Lamb-Wave Based Structural Health Monitoring in Polymer Composites (eBook)

Preface 6
Contents 8
Part I Introduction 16
1 Motivation 17
1.1 Why Structural Health Monitoring? 17
1.2 What Should a SHM System Be Capable of? 19
1.3 What is the Foundation of the SHM Research Presented? 19
1.4 What Are the Challenges? 22
References 23
2 Objectives 24
Part II Foundations 27
3 Wave Propagation in Elastic Solids: An Analytical Approach 28
3.1 Introduction 28
3.2 Isotropic Solids 30
3.2.1 Lamé–Navier Equations 30
3.2.2 Waves in Infinite Solids 30
3.2.3 Waves in Thin-Walled Solids 31
3.2.3.1 Stress Boundary Conditions 35
3.2.3.2 Rayleigh–Lamb Wave Equation 36
3.2.3.3 Determination of the Displacement Field 37
3.2.3.4 Group Velocity 38
3.3 Anisotropic Solids 39
3.3.1 General Fundamentals 39
3.3.2 Wave Propagation Without Decoupling of Lamb Waves and Shear Horizontal Waves 41
Dispersion Relation 45
3.3.3 Wave Propagation with Decoupling of Lamb Waves and Shear Horizontal Waves 46
3.3.3.1 Dispersion Relation 49
3.4 Layered Anisotropic Solids 51
3.4.1 Transfer-Matrix Method 52
Separation of the Symmetric and Antisymmetric Wave Modes 55
3.4.2 Global-Matrix Method 58
Separation of the Symmetric and Antisymmetric Wave Modes 60
3.4.3 Stiffness-Matrix Method 60
Dispersion of a Quasi-Isotropic Specimen 63
3.4.3.1 Summary 64
Appendix 1: Characteristic Polynomial of the Christoffel Equation 65
Appendix 2: Summary of Stresses and Displacements of a Single Anisotropic Layer in a System of Equations 66
Appendix 3: Separated Dispersion Relations for the Symmetric and Antisymmetric Wave Modes 69
References 72
4 Fundamental Principles of the Finite Element Method 74
4.1 Governing Equations 74
4.2 Constitutive Equations 76
4.3 Weak Form of the Equations of Motion 77
4.4 Finite Element Equations 79
4.5 h-Version of the Finite Element Method 86
4.6 Time-Integration Methods 87
4.6.1 Explicit Time Integration: Central Difference Method 88
4.6.2 Implicit Time Integration: Newmark Method 90
4.6.3 Mass Lumping Techniques 93
4.6.3.1 Nodal Quadrature Technique 93
4.6.3.2 Row Sum Technique 94
4.6.3.3 Diagonal Scaling: HRZ Lumping Technique 95
4.7 Geometry Approximation 95
4.7.1 Subparametric Mapping 96
4.7.2 Isoparametric Mapping 97
4.7.3 Superparametric Mapping 98
4.7.3.1 Blending Function Method 98
References 100
5 Experimental Methods 102
5.1 Requirements for a Measurement System for High-Resolution Wave Field Recording of Lamb Waves 102
5.1.1 Measurement Principle 103
5.1.2 Flexibility 104
5.1.3 Speed 104
5.2 Ultrasonic Scanning Technique 104
5.3 Imaging Methods 106
5.4 3D Laser Vibrometry 108
5.4.1 Description of the Measurement Platform 108
5.4.2 Determining the Three-Dimensional Displacement 111
5.4.3 Sensitivity of In-Plane Displacement Measurements 114
5.5 Conclusion 121
References 122
Part III Efficient Numerical Methods for Wave Propagation Analysis 123
References 124
6 Higher Order Finite Element Methods 126
6.1 Higher Order Finite Element Methods: One-Dimensional Case 126
6.1.1 p-Version of the Finite Element Method 127
6.1.2 The Spectral Element Method 129
6.1.3 The Isogeometric Analysis 133
6.1.3.1 B-Spline Curve 133
6.1.3.2 Nonuniform Rational B-Spline Curve 134
6.2 Comparison of the Properties of Different Higher Order Finite Element Approaches 136
6.2.1 Hierarchic Basis Functions 136
