Full-Field Measurements and Identification in Solid Mechanics

This timely book presents cutting-edge developments by experts in the field on the rapidly developing and scientifically challenging area of full-field measurement techniques used in solid mechanics ? including photoelasticity, grid methods, deflectometry, holography, speckle interferometry and digital image correlation. The evaluation of strains and the use of the measurements in subsequent parameter identification techniques to determine material properties are also presented.

Since parametric identification techniques require a close coupling of theoretical models and experimental measurements, the book focuses on specific modeling approaches that include finite element model updating, the equilibrium gap method, constitutive equation gap method, virtual field method and reciprocity gap method. In the latter part of the book, the authors discuss two particular applications of selected methods that are of special interest to many investigators: the analysis of localized phenomenon and connections between microstructure and constitutive laws. The final chapter highlights infrared measurements and their use in the mechanics of materials.

Written and edited by knowledgeable scientists, experts in their fields, this book will be a valuable resource for all students, faculties and scientists seeking to expand their understanding of an important, growing research area

Foreword xv
Michael A. SUTTON

Introduction xvii
Michel GRÉDIAC and François HILD

Chapter 1. Basics of Metrology and Introduction to Techniques 1
André CHRYSOCHOOS and Yves SURREL

1.1. Introduction 1

1.2. Terminology: international vocabulary of metrology 2

1.2.1. Absolute or differential measurement 2

1.2.2. Main concepts 4

1.3. Spatial aspect 11

1.3.1. Spatial frequency 11

1.3.2. Spatial filtering 16

1.4. Classification of optical measurement techniques 18

1.4.1. White light measurement methods 19

1.4.2. Interference methods 21

1.4.3. Sensitivity vector 23

1.4.4. Synthetic sensitivity vectors 23

1.4.5. The different types of interferometric measurements 24

1.4.6. Holography, digital holography 27

1.4.7. Conclusion 28

1.5. Bibliography 29

Chapter 2. Photoelasticity 31
Fabrice BRÉMAND and Jean-Christophe DUPRÉ

2.1. Introduction 31

2.2. Concept of light polarization 32

2.3. Birefringence phenomenon 33

2.4. The law of optico-mechanics 34

2.5. Several types of polariscopes 35

2.5.1. Plane polariscope 35

2.5.2. Circular polariscope 38

2.5.3. White light polariscope 40

2.5.4. Photoelastic coating 40

2.6. Measurement of photoelastic constant C 42

2.7. Analysis by image processing 43

2.7.1. Using a plane polariscope 43

2.7.2. Using a circular polariscope 47

2.7.3. Using color images 48

2.8. Post-processing of photoelastic parameters 48

2.8.1. Drawing of isostatics or stress trajectories 48

2.8.2. Particular points 48

2.8.3. Stress separation and integration of the equilibrium equations 49

2.8.4. Comparison between experimentation and numerical modeling 50

2.9. Three-dimensional photoelasticity 50

2.9.1. The method of stress freezing and mechanical slicing 51

2.9.2. Optical slicing 52

2.9.3. Application example 56

2.10. Conclusion 57

2.11. Bibliography 57

Chapter 3. Grid Method, Moiré and Deflectometry 61
Jérôme MOLIMARD and Yves SURREL

