Ultrasonic non-destructive evaluation (NDE) plays an increasingly important role in determining properties and detecting defects in composite materials, and the analysis of wave behavior is crucial to effectively using NDE techniques. The complexity of elastic wave propagation in anisotropic media has led to a reliance on numerical methods of analysis-methods that are often quite time-consuming and whose results yield even further difficulties in extracting explicit phenomena and characteristics.Innovative and insightful, Elastic Waves in Anisotropic Laminates establishes a set of high-performance, analytical-numerical methods for elastic wave analysis of anisotropic layered structures. The treatment furnishes a comprehensive introduction, sound theoretical development, and applications to smart materials, plates, and shells. The techniques, detailed in both the time and frequency domains, include methods that combine the finite element method (FEM) with the Fourier transform approach and the strip element method (SEM). These -methods can also be used for expediently finding the Green's function for anisotropic laminates useful for inverse problems related to wave propagation, and methods for inverse analyses, including conjugate gradient methods, and genetic algorithms are also introduced.The text is complemented by many examples generated using software codes based on the techniques developed. Filled with charts and illustrations, Elastic Waves in Anisotropic Laminates is accessible even to readers from non-engineering backgrounds and offers a unique opportunity to discover methods that can lead to an understanding of the dynamic characteristics and wave motion behaviors of advanced composite materials.
Author(s): G.R. Liu, Z. C. Xi
Edition: 1
Publisher: CRC Press
Year: 2001
Language: English
Pages: 454
Title Page......Page 1
Preface......Page 3
Authors......Page 9
Contents......Page 11
1.1 Introduction......Page 18
1.1.1 Statement of the Problem......Page 21
1.1.4 Equation of Motion for Free Wave Motion......Page 22
1.3 Free Wave Motion in Infinite Bars......Page 23
1.4 Free Wave Motion in a Finite Bar......Page 27
1.5 Forced Wave Motion in an Infinite Bar......Page 28
1.6 Forced Wave Motion in a Finite Bar......Page 32
1.7 Transient Waves in an Infinite Bar......Page 34
1.8 Remarks......Page 36
2.1 Introduction......Page 38
2.2 Element of Linear Property Variation......Page 39
2.3 Boundary and Continuity Conditions......Page 41
2.4 Transient Response......Page 43
2.5 Evaluation of Confluent Hypergeometric Function......Page 45
2.5.1 Integral of Gamma Function......Page 46
2.5.2 Integral of Confluent Hypergeometric Function......Page 47
2.5.3 Interval Division and Error Control......Page 49
2.6.1 Confluent Hypergeometric Function......Page 51
2.6.2 Wave Fields in FGM Plates......Page 52
2.7 Remarks......Page 57
3.1 Introduction......Page 59
3.2.1 Constitutive Equations......Page 60
3.2.4 Continuity Equations Between Layers......Page 61
3.3.1 Bulk Waves in 3D Anisotropic Solids......Page 62
3.3.2 Lamb Waves in Laminates......Page 64
3.4 Strain Energy Distribution......Page 69
3.5.1 Dispersion and Anisotropy of Phase Velocities......Page 70
3.5.2 Strain Energy Distribution......Page 75
3.6 Remarks......Page 79
4.1 Introduction......Page 80
4.2 Basic Equations......Page 81
4.2.1 Strain-Displacement Relation......Page 82
4.2.2 Stress-Strain Relation......Page 83
4.2.3 Equation of Motion......Page 84
4.4 Displacement in the Wavenumber Domain......Page 85
4.5 A Technique for the Inverse Fourier Integration......Page 89
4.6 Response in Time Domain......Page 93
