Full-Field Measurements and Identification in Solid Mechanics

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.

• Foreword xvMichael A. SUTTONIntroduction xviiMichel GRÉDIAC and François HILDChapter 1. Basics of Metrology and Introduction to Techniques 1André CHRYSOCHOOS and Yves SURREL1.1. Introduction 11.2. Terminology: international vocabulary of metrology 21.2.1. Absolute or differential measurement 21.2.2. Main concepts 41.3. Spatial aspect 111.3.1. Spatial frequency 111.3.2. Spatial filtering 161.4. Classification of optical measurement techniques 181.4.1. White light measurement methods 191.4.2. Interference methods 211.4.3. Sensitivity vector 231.4.4. Synthetic sensitivity vectors 231.4.5. The different types of interferometric measurements 241.4.6. Holography, digital holography 271.4.7. Conclusion 281.5. Bibliography 29Chapter 2. Photoelasticity 31Fabrice BRÉMAND and Jean-Christophe DUPRÉ2.1. Introduction 312.2. Concept of light polarization 322.3. Birefringence phenomenon 332.4. The law of optico-mechanics 342.5. Several types of polariscopes 352.5.1. Plane polariscope 352.5.2. Circular polariscope 382.5.3. White light polariscope 402.5.4. Photoelastic coating 402.6. Measurement of photoelastic constant C 422.7. Analysis by image processing 432.7.1. Using a plane polariscope 432.7.2. Using a circular polariscope 472.7.3. Using color images 482.8. Post-processing of photoelastic parameters 482.8.1. Drawing of isostatics or stress trajectories 482.8.2. Particular points 482.8.3. Stress separation and integration of the equilibrium equations 492.8.4. Comparison between experimentation and numerical modeling 502.9. Three-dimensional photoelasticity 502.9.1. The method of stress freezing and mechanical slicing 512.9.2. Optical slicing 522.9.3. Application example 562.10. Conclusion 572.11. Bibliography 57Chapter 3. Grid Method, Moiré and Deflectometry 61Jérôme MOLIMARD and Yves SURREL3.1. Introduction 613.2. Principle 613.3. Surface encoding 633.4. Moiré 643.5. Phase detection 663.5.1. Global extraction procedure 663.5.2. Local phase detection: phase shifting 673.5.3. Measuring both components of the displacement 703.6. Sensitivity to out-of-plane displacements 713.7. Grid defects 723.8. Large deformation/large strain 733.8.1. Explicit method 733.8.2. Implicit method 743.8.3. Large strain 743.9. Fringe projection 753.10. Deflectometry 783.11. Examples 813.11.1. Off-axis tensile test of a unidirectional composite coupon 813.11.2. Rigid body displacement 833.11.3. SEM measurement 843.11.4. Characterization of lens distortion 853.12. Conclusion 883.13. Bibliography 89Chapter 4. Digital Holography Methods 93Pascal PICART and Paul SMIGIELSKI4.1. Introduction 934.2. Basics of wave optics 944.2.1. Light diffraction 954.2.2. Interference 964.3. Basics of digital holography 974.3.1. Recording the hologram 974.3.2. Numerical reconstruction with the discrete Fresnel transform 994.3.3. Numerical reconstruction using convolution with adjustable magnification 1004.3.4. Sensitivity vector 1014.4. Basics of digital holographic interferometry 1034.4.1. Phase difference 1034.4.2. Spatial filtering of the phase and phase unwrapping 1044.5. Digital holographic interferometry with spatial multiplexing 1044.5.1. Principle 1044.5.2. Theory 1054.5.3. Experimental set-up 1054.5.4. Application to synthetic concrete subjected to three-point bending 1074.6. Digital color holography applied to three-dimensional measurements 1124.6.1. Recording digital color holograms 1124.6.2. Application to composite material subjected to a short beam test 1134.7. Conclusion 1184.8. Acknowledgment 1194.9. Bibliography 119Chapter 5. Elementary Speckle Interferometry 125Pierre JACQUOT, Pierre SLANGEN and Dan BORZA5.1. Introduction 1255.2. What is speckle interferometry? 1265.2.1. Simplified principle – correlation fringes 1285.2.2. Speckle field and specklegram statistics in a nutshell 1295.2.3. Speckle field transformation – small perturbation theory 1315.2.4. Phase change-deformation law – sensitivity vector 1325.2.5. Success or failure of experiments – central role of decorrelation 1335.3. Optical point of view 1345.4. Mechanical point of view: specific displacement field components 1365.4.1. Measurement of the out-of-plane component 1365.4.2. Measurement of the in-plane component [LEE 70] 1375.4.3. 