3.1. Introduction 61

3.2. Principle 61

3.3. Surface encoding 63

3.4. Moiré 64

3.5. Phase detection 66

3.5.1. Global extraction procedure 66

3.5.2. Local phase detection: phase shifting 67

3.5.3. Measuring both components of the displacement 70

3.6. Sensitivity to out-of-plane displacements 71

3.7. Grid defects 72

3.8. Large deformation/large strain 73

3.8.1. Explicit method 73

3.8.2. Implicit method 74

3.8.3. Large strain 74

3.9. Fringe projection 75

3.10. Deflectometry 78

3.11. Examples 81

3.11.1. Off-axis tensile test of a unidirectional composite coupon 81

3.11.2. Rigid body displacement 83

3.11.3. SEM measurement 84

3.11.4. Characterization of lens distortion 85

3.12. Conclusion 88

3.13. Bibliography 89

Chapter 4. Digital Holography Methods 93
Pascal PICART and Paul SMIGIELSKI

4.1. Introduction 93

4.2. Basics of wave optics 94

4.2.1. Light diffraction 95

4.2.2. Interference 96

4.3. Basics of digital holography 97

4.3.1. Recording the hologram 97

4.3.2. Numerical reconstruction with the discrete Fresnel transform 99

4.3.3. Numerical reconstruction using convolution with adjustable magnification 100

4.3.4. Sensitivity vector 101

4.4. Basics of digital holographic interferometry 103

4.4.1. Phase difference 103

4.4.2. Spatial filtering of the phase and phase unwrapping 104

4.5. Digital holographic interferometry with spatial multiplexing 104

4.5.1. Principle 104

4.5.2. Theory 105

4.5.3. Experimental set-up 105

4.5.4. Application to synthetic concrete subjected to three-point bending 107

4.6. Digital color holography applied to three-dimensional measurements 112

4.6.1. Recording digital color holograms 112

4.6.2. Application to composite material subjected to a short beam test 113

4.7. Conclusion 118

4.8. Acknowledgment 119

4.9. Bibliography 119

Chapter 5. Elementary Speckle Interferometry 125
Pierre JACQUOT, Pierre SLANGEN and Dan BORZA

5.1. Introduction 125

5.2. What is speckle interferometry? 126

5.2.1. Simplified principle – correlation fringes 128

5.2.2. Speckle field and specklegram statistics in a nutshell 129

5.2.3. Speckle field transformation – small perturbation theory 131

5.2.4. Phase change-deformation law – sensitivity vector 132

5.2.5. Success or failure of experiments – central role of decorrelation 133

5.3. Optical point of view 134

5.4. Mechanical point of view: specific displacement field components 136

5.4.1. Measurement of the out-of-plane component 136

5.4.2. Measurement of the in-plane component [LEE 70] 137

5.4.3. 3C-3D: three components attached to three-dimensional objects 138

5.4.4. Partial derivatives of the displacement – shearography 139

5.4.5. Shape measurement and other considerations 140

5.5. Phase extraction 141

5.5.1. One-image methods 141

5.5.2. Phase-shifting methods 142

5.5.3. Advanced methods 143

5.5.4. Phase unwrapping 144

5.6. Dynamic deformations and vibrations 46

5.7. Setup calibration 148

5.7.1. Specifying the material point in object coordinates 149

5.7.2. Determination of the sensitivity vector 149

5.8. Specifications and limits 150

5.9. Final remarks, outlook and trends 151

5.10. Bibliography 153

Chapter 6. Digital Image Correlation 157
Michel BORNERT, François HILD, Jean-José ORTEU and Stéphane ROUX