6.2.2 Nodal Basis Functions 137
6.3 Multivariate Basis Functions 138
6.4 Benchmark Problems 138
6.4.1 p-FEM: Modal Analysis of a Three-Dimensional Piezoelectric Disc 139
6.4.1.1 Short-Circuited Electrodes 140
6.4.1.2 Open Electrodes 140
6.4.1.3 Eigenfrequencies and Mode Shapes of a Circular Disc 141
6.4.2 Spectral Element Method: Wave Propagation Analysis in a Two-Dimensional Porous Plate 144
6.4.3 Isogeometric Analysis: Wave Propagation Analysis in a Three-Dimensional Perforated Plate 151
6.5 Convergence Studies 153
6.5.1 Numerical Model 153
6.5.2 Signal Analysis 155
6.5.3 Polynomial Degree in x1 155
6.5.4 Polynomial Degree in x2 156
6.6 Industrial Applications 160
6.6.1 Stiffened Composite Plate 160
6.6.2 Rotor Blade of a Wind Turbine 163
References 165
7 Hybrid Simulation Methods: Combining Finite Element Methods and Analytical Solutions 169
7.1 The Semi-Analytical Finite Element Method 169
7.1.1 Motivation 170
7.1.2 Theoretical Principles 170
7.1.3 Plate with Infinite Dimensions 174
7.1.4 Dispersion Curves for Undamped Media 176
7.1.4.1 Phase and Group Velocities of Guided Waves 178
7.1.4.2 Verification 179
7.1.5 Interaction of Guided Waves with Perturbations 182
7.1.5.1 General approach 182
7.1.5.2 Verification 184
7.1.6 Force Response Analysis 187
7.1.6.1 General approach 187
7.1.6.2 Verification 188
7.1.7 Summary 190
7.2 Coupling of Analytical Solutions and the Spectral Element Method in the Frequency Domain 190
7.2.1 Motivation 190
7.2.2 Definition of the Problem 191
7.2.3 Analytical Solution to the Wave Propagation Problem in Isotropic Plates 193
7.2.3.1 Analytical Approach 193
7.2.3.2 Two-Dimensional Problem 194
7.2.4 Coupling Boundary Conditions 199
7.2.5 Numerical Results 200
References 203
8 Damping Boundary Conditions for a Reduced Solution Domain Size and Effective Numerical Analysis of HeterogeneousWaveguides 207
8.1 Objective 207
8.2 Non-reflecting Boundary Conditions 209
8.2.1 Basic Principles 210
8.2.1.1 Damping Factor 212
8.2.1.2 Direction of Dashpot Elements 213
8.2.1.3 Number of Dashpot Element Layers 214
8.2.2 Numerical Example 215
8.3 Parametric Studies of Wave Propagation in Cellular Materials 217
8.3.1 Sandwich Panel with a Foam Core 217
8.3.1.1 Numerical Model 217
8.3.1.2 Influence of the Excitation Frequency 219
8.3.1.3 Influence of Geometrical Dimensions 219
8.3.2 Sandwich Panel with a Honeycomb Core 220
8.3.2.1 Numerical Model 220
8.3.2.2 Influence of the Excitation Frequency 221
8.3.2.3 Influence of Geometrical Dimensions 221
References 223
9 The Finite Cell Method: A Higher Order Fictitious Domain Approach for Wave Propagation Analysis in Heterogeneous Structures 225
9.1 Motivation 225
9.2 Fictitious Domain Concept 228
9.3 Finite Cell Equations 230
9.4 Numerical Integration 232
9.4.1 Adaptive Quadrature Scheme 232
9.4.2 Improved Integration Algorithms 235
9.5 Geometry Description 237
9.5.1 Implicit Functions 237
9.5.2 Boundary Representation (B-Rep) 238
9.5.3 CT-Scan 238
9.6 Numerical Results: Wave Propagation Analysis in a Two-Dimensional Porous Plate 239
9.7 Note on the Extension to Unstructured Discretizations 244
References 245
10 A Minimal Model for Fast Approximation of Lamb Wave Propagation in Complex Aircraft Parts 248
10.1 Lamb Wave Simulation and Its Applications 248
10.2 Minimal Model 250
