Cover image for Spectral finite element method : wave propagation, diagnostics and control in anisotropic and inhomogenous structures
Title:
Spectral finite element method : wave propagation, diagnostics and control in anisotropic and inhomogenous structures
Personal Author:
Series:
Computational fluid and solid mechanics
Publication Information:
London : Springer, 2008
ISBN:
9781846283550

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30000010156634 QA935 G66 2008 Open Access Book Book
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30000010195479 QA935 G66 2008 Open Access Book Book
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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 Introductionp. 1
1.1 Solution Methods for Wave Propagation Problemsp. 1
1.2 Fourier Analysisp. 6
1.2.1 Continuous Fourier Transformsp. 6
1.2.2 Fourier Seriesp. 9
1.2.3 Discrete Fourier Transformp. 11
1.3 Spectral Analysisp. 15
1.4 What is the Spectral Element Method?p. 19
1.5 Outline and Scope of Bookp. 21
2 Introduction to the Theory of Anisotropic and Inhomogeneous Materialsp. 23
2.1 Introduction to Composite Materialsp. 23
2.2 Theory of Laminated Compositesp. 24
2.2.1 Micromechanical Analysis of a Laminap. 25
2.2.2 Strength of Materials Approach to Determination of Elastic Modulip. 25
2.2.3 Stress-Strain Relations for a Laminap. 29
2.2.4 Stress-Strain Relation for a Lamina with Arbitrary Orientation of Fibersp. 31
2.3 Introduction to Smart Compositesp. 34
2.4 Modeling Inhomogeneous Materialsp. 38
3 Idealization of Wave Propagation and Solution Techniquesp. 41
3.1 General Form of the Wave Equationsp. 41
3.2 Characteristics of Waves in Anisotropic Mediap. 42
3.3 General Form of Inhomogeneous Wave Equationsp. 43
3.4 Basic Properties and Solution Techniquesp. 43
3.5 Spectral Finite Element Discretizationp. 44
3.6 Efficient Computation of the Wavenumber and Wave Amplitudep. 48
3.6.1 Method 1: The Companion Matrix and the SVD Techniquep. 49
3.6.2 Method 2: Linearization of PEPp. 50
3.7 Spectral Element Formulation for Isotropic Materialp. 51
3.7.1 Spectral Element for Rodsp. 51
3.7.2 Spectral Element for Beamsp. 53
4 Wave Propagation in One-dimensional Anisotropic Structuresp. 55
4.1 Wave Propagation in Laminated Composite Thin Rods and Beamsp. 55
4.1.1 Governing Equations and PEPp. 56
4.1.2 Spectrum and Dispersion Relationsp. 58
4.2 Spectral Element Formulationp. 59
4.2.1 Finite Length Elementp. 59
4.2.2 Throw-off Elementp. 61
4.3 Numerical Results and Discussionsp. 61
4.3.1 Impact on a Cantilever Beamp. 61
4.3.2 Effect of the Axial-Flexural Couplingp. 63
4.3.3 Wave Transmission and Scattering Through an Angle-jointp. 66
4.4 Wave Propagation in Laminated Composite Thick Beams: Poisson's Contraction and Shear Deformation Modelsp. 69
4.4.1 Wave Motion in a Thick Composite Beamp. 70
4.4.2 Coupled Axial-Flexural Shear and Thickness Contractional Modesp. 72
4.4.3 Correction Factors at High Frequency Limitp. 74
4.4.4 Coupled Axial-Flexural Shear Without the Thickness Contractional Modesp. 76
4.4.5 Modeling Spatially Distributed Dynamic Loadsp. 79
4.5 Modeling Damping Using Spectral Elementp. 81
4.5.1 Proportional Damping Through a Discretized Finite Element Modelp. 81
4.5.2 Proportional Damping Through the Wave Equationp. 83
4.6 Numerical Results and Discussionsp. 88
4.6.1 Comparison of Response with Standard FEMp. 91
4.6.2 Presence of Axial-Flexural Shear Couplingp. 93
4.6.3 Parametric Studies on a Cantilever Beamp. 96
4.6.4 Response of a Beam with Ply-dropsp. 96
4.7 Layered Composite Thin-walled Tubesp. 99
4.7.1 Linear Wave Motion in Composite Tubep. 102
4.8 Spectral Finite Element Modelp. 107
4.8.1 Short and Long Wavelength Limits for Thin Shell and Limitations of the Proposed Modelp. 107
4.8.2 Comparison with Analytical Solutionp. 114
