In recent times, the use of composites and functionally graded materials (FGMs) in structural applications has increased. FGMs allow the user to design materials for a specified functionality and therefore have numerous uses in structural engineering. However, the behaviour of these structures under high-impact loading is not well understood. Spectral Finite Element Method: Wave Propagation, Health Monitoring and Control in Composite and Functionally Graded Structures focuses on some of the wave propagation and transient dynamics problems with this complex media which had previously been thought unmanageable. By using state-off-the-art computational power, the Spectral Finite Element Method (SFEM) can solve many practical engineering problems. This book is the first to apply SFEM to inhomogeneous and anisotropic structures in a unified and systematic manner. The authors discuss the different types of SFEM for regular and damaged 1-D and 2-D waveguides, various solution techniques, different methods of detecting the presence of damages and their locations, and different methods available to actively control the wave propagation responses. The theory is supported by tables, figures and graphs; all the numerical examples are so designed to bring out the essential wave behaviour in these complex structures. Some case studies based on real-world problems are also presented. This book is intended for senior undergraduate students and graduate students studying wave propagation in structures, smart structures, spectral finite element method and structural health monitoring. Readers will gain a complete understanding of how to formulate a spectral finite element; learn about wave behaviour in inhomogeneous and anisotropic media; and, discover how to design some diagnostic tools for monitoring the health or integrity of a structure. This important contribution to the engineering mechanics research community will also be of value to researchers and practicing engineers in structural integrity.

Introduction	p. 1
Solution Methods for Wave Propagation Problems	p. 1
Fourier Analysis	p. 6
Continuous Fourier Transforms	p. 6
Fourier Series	p. 9
Discrete Fourier Transform	p. 11
Spectral Analysis	p. 15
What is the Spectral Element Method?	p. 19
Outline and Scope of Book	p. 21
Introduction to the Theory of Anisotropic and Inhomogeneous Materials	p. 23
Introduction to Composite Materials	p. 23
Theory of Laminated Composites	p. 24
Micromechanical Analysis of a Lamina	p. 25
Strength of Materials Approach to Determination of Elastic Moduli	p. 25
Stress-Strain Relations for a Lamina	p. 29
Stress-Strain Relation for a Lamina with Arbitrary Orientation of Fibers	p. 31
Introduction to Smart Composites	p. 34
Modeling Inhomogeneous Materials	p. 38
Idealization of Wave Propagation and Solution Techniques	p. 41
General Form of the Wave Equations	p. 41
Characteristics of Waves in Anisotropic Media	p. 42
General Form of Inhomogeneous Wave Equations	p. 43
Basic Properties and Solution Techniques	p. 43
Spectral Finite Element Discretization	p. 44
Efficient Computation of the Wavenumber and Wave Amplitude	p. 48
Method 1: The Companion Matrix and the SVD Technique	p. 49
Method 2: Linearization of PEP	p. 50
Spectral Element Formulation for Isotropic Material	p. 51
Spectral Element for Rods	p. 51
Spectral Element for Beams	p. 53
Wave Propagation in One-dimensional Anisotropic Structures	p. 55
Wave Propagation in Laminated Composite Thin Rods and Beams	p. 55
Governing Equations and PEP	p. 56
Spectrum and Dispersion Relations	p. 58
Spectral Element Formulation	p. 59
Finite Length Element	p. 59
Throw-off Element	p. 61
Numerical Results and Discussions	p. 61
Impact on a Cantilever Beam	p. 61
Effect of the Axial-Flexural Coupling	p. 63
Wave Transmission and Scattering Through an Angle-joint	p. 66
Wave Propagation in Laminated Composite Thick Beams: Poisson's Contraction and Shear Deformation Models	p. 69
Wave Motion in a Thick Composite Beam	p. 70
Coupled Axial-Flexural Shear and Thickness Contractional Modes	p. 72
