Experimental Mechanics of Solids

Cesar & Federico Sciammarella, University of Illinois, USA Cesar A Sciammarella is Adjunct Professor in the Department of Mechanical Engineering, University of Illinois, USA. In the past he has worked as a consultant for companies including: General Motors, Goodyear, Honeywell Corporation, Rand Corporation, Rockwell International, Sundstran, Uniroyal Tires, IBM, Tryodyne, Samsung, Case Corporation. A renowned experimentalist, his research currently focuses on developing techniques in solid mechanics and he has spoken at many conferences and published prolifically in journals which include Strain; Optical Engineering; SEM Conference on Experimental Mechanics and Journal of Strain Analysis for Engineering Design. Federico Sciammarella is Assistant Professor in the Department of Mechanical Engineering, University of Illinois. His research interests centre upon using optical methods for characterization of materials and structures including failure analysis. Over the past five years he has written multiple journal and conference research papers.

About the Authors xvii Preface xix Foreword xxi 1 Continuum Mechanics Historical Background 1 1.1 Definition of the Concept of Stress 4 1.2 Transformation of Coordinates 5 1.3 Stress Tensor Representation 6 1.4 Principal Stresses 8 1.5 Principal Stresses in Two Dimensions 10 1.6 The Equations of Equilibrium 11 1.7 Strain Tensor 13 1.8 Stress Strain Relations 15 1.9 Equations of Compatibility 18 References 19 2 Theoretical Stress Analysis Basic Formulation of Continuum Mechanics. Theory of Elasticity 21 2.1 Introduction 21 2.2 Fundamental Assumptions 21 2.3 General Problem 22 2.4 St. Venant s Principle 25 2.5 Plane Stress, Plane Strain 28 2.6 Plane Stress Solution of a Simply Supported Beam with a Uniform Load 30 2.7 Solutions in Plane Strain and in Plane Stress 33 2.8 The Plane Problem in Polar Coordinates 35 2.9 Thick Wall Cylinders 36 References 39 3 Strain Gages Introduction to Electrical Strain Gages 41 3.1 Strain Measurements Point Methods 41 3.2 Electrical Strain Gages 42 3.3 Basics of Electrical Strain Gages 43 3.4 Gage Factor 45 3.5 Basic Characteristics of Electrical Strain Gages 48 3.6 Errors Due to the Transverse Sensitivity 54 3.7 Errors Due to Misalignment of Strain Gages 58 3.8 Reinforcing Effect of the Gage 60 3.9 Effect of the Resistance to Ground 61 3.10 Linearity of the Gages. Hysteresis 63 3.11 Maximum Deformations 64 3.12 Stability in Time 64 3.13 Heat Generation and Dissipation 64 3.14 Effect of External Ambient Pressure 65 3.15 Dynamic Effects 67 References 71 4 Strain Gages Instrumentation TheWheatstone Bridge 75 4.1 Introduction 75 References 109 5 Strain Gage Rosettes: Selection, Application and Data Reduction 111 5.1 Introduction 111 5.2 Errors, Corrections, and Limitations for Rosettes 119 5.3 Applications of Gages to Load Cells 119 References 121 6 Optical Methods Introduction 123 6.1 Historical Perspective and Overview 123 6.2 Fundamental Basic Definitions of Optics 127 6.3 The Electromagnetic Theory of Light 128 6.4 Properties of Polarized Light 137 6.5 The Jones Vector Representation 138 6.6 Light Intensity 141 6.7 Refraction of the Light 141 6.8 Geometrical Optics. Lenses and Mirrors 146 References 154 7 Optical Methods Interference and Diffraction of Light 155 7.1 Connecting Light Interference with Basic Optical Concepts 155 7.2 Light Sources 155 7.3 Interference 161 7.4 Interferometers 166 7.5 Diffraction of the Light 171 References 181 8 Optical Methods Fourier Transform 183 8.1 Introduction 183 8.2 Simple Properties 185 8.3 Transition to Two Dimensions 187 8.4 Special Functions 188 8.5 Applications