6.1. Background 157

6.2. Surface and volume digital image correlation 158

6.2.1. Images 158

6.2.2. Texture of images 159

6.2.3. Guiding principles 161

6.2.4. Correlation coefficients 163

6.2.5. Subpixel interpolation 164

6.2.6. Local approaches 166

6.2.7. Optimization algorithms 168

6.2.8. Global approaches 169

6.3. Errors and uncertainties 172

6.3.1. Main error sources 172

6.3.2. Uncertainty and spatial resolution 173

6.3.3. Noise sensitivity 174

6.4. Stereo-correlation or 3D-DIC 175

6.4.1. The stereovision technique 176

6.4.2. 3D displacement measurement by stereo-correlation 180

6.4.3. Computation of surface strains from 3D displacements 181

6.4.4. Applications 182

6.5. Conclusions 182

6.6. Bibliography 183

Chapter 7. From Displacement to Strain 191
Pierre FEISSEL

7.1. Introduction 191

7.2. From measurement to strain 191

7.2.1. Three related steps 191

7.2.2. Framework for the differentiation of displacement measurements 192

7.2.3. The main families of methods for differentiating data 194

7.2.4. Quality of the reconstruction 195

7.3. Differentiation: difficulties illustrated for a one-dimensional example 197

7.3.1. A simple one-dimensional example 197

7.3.2. Finite differences 198

7.3.3. Global least squares – polynomial basis 199

7.3.4. Filtering through a convolution kernel 200

7.4. Approximation methods 203

7.4.1. General presentation 203

7.4.2. Global least squares – Finite element basis 204

7.4.3. Local least squares – polynomial basis 206

7.4.4. Three converging points of view 207

7.5. Behavior of the reconstruction methods 209

7.5.1. Splitting the reconstruction error 209

7.5.2. Estimation of approximation error 210

7.5.3. Estimation of random error 211

7.6. Selection criterion for the filtering parameters 214

7.6.1. Constant signal-to-noise ratio 214

7.6.2. A pragmatic criterion 216

7.7. Taking the time dimension into consideration 218

7.8. Concluding remarks 220

7.9. Bibliography 220

Chapter 8. Introduction to Identification Methods 223
Marc BONNET

8.1. Introduction 223

8.2. Identification and inversion: a conceptual overview 223

8.2.1. Inversion 223

8.2.2. Constitutive parameter identification 230

8.3. Numerical methods based on optimization 232

8.3.1. Gradient-based methods 232

8.3.2. Other methods 236

8.4. Methods specifically designed for full-field measurements: an overview 237

8.4.1. Finite element model updating 237

8.4.2. Constitutive relation error 238

8.4.3. Methods based on equilibrium satisfaction 239

8.4.4. Reciprocity gap 241

8.5. Conclusion 242

8.6. Bibliography 242

Chapter 9. Parameter Identification from Mechanical Field Measurements using Finite Element Model Updating Strategies 247
Emmanuel PAGNACCO, Anne-Sophie CARO-BRETELLE and Patrick IENNY

9.1. Introduction 247

9.2. Finite element method 249

9.2.1. Principles of the method 249

9.2.2. The “direct mechanical problem” and finite element analysis 252

9.3. Updating a finite element model for parameter identification 254

9.3.1. Theory 254

9.3.2. Objective functions and minimization procedure 256

9.3.3. Structural sensitivities 262

9.4. Applications, results and accuracy 264

9.4.1. Full-field measurements for the FEMU method 264

9.4.2. Application to the material behavior 265

9.4.3. Identification accuracy 267

9.5. Conclusion 268

9.6. Bibliography 269

Chapter 10. Constitutive Equation Gap 275
Stéphane PAGANO and Marc BONNET

10.1. Introduction 275

10.2. CEG in the linear elastic case: heterogeneous behavior and full-field measurement 276

10.2.1. First variant: exact enforcement of kinematic measurements 278

10.2.2. Second variant: enforcement of measurements by kinematic penalization 283

10.2.3. Comments 283

10.2.4. Some numerical examples 284

10.3. Extension to elastoplasticity 288

10.3.1. Formulation 288

10.3.2. Numerical method 290

10.4. Formulations based on the Legendre–Fenchel transform 293

10.5. Suitable formulations for dynamics or vibration 295

10.6. Conclusions 297

10.7. Bibliography 298

Chapter 11. The Virtual Fields Method 301
Michel GRÉDIAC, Fabrice PIERRON, Stéphane AVRIL, Evelyne TOUSSAINT and Marco ROSSI

11.1. Introduction 301

11.2. General principle 301

11.3. Constitutive equations depending linearly on the parameters: determination of the virtual fields 303

11.3.1. Introduction 303

11.3.2. Developing the PVW 303

11.3.3. Special virtual fields 305

11.3.4. Virtual fields optimized with respect to measurement noise 307

11.3.5. Virtual fields defined by subdomains 309

11.3.6. Examples 311

11.3.7. Plate bending 313

11.3.8. Large deformations: example of hyperelasticity 319

11.4. Case of constitutive equations that do not linearly depend on the constitutive parameters 321

11.4.1. Introduction 321

11.4.2. Elastoplasticity 321

11.4.3. Hyperelastic behavior 324

11.5. Conclusion 325

11.6. Bibliography 326

Chapter 12. Equilibrium Gap Method 331
Fabien AMIOT, Jean-Noël PÉRIÉ and Stéphane ROUX

12.1. Theoretical basis 331

12.1.1. Homogeneous elastic medium 332

12.1.2. Heterogeneous elastic medium 334

12.1.3. Incremental formulation 334

12.2. Finite difference implementation 335

12.3. Finite element implementation 337

12.4. Application to beam theory: local buckling 340

12.4.1. Application to beam theory 340

12.4.2. Loading identification 342

12.4.3. Identification of a heterogeneous stiffness field 343

12.5. Simultaneous identification of stiffness and loading fields 345

12.6. Spectral sensitivity and reconditioning 347

12.7. Damage 349

12.8. Application to a biaxial test carried out on a composite material 351

12.8.1. Damage modeling 352

12.8.2. Adapted expression of the reconditioned equilibrium gap 354

12.8.3. Application to a biaxial test 355

12.9. Exploitation of measurement uncertainty 358

12.10. Conclusions 359

12.11. Bibliography 360

Chapter 13. Reciprocity Gap Method 363
Stéphane ANDRIEUX, Huy Duong BUI and Andrei CONSTANTINESCU

13.1. Introduction 363

13.2. The reciprocity gap method 365

13.2.1. Definition of the reciprocity gap 367

13.2.2. Fundamental property of the reciprocity gap 367

13.3. Identification of cracks in electrostatics 368

13.3.1. Identification formulas for the plane of the crack(s) 370

13.3.2. Complete identification of cracks 371

13.4. Crack identification in thermoelasticity using displacement measurements 373

13.5. Conclusions and perspectives 377

13.6. Bibliography 378

Chapter 14. Characterization of Localized Phenomena 379
Jacques DESRUES and Julien RÉTHORÉ