Correction Factors at High Frequency Limit	p. 74
Coupled Axial-Flexural Shear Without the Thickness Contractional Modes	p. 76
Modeling Spatially Distributed Dynamic Loads	p. 79
Modeling Damping Using Spectral Element	p. 81
Proportional Damping Through a Discretized Finite Element Model	p. 81
Proportional Damping Through the Wave Equation	p. 83
Numerical Results and Discussions	p. 88
Comparison of Response with Standard FEM	p. 91
Presence of Axial-Flexural Shear Coupling	p. 93
Parametric Studies on a Cantilever Beam	p. 96
Response of a Beam with Ply-drops	p. 96
Layered Composite Thin-walled Tubes	p. 99
Linear Wave Motion in Composite Tube	p. 102
Spectral Finite Element Model	p. 107
Short and Long Wavelength Limits for Thin Shell and Limitations of the Proposed Model	p. 107
Comparison with Analytical Solution	p. 114
Numerical Simulations	p. 116
Time Response Under Short Impulse Load and the Effect of Fiber Orientations	p. 116
Wave Propagation in One-dimensional Inhomogeneous Structures	p. 123
Length-wise Functionally Graded Rod	p. 124
Development of Spectral Finite Elements	p. 126
Smoothing of Reflected Pulse	p. 132
Depth-wise Functionally Graded Beam	p. 135
Spectral Finite Element Formulation	p. 137
The Spectrum and Dispersion Relation	p. 137
Effect of Gradation on the Cut-off Frequencies	p. 139
Computation of the Temperature Field	p. 142
Wave Propagation Analysis: Depth-wise Graded Beam (HMT)	p. 142
Validation of the Formulated SFE	p. 143
Lamb Wave Propagation in FSDT and HMT Beams	p. 148
Effect of Gradation on Stress Waves	p. 151
Coupled Thermoelastic Wave Propagation	p. 153
Length-wise Graded Beam: FSDT	p. 157
Spectral Finite Element Formulation	p. 158
Effect of Gradation on the Spectrum and Dispersion Relation	p. 159
Effect of Gradation on the Cut-off Frequencies	p. 160
Numerical Examples	p. 162
Effect of the Inhomogeneity	p. 162
Elimination of the Reflection from Material Boundary	p. 165
Wave Propagation in Two-dimensional Anisotropic Structures	p. 171
Two-dimensional Initial Boundary Value Problem	p. 172
Spectral Element for Doubly Bounded Media	p. 176
Finite Layer Element (FLE)	p. 177
Infinite Layer Element (ILE)	p. 178
Expressions for Stresses and Strains	p. 178
Prescription of Boundary Conditions	p. 179
Determination of Lamb Wave Modes	p. 179
Numerical Examples	p. 181
Propagation of Surface and Interface Waves	p. 181
Propagation of Lamb Wave	p. 185
Wave Propagation in Two-dimensional Inhomogeneous Structures	p. 195
SLE Formulation: Inhomogeneous Media	p. 195
Exact Formulation	p. 196
Numerical Examples	p. 201
Propagation of Stress Waves	p. 201
Propagation of Lamb Waves	p. 204
SLE Formulation: Thermoelastic Analysis	p. 208
Inhomogeneous Anisotropic Material	p. 209
Discussion on the Properties of Wavenumbers	p. 212
Finite Layer Element (FLE)	p. 215
Infinite Layer Element (ILE)	p. 216
Homogeneous Anisotropic Material	p. 217
Numerical Examples	p. 217
Effect of the Relaxation Parameters - Symmetric Ply-layup	p. 217
Interfacial Waves: Thermal and Mechanical Loading	p. 220
Propagation of Stress Waves	p. 221
Propagation of Thermal Waves	p. 226
Effect of Inhomogeneity	p. 227
Wave Motion in Anisotropic and Inhomogeneous Plate	p. 229
SPE Formulation: CLPT	p. 230
Computation of Wavenumber: Anisotropic Plate	p. 234
Computation of Wavenumber: Inhomogeneous Plate	p. 237
The Finite Plate Element	p. 241
Semi-infinite or Throw-off Plate Element	p. 242
Numerical Examples	p. 243
Wave Propagation in Plate with Ply-drop	p. 243
Propagation of Lamb waves	p. 246
Solution of Inverse Problems: Source and System Identification	p. 249
Force Identification	p. 249
Force Reconstruction from Truncated Response	p. 250
Material Property Identification	p. 253
Estimation of Material Properties: Inhomogeneous Layer	p. 254
Application of SFEM to SHM: Simplified Damage Models	p. 259
Various Damage Identification Techniques	p. 259
Techniques for Modeling Delamination	p. 260
Modeling Issues in Structural Health Monitoring	p. 261