4.7 Poles and Complex Paths......Page 94
4.8 Examples......Page 96
4.9 Remarks......Page 99
5.1 Introduction......Page 101
5.2 Dispersion Equation......Page 102
5.3 Group Velocities......Page 107
5.4 Phase Velocity Surface......Page 110
5.6 Phase Wave Surface......Page 111
5.8 Group Slowness Surface......Page 114
5.9 Group Wave Surface......Page 115
5.10.1 Dispersion Curves......Page 116
5.10.2 Results for Graphite/Epoxy Laminates......Page 117
5.10.3 Results for a Hybrid Composite Laminate......Page 119
5.11 Remarks......Page 120
6.1 Introduction......Page 123
6.2 System Equation......Page 124
6.3 Examples......Page 130
6.4 Remarks......Page 137
7.2 System Equation......Page 138
7.3 Examples of Harmonic Waves in Bars......Page 144
7.3.1 Results for a Clamped Bar......Page 145
7.3.2 Results for a Free Bar......Page 147
7.3.3 Anisotropic Laminated Bar......Page 148
7.4 Edge Waves in Semi-Infinite Laminates......Page 149
7.4.1 Verification of Results for Edge Waves......Page 151
7.4.2 Effect of Poisson’s Ratio on Edge Waves......Page 152
7.4.4 Results for Anisotropic Semi-Infinite Laminates......Page 155
7.5 Remarks......Page 157
8.1 Introduction......Page 159
8.2 HNM Formulation......Page 161
8.3 Equation in Wavenumber Domain......Page 164
8.4 Displacement in Wavenumber Domain......Page 165
8.5.2 Two-Dimensional Response......Page 167
8.6.1 Settings of the Parameters......Page 168
8.6.2 Results for an Isotropic Plate......Page 170
8.6.3 Results for a Hybrid Laminate......Page 171
8.7 Response to Point Time-Step Load......Page 173
8.7.1 Results for an Isotropic Plate......Page 174
8.7.2 Results for a Hybrid Laminate......Page 175
8.8.1 Problems......Page 177
8.8.2 Technique......Page 178
8.8.3 Application......Page 183
8.9 Response to Transient Load of Arbitrary Time Function......Page 184
8.10 Remarks......Page 187
9.1 Introduction......Page 188
9.2 Dynamic System Equation......Page 189
9.3 Dispersion Relation......Page 190
9.4 Group Velocity......Page 192
9.6 Two-Dimensional Problem......Page 193
9.7 Computational Procedure......Page 194
9.8 Dispersion Curves......Page 195
9.9.1 Results for Vertical Transient Loads......Page 200
9.9.2 Results for a Shear Load in the x Direction......Page 204
9.9.3 Results for a Shear Load in the y Direction......Page 205
9.9.4 Results for a Line Time-Pulse Load......Page 207
9.10 Remarks......Page 209
10.1 Introduction......Page 211
10.2 Basic Equations......Page 212
10.3 Approximated Governing Equations......Page 215
10.4 Equations in Transform Domain......Page 221
10.5 Characteristics of Waves in FGPM Plates......Page 223
10.6 Transient Response Analysis......Page 225
10.7 Interdigital Electrodes Excitation......Page 226
10.8 Displacement and Electrostatic Potential Response......Page 227
10.9 Computation Procedure......Page 228
10.10 Dispersion Curves......Page 230
10.11 Excitation of Time-Step Shear Force in y Direction......Page 237
10.12 Excitation of a Line Electrode......Page 238
10.13 Excitation of Interdigital Electrodes......Page 240
10.14 Remarks......Page 242
11.2 System Equations......Page 244
11.2.1 Strain-Displacement Relation......Page 245
11.2.2 Stress-Strain Relation......Page 246
11.2.3 Equation of Motion......Page 247
11.2.4 Strip Element Method Equation......Page 248
11.3 SEM for Static Problems (Flamant’s Problem)......Page 252
11.4 SEM for Dynamic Problems......Page 253
11.4.1 Harmonic Waves in 2D Space......Page 256
11.4.2 Harmonic Waves in Half-Space (Lamb’s Problem)......Page 258
11.5 Remarks......Page 261
12.1 Introduction......Page 263