to Diffraction Problems 191 8.6 Diffraction Patterns of Gratings 193 8.7 Angular Spectrum 195 8.8 Utilization of the FT in the Analysis of Diffraction Gratings 199 References 205 9 Optical Methods Computer Vision 207 9.1 Introduction 207 9.2 Study of Lens Systems 208 9.3 Lens System, Coordinate Axis and Basic Layout 210 9.4 Diffraction Effect on Images 211 9.5 Analysis of the Derived Pupil Equations for Coherent Illumination 216 9.6 Imaging with Incoherent Illumination 217 9.7 Digital Cameras 230 9.8 Illumination Systems 242 9.9 Imaging Processing Systems 245 9.10 Getting High Quality Images 246 References 249 10 Optical Methods Discrete Fourier Transform 251 10.1 Extension to Two Dimensions 253 10.2 The Whittaker-Shannon Theorem 257 10.3 General Representation of the Signals Subjected to Analysis 261 10.4 Computation of the Phase of the Fringes 271 10.5 Fringe Patterns Singularities 276 10.6 Extension of the Fringes beyond Boundaries 279 References 283 11 Photoelasticity Introduction 285 11.1 Introduction 285 11.2 Derivation of the Fundamental Equations 286 11.3 Wave Plates 291 11.4 Polarizers 293 11.5 Instrument Matrices 294 11.6 Polariscopes 296 11.7 Artificial Birefringence 304 11.8 Polariscopes 307 11.9 Equations of the Intensities of the Plane Polariscope and the Circular Polariscope for a Stressed Plate 309 References 311 12 Photoelasticity Applications 313 12.1 Calibration Procedures of a Photoelastic Material 313 12.2 Interpretation of the Fringe Patterns 319 12.3 Determination of the Fringe Order 319 12.4 Relationship between Retardation Changes of Path and Sign of the Stress Differences 327 12.5 Isoclinics and Lines of Principal Stress Trajectories 328 12.6 Utilization of White Light in Photoelasticity 333 12.7 Determination of the Sign of the Boundary Stresses 338 12.8 Phase Stepping Techniques 342 12.9 RGB Photoelasticity 343 12.10 Reflection Photoelasticity 355 12.11 Full Field Analysis 364 12.12 Three Dimensional Analysis 366 12.13 Integrated Photoelasticity 375 12.14 Dynamic Photoelasticity 380 References 383 13 Techniques that Measure Displacements 387 13.1 Introduction 387 13.2 Formation of Moir'e Patterns. One Dimensional Case 388 13.3 Formation of Moir'e Patterns. Two Dimensional Case 390 13.4 Relationship of the Displacement Vector and the Strain Tensor Components 393 13.5 Properties of the Moire Fringes (Isothetic Lines) 395 13.6 Sections of the Surface of Projected Displacements 396 13.7 Singular Points and Singular Lines 401 13.8 Digital Moir'e 402 13.9 Equipment Required to Apply the Moir'e Method for Displacement and Strain Determination Utilizing Incoherent Illumination 412 13.10 Strain Analysis at the Sub-Micrometer Scale 419 13.11 Three Dimensional Moir'e 424 13.12 Dynamic Moir'e 426 References 432 14 Moir'e Method. Coherent Ilumination 435 14.1 Introduction 435 14.2 Moir'e Interferometry 435 14.3 Optical Developments to Obtain Displacement, Contours and Strain Information 439 14.4 Determination of All the Components of the Displacement Vector 3-D Interferometric Moir'e 446 14.5 Application of Moir'e Interferometry to High Temperature Fracture Analysis 451 References 456 15 Shadow Moir'e & Projection Moir'e The Basic Relationships 459 15.1 Introduction 459 15.2 Basic Equation of Shadow Moir'e 460 15.3 Basic Differential Geometry Properties of Surfaces 461 15.4 Connection between Differential Geometry and Moir'e 463 15.5 Projective Geometry and Projection Moir'e 467 15.6 Epipolar Model of the Two Projectors and One Camera System 469 15.7 Approaches to Extend the Moir'e Method to More General Conditions of Projection and Observation 471 15.8 Summary of the Chapter 482 