasonic Vibrations ..................................... 345
18.1 Introduction .......................................... 345
18.2 Practical Insights into the Ultrasonic Vibrations
Problem ............................................... 350
18.3 Concluding Remarks .................................... 357
18.4 Exercises ............................................. 357
18.5 References ............................................ 358
19 Guided Wave Array Transducers .............................. 359
19.1 Introduction .......................................... 359
19.2 Analytical Development ................................ 360
19.2.1 Linear Comb Array Solution ..................... 361
19.2.2 Annular Array Solution ......................... 366
19.3 Phased Transducer Arrays for Mode Selection ........... 370
19.3.1 Phased Array Analytical Development ............ 370
19.3.2 Phased Array Analysis .......................... 371
19.4 Concluding Remarks .................................... 376
19.5 Exercises ............................................. 376
19.6 References ............................................ 377
20 Introduction to Guided Wave Nonlinear Methods .............. 378
20.1 Introduction .......................................... 378
20.2 Bulk Waves in Weakly Nonlinear Elastic Media .......... 379
20.3 Measurement of the Second Harmonic .................... 380
20.4 Second Harmonic Generation Related to Microstructure .. 383
20.5 Weakly Nonlinear Wave Equation ........................ 384
20.6 Higher Harmonic Generation in Plates .................. 388
20.6.1 Synchronism .................................... 388
20.6.2 Power Flux ..................................... 391
20.6.3 Group Velocity Matching ........................ 393
20.6.4 Sample Laboratory Experiments .................. 393
20.7 Applications of Higher Harmonic Generation by Guided
Waves ................................................. 399
20.8 Exercises ............................................. 399
20.9 References ............................................ 400
21 Guided Wave Imaging Methods ................................ 402
21.1 Introduction .......................................... 402
21.2 Guided Wave through Transmission Dual Probe Imaging ... 402
21.3 Defect Locus Map ...................................... 407
21.4 Guided Wave Tomographic Imaging ....................... 408
21.5 Guided Wave Phased Array in Plates .................... 412
21.6 Long-Range Ultrasonic Guided Wave Pipe Inspection
Images ................................................ 417
21.7 Exercises ............................................. 418
21.8 References ............................................ 419
Appendix A - Ultrasonic Nondestructive Testing Principles,
Analysis, and Display Technology ........................... 421
A.l Some Physical Principles .............................. 421
A.2 Wave Interference ..................................... 425
A.3 Computational Model for a Single Point Source ......... 425
A.4 Directivity Function for a Cylindrical Element ........ 430
A.5 Ultrasonic Field Presentations ........................ 432
A.6 Near-Field Calculations ............................... 433
A.7 Angle-of-Divergence Calculations ...................... 434
A.8 Ultrasonic Beam Control ............................... 435
A.9 A Note on Ultrasonic Field Solution Techniques ........ 435
A.10 Time and Frequency Domain Analysis .................... 436
A.11 Pulsed Ultrasonic Field Effects ....................... 436
A.12 Introduction to Display Technology .................... 440
A.13 Amplitude Reduction of an Ultrasonic Waveform ......... 441
A.14 Resolution and Penetration Principles ................. 441
A.14.1 Axial Resolution ............................... 441
A.14.2 Lateral Resolution .................................. 442
A.15 Phased Arrays and Beam Focusing ....................... 443
A.16 Exercises ............................................. 443
A.17 References ............................................ 444
Appendix В - Basic Formulas and Concepts in the Theory of
Elasticity ................................................. 445
B.l Introduction .......................................... 445
B.2 Nomenclature .......................................... 445
B.3 Stress, Strain, and Constitutive Equations ............ 448
B.4 Elastic Constant Relationships ........................ 448
B.5 Vector and Tensor Transformation ...................... 449
B.6 Principal Stresses and Strains ........................ 449
B.7 The Strain Displacement Equations ..................... 450
B.8 Derivation of the Governing Wave Equation ............. 452
B.9 Anisotropic Elastic Constants ......................... 452
B.10 Exercises ............................................. 455
B.11 References ............................................ 455
Appendix С - Physically Based Signal Processing Concepts
for Guided Waves ........................................... 456
C.I General Concepts ...................................... 456
C.2 The Fast Fourier Transform (FFT) ...................... 457
C.2.1 Example FFT Use: Analytic Envelope ............. 460
C.2.2 Example FFT Use: Feature Source for Pattern
Recognition .................................... 462
C.2.3 Discrete Fourier Transform Properties .......... 462
C.3 The Short Time Fourier Transform (STFFT) .............. 463
C.3.1 Example: STFFT to Dispersion Curves ............ 466
C.4 The 2-D Fourier Transform (2DFFT) ..................... 467
C.5 The Wavelet Transform (WT) ............................ 472
C.6 Exercises ............................................. 477
C.7 References ............................................ 477
Appendix D - Guided Wave Mode and Frequency Selection Tips .... 478
D.l Introduction .......................................... 478
D.2 Mode and Frequency Selection Considerations ........... 480
D.2.1 A Surface-Breaking Defect ...................... 481
D.2.2 Mild Corrosion and Wall Thinning ............... 482
D.2.3 Transverse Crack Detection in the Head of
a Rail ......................................... 485
D.2.4 Repair Patch Bonded to an Aluminum Layer ....... 487
D.2.5 Water-Loaded Structures ........................ 487
D.2.6 Frequency and Other Tuning Possibilities ....... 489
D.2.7 Ice Detection with Ultrasonic Guided Waves ..... 491
D.2.8 Deicing ........................................ 492
D.2.9 Real-Time Phased Array Focusing in Pipe ........ 493
D.2.10 Aircraft, Lap Splice, Tear Strap, and Skin-
to-Core Delamination Inspection Potential ...... 495
D.2.11 Coating Delamination and Axial Crack
Detection ...................................... 498
D.2.12 Multilayer Structures .......................... 502
D.2.13 Concluding Remarks ............................. 502
D.3 Exercises ............................................. 503
D.4 References ............................................ 505
Index ......................................................... 507
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