Experimental Mechanics of Solids

Cesar & Federico Sciammarella, University of Illinois, USACesar 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 xixForeword xxi1 Continuum Mechanics – Historical Background 11.1 Definition of the Concept of Stress 41.2 Transformation of Coordinates 51.3 Stress Tensor Representation 61.4 Principal Stresses 81.5 Principal Stresses in Two Dimensions 101.6 The Equations of Equilibrium 111.7 Strain Tensor 131.8 Stress – Strain Relations 151.9 Equations of Compatibility 18References 192 Theoretical Stress Analysis – Basic Formulation of Continuum Mechanics. Theory of Elasticity 212.1 Introduction 212.2 Fundamental Assumptions 212.3 General Problem 222.4 St. Venant’s Principle 252.5 Plane Stress, Plane Strain 282.6 Plane Stress Solution of a Simply Supported Beam with a Uniform Load 302.7 Solutions in Plane Strain and in Plane Stress 332.8 The Plane Problem in Polar Coordinates 352.9 Thick Wall Cylinders 36References 393 Strain Gages – Introduction to Electrical Strain Gages 413.1 Strain Measurements – Point Methods 413.2 Electrical Strain Gages 423.3 Basics of Electrical Strain Gages 433.4 Gage Factor 453.5 Basic Characteristics of Electrical Strain Gages 483.6 Errors Due to the Transverse Sensitivity 543.7 Errors Due to Misalignment of Strain Gages 583.8 Reinforcing Effect of the Gage 603.9 Effect of the Resistance to Ground 613.10 Linearity of the Gages. Hysteresis 633.11 Maximum Deformations 643.12 Stability in Time 643.13 Heat Generation and Dissipation 643.14 Effect of External Ambient Pressure 653.15 Dynamic Effects 67References 714 Strain Gages Instrumentation – TheWheatstone Bridge 754.1 Introduction 75References 1095 Strain Gage Rosettes: Selection, Application and Data Reduction 1115.1 Introduction 1115.2 Errors, Corrections, and Limitations for Rosettes 1195.3 Applications of Gages to Load Cells 119References 1216 Optical Methods – Introduction 1236.1 Historical Perspective and Overview 1236.2 Fundamental Basic Definitions of Optics 1276.3 The Electromagnetic Theory of Light 1286.4 Properties of Polarized Light 1376.5 The Jones Vector Representation 1386.6 Light Intensity 1416.7 Refraction of the Light 1416.8 Geometrical Optics. Lenses and Mirrors 146References 1547 Optical Methods – Interference and Diffraction of Light 1557.1 Connecting Light Interference with Basic Optical Concepts 1557.2 Light Sources 1557.3 Interference 1617.4 Interferometers 1667.5 Diffraction of the Light 171References 1818 Optical Methods – Fourier Transform 1838.1 Introduction 1838.2 Simple Properties 1858.3 Transition to Two Dimensions 1878.4 Special Functions 1888.5 Applications to Diffraction Problems 1918.6 Diffraction Patterns of Gratings 1938.7 Angular Spectrum 1958.8 Utilization of the FT in the Analysis of Diffraction Gratings 199References 2059 Optical Methods – Computer Vision 2079.1 Introduction 2079.2 Study of Lens Systems 2089.3 Lens System, Coordinate Axis and Basic Layout 2109.4 Diffraction Effect on Images 2119.5 Analysis of the Derived Pupil Equations for Coherent Illumination 2169.6 Imaging with Incoherent Illumination 2179.7 Digital Cameras 2309.8 Illumination Systems 2429.9 Imaging Processing Systems 2459.10 Getting High Quality Images 246References 24910 Optical Methods – Discrete Fourier Transform 25110.1 Extension to Two Dimensions 25310.2 The Whittaker-Shannon Theorem 25710.3 General Representation of the Signals Subjected to Analysis 26110.4 Computation of the Phase of the Fringes 27110.5 Fringe Patterns Singularities 27610.6 Extension of the Fringes beyond Boundaries 279References 28311 Photoelasticity – Introduction 28511.1 Introduction 28511.2 Derivation of the Fundamental Equations 28611.3 Wave Plates 29111.4 Polarizers 29311.5 Instrument Matrices 29411.6 Polariscopes 29611.7 Artificial Birefringence 30411.8 Polariscopes 30711.9 Equations of the Intensities of the Plane Polariscope and the Circular Polariscope for a Stressed Plate 309References 31112 Photoelasticity Applications 31312.1 Calibration Procedures of a Photoelastic Material 31312.2 Interpretation of the Fringe Patterns 31912.3 Determination of the Fringe Order 31912.4 Relationship between Retardation Changes of Path and Sign of the Stress Differences 32712.5 Isoclinics and Lines of Principal Stress Trajectories 32812.6 Utilization of White Light in Photoelasticity 33312.7 Determination of the Sign of the Boundary Stresses 33812.8 Phase Stepping Techniques 34212.9 RGB Photoelasticity 34312.10 Reflection Photoelasticity 35512.11 Full Field Analysis 36412.12 Three Dimensional Analysis 36612.13 Integrated Photoelasticity 37512.14 Dynamic Photoelasticity 380References 38313 Techniques that Measure Displacements 