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Summary
Summary
Wave propagation is an exciting ?eld having applications cutting across many disciplines. In the ?eld of structural engineering and smart structures, wave propagation based tools have found increasing applications especially in the areaofstructuralhealthmonitoringandactivecontrolofvibrationsandnoise. Inaddition,therehasbeentremendousprogressintheareaofmaterialscience, wherein a new class of structural materials is designed to meet the parti- lar application. In most cases, these materials are not isotropic as in metallic structures. They are either anisotropic (as in the case of laminated composite structures) or inhomogeneous (as in the case of functionally graded mate- als). Analysis of these structures is many orders more complex than that of isotropic structures. For many scientists/engineers, a clear di?erence between structural dynamics and wave propagation is not evident. Traditionally, a structural designer will not be interested in the behavior of structures beyond certain frequencies, which are essentially at the lower end of the frequency scale. For such situations, available general purpose ?nite element code will satisfy the designer's requirement. However, currently, structures are required tobedesignedtosustainverycomplexandharshloadingenvironments. These loadings are essentially multi-modal phenomena and their analysis falls under the domain of wave propagation rather than structural dynamics. Evaluation of the structural integrity of anisotropic and inhomogeneous structures s- jected to such loadings is a complex process. The currently available analysis tools are highly inadequate to handle the modeling of these structures. In this book, we present a technique called the "Spectral Finite Element Method",whichwebelievewilladdresssomeoftheshortcomingsoftheexistinganalysis tools.
Author Notes
Prof. S. Gopalakrishnan is an Associate Professor at the Indian Institute of Science, Bangalore, India. He has a decade of experience in applying wave based techniques for solving various structural engineering related problems. He is internationally recognized as one of the experts in the field, and is one of the few people responsible for popularizing the use of SFEM through his research publications and presentations.
Dr A. Chakraborty is a Senior Researcher at General Motors India.
Dr Roy Mahapatra is an Assistant Professor at the Indian Institute of Science, Bangalore, India. His research activities are related to the mechanics and dynamics of solid-state engineering materials and structures, and the study of complex systems.
Table of Contents
1 Introduction | p. 1 |
1.1 Solution Methods for Wave Propagation Problems | p. 1 |
1.2 Fourier Analysis | p. 6 |
1.2.1 Continuous Fourier Transforms | p. 6 |
1.2.2 Fourier Series | p. 9 |
1.2.3 Discrete Fourier Transform | p. 11 |
1.3 Spectral Analysis | p. 15 |
1.4 What is the Spectral Element Method? | p. 19 |
1.5 Outline and Scope of Book | p. 21 |
2 Introduction to the Theory of Anisotropic and Inhomogeneous Materials | p. 23 |
2.1 Introduction to Composite Materials | p. 23 |
2.2 Theory of Laminated Composites | p. 24 |
2.2.1 Micromechanical Analysis of a Lamina | p. 25 |
2.2.2 Strength of Materials Approach to Determination of Elastic Moduli | p. 25 |
2.2.3 Stress-Strain Relations for a Lamina | p. 29 |
2.2.4 Stress-Strain Relation for a Lamina with Arbitrary Orientation of Fibers | p. 31 |
2.3 Introduction to Smart Composites | p. 34 |
2.4 Modeling Inhomogeneous Materials | p. 38 |
3 Idealization of Wave Propagation and Solution Techniques | p. 41 |
