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): Gui-Rong Liu, Z. C. Xi
Edition: 1
Publisher: CRC Press LLC
Year: 2002
Language: English
Pages: 455
Title Page......Page 2
Preface......Page 4
Authors......Page 10
Contents......Page 12
1.1 Introduction......Page 19
1.1.1 Statement of the Problem......Page 22
1.1.4 Equation of Motion for Free Wave Motion......Page 23
1.3 Free Wave Motion in Infinite Bars......Page 24
1.4 Free Wave Motion in a Finite Bar......Page 28
1.5 Forced Wave Motion in an Infinite Bar......Page 29
1.6 Forced Wave Motion in a Finite Bar......Page 33
1.7 Transient Waves in an Infinite Bar......Page 35
1.8 Remarks......Page 37
2.1 Introduction......Page 39
2.2 Element of Linear Property Variation......Page 40
2.3 Boundary and Continuity Conditions......Page 42
2.4 Transient Response......Page 44
2.5 Evaluation of Confluent Hypergeometric Function......Page 46
2.5.1 Integral of Gamma Function......Page 47
2.5.2 Integral of Confluent Hypergeometric Function......Page 48
2.5.3 Interval Division and Error Control......Page 50
2.6.1 Confluent Hypergeometric Function......Page 52
2.6.2 Wave Fields in FGM Plates......Page 53
2.7 Remarks......Page 58
3.1 Introduction......Page 60
3.2.1 Constitutive Equations......Page 61
3.2.4 Continuity Equations Between Layers......Page 62
3.3.1 Bulk Waves in 3D Anisotropic Solids......Page 63
3.3.2 Lamb Waves in Laminates......Page 65
3.4 Strain Energy Distribution......Page 70
3.5.1 Dispersion and Anisotropy of Phase Velocities......Page 71
3.5.2 Strain Energy Distribution......Page 76
3.6 Remarks......Page 80
4.1 Introduction......Page 81
4.2 Basic Equations......Page 82
4.2.1 Strain-Displacement Relation......Page 83
4.2.2 Stress-Strain Relation......Page 84
4.2.3 Equation of Motion......Page 85
4.4 Displacement in the Wavenumber Domain......Page 86
4.5 A Technique for the Inverse Fourier Integration......Page 90
4.6 Response in Time Domain......Page 94
4.7 Poles and Complex Paths......Page 95
4.8 Examples......Page 97
4.9 Remarks......Page 100
5.1 Introduction......Page 102
5.2 Dispersion Equation......Page 103
5.3 Group Velocities......Page 108
5.4 Phase Velocity Surface......Page 111
5.6 Phase Wave Surface......Page 112
5.8 Group Slowness Surface......Page 115
5.9 Group Wave Surface......Page 116
5.10.1 Dispersion Curves......Page 117
5.10.2 Results for Graphite/Epoxy Laminates......Page 118
5.10.3 Results for a Hybrid Composite Laminate......Page 120
5.11 Remarks......Page 121
6.1 Introduction......Page 124
6.2 System Equation......Page 125
6.3 Examples......Page 131
6.4 Remarks......Page 138
7.2 System Equation......Page 139
7.3 Examples of Harmonic Waves in Bars......Page 145
7.3.1 Results for a Clamped Bar......Page 146
7.3.2 Results for a Free Bar......Page 148
7.3.3 Anisotropic Laminated Bar......Page 149
7.4 Edge Waves in Semi-Infinite Laminates......Page 150
7.4.1 Verification of Results for Edge Waves......Page 152
7.4.2 Effect of Poisson’s Ratio on Edge Waves......Page 153
7.4.4 Results for Anisotropic Semi-Infinite Laminates......Page 156
7.5 Remarks......Page 158
8.1 Introduction......Page 160
8.2 HNM Formulation......Page 162
8.3 Equation in Wavenumber Domain......Page 165
8.4 Displacement in Wavenumber Domain......Page 166
8.5.2 Two-Dimensional Response......Page 168
8.6.1 Settings of the Parameters......Page 169
8.6.2 Results for an Isotropic Plate......Page 171
8.6.3 Results for a Hybrid Laminate......Page 172
8.7 Response to Point Time-Step Load......Page 174
8.7.1 Results for an Isotropic Plate......Page 175
8.7.2 Results for a Hybrid Laminate......Page 176
8.8.1 Problems......Page 178
8.8.2 Technique......Page 179
8.8.3 Application......Page 184
8.9 Response to Transient Load of Arbitrary Time Function......Page 185
8.10 Remarks......Page 188
9.1 Introduction......Page 189
9.2 Dynamic System Equation......Page 190
9.3 Dispersion Relation......Page 191
9.4 Group Velocity......Page 193
9.6 Two-Dimensional Problem......Page 194
9.7 Computational Procedure......Page 195
9.8 Dispersion Curves......Page 196
9.9.1 Results for Vertical Transient Loads......Page 201
9.9.2 Results for a Shear Load in the x Direction......Page 205
9.9.3 Results for a Shear Load in the y Direction......Page 206
9.9.4 Results for a Line Time-Pulse Load......Page 208
9.10 Remarks......Page 210
10.1 Introduction......Page 212
10.2 Basic Equations......Page 213
10.3 Approximated Governing Equations......Page 216
10.4 Equations in Transform Domain......Page 222
10.5 Characteristics of Waves in FGPM Plates......Page 224