3C-3D: three components attached to three-dimensional objects 1385.4.4. Partial derivatives of the displacement – shearography 1395.4.5. Shape measurement and other considerations 1405.5. Phase extraction 1415.5.1. One-image methods 1415.5.2. Phase-shifting methods 1425.5.3. Advanced methods 1435.5.4. Phase unwrapping 1445.6. Dynamic deformations and vibrations 465.7. Setup calibration 1485.7.1. Specifying the material point in object coordinates 1495.7.2. Determination of the sensitivity vector 1495.8. Specifications and limits 1505.9. Final remarks, outlook and trends 1515.10. Bibliography 153Chapter 6. Digital Image Correlation 157Michel BORNERT, François HILD, Jean-José ORTEU and Stéphane ROUX6.1. Background 1576.2. Surface and volume digital image correlation 1586.2.1. Images 1586.2.2. Texture of images 1596.2.3. Guiding principles 1616.2.4. Correlation coefficients 1636.2.5. Subpixel interpolation 1646.2.6. Local approaches 1666.2.7. Optimization algorithms 1686.2.8. Global approaches 1696.3. Errors and uncertainties 1726.3.1. Main error sources 1726.3.2. Uncertainty and spatial resolution 1736.3.3. Noise sensitivity 1746.4. Stereo-correlation or 3D-DIC 1756.4.1. The stereovision technique 1766.4.2. 3D displacement measurement by stereo-correlation 1806.4.3. Computation of surface strains from 3D displacements 1816.4.4. Applications 1826.5. Conclusions 1826.6. Bibliography 183Chapter 7. From Displacement to Strain 191Pierre FEISSEL7.1. Introduction 1917.2. From measurement to strain 1917.2.1. Three related steps 1917.2.2. Framework for the differentiation of displacement measurements 1927.2.3. The main families of methods for differentiating data 1947.2.4. Quality of the reconstruction 1957.3. Differentiation: difficulties illustrated for a one-dimensional example 1977.3.1. A simple one-dimensional example 1977.3.2. Finite differences 1987.3.3. Global least squares – polynomial basis 1997.3.4. Filtering through a convolution kernel 2007.4. Approximation methods 2037.4.1. General presentation 2037.4.2. Global least squares – Finite element basis 2047.4.3. Local least squares – polynomial basis 2067.4.4. Three converging points of view 2077.5. Behavior of the reconstruction methods 2097.5.1. Splitting the reconstruction error 2097.5.2. Estimation of approximation error 2107.5.3. Estimation of random error 2117.6. Selection criterion for the filtering parameters 2147.6.1. Constant signal-to-noise ratio 2147.6.2. A pragmatic criterion 2167.7. Taking the time dimension into consideration 2187.8. Concluding remarks 2207.9. Bibliography 220Chapter 8. Introduction to Identification Methods 223Marc BONNET8.1. Introduction 2238.2. Identification and inversion: a conceptual overview 2238.2.1. Inversion 2238.2.2. Constitutive parameter identification 2308.3. Numerical methods based on optimization 2328.3.1. Gradient-based methods 2328.3.2. Other methods 2368.4. Methods specifically designed for full-field measurements: an overview 2378.4.1. Finite element model updating 2378.4.2. Constitutive relation error 2388.4.3. Methods based on equilibrium satisfaction 2398.4.4. Reciprocity gap 2418.5. Conclusion 2428.6. Bibliography 242Chapter 9. Parameter Identification from Mechanical Field Measurements using Finite Element Model Updating Strategies 247Emmanuel PAGNACCO, Anne-Sophie CARO-BRETELLE and Patrick IENNY9.1. Introduction 2479.2. Finite element method 2499.2.1. Principles of the method 2499.2.2. The “direct mechanical problem” and finite element analysis 2529.3. Updating a finite element model for parameter identification 2549.3.1. Theory 2549.3.2. Objective functions and minimization procedure 2569.3.3. Structural sensitivities 2629.4. Applications, results and accuracy 2649.4.1. Full-field measurements for the FEMU method 2649.4.2. Application to the material behavior 2659.4.3. Identification accuracy 2679.5. Conclusion 2689.6. Bibliography 269Chapter 10. Constitutive Equation Gap 275Stéphane PAGANO and Marc BONNET10.1. Introduction 27510.2. CEG in the linear elastic case: heterogeneous behavior and full-field measurement 27610.2.1. First variant: exact enforcement of kinematic measurements 27810.2.2. Second variant: enforcement of measurements by kinematic penalization 28310.2.3. Comments 28310.2.4. Some numerical examples 28410.3. Extension to elastoplasticity 28810.3.1. Formulation 28810.3.2. Numerical method 29010.4. Formulations based on the Legendre–Fenchel transform 29310.5. Suitable formulations for dynamics or vibration 29510.6. Conclusions 29710.7. Bibliography 298Chapter 11. The Virtual Fields Method 301Michel GRÉDIAC, Fabrice PIERRON, Stéphane AVRIL, Evelyne TOUSSAINT and Marco ROSSI11.1. Introduction 30111.2. General principle 30111.3. Constitutive equations depending linearly on the parameters: determination of the virtual fields 30311.3.1. Introduction 30311.3.2. Developing the PVW 30311.3.3. Special virtual fields 30511.3.4. Virtual fields optimized with respect to measurement noise 30711.3.5. Virtual fields defined by subdomains 30911.3.6. Examples 31111.3.7. Plate bending 31311.3.8. Large deformations: example of hyperelasticity 31911.4. Case of constitutive equations that do not linearly depend on the constitutive parameters 32111.4.1. Introduction 