Elastic Waves in Anisotropic LaminatesUltrasonic 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. |
Spis treści
Fundamentals of Waves in Elastic Solids | 1 |
12 Formulation of Longitudinal Wave in a Bar | 4 |
112 StrainDisplacement Relation | 5 |
13 Free Wave Motion in Infinite Bars | 6 |
14 Free Wave Motion in a Finite Bar | 10 |
15 Forced Wave Motion in an Infinite Bar | 11 |
16 Forced Wave Motion in a Finite Bar | 15 |
17 Transient Waves in an Infinite Bar | 17 |
108 Displacement and Electrostatic Potential Response | 217 |
109 Computation Procedure | 218 |
1010 Dispersion Curves | 220 |
1011 Excitation of TimeStep Shear Force in y Direction | 227 |
1012 Excitation of a Line Electrode | 228 |
1013 Excitation of Interdigital Electrodes | 230 |
1014 Remarks | 232 |
Strip Element Method for Stress Waves in Anisotropic Solids | 235 |
18 Remarks | 19 |
Waves in Plates of functionally Graded Material | 21 |
22 Element of Linear Property Variation | 22 |
23 Boundary and Continuity Conditions | 24 |
24 Transient Response | 26 |
25 Evaluation of Confluent Hypergeometric Function | 28 |
251 Integral of Gamma Function | 29 |
252 Integral of Confluent Hypergeometric Function | 30 |
253 Interval Division and Error Control | 32 |
26 Examples | 34 |
262 Wave Fields in FGM Plates | 35 |
27 Remarks | 40 |
Free Wave Motion in Anisotropic Laminates | 43 |
32 Basic Equations | 44 |
322 Equation of Motion | 45 |
325 Boundary Conditions | 46 |
332 Lamb Waves in Laminates | 48 |
34 Strain Energy Distribution | 53 |
35 Examples | 54 |
352 Strain Energy Distribution | 59 |
36 Remarks | 63 |
Forced Wave Motion in Composite Laminates | 65 |
42 Basic Equations | 66 |
421 StrainDisplacement Relation | 67 |
422 StressStrain Relation | 68 |
423 Equation of Motion | 69 |
43 Boundary and Interface Conditions | 70 |
45 A Technique for the Inverse Fourier Integration | 74 |
46 Response in Time Domain | 78 |
47 Poles and Complex Paths | 79 |
48 Examples | 81 |
49 Remarks | 84 |
Characteristics of Waves in Composite Laminates | 87 |
52 Dispersion Equation | 88 |
53 Group Velocities | 93 |
54 Phase Velocity Surface | 96 |
55 Phase Slowness Surface | 97 |
57 Group Velocity Surface | 100 |
59 Group Wave Surface | 101 |
510 Examples | 102 |
5102 Results for GraphiteEpoxy Laminates | 103 |
5103 Results for a Hybrid Composite Laminate | 105 |
511 Remarks | 106 |
Free Wave Motion in Anisotropic Laminated Bars Finite Strip Element Method | 109 |
62 System Equation | 110 |
63 Examples | 116 |
64 Remarks | 123 |
Free Wave Motion in Composite Laminated Bars SemiExact Method | 125 |
73 Examples of Harmonic Waves in Bars | 131 |
731 Results for a Clamped Bar | 132 |
732 Results for a Free Bar | 134 |
733 Anisotropic Laminated Bar | 135 |
74 Edge Waves in SemiInfinite Laminates | 136 |
741 Verification of Results for Edge Waves | 138 |
742 Effect of Poissons Ratio on Edge Waves | 139 |
743 Higher Modes of Edge Waves | 142 |
75 Remarks | 144 |
Transient Waves in Composite Laminates | 147 |
82 HNM Formulation | 149 |
83 Equation in Wavenumber Domain | 152 |
84 Displacement in Wavenumber Domain | 153 |
85 Response in SpaceTime Domain | 155 |
853 Computational Procedure | 156 |
862 Results for an Isotropic Plate | 158 |
863 Results for a Hybrid Laminate | 159 |
87 Response to Point TimeStep Load | 161 |
871 Results for an Isotropic Plate | 162 |
872 Results for a Hybrid Laminate | 163 |
88 Techniques for Inverse Fourier Integral | 165 |
882 Technique | 166 |
883 Application | 171 |
89 Response to Transient Load of Arbitrary Time Function | 172 |
810 Remarks | 175 |
Waves in Functionally Graded Plates | 177 |
92 Dynamic System Equation | 178 |
93 Dispersion Relation | 179 |
94 Group Velocity | 181 |
95 Response Analysis | 182 |
97 Computational Procedure | 183 |
98 Dispersion Curves | 184 |
99 Transient Response to Line TimeStep Loads | 189 |
992 Results for a Shear Load in the x Direction | 193 |
993 Results for a Shear Load in the y Direction | 194 |
994 Results for a Line TimePulse Load | 196 |
910 Remarks | 198 |
Waves in Anisotropic functionally Graded Piezoelectric Plates | 201 |
102 Basic Equations | 202 |
103 Approximated Governing Equations | 205 |
104 Equations in Transform Domain | 211 |
105 Characteristics of Waves in FGPM Plates | 213 |
106 Transient Response Analysis | 215 |
107 Interdigital Electrodes Excitation | 216 |
1121 StrainDisplacement Relation | 236 |
1122 StressStrain Relation | 237 |
1123 Equation of Motion | 238 |
1124 Strip Element Method Equation | 239 |
113 SEM for Static Problems Flamants Problem | 243 |
114 SEM for Dynamic Problems | 244 |
1141 Harmonic Waves in 2D Space | 247 |