12.2 Governing Differential Equations......Page 265
12.3 Particular Solution......Page 266
12.4 General Solution......Page 270
12.5 Application of the SEM to Cracked Laminates......Page 272
12.6 Solution in the Time Domain......Page 273
12.7 Examples of Scattered Wave Fields......Page 274
12.7.1 Response in Frequency Domain......Page 275
12.7.2 Response in Time Domain......Page 279
12.8.1 SEM Formulation......Page 281
12.8.3 Detection of a Crack in an Isotropic Plate......Page 285
12.8.4 Detection of a Crack in an Anisotropic Laminate......Page 292
12.9.1 SEM Formulation......Page 294
12.9.2 Technique for Crack Detection......Page 296
12.9.4 Dependency of Loading Position......Page 298
12.9.5 Determination of Crack Length......Page 299
12.10.1 SEM Formulation......Page 302
12.10.2 Technique for Crack Detection......Page 304
12.10.4 Dependency of Loading Position......Page 305
12.10.5 Determination of Crack Length......Page 306
12.11.1 Technique for Crack Detection......Page 308
12.11.2 Wave Scattering by Arbitrary Interior Vertical Cracks......Page 309
12.11.3 Dependency of Loading Position......Page 313
12.11.4 Determination of Crack Length......Page 315
12.11.5 Determination of the Crack Depth......Page 317
12.12 Remarks......Page 320
13.1 Introduction......Page 321
13.2 Application of the SEM to Plates Containing Flaws......Page 322
13.3 Examples for Wave Scattering in Laminates......Page 324
13.4 SH Waves in Sandwich Plates......Page 329
13.5 Strip Element Equation for SH Waves......Page 331
13.6 Particular Solution......Page 333
13.8 General Solution......Page 334
13.9 SH Waves Scattered by Flaws......Page 335
13.10 Remarks......Page 340
14.1 Introduction......Page 341
14.2 Governing Equation......Page 342
14.3 Strip Element Equation......Page 344
14.4 Assembly of Element Equations......Page 357
14.5.1 System Equations......Page 358
14.5.2 Complementary Solution......Page 359
14.5.3 Particular Solution......Page 360
14.5.4 Imposition of Boundary Conditions......Page 363
14.5.5 Examples for Static Problems......Page 366
14.6.1 System Equation in Frequency Domain......Page 370
14.6.2 Complementary Solution......Page 371
14.6.3 Particular Solution......Page 372
14.6.4 The General Solution......Page 375
14.6.5 Solution in the Time Domain......Page 376
14.6.6 Results for Anisotropic Laminated Plates......Page 377
14.6.7 Effect of Rotatory Inertia......Page 385
14.7 Concluding Remarks......Page 394
15.1 Introduction......Page 395
15.2 Basic Equations......Page 396
15.3 Dispersion Relations......Page 398
15.4 Examples......Page 401
15.5 Remarks......Page 406
16.1 Introduction......Page 407
16.2 Basic Equations......Page 408
16.3 Axisymmetric Strip Element......Page 410
16.4 Examples......Page 413
16.5 Remarks......Page 414
17.1 Introduction......Page 416
17.2 Two-Dimensional Line Load......Page 418
17.2.1 Green’s Function......Page 419
17.2.2 Time-Step Response Function......Page 420
17.3 Two-Dimensional Extended Line Load......Page 421
17.3.1 Identification of Loading Time Function......Page 422
17.3.2 Identification of Loading Distribution Function......Page 423
17.4 Three-Dimensional Concentrated Load......Page 424
17.5.1 Identification of Time-Function for Concentrated Line Load......Page 426
17.5.2 Identification for Extended Line Loads......Page 427
17.5.3 Concentrated Three-Dimensional Loads......Page 429
18.1 Introduction......Page 436
18.2 Inverse Operation......Page 438
18.3 Uniform-Micro Genetic Algorithms......Page 439
18.4.1 Inverse Procedure......Page 440
18.4.2 Effects of Noise......Page 442
18.5 Remarks......Page 445
References......Page 446