10.6 Transient Response Analysis......Page 226
10.7 Interdigital Electrodes Excitation......Page 227
10.8 Displacement and Electrostatic Potential Response......Page 228
10.9 Computation Procedure......Page 229
10.10 Dispersion Curves......Page 231
10.11 Excitation of Time-Step Shear Force in y Direction......Page 238
10.12 Excitation of a Line Electrode......Page 239
10.13 Excitation of Interdigital Electrodes......Page 241
10.14 Remarks......Page 243
11.2 System Equations......Page 245
11.2.1 Strain-Displacement Relation......Page 246
11.2.2 Stress-Strain Relation......Page 247
11.2.3 Equation of Motion......Page 248
11.2.4 Strip Element Method Equation......Page 249
11.3 SEM for Static Problems (Flamant’s Problem)......Page 253
11.4 SEM for Dynamic Problems......Page 254
11.4.1 Harmonic Waves in 2D Space......Page 257
11.4.2 Harmonic Waves in Half-Space (Lamb’s Problem)......Page 259
11.5 Remarks......Page 262
12.1 Introduction......Page 264
12.2 Governing Differential Equations......Page 266
12.3 Particular Solution......Page 267
12.4 General Solution......Page 271
12.5 Application of the SEM to Cracked Laminates......Page 273
12.6 Solution in the Time Domain......Page 274
12.7 Examples of Scattered Wave Fields......Page 275
12.7.1 Response in Frequency Domain......Page 276
12.7.2 Response in Time Domain......Page 280
12.8.1 SEM Formulation......Page 282
12.8.3 Detection of a Crack in an Isotropic Plate......Page 286
12.8.4 Detection of a Crack in an Anisotropic Laminate......Page 293
12.9.1 SEM Formulation......Page 295
12.9.2 Technique for Crack Detection......Page 297
12.9.4 Dependency of Loading Position......Page 299
12.9.5 Determination of Crack Length......Page 300
12.10.1 SEM Formulation......Page 303
12.10.2 Technique for Crack Detection......Page 305
12.10.4 Dependency of Loading Position......Page 306
12.10.5 Determination of Crack Length......Page 307
12.11.1 Technique for Crack Detection......Page 309
12.11.2 Wave Scattering by Arbitrary Interior Vertical Cracks......Page 310
12.11.3 Dependency of Loading Position......Page 314
12.11.4 Determination of Crack Length......Page 316
12.11.5 Determination of the Crack Depth......Page 318
12.12 Remarks......Page 321
13.1 Introduction......Page 322
13.2 Application of the SEM to Plates Containing Flaws......Page 323
13.3 Examples for Wave Scattering in Laminates......Page 325
13.4 SH Waves in Sandwich Plates......Page 330
13.5 Strip Element Equation for SH Waves......Page 332
13.6 Particular Solution......Page 334
13.8 General Solution......Page 335
13.9 SH Waves Scattered by Flaws......Page 336
13.10 Remarks......Page 341
14.1 Introduction......Page 342
14.2 Governing Equation......Page 343
14.3 Strip Element Equation......Page 345
14.4 Assembly of Element Equations......Page 358
14.5.1 System Equations......Page 359
14.5.2 Complementary Solution......Page 360
14.5.3 Particular Solution......Page 361
14.5.4 Imposition of Boundary Conditions......Page 364
14.5.5 Examples for Static Problems......Page 367
14.6.1 System Equation in Frequency Domain......Page 371
14.6.2 Complementary Solution......Page 372
14.6.3 Particular Solution......Page 373
14.6.4 The General Solution......Page 376
14.6.5 Solution in the Time Domain......Page 377
14.6.6 Results for Anisotropic Laminated Plates......Page 378
14.6.7 Effect of Rotatory Inertia......Page 386
14.7 Concluding Remarks......Page 395
15.1 Introduction......Page 396
15.2 Basic Equations......Page 397
15.3 Dispersion Relations......Page 399
15.4 Examples......Page 402
15.5 Remarks......Page 407
16.1 Introduction......Page 408
16.2 Basic Equations......Page 409
16.3 Axisymmetric Strip Element......Page 411
16.4 Examples......Page 414
16.5 Remarks......Page 415
17.1 Introduction......Page 417
17.2 Two-Dimensional Line Load......Page 419
17.2.1 Green’s Function......Page 420
17.2.2 Time-Step Response Function......Page 421
17.3 Two-Dimensional Extended Line Load......Page 422
17.3.1 Identification of Loading Time Function......Page 423
17.3.2 Identification of Loading Distribution Function......Page 424
17.4 Three-Dimensional Concentrated Load......Page 425
17.5.1 Identification of Time-Function for Concentrated Line Load......Page 427
17.5.2 Identification for Extended Line Loads......Page 428
17.5.3 Concentrated Three-Dimensional Loads......Page 430
18.1 Introduction......Page 437
18.2 Inverse Operation......Page 439
18.3 Uniform-Micro Genetic Algorithms......Page 440
18.4.1 Inverse Procedure......Page 441
18.4.2 Effects of Noise......Page 443
18.5 Remarks......Page 446
References......Page 447
