Preface ........................................................ xi
Introduction .................................................. xiv
1 Fundamentals of Composite Elastic Properties ................. 3
1.1 Introduction ............................................ 3
1.2 Mechanical Relationships ................................ 3
1.2.1 Stress ........................................... 3
1.2.2 Strain ........................................... 5
1.2.3 Constitutive equation ............................ 6
1.3 Generalized Hooke's Law ................................ 10
1.4 Stress-Strain Relations for Media of Higher
Symmetries ............................................. 13
1.4.1 Monoclinic symmetry ............................. 13
1.4.2 Orthotropic symmetry ............................ 14
1.4.3 Transversely isotropic symmetry ................. 14
1.4.4 Cubic symmetry .................................. 16
1.4.5 Isotropic material .............................. 16
1.5 Relation between Stiffness and Compliance .............. 17
1.5.1 Engineering constants for orthotropic
materials ....................................... 18
1.6 Determination of the Full Compliance Matrix ............ 18
1.7 Material Coordinate System Transformations ............. 19
1.7.1 Property matrices in a rotated coordinate
system .......................................... 20
1.7.2 Rotation about the x3 axis ...................... 22
1.8 Analysis in a Planar Geometry .......................... 23
1.8.1 Two-dimensional stress-strain relations ........ 23
1.8.2 Rotation about the principal axis ............... 25
1.8.3 Boron-epoxy composite: An example ............... 27
1.9 Experimental Determination of Stiffness ................ 30
Bibliography ................................................ 34
2 Elastic Waves in Anisotropic Media .......................... 35
2.1 Equations of Motion .................................... 35
2.1.1 Introduction .................................... 35
2.1.2 Differential approach ........................... 37
2.1.3 Integral principle .............................. 38
2.2 Plane Wave Propagation in Bulk Materials ............... 42
2.3 Determination of the Polarization Vectors .............. 45
2.4 Example: Plane Waves in an Orthotropic Material ........ 48
2.5 Energy Flux and Group Velocity ......................... 55
2.6 Relation between the Phase and Group Velocities ........ 58
2.7 Measurement of the Group Velocity ...................... 61
2.8 Examples ............................................... 62
2.9 Phase Velocity, Group Velocity, and Slowness ........... 66
2.9.1 Slowness ........................................ 66
2.9.2 Examples for a graphite-epoxy composite ......... 67
2.9.3 Phase and group velocities in symmetry planes
of orthotropic materials ........................ 72
2.9.4 Phase and group velocities in non-symmetry
planes of orthotropic and transversely
isotropic materials ............................. 75
Bibliography ................................................ 80
3 Bulk Ultrasonic Techniques for Evaluation of Elastic
Properties .................................................. 81
3.1 Bulk Wave Refraction Method for Phase Velocity
Measurement ............................................ 82
3.1.1 Introduction .................................... 82
3.1.2 Delay times for phase and group velocities in
a solid layer of general anisotropy ............. 83
3.1.3 Phase velocity measurement ...................... 90
3.1.4 Double through-transmission phase velocity
measurement ..................................... 91
3.1.5 Self-reference method ........................... 91
3.1.6 Multiple reflection method ...................... 94
3.1.7 Comments on accuracy, phase correction, and
applicability of the plane wave approximation
in angle-beam, self-referenced, through-
transmission method ............................ 102
3.2 Examples of Bulk Wave Refraction Velocity
Measurements .......................................... 105
3.2.1 Experimental apparatus ......................... 105
3.2.2 Double through-transmission method for
graphite/epoxy composites ...................... 106
3.3 Critical Angle Measurement of Elastic Constants ....... 110
3.3.1 Introduction ................................... 110
3.3.2 Concept of the method .......................... 111
3.3.3 Critical angle measurements .................... 115
3.3.4 Experimental results for graphite/epoxy
composites ..................................... 117
3.4 Determination of Elastic Constants from Phase
Velocity Data ......................................... 119
3.4.1 Reconstruction of elastic constants ............ 119
3.4.2 Stability of the reconstruction algorithm ...... 121
3.5 Group Velocity Measurements ........................... 124
3.5.1 Reconstruction of elastic constants from
group velocity data ............................ 125
3.5.2 Determination of elastic constants from group
velocity data in a symmetry plane .............. 126
3.5.3 Comparison of the reconstruction results from
group and phase velocity data in symmetry
planes ......................................... 129
3.5.4 Reconstruction from group velocity data in
non-symmetry planes ............................ 131
Bibliography ............................................... 134
4 Reflection and Refraction of Waves at a Planar Composite
Interface .................................................. 137
4.1 Introduction .......................................... 137
4.2 Background ............................................ 138