Modeling Wave Scattering due to Multiple Delaminations and Inclusions	p. 262
Spectral Element with Embedded Delamination	p. 265
Modeling Distributed Contact Between Delaminated Surfaces	p. 269
Numerical Studies on Wave Scattering due to Single Delamination	p. 271
Comparison with 2-D FEM	p. 271
Identification of Delamination Location from Scattered Wave	p. 273
Effect of Delamination at Ply-drops	p. 274
Sensitivity of the Delaminated Configuration	p. 276
A Sublaminate-wise Constant Shear Kinematics Model	p. 279
Spectral Elements with Embedded Transverse Crack	p. 284
Element-internal Discretization and Kinematic Assumptions	p. 284
Modeling Dynamic Contact Between Crack Surfaces	p. 288
Modeling Surface-breaking Cracks	p. 290
Distributed Constraints at the Interfaces Between Sublaminates and Hanging Laminates	p. 291
Numerical Simulations	p. 293
Comparison with 2-D FEM	p. 293
Identification of Crack Location from Scattered Wave	p. 294
Sensitivity of the Crack Configuration	p. 296
Spectral Finite Element Model for Damage Estimation	p. 297
Spectral Element with Embedded Degraded Zone	p. 300
Numerical Simulations	p. 301
Application of SFEM to SHM: Efficient Damage Detection Techniques	p. 307
Strategies for Identification of Damage in Composites	p. 307
Spectral Power Flow	p. 311
Properties of Spectral Power	p. 312
Measurement of Wave Scattering due to Delaminations and Inclusions Using Spectral Power	p. 314
Power Flow Studies on Wave Scattering	p. 314
Wave Scattering due to Single Delamination	p. 314
Wave Scattering due to Length-wise Multiple Delaminations	p. 316
Wave Scattering due to Depth-wise Multiple Delaminations	p. 317
Wave Scattering due to Strip Inclusion	p. 319
Power Flow in a Semi-infinite Strip Inclusion with Bounded Media: Effect of Change in the Material Properties	p. 319
Effect of Change in the Material Properties of a Strip Inclusion	p. 321
Damage Force Indicator for SFEM	p. 323
Numerical Simulation of Global Identification Process	p. 327
Effect of Single Delamination	p. 327
Effect of Multiple Delaminations	p. 329
Sensitivity of Damage Force Indicator due to Variation in Delamination Size	p. 330
Sensitivity of Damage Force Indicator due to Variation in Delamination Depth	p. 331
Genetic Algorithm (GA) for Delamination Identification	p. 337
Objective Functions in GA for Delamination Identification	p. 338
Displacement-based Objective Functions	p. 338
Power-based Objective Functions	p. 343
Case Studies with a Cantilever Beam	p. 346
Identification of Delamination Location	p. 346
Identification of Delamination Size	p. 348
Identification of Delamination Location and Size	p. 349
Identification of Delamination Location, Size and Depth	p. 349
Effect of Delamination Near the Boundary	p. 350
Neural Network Integrated with SFEM	p. 352
Numerical Results and Discussion	p. 357
Spectral Finite Element Method for Active Wave Control	p. 365
Challenges in Designing Active Broadband Control Systems	p. 365
Strategies for Vibration and Wave Control	p. 366
Active LAC of Structural Waves	p. 371
Externally Mounted Passive/Active Devices	p. 372
Modeling Distributed Transducer Devices	p. 377
Plane Stress Constitutive Model of Stacked and Layered Piezoelectric Composite	p. 378
Constitutive Model for Piezoelectric Fiber Composite (PFC)	p. 381
Design Steps for Broadband Control	p. 391
Active Spectral Finite Element Model	p. 394
Spectral Element for Finite Beams	p. 394
Sensor Element	p. 395
Actuator Element	p. 395
Numerical Implementation	p. 397
Effect of Broadband Distributed Actuator Dynamics	p. 398
Active Control of Multiple Waves in Helicopter Gearbox Support Struts	p. 402
Active Strut System	p. 404
Numerical Simulations	p. 405
Optimal Control Based on ASFEM and Power Flow	p. 415
Linear Quadratic Optimal Control Using Spectral Power	p. 416
Broadband Control of a Three-member Composite Beam Network	p. 417
References	p. 423
Index	p. 439
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Spectral Finite Element Method

9781846283550

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