3.1 General Form of the Wave Equations | p. 41 |
3.2 Characteristics of Waves in Anisotropic Media | p. 42 |
3.3 General Form of Inhomogeneous Wave Equations | p. 43 |
3.4 Basic Properties and Solution Techniques | p. 43 |
3.5 Spectral Finite Element Discretization | p. 44 |
3.6 Efficient Computation of the Wavenumber and Wave Amplitude | p. 48 |
3.6.1 Method 1: The Companion Matrix and the SVD Technique | p. 49 |
3.6.2 Method 2: Linearization of PEP | p. 50 |
3.7 Spectral Element Formulation for Isotropic Material | p. 51 |
3.7.1 Spectral Element for Rods | p. 51 |
3.7.2 Spectral Element for Beams | p. 53 |
4 Wave Propagation in One-dimensional Anisotropic Structures | p. 55 |
4.1 Wave Propagation in Laminated Composite Thin Rods and Beams | p. 55 |
4.1.1 Governing Equations and PEP | p. 56 |
4.1.2 Spectrum and Dispersion Relations | p. 58 |
4.2 Spectral Element Formulation | p. 59 |
4.2.1 Finite Length Element | p. 59 |
4.2.2 Throw-off Element | p. 61 |
4.3 Numerical Results and Discussions | p. 61 |
4.3.1 Impact on a Cantilever Beam | p. 61 |
4.3.2 Effect of the Axial-Flexural Coupling | p. 63 |
4.3.3 Wave Transmission and Scattering Through an Angle-joint | p. 66 |
4.4 Wave Propagation in Laminated Composite Thick Beams: Poisson's Contraction and Shear Deformation Models | p. 69 |
4.4.1 Wave Motion in a Thick Composite Beam | p. 70 |
4.4.2 Coupled Axial-Flexural Shear and Thickness Contractional Modes | p. 72 |
4.4.3 Correction Factors at High Frequency Limit | p. 74 |
4.4.4 Coupled Axial-Flexural Shear Without the Thickness Contractional Modes | p. 76 |
4.4.5 Modeling Spatially Distributed Dynamic Loads | p. 79 |
4.5 Modeling Damping Using Spectral Element | p. 81 |
4.5.1 Proportional Damping Through a Discretized Finite Element Model | p. 81 |
4.5.2 Proportional Damping Through the Wave Equation | p. 83 |
4.6 Numerical Results and Discussions | p. 88 |
4.6.1 Comparison of Response with Standard FEM | p. 91 |
4.6.2 Presence of Axial-Flexural Shear Coupling | p. 93 |
4.6.3 Parametric Studies on a Cantilever Beam | p. 96 |
4.6.4 Response of a Beam with Ply-drops | p. 96 |
4.7 Layered Composite Thin-walled Tubes | p. 99 |
4.7.1 Linear Wave Motion in Composite Tube | p. 102 |
4.8 Spectral Finite Element Model | p. 107 |
4.8.1 Short and Long Wavelength Limits for Thin Shell and Limitations of the Proposed Model | p. 107 |
4.8.2 Comparison with Analytical Solution | p. 114 |
4.9 Numerical Simulations | p. 116 |
4.9.1 Time Response Under Short Impulse Load and the Effect of Fiber Orientations | p. 116 |
5 Wave Propagation in One-dimensional Inhomogeneous Structures | p. 123 |
5.1 Length-wise Functionally Graded Rod | p. 124 |
5.1.1 Development of Spectral Finite Elements | p. 126 |
5.1.2 Smoothing of Reflected Pulse | p. 132 |
5.2 Depth-wise Functionally Graded Beam | p. 135 |
5.2.1 Spectral Finite Element Formulation | p. 137 |
5.2.2 The Spectrum and Dispersion Relation | p. 137 |
5.2.3 Effect of Gradation on the Cut-off Frequencies | p. 139 |
5.2.4 Computation of the Temperature Field | p. 142 |
5.3 Wave Propagation Analysis: Depth-wise Graded Beam (HMT) | p. 142 |
5.3.1 Validation of the Formulated SFE | p. 143 |
5.3.2 Lamb Wave Propagation in FSDT and HMT Beams | p. 148 |
5.3.3 Effect of Gradation on Stress Waves | p. 151 |
5.3.4 Coupled Thermoelastic Wave Propagation | p. 153 |
5.4 Length-wise Graded Beam: FSDT | p. 157 |
5.4.1 Spectral Finite Element Formulation | p. 158 |
5.4.2 Effect of Gradation on the Spectrum and Dispersion Relation | p. 159 |
5.4.3 Effect of Gradation on the Cut-off Frequencies | p. 160 |