4.2.1 Snell's law .................................... 138
4.2.2 Isotropic media analysis using slowness
surfaces ....................................... 140
4.3 Scattering at an Interface between Two Generally
Anisotropic Media ..................................... 142
4.3.1 Introduction ................................... 142
4.3.2 Snell's law (anisotropic case) ................. 143
4.3.3 Determination of the slowness vectors of
reflected and refracted waves .................. 147
4.3.4 Calculation of the wave and polarization
vectors ........................................ 149
4.3.5 Example: Refracted waves in isotropic solids
with incident waves from a fluid ............... 155
4.4 Determination of Reflection and Transmission
Coefficients .......................................... 157
4.4.1 Boundary conditions, amplitude coefficients .... 158
4.4.2 Energy conversion coefficients ................. 161
4.5 Examples for Graphite/Epoxy Composite ................. 164
4.5.1 Fluid/composite interface ...................... 165
4.5.2 Isotropic wedge/composite interface ............ 167
4.5.3 Composite/composite interface .................. 171
4.6 Geometrical Considerations on Reflection and
Refraction, Grazing and Critical Angles ............... 172
Bibliography ............................................... 177
5 Guided Waves in Plates and Rods ............................ 179
5.1 Introduction .......................................... 179
5.2 Guided Waves in a Uniaxial Laminate ................... 180
5.2.1 Preliminaries .................................. 180
5.2.2 Plate Wave Solutions ........................... 184
5.3 Leaky Guided Waves in a Fluid-Loaded Plate ............ 192
5.4 Fluid-Solid Plate Reflection Coefficient .............. 201
5.5 Waves in Composite Rods ............................... 205
Bibliography ............................................... 221
6 Elastic Waves in Multilayer Composites ..................... 225
6.1 Introduction .......................................... 225
6.2 Transfer Matrix ....................................... 228
6.3 Stiffness Matrix ...................................... 233
6.3.1 General formulation of the stiffness matrix
for an anisotropic layer ....................... 233
6.3.2 Global stiffness matrix ........................ 239
6.3.3 Relation between transfer and stiffness
matrices ....................................... 239
6.4 Scattering Coefficients for a Fluid-Loaded Composite
Laminate .............................................. 240
6.5 Experimental Phenomenology ............................ 244
6.6 Extensions of the Stiffness Matrix .................... 258
6.6.1 Higher symmetry lamina ......................... 258
6.6.2 Stiffness matrix for a fluid ................... 259
6.6.3 Stiffness matrix for imperfect boundary
conditions ..................................... 260
6.6.4 Stiffness matrix computations for a
monoclinic lamina .............................. 262
6.6.5 Simple asymptotic method to compute stiffness
matrix ......................................... 263
Bibliography ................................... 267
7 Waves in Periodically Layered Composites ................... 270
7.1 Introduction and В ackground .......................... 270
7.2 A Simple Illustration ................................. 275
7.3 Floquet Analysis for Anisotropic Periodic Plates ...... 278
7.4 Homogenization of Periodically Layered Composites ..... 282
7.4.1 Floquet wave spectrum and signal distortion .... 282
7.4.2 Homogenization approach ........................ 283
7.4.3 Homogenization domain estimation for an
anisotropic cell ............................... 285
7.5 Lamina Moduli Measurement by Roquet waves ............. 286
7.5.1 Problem statement .............................. 286
7.5.2 Floquet wave lamina moduli determination ....... 287
7.5.3 Effective property determination: Static
limit .......................................... 289
7.6 Computation of the Guided Wave Spectrum ............... 291
Bibliography ............................................... 292
8 Measurement of Scattering Coefficients ..................... 294
8.1 Introduction .......................................... 294
8.2 Scattering Coefficient Integral ....................... 296
8.2.1 Uniform asymptotics ............................ 301
8.3 Computational Results ................................. 306
8.4 Complex Transducer Points ............................. 309
8.4.1 Two-dimensional voltage calculation ............ 313
8.4.2 Synthetic aperture scanning .................... 315
8.4.3 Focused beams .................................. 317
8.4.4 Three-dimensional effects on the receiver
voltage ........................................ 318
8.5 Experimental Results .................................. 321
8.6 Elastic Stiffness Reconstruction ...................... 324
Bibliography ............................................... 329
9 Air-Coupled Ultrasonics .................................... 332
9.1 Introduction .......................................... 332
9.2 Transduction and Other Challenges ..................... 333
9.3 Material Characterization in Air ...................... 337
9.4 Focusing .............................................. 348
9.5 Techniques and Applications ........................... 351
Bibliography ............................................... 367
Index ......................................................... 369
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