14.1. Introduction 379

14.2. Definitions and properties of the localized phenomena being considered 380

14.3. Available methods for the experimental characterization of localized phenomena 386

14.3.1. Direct observation 386

14.3.2. Recording the coordinates of predefined markers 387

14.3.3. False relief photogrammetry 387

14.3.4. Digital image correlation 387

14.3.5. Digital volume correlation 388

14.3.6. X-ray tomography 388

14.4. Localization kinematics: a case study 390

14.4.1. Emergence and development of shear bands in a sand specimen under plane strain revealed by stereophotogrammetry 390

14.4.2. Comparison of stereophotogrammetry and digital image correlation for a biaxial test of a soft clay-rock specimen 391

14.4.3. The contribution of digital volume correlation to the detection of localization in isochoric shearing 393

14.4.4. Characterization of severe discontinuities: stereophotogrammetry and correlation 393

14.4.5. Localization on the grain scale: the contribution of discrete DVC 394

14.4.6. A fatigue crack in steel 395

14.4.7. Piobert–Lüders band in steel 395

14.4.8. Portevin–Le Châtelier band 396

14.5. The use of enriched kinematics 397

14.5.1. Displacement discontinuity 398

14.5.2. Strain discontinuity 399

14.6. Localization of the discontinuity zone 399

14.6.1. The use of strain fields 400

14.6.2. The use of correlation residuals 400

14.7. Identification of fracture parameters 401

14.8. Conclusion 405

14.9. Bibliography 406

Chapter 15. From Microstructure to Constitutive Laws 411
Jérôme CRÉPIN and Stéphane ROUX

15.1. Introduction 411

15.2. General problem 411

15.2.1. How can we appreciate spatial heterogeneity? 411

15.2.2. Phase segmentation 413

15.2.3. Inverse problem 413

15.2.4. Statistical description/morphological model 414

15.2.5. Coupling of identification with an exogenous field 417

15.3. Examples of local field characterization 418

15.3.1. EBSD analysis and orientation imaging microscopy 419

15.4. First example: elastic medium with microstructure 423

15.4.1. Glass wool 423

15.4.2. Identification 426

15.5. Second example: crystal plasticity 427

15.5.1. Multiscale approach for identification of material mechanical behavior 428

15.5.2. Methodology 430

15.5.3. Numerical simulation of mechanical behavior 431

15.6. Conclusions 434

15.7. Bibliography 435

Chapter 16. Thermographic Analysis of Material Behavior 439
Jean-Christophe BATSALE, André CHRYSOCHOOS, Hervé PRON and Bertrand WATTRISSE

16.1. Introduction 439

16.2. Thermomechanical framework 441

16.2.1. Constitutive equations 441

16.2.2. Heat equation 443

16.2.3. Energy balance over a load-unload cycle 444

16.3. Metrological considerations 446

16.3.1. Physics of radiation preliminaries 447

16.3.2. Calibration 448

16.3.3. Thermal noise and thermal drift 452

16.4. Heat diffusion models and identification methods 454

16.4.1. Diffusion equation for thin plates 454

16.4.2. Diffusion equation for straight beams 455

16.4.3. Diffusion equation for a monotherm material volume element 456

16.4.4. Integral transforms and quadrupole method related to thick media 457

16.5. Concluding comments and prospects 463

16.6. Bibliography 464

List of Authors 469

Index 475

Dr. Michel Grédiac received an M.S. degree in mechanical engineering from the "Ecole Nationale Supérieure d'Arts et Métiers" in Paris and a Ph.D. degree in mechanical engineering from the University of Lyon in 1991. He was appointed as research professor at the Ecole Nationale Supérieure des Mines de Saint-Etienne and promoted to full professor at the University of Clermont-Ferrand in 1997. In 2003, Dr. Michel Grédiac founded a network named "Full-field measurements and identification in solid mechanics" gathering 25 French research groups devoted to the development and the use of full-field measurement techniques in experimental solids mechanics. He was the head of this network till 2010.

Dr. François Hild graduated from École Normale Supérieure de Cachan in 1989.  He received his Ph.D. in mechanical engineering from the University of Paris 6 in 1992, and from the University of California in 1995.  He received his habilitation from the University of Paris 6 in 1998. Since 2003, he is a Research Professor at the Laboratory of Mechanics and Technology in Cachan (France), and is the current head of the Mechanics and Materials division.