5.5 Numerical Examples | p. 162 |
5.5.1 Effect of the Inhomogeneity | p. 162 |
5.5.2 Elimination of the Reflection from Material Boundary | p. 165 |
6 Wave Propagation in Two-dimensional Anisotropic Structures | p. 171 |
6.1 Two-dimensional Initial Boundary Value Problem | p. 172 |
6.2 Spectral Element for Doubly Bounded Media | p. 176 |
6.2.1 Finite Layer Element (FLE) | p. 177 |
6.2.2 Infinite Layer Element (ILE) | p. 178 |
6.2.3 Expressions for Stresses and Strains | p. 178 |
6.2.4 Prescription of Boundary Conditions | p. 179 |
6.2.5 Determination of Lamb Wave Modes | p. 179 |
6.3 Numerical Examples | p. 181 |
6.3.1 Propagation of Surface and Interface Waves | p. 181 |
6.3.2 Propagation of Lamb Wave | p. 185 |
7 Wave Propagation in Two-dimensional Inhomogeneous Structures | p. 195 |
7.1 SLE Formulation: Inhomogeneous Media | p. 195 |
7.1.1 Exact Formulation | p. 196 |
7.2 Numerical Examples | p. 201 |
7.2.1 Propagation of Stress Waves | p. 201 |
7.2.2 Propagation of Lamb Waves | p. 204 |
7.3 SLE Formulation: Thermoelastic Analysis | p. 208 |
7.3.1 Inhomogeneous Anisotropic Material | p. 209 |
7.3.2 Discussion on the Properties of Wavenumbers | p. 212 |
7.3.3 Finite Layer Element (FLE) | p. 215 |
7.3.4 Infinite Layer Element (ILE) | p. 216 |
7.3.5 Homogeneous Anisotropic Material | p. 217 |
7.4 Numerical Examples | p. 217 |
7.4.1 Effect of the Relaxation Parameters - Symmetric Ply-layup | p. 217 |
7.4.2 Interfacial Waves: Thermal and Mechanical Loading | p. 220 |
7.4.3 Propagation of Stress Waves | p. 221 |
7.4.4 Propagation of Thermal Waves | p. 226 |
7.4.5 Effect of Inhomogeneity | p. 227 |
7.5 Wave Motion in Anisotropic and Inhomogeneous Plate | p. 229 |
7.5.1 SPE Formulation: CLPT | p. 230 |
7.5.2 Computation of Wavenumber: Anisotropic Plate | p. 234 |
7.5.3 Computation of Wavenumber: Inhomogeneous Plate | p. 237 |
7.5.4 The Finite Plate Element | p. 241 |
7.5.5 Semi-infinite or Throw-off Plate Element | p. 242 |
7.6 Numerical Examples | p. 243 |
7.6.1 Wave Propagation in Plate with Ply-drop | p. 243 |
7.6.2 Propagation of Lamb waves | p. 246 |
8 Solution of Inverse Problems: Source and System Identification | p. 249 |
8.1 Force Identification | p. 249 |
8.1.1 Force Reconstruction from Truncated Response | p. 250 |
8.2 Material Property Identification | p. 253 |
8.2.1 Estimation of Material Properties: Inhomogeneous Layer | p. 254 |
9 Application of SFEM to SHM: Simplified Damage Models | p. 259 |
9.1 Various Damage Identification Techniques | p. 259 |
9.1.1 Techniques for Modeling Delamination | p. 260 |
9.1.2 Modeling Issues in Structural Health Monitoring | p. 261 |
9.2 Modeling Wave Scattering due to Multiple Delaminations and Inclusions | p. 262 |
9.3 Spectral Element with Embedded Delamination | p. 265 |
9.3.1 Modeling Distributed Contact Between Delaminated Surfaces | p. 269 |
9.4 Numerical Studies on Wave Scattering due to Single Delamination | p. 271 |
9.4.1 Comparison with 2-D FEM | p. 271 |
9.4.2 Identification of Delamination Location from Scattered Wave | p. 273 |
9.4.3 Effect of Delamination at Ply-drops | p. 274 |
9.4.4 Sensitivity of the Delaminated Configuration | p. 276 |
9.5 A Sublaminate-wise Constant Shear Kinematics Model | p. 279 |
9.6 Spectral Elements with Embedded Transverse Crack | p. 284 |
9.6.1 Element-internal Discretization and Kinematic Assumptions | p. 284 |
9.6.2 Modeling Dynamic Contact Between Crack Surfaces | p. 288 |
9.6.3 Modeling Surface-breaking Cracks | p. 290 |
9.6.4 Distributed Constraints at the Interfaces Between Sublaminates and Hanging Laminates | p. 291 |
9.7 Numerical Simulations | p. 293 |