32111.4.2. Elastoplasticity 32111.4.3. Hyperelastic behavior 32411.5. Conclusion 32511.6. Bibliography 326Chapter 12. Equilibrium Gap Method 331Fabien AMIOT, Jean-Noël PÉRIÉ and Stéphane ROUX12.1. Theoretical basis 33112.1.1. Homogeneous elastic medium 33212.1.2. Heterogeneous elastic medium 33412.1.3. Incremental formulation 33412.2. Finite difference implementation 33512.3. Finite element implementation 33712.4. Application to beam theory: local buckling 34012.4.1. Application to beam theory 34012.4.2. Loading identification 34212.4.3. Identification of a heterogeneous stiffness field 34312.5. Simultaneous identification of stiffness and loading fields 34512.6. Spectral sensitivity and reconditioning 34712.7. Damage 34912.8. Application to a biaxial test carried out on a composite material 35112.8.1. Damage modeling 35212.8.2. Adapted expression of the reconditioned equilibrium gap 35412.8.3. Application to a biaxial test 35512.9. Exploitation of measurement uncertainty 35812.10. Conclusions 35912.11. Bibliography 360Chapter 13. Reciprocity Gap Method 363Stéphane ANDRIEUX, Huy Duong BUI and Andrei CONSTANTINESCU13.1. Introduction 36313.2. The reciprocity gap method 36513.2.1. Definition of the reciprocity gap 36713.2.2. Fundamental property of the reciprocity gap 36713.3. Identification of cracks in electrostatics 36813.3.1. Identification formulas for the plane of the crack(s) 37013.3.2. Complete identification of cracks 37113.4. Crack identification in thermoelasticity using displacement measurements 37313.5. Conclusions and perspectives 37713.6. Bibliography 378Chapter 14. Characterization of Localized Phenomena 379Jacques DESRUES and Julien RÉTHORÉ14.1. Introduction 37914.2. Definitions and properties of the localized phenomena being considered 38014.3. Available methods for the experimental characterization of localized phenomena 38614.3.1. Direct observation 38614.3.2. Recording the coordinates of predefined markers 38714.3.3. False relief photogrammetry 38714.3.4. Digital image correlation 38714.3.5. Digital volume correlation 38814.3.6. X-ray tomography 38814.4. Localization kinematics: a case study 39014.4.1. Emergence and development of shear bands in a sand specimen under plane strain revealed by stereophotogrammetry 39014.4.2. Comparison of stereophotogrammetry and digital image correlation for a biaxial test of a soft clay-rock specimen 39114.4.3. The contribution of digital volume correlation to the detection of localization in isochoric shearing 39314.4.4. Characterization of severe discontinuities: stereophotogrammetry and correlation 39314.4.5. Localization on the grain scale: the contribution of discrete DVC 39414.4.6. A fatigue crack in steel 39514.4.7. Piobert–Lüders band in steel 39514.4.8. Portevin–Le Châtelier band 39614.5. The use of enriched kinematics 39714.5.1. Displacement discontinuity 39814.5.2. Strain discontinuity 39914.6. Localization of the discontinuity zone 39914.6.1. The use of strain fields 40014.6.2. The use of correlation residuals 40014.7. Identification of fracture parameters 40114.8. Conclusion 40514.9. Bibliography 406Chapter 15. From Microstructure to Constitutive Laws 411Jérôme CRÉPIN and Stéphane ROUX15.1. Introduction 41115.2. General problem 41115.2.1. How can we appreciate spatial heterogeneity? 41115.2.2. Phase segmentation 41315.2.3. Inverse problem 41315.2.4. Statistical description/morphological model 41415.2.5. Coupling of identification with an exogenous field 41715.3. Examples of local field characterization 41815.3.1. EBSD analysis and orientation imaging microscopy 41915.4. First example: elastic medium with microstructure 42315.4.1. Glass wool 42315.4.2. Identification 42615.5. Second example: crystal plasticity 42715.5.1. Multiscale approach for identification of material mechanical behavior 42815.5.2. Methodology 43015.5.3. Numerical simulation of mechanical behavior 43115.6. Conclusions 43415.7. Bibliography 435Chapter 16. Thermographic Analysis of Material Behavior 439Jean-Christophe BATSALE, André CHRYSOCHOOS, Hervé PRON and Bertrand WATTRISSE16.1. Introduction 43916.2. Thermomechanical framework 44116.2.1. Constitutive equations 44116.2.2. Heat equation 44316.2.3. Energy balance over a load-unload cycle 44416.3. Metrological considerations 44616.3.1. Physics of radiation preliminaries 44716.3.2. Calibration 44816.3.3. Thermal noise and thermal drift 45216.4. Heat diffusion models and identification methods 45416.4.1. Diffusion equation for thin plates 45416.4.2. Diffusion equation for straight beams 45516.4.3. Diffusion equation for a monotherm material volume element 45616.4.4. Integral transforms and quadrupole method related to thick media 45716.5. Concluding comments and prospects 46316.6. Bibliography 464List of Authors 469Index 475