1142 Harmonic Waves in HalfSpace Lambs Problem | 249 |
115 Remarks | 252 |
Wave Scattering by Cracks in Composite Laminates | 255 |
122 Governing Differential Equations | 257 |
123 Particular Solution | 258 |
124 General Solution | 262 |
125 Application of the SEM to Cracked Laminates | 264 |
126 Solution in the Time Domain | 265 |
127 Examples of Scattered Wave Fields | 266 |
1271 Response in Frequency Domain | 267 |
1272 Response in Time Domain | 271 |
128 Characterization of Horizontal Cracks | 273 |
1282 Technique for Crack Detection | 277 |
1284 Detection of a Crack in an Anisotropic Laminate | 284 |
129 Characterization of a Vertical SurfaceBreaking Crack | 286 |
1292 Technique for Crack Detection | 288 |
1293 Frequency Dependency | 290 |
1295 Determination of Crack Length | 291 |
1210 Characterization of Middle Interior Vertical Cracks | 294 |
12102 Technique for Crack Detection | 296 |
12103 Dependency of Load Frequency | 297 |
12105 Determination of Crack Length | 298 |
1211 Characterization of Arbitrary Interior Vertical Cracks | 300 |
12112 Wave Scattering by Arbitrary Interior Vertical Cracks | 301 |
12113 Dependency of Loading Position | 305 |
12114 Determination of Crack Length | 307 |
12115 Determination of the Crack Depth | 309 |
1212 Remarks | 312 |
Wave Scattering by Flaws in Composite Laminates | 313 |
132 Application of the SEM to Plates Containing Flaws | 314 |
133 Examples for Wave Scattering in Laminates | 316 |
134 SH Waves in Sandwich Plates | 321 |
135 Strip Element Equation for SH Waves | 323 |
136 Particular Solution | 325 |
137 Complementary Solution | 326 |
139 SH Waves Scattered by Flaws | 327 |
1310 Remarks | 332 |
Bending Waves in Anisotropic Laminated Plates | 333 |
142 Governing Equation | 334 |
143 Strip Element Equation | 336 |
144 Assembly of Element Equations | 349 |
145 Static Problems for Orthotropic Laminated Plates | 350 |
1452 Complementary Solution | 351 |
1453 Particular Solution | 352 |
1454 Imposition of Boundary Conditions | 355 |
1455 Examples for Static Problems | 358 |
146 Wave Motion in Anisotropic Laminated Plates | 362 |
1462 Complementary Solution | 363 |
1463 Particular Solution | 364 |
1464 The General Solution | 367 |
1465 Solution in the Time Domain | 368 |
1466 Results for Anisotropic Laminated Plates | 369 |
1467 Effect of Rotatory Inertia | 377 |
146 Concluding Remarks | 386 |
Characteristics of Waves in Composite Cylinders | 387 |
152 Basic Equations | 388 |
153 Dispersion Relations | 390 |
154 Examples | 393 |
155 Remarks | 398 |
Wave Scattering by Cracks in Composite Cylinders | 399 |
162 Basic Equations | 400 |
163 Axisymmetric Strip Element | 402 |
164 Examples | 405 |
165 Remarks | 406 |
Inverse Identification of Impact Loads Using Elastic Waves | 409 |
172 TwoDimensional Line Load | 411 |
1721 Greens Function | 412 |
1722 TimeStep Response Function | 413 |
173 TwoDimensional Extended Line Load | 414 |
1731 Identification of Loading Time Function | 415 |
1732 Identification of Loading Distribution Function | 416 |
1733 Identification of Both Time History and Distribution Functions | 417 |
175 Examples | 419 |
1752 Identification for Extended Line Loads | 420 |
1753 Concentrated ThreeDimensional Loads | 422 |
Inverse Determination of Material Constants of Composite Laminates | 429 |
182 Inverse Operation | 431 |
183 UniformMicro Genetic Algorithms | 432 |
184 Examples | 433 |
1842 Effects of Noise | 435 |
185 Remarks | 438 |
| 439 | |
| 449 | |
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amplitude approximate boundary conditions carbon/epoxy Chapter complex path composite laminates computed crack length cylinders d₁ denoted Dimensionless Dimensionless Displacement dimensionless frequency dispersion displacement field displacement response displacement vector distribution eigenvalues eigenvectors elastic FGPM plate finite element finite element method flaw Fourier transform given Green's function group velocity harmonic waves hybrid laminate integral inverse isotropic plate k₁ laminated bar layer element LiTaO3 plate lower surface material constants material properties matrix mode shapes natural frequencies nonreflecting boundary obtained oscillations parameters permission phase velocity piezoelectric plane wave plate excited plate subjected Poisson's ratio present method problem procedure Rayleigh wave respectively sandwich plate Scanning results semi-exact method shear shown in Figure SiC-C plate stress strip element method surface wave system equation t₁ technique thickness direction time-step load upper surface vector wave modes wave propagation wave surfaces wavelength wavenumber wavenumber domain x/H FIGURE ди ду дх