38713.1 Introduction 38713.2 Formation of Moir´e Patterns. One Dimensional Case 38813.3 Formation of Moir´e Patterns. Two Dimensional Case 39013.4 Relationship of the Displacement Vector and the Strain Tensor Components 39313.5 Properties of the Moire Fringes (Isothetic Lines) 39513.6 Sections of the Surface of Projected Displacements 39613.7 Singular Points and Singular Lines 40113.8 Digital Moir´e 40213.9 Equipment Required to Apply the Moir´e Method for Displacement and Strain Determination Utilizing Incoherent Illumination 41213.10 Strain Analysis at the Sub-Micrometer Scale 41913.11 Three Dimensional Moir´e 42413.12 Dynamic Moir´e 426References 43214 Moir´e Method. Coherent Ilumination 43514.1 Introduction 43514.2 Moir´e Interferometry 43514.3 Optical Developments to Obtain Displacement, Contours and Strain Information 43914.4 Determination of All the Components of the Displacement Vector 3-D Interferometric Moir´e 44614.5 Application of Moir´e Interferometry to High Temperature Fracture Analysis 451References 45615 Shadow Moir´e & Projection Moir´e – The Basic Relationships 45915.1 Introduction 45915.2 Basic Equation of Shadow Moir´e 46015.3 Basic Differential Geometry Properties of Surfaces 46115.4 Connection between Differential Geometry and Moir´e 46315.5 Projective Geometry and Projection Moir´e 46715.6 Epipolar Model of the Two Projectors and One Camera System 46915.7 Approaches to Extend the Moir´e Method to More General Conditions of Projection and Observation 47115.8 Summary of the Chapter 482References 48216 Moir´e Contouring Applications 48516.1 Introduction 48516.2 Basic Principles of Optical Contouring Measuring Devices 48616.3 Contouring Methods that Utilize Projected Carriers 48616.4 Parallax Determination in an Area 48916.5 Mathematical Modeling of the Parallax Determination in an Area 49016.6 Limitations of the Contouring Model 49216.7 Applications of the Contouring Methods 49416.8 Double Projector System with Slope and Depth-of-Focus Corrections 50616.9 Sensitivity Limits for Contouring Methods 518References 52017 Reflection Moir´e 52317.1 Introduction 52317.2 Incoherent Illumination. Derivation of the Fundamental Relationship 52317.3 Interferometric Reflection Moir´e 52617.4 Analysis of the Sensitivity that can be Achieved with the Described Setups 53017.5 Determination of the Deflection of Surfaces Using Reflection Moir´e 53117.6 Applications of the Reflection Moir´e Method 53217.7 Reflection Moir´e Application – Analysis of a Shell 539References 54518 Speckle Patterns and Their Properties 54718.1 Introduction 54718.2 First Order Statistics 55018.3 Three Dimensional Structure of Speckle Patterns 55818.4 Sensor Effect on Speckle Statistics 56018.5 Utilization of Speckles to Measure Displacements. Speckle Interferometry 56218.6 Decorrelation Phenomena 56418.7 Model for the Formation of the Interference Fringes 56718.8 Integrated Regime. Metaspeckle 56918.9 Sensitivity Vector 57218.10 Speckle Techniques Set-Ups 57318.11 Out-of-Plane Interferometer 57618.12 Shear Interferometry (Shearography) 57718.13 Contouring Interferometer 57818.14 Double Viewing. Duffy Double Aperture Method 579References 58119 Speckle 2 58319.1 Speckle Photography 58319.2 Point-Wise Observation of the Speckle Field 58419.3 Global View 58519.4 Different Set-Ups for Speckle Photography 58919.5 Applications of Speckle Interferometry 59019.6 High Temperature Strain Measurement 59319.7 Four Beam Interferometer Sensitive to in Plane Displacements 597References 60620 Digital Image Correlation (DIC) 60720.1 Introduction 60720.2 Process to Obtain the Displacement Information 60820.3 Basic Formulation of the Problem 61020.4 Introduction of Smoothing Functions to Solve the Optimization Problem 61320.5 Determination of the Components of the Displacement Vector 61820.6 Important Factors that Influence the Packages of DIC 61920.7 Evaluation of the DIC Method 62120.8 Double Viewing DIC. Stereo Vision 627References 62821 Holographic Interferometry 63121.1 Holography 63121.2 Basic Elements of the Holographic Process 63221.3 Properties of Holograms 63421.4 Set up to Record Holograms 63621.5 Holographic Interferometry 64121.6 Derivation of the Equation of the Sensitivity Vector 64421.7 Measuring Displacements 64621.8 Holographic Moir´e 65121.9 Lens Holography 65821.10 Holographic Moir´e. Real Time Observation 66121.11 Displacement Analysis of Curved Surfaces 66521.12 Holographic Contouring 66921.13 Measurement of Displacements in 3D of Transparent Bodies 67521.14 Fiber Optics Version of the Holographic Moir´e System 675References 67722 Digital and Dynamic Holography 68122.1 Digital Holography 68122.2 Determination of Strains from 3D