4.9 Numerical Simulationsp. 116
4.9.1 Time Response Under Short Impulse Load and the Effect of Fiber Orientationsp. 116
5 Wave Propagation in One-dimensional Inhomogeneous Structuresp. 123
5.1 Length-wise Functionally Graded Rodp. 124
5.1.1 Development of Spectral Finite Elementsp. 126
5.1.2 Smoothing of Reflected Pulsep. 132
5.2 Depth-wise Functionally Graded Beamp. 135
5.2.1 Spectral Finite Element Formulationp. 137
5.2.2 The Spectrum and Dispersion Relationp. 137
5.2.3 Effect of Gradation on the Cut-off Frequenciesp. 139
5.2.4 Computation of the Temperature Fieldp. 142
5.3 Wave Propagation Analysis: Depth-wise Graded Beam (HMT)p. 142
5.3.1 Validation of the Formulated SFEp. 143
5.3.2 Lamb Wave Propagation in FSDT and HMT Beamsp. 148
5.3.3 Effect of Gradation on Stress Wavesp. 151
5.3.4 Coupled Thermoelastic Wave Propagationp. 153
5.4 Length-wise Graded Beam: FSDTp. 157
5.4.1 Spectral Finite Element Formulationp. 158
5.4.2 Effect of Gradation on the Spectrum and Dispersion Relationp. 159
5.4.3 Effect of Gradation on the Cut-off Frequenciesp. 160
5.5 Numerical Examplesp. 162
5.5.1 Effect of the Inhomogeneityp. 162
5.5.2 Elimination of the Reflection from Material Boundaryp. 165
6 Wave Propagation in Two-dimensional Anisotropic Structuresp. 171
6.1 Two-dimensional Initial Boundary Value Problemp. 172
6.2 Spectral Element for Doubly Bounded Mediap. 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 Strainsp. 178
6.2.4 Prescription of Boundary Conditionsp. 179
6.2.5 Determination of Lamb Wave Modesp. 179
6.3 Numerical Examplesp. 181
6.3.1 Propagation of Surface and Interface Wavesp. 181
6.3.2 Propagation of Lamb Wavep. 185
7 Wave Propagation in Two-dimensional Inhomogeneous Structuresp. 195
7.1 SLE Formulation: Inhomogeneous Mediap. 195
7.1.1 Exact Formulationp. 196
7.2 Numerical Examplesp. 201
7.2.1 Propagation of Stress Wavesp. 201
7.2.2 Propagation of Lamb Wavesp. 204
7.3 SLE Formulation: Thermoelastic Analysisp. 208
7.3.1 Inhomogeneous Anisotropic Materialp. 209
7.3.2 Discussion on the Properties of Wavenumbersp. 212
7.3.3 Finite Layer Element (FLE)p. 215
7.3.4 Infinite Layer Element (ILE)p. 216
7.3.5 Homogeneous Anisotropic Materialp. 217
7.4 Numerical Examplesp. 217
7.4.1 Effect of the Relaxation Parameters - Symmetric Ply-layupp. 217
7.4.2 Interfacial Waves: Thermal and Mechanical Loadingp. 220
7.4.3 Propagation of Stress Wavesp. 221
7.4.4 Propagation of Thermal Wavesp. 226
7.4.5 Effect of Inhomogeneityp. 227
7.5 Wave Motion in Anisotropic and Inhomogeneous Platep. 229
7.5.1 SPE Formulation: CLPTp. 230
7.5.2 Computation of Wavenumber: Anisotropic Platep. 234
7.5.3 Computation of Wavenumber: Inhomogeneous Platep. 237
7.5.4 The Finite Plate Elementp. 241
7.5.5 Semi-infinite or Throw-off Plate Elementp. 242
7.6 Numerical Examplesp. 243
7.6.1 Wave Propagation in Plate with Ply-dropp. 243
7.6.2 Propagation of Lamb wavesp. 246
8 Solution of Inverse Problems: Source and System Identificationp. 249
8.1 Force Identificationp. 249
8.1.1 Force Reconstruction from Truncated Responsep. 250
8.2 Material Property Identificationp. 253
8.2.1 Estimation of Material Properties: Inhomogeneous Layerp. 254
9 Application of SFEM to SHM: Simplified Damage Modelsp. 259
9.1 Various Damage Identification Techniquesp. 259
9.1.1 Techniques for Modeling Delaminationp. 260
9.1.2 Modeling Issues in Structural Health Monitoringp. 261
9.2 Modeling Wave Scattering due to Multiple Delaminations and Inclusionsp. 262
9.3 Spectral Element with Embedded Delaminationp. 265
9.3.1 Modeling Distributed Contact Between Delaminated Surfacesp. 269
9.4 Numerical Studies on Wave Scattering due to Single Delaminationp. 271
9.4.1 Comparison with 2-D FEMp. 271
9.4.2 Identification of Delamination Location from Scattered Wavep. 273
9.4.3 Effect of Delamination at Ply-dropsp. 274