9.7.1 Comparison with 2-D FEM | p. 293 |
9.7.2 Identification of Crack Location from Scattered Wave | p. 294 |
9.7.3 Sensitivity of the Crack Configuration | p. 296 |
9.8 Spectral Finite Element Model for Damage Estimation | p. 297 |
9.8.1 Spectral Element with Embedded Degraded Zone | p. 300 |
9.9 Numerical Simulations | p. 301 |
10 Application of SFEM to SHM: Efficient Damage Detection Techniques | p. 307 |
10.1 Strategies for Identification of Damage in Composites | p. 307 |
10.2 Spectral Power Flow | p. 311 |
10.2.1 Properties of Spectral Power | p. 312 |
10.2.2 Measurement of Wave Scattering due to Delaminations and Inclusions Using Spectral Power | p. 314 |
10.3 Power Flow Studies on Wave Scattering | p. 314 |
10.3.1 Wave Scattering due to Single Delamination | p. 314 |
10.3.2 Wave Scattering due to Length-wise Multiple Delaminations | p. 316 |
10.3.3 Wave Scattering due to Depth-wise Multiple Delaminations | p. 317 |
10.4 Wave Scattering due to Strip Inclusion | p. 319 |
10.4.1 Power Flow in a Semi-infinite Strip Inclusion with Bounded Media: Effect of Change in the Material Properties | p. 319 |
10.4.2 Effect of Change in the Material Properties of a Strip Inclusion | p. 321 |
10.5 Damage Force Indicator for SFEM | p. 323 |
10.6 Numerical Simulation of Global Identification Process | p. 327 |
10.6.1 Effect of Single Delamination | p. 327 |
10.6.2 Effect of Multiple Delaminations | p. 329 |
10.6.3 Sensitivity of Damage Force Indicator due to Variation in Delamination Size | p. 330 |
10.6.4 Sensitivity of Damage Force Indicator due to Variation in Delamination Depth | p. 331 |
10.7 Genetic Algorithm (GA) for Delamination Identification | p. 337 |
10.7.1 Objective Functions in GA for Delamination Identification | p. 338 |
10.7.2 Displacement-based Objective Functions | p. 338 |
10.7.3 Power-based Objective Functions | p. 343 |
10.8 Case Studies with a Cantilever Beam | p. 346 |
10.8.1 Identification of Delamination Location | p. 346 |
10.8.2 Identification of Delamination Size | p. 348 |
10.8.3 Identification of Delamination Location and Size | p. 349 |
10.8.4 Identification of Delamination Location, Size and Depth | p. 349 |
10.8.5 Effect of Delamination Near the Boundary | p. 350 |
10.9 Neural Network Integrated with SFEM | p. 352 |
10.10 Numerical Results and Discussion | p. 357 |
11 Spectral Finite Element Method for Active Wave Control | p. 365 |
11.1 Challenges in Designing Active Broadband Control Systems | p. 365 |
11.1.1 Strategies for Vibration and Wave Control | p. 366 |
11.1.2 Active LAC of Structural Waves | p. 371 |
11.2 Externally Mounted Passive/Active Devices | p. 372 |
11.3 Modeling Distributed Transducer Devices | p. 377 |
11.3.1 Plane Stress Constitutive Model of Stacked and Layered Piezoelectric Composite | p. 378 |
11.3.2 Constitutive Model for Piezoelectric Fiber Composite (PFC) | p. 381 |
11.3.3 Design Steps for Broadband Control | p. 391 |
11.4 Active Spectral Finite Element Model | p. 394 |
11.4.1 Spectral Element for Finite Beams | p. 394 |
11.4.2 Sensor Element | p. 395 |
11.4.3 Actuator Element | p. 395 |
11.4.4 Numerical Implementation | p. 397 |
11.5 Effect of Broadband Distributed Actuator Dynamics | p. 398 |
11.6 Active Control of Multiple Waves in Helicopter Gearbox Support Struts | p. 402 |
11.6.1 Active Strut System | p. 404 |
11.6.2 Numerical Simulations | p. 405 |
11.7 Optimal Control Based on ASFEM and Power Flow | p. 415 |
11.7.1 Linear Quadratic Optimal Control Using Spectral Power | p. 416 |
11.7.2 Broadband Control of a Three-member Composite Beam Network | p. 417 |
References | p. 423 |
Index | p. 439 |