References 482 16 Moir'e Contouring Applications 485 16.1 Introduction 485 16.2 Basic Principles of Optical Contouring Measuring Devices 486 16.3 Contouring Methods that Utilize Projected Carriers 486 16.4 Parallax Determination in an Area 489 16.5 Mathematical Modeling of the Parallax Determination in an Area 490 16.6 Limitations of the Contouring Model 492 16.7 Applications of the Contouring Methods 494 16.8 Double Projector System with Slope and Depth-of-Focus Corrections 506 16.9 Sensitivity Limits for Contouring Methods 518 References 520 17 Reflection Moir'e 523 17.1 Introduction 523 17.2 Incoherent Illumination. Derivation of the Fundamental Relationship 523 17.3 Interferometric Reflection Moir'e 526 17.4 Analysis of the Sensitivity that can be Achieved with the Described Setups 530 17.5 Determination of the Deflection of Surfaces Using Reflection Moir'e 531 17.6 Applications of the Reflection Moir'e Method 532 17.7 Reflection Moir'e Application Analysis of a Shell 539 References 545 18 Speckle Patterns and Their Properties 547 18.1 Introduction 547 18.2 First Order Statistics 550 18.3 Three Dimensional Structure of Speckle Patterns 558 18.4 Sensor Effect on Speckle Statistics 560 18.5 Utilization of Speckles to Measure Displacements. Speckle Interferometry 562 18.6 Decorrelation Phenomena 564 18.7 Model for the Formation of the Interference Fringes 567 18.8 Integrated Regime. Metaspeckle 569 18.9 Sensitivity Vector 572 18.10 Speckle Techniques Set-Ups 573 18.11 Out-of-Plane Interferometer 576 18.12 Shear Interferometry (Shearography) 577 18.13 Contouring Interferometer 578 18.14 Double Viewing. Duffy Double Aperture Method 579 References 581 19 Speckle 2 583 19.1 Speckle Photography 583 19.2 Point-Wise Observation of the Speckle Field 584 19.3 Global View 585 19.4 Different Set-Ups for Speckle Photography 589 19.5 Applications of Speckle Interferometry 590 19.6 High Temperature Strain Measurement 593 19.7 Four Beam Interferometer Sensitive to in Plane Displacements 597 References 606 20 Digital Image Correlation (DIC) 607 20.1 Introduction 607 20.2 Process to Obtain the Displacement Information 608 20.3 Basic Formulation of the Problem 610 20.4 Introduction of Smoothing Functions to Solve the Optimization Problem 613 20.5 Determination of the Components of the Displacement Vector 618 20.6 Important Factors that Influence the Packages of DIC 619 20.7 Evaluation of the DIC Method 621 20.8 Double Viewing DIC. Stereo Vision 627 References 628 21 Holographic Interferometry 631 21.1 Holography 631 21.2 Basic Elements of the Holographic Process 632 21.3 Properties of Holograms 634 21.4 Set up to Record Holograms 636 21.5 Holographic Interferometry 641 21.6 Derivation of the Equation of the Sensitivity Vector 644 21.7 Measuring Displacements 646 21.8 Holographic Moir'e 651 21.9 Lens Holography 658 21.10 Holographic Moir'e. Real Time Observation 661 21.11 Displacement Analysis of Curved Surfaces 665 21.12 Holographic Contouring 669 21.13 Measurement of Displacements in 3D of Transparent Bodies 675 21.14 Fiber Optics Version of the Holographic Moir'e System 675 References 677 22 Digital and Dynamic Holography 681 22.1 Digital Holography 681 22.2 Determination of Strains from 3D Holographic Moir'e Interferograms 685 22.3 Introduction to Dynamic Holographic Interferometry 689 22.4 Vibration Analysis 693 22.5 Experimental Set up for Time Average Holography 695 22.6 Investigation on Fracture Behavior of Turbine Blades Under Self-Exciting Modes 700 22.7 Dynamic Holographic Interferometry. Impact Analysis. Wave Propagation 708 22.8 Applications of Dynamic Holographic Interferometry 712 References 721 Index 723