9.4.4 Sensitivity of the Delaminated Configurationp. 276
9.5 A Sublaminate-wise Constant Shear Kinematics Modelp. 279
9.6 Spectral Elements with Embedded Transverse Crackp. 284
9.6.1 Element-internal Discretization and Kinematic Assumptionsp. 284
9.6.2 Modeling Dynamic Contact Between Crack Surfacesp. 288
9.6.3 Modeling Surface-breaking Cracksp. 290
9.6.4 Distributed Constraints at the Interfaces Between Sublaminates and Hanging Laminatesp. 291
9.7 Numerical Simulationsp. 293
9.7.1 Comparison with 2-D FEMp. 293
9.7.2 Identification of Crack Location from Scattered Wavep. 294
9.7.3 Sensitivity of the Crack Configurationp. 296
9.8 Spectral Finite Element Model for Damage Estimationp. 297
9.8.1 Spectral Element with Embedded Degraded Zonep. 300
9.9 Numerical Simulationsp. 301
10 Application of SFEM to SHM: Efficient Damage Detection Techniquesp. 307
10.1 Strategies for Identification of Damage in Compositesp. 307
10.2 Spectral Power Flowp. 311
10.2.1 Properties of Spectral Powerp. 312
10.2.2 Measurement of Wave Scattering due to Delaminations and Inclusions Using Spectral Powerp. 314
10.3 Power Flow Studies on Wave Scatteringp. 314
10.3.1 Wave Scattering due to Single Delaminationp. 314
10.3.2 Wave Scattering due to Length-wise Multiple Delaminationsp. 316
10.3.3 Wave Scattering due to Depth-wise Multiple Delaminationsp. 317
10.4 Wave Scattering due to Strip Inclusionp. 319
10.4.1 Power Flow in a Semi-infinite Strip Inclusion with Bounded Media: Effect of Change in the Material Propertiesp. 319
10.4.2 Effect of Change in the Material Properties of a Strip Inclusionp. 321
10.5 Damage Force Indicator for SFEMp. 323
10.6 Numerical Simulation of Global Identification Processp. 327
10.6.1 Effect of Single Delaminationp. 327
10.6.2 Effect of Multiple Delaminationsp. 329
10.6.3 Sensitivity of Damage Force Indicator due to Variation in Delamination Sizep. 330
10.6.4 Sensitivity of Damage Force Indicator due to Variation in Delamination Depthp. 331
10.7 Genetic Algorithm (GA) for Delamination Identificationp. 337
10.7.1 Objective Functions in GA for Delamination Identificationp. 338
10.7.2 Displacement-based Objective Functionsp. 338
10.7.3 Power-based Objective Functionsp. 343
10.8 Case Studies with a Cantilever Beamp. 346
10.8.1 Identification of Delamination Locationp. 346
10.8.2 Identification of Delamination Sizep. 348
10.8.3 Identification of Delamination Location and Sizep. 349
10.8.4 Identification of Delamination Location, Size and Depthp. 349
10.8.5 Effect of Delamination Near the Boundaryp. 350
10.9 Neural Network Integrated with SFEMp. 352
10.10 Numerical Results and Discussionp. 357
11 Spectral Finite Element Method for Active Wave Controlp. 365
11.1 Challenges in Designing Active Broadband Control Systemsp. 365
11.1.1 Strategies for Vibration and Wave Controlp. 366
11.1.2 Active LAC of Structural Wavesp. 371
11.2 Externally Mounted Passive/Active Devicesp. 372
11.3 Modeling Distributed Transducer Devicesp. 377
11.3.1 Plane Stress Constitutive Model of Stacked and Layered Piezoelectric Compositep. 378
11.3.2 Constitutive Model for Piezoelectric Fiber Composite (PFC)p. 381
11.3.3 Design Steps for Broadband Controlp. 391
11.4 Active Spectral Finite Element Modelp. 394
11.4.1 Spectral Element for Finite Beamsp. 394
11.4.2 Sensor Elementp. 395
11.4.3 Actuator Elementp. 395
11.4.4 Numerical Implementationp. 397
11.5 Effect of Broadband Distributed Actuator Dynamicsp. 398
11.6 Active Control of Multiple Waves in Helicopter Gearbox Support Strutsp. 402
11.6.1 Active Strut Systemp. 404
11.6.2 Numerical Simulationsp. 405
11.7 Optimal Control Based on ASFEM and Power Flowp. 415
11.7.1 Linear Quadratic Optimal Control Using Spectral Powerp. 416
11.7.2 Broadband Control of a Three-member Composite Beam Networkp. 417
Referencesp. 423
Indexp. 439