10.2.1 Interaction of Lamb Waves with Discontinuities 251
10.2.2 Ray Tracing 253
10.2.3 Signal Synthesis 256
10.3 Experimental Results and Comparison 258
10.3.1 Aluminum Plate 259
10.3.2 Aluminum Plate with Cutout 262
10.4 Discussion 264
10.5 Conclusion 266
References 266
Part IV Continuous Mode Conversion 269
11 Continuous Mode Conversion in Experimental Observations 270
11.1 Mode Conversion in Polymer Composites 270
11.2 Occurrence and Characteristics of Continuous ModeConversion 272
11.2.1 CFRP Plates Made of Unidirectional Layers 272
11.2.2 CFRP Plates Made of Woven Layers 273
11.3 Physical Reasons of Continuous Mode Conversion in Woven Layers 274
11.3.1 Experimental Tensile Tests 274
11.3.2 Finite Element Modeling 276
11.3.3 Numerical Tensile Tests 277
11.3.4 Lamb Wave Simulation 278
11.3.5 Frequency Dependence of Coupling-Induced Mode Conversion 279
11.4 Conclusion 281
References 281
12 Material Modeling of Polymer Composites for Numerical Investigations of Continuous Mode Conversion 283
12.1 Analysis of the Wave Behavior in Simplified Models with Reference to the Continuous Mode Conversion 283
12.1.1 Aluminum Plates with Changes in Cross Section 285
12.1.1.1 Plate with Obstacles 285
12.1.1.2 Plate with Notches 287
12.1.1.3 Intermediate Results 289
12.1.2 Conventional Material Modeling of CFRP 289
12.1.2.1 General Rule of Mixture 290
12.1.2.2 Semiempirical Homogenization Method of Halpin and Tsai 291
12.1.3 Fiber–Matrix Models 291
12.1.4 Intermediate Results 294
12.2 Numerical Realization of the Continuous Mode ConversionEffect 295
12.2.1 Enhanced FE-Material Modeling of UD-Layers 295
12.2.2 Wave Propagation in UD-Layers Using the Enhanced FE-Material Modeling 297
12.3 Conclusion 299
References 301
Part V Signal Processing 302
13 Localization of Damaging Events and Damage in Anisotropic Plates by Migration Technique 303
13.1 Impact Localization 303
13.1.1 Migration Method for Isotropic Materials 303
13.1.2 Enhanced Migration Method for AnisotropicMaterials 305
13.1.2.1 Determination of the Wave Velocity 306
13.1.2.2 Determination by Experimental Data 306
13.1.2.3 Determination by Material Data 306
13.1.2.4 Parametrization of Wave Propagation 309
13.1.2.5 Formulation of the Enhanced Migration Method for Anisotropic Material 309
13.1.3 Solution Procedure 310
13.2 Damage Localization 311
13.3 Experimental Verification 312
13.3.1 Impact Localization 312
13.3.1.1 Experimental Setup 312
13.3.1.2 Shape of the Wave Front 313
13.3.1.3 Experimental Results and Data Analysis 314
13.3.1.4 Signal Analysis 314
13.3.1.5 Optimization 318
13.3.1.6 Results 318
13.3.1.7 Discussion of the Results 319
13.3.1.8 Influence of Time Resolution 321
13.3.1.9 Influence of the Wavelet Transform 321
13.3.1.10 Influence of the Reference Signal 322
13.3.1.11 Influence of the Number of Active Sensors 323
13.3.1.12 Quality of Results 324
13.3.2 Defect Localization 325
13.3.2.1 Experimental Setup 325
13.3.2.2 Determination of the Wave Propagation Pattern 327
13.3.2.3 Experimental Results and Evaluation 327
13.3.2.4 Results 328
13.3.2.5 Discussion of the Results 328
References 331
14 Time-of-Flight Calculation in Complex Structures 333
14.1 Requirements for Time-of-Flight Calculation 334
14.2 Algorithms for Time-of-Flight Calculation 336
14.2.1 Raytracing 336
14.2.2 Dijkstra Algorithm 337
14.2.3 Path Calculation in Segmented Areas 338
14.2.4 Bellman-Ford Algorithm 338
14.2.5 Floyd-Warshall Algorithm 339
14.2.6 Front Propagation Algorithms 339
14.3 Time-of-Flight Calculation for Lamb Waves in Complex Structures 340
14.3.1 Discretization of the Structure 341
14.3.2 Considerations for the Algorithm for the Time-of-Flight Calculation 345
14.3.2.1 Efficiency 345
14.3.2.2 Memory 345
14.3.2.3 Parallel Implementation 346
14.3.2.4 Classification of Nodes 346
14.3.2.5 Covered Nodes per Iteration 347
14.3.2.6 Number of Initial Sources 347
14.3.3 Algorithm for Time-of-Flight Calculation 347
14.4 Influences on the Time-of-Flight Calculation 349
14.4.1 Discretization 349
14.4.2 Non-isotropic Material Properties Inside the Elements 354
14.4.3 Non-convex Velocity Distributions Insidethe Elements 356
14.4.4 Subsequent Determination of Fastest Paths 357
References 358
15 The Determination of Dispersion Curves from Measurements by the Matrix Pencil Method 360
15.1 Introduction 360
15.2 Dispersion Relations via Mode Decomposition in the Wavenumber Domain 361
15.3 The Matrix Pencil Method 363
15.4 Experimental Setup 365
15.5 From Laser Vibrometer Data to Matrix Pencil Data 366
15.6 Numerical Results and Detection of BackwardPropagating Waves 368
15.7 Conclusions 372
References 373
16 Damage Identification by Dynamic Load Monitoring 374
16.1 Introduction 375
16.2 Dynamic Load Monitoring as a Minimization Problem 377
16.3 Numerical Solution of the Tikhonov Minimization Problem 386
16.4 Numerical Results 391
References 397
Part VI SHM: Systems 399
17 Mode Selective Actuator-Sensor-Systems 401
17.1 Introduction 401
17.2 Analytical Model for Mode SelectiveActuator-Sensor-Systems 406
17.2.1 Mode Tuning: 2D Problem 406
17.2.2 Acoustic Wave Field: 3D Problem 413
17.3 Experimental Verification of Mode Selective Actuator-Sensor-Systems 418
17.3.1 Manufacturing Technologies of Mode Selective Transducers 418
17.3.2 Experimental Setup 421
17.3.3 Experimental Results Regarding Mode Tuning 422
17.3.4 Experimental Results Regarding Acoustic Wave Field 425
Appendix 427
References 429
18 Virtual Sensors for SHM 431
18.1 Introduction 431
18.2 Leaky Guided Waves 432
18.3 Displacement Ratios 433
18.4 Adaption of Wave Radiation 436
18.5 Sensor Model 437
18.6 Experimental Validation 439
References 441
19 Lamb Wave Generation, Propagation, and Interactions in CFRP Plates 442
19.1 Experimental Setup 442
19.2 Characterization of Piezo-Actuators and Their Wave Fields 443
19.3 Velocity and Attenuation Measurement of Lamb Waves 447
19.3.1 Methods of Dispersion Curves Determination 447
19.3.2 Comparison of Lamb Wave Velocities in DifferentPlates 448
19.4 Attenuation of Lamb Waves 450
19.5 Interactions with Inhomogeneities 452
19.5.1 Non-Mode Converting Inhomogeneities 452
19.5.2 Mode Converting Inhomogeneities 452
19.5.2.1 Interactions with Asymmetric Wall Thickness Changes 452
19.5.2.2 Interactions with Stringers 454
19.5.2.3 Interactions with Stringer Defects 456
19.5.2.4 Continuous Mode Conversion 457
19.6 Conclusion 458
References 459
20 Structural Health Monitoring on the SARISTU Full Scale Door Surround Structure 461
20.1 Introduction 461
20.2 Integration of a SHM Network in the Structure 462
20.3 A Probability-Based Diagnostic Imaging Approach 464
20.4 Damage Assessment 465
20.5 Conclusion 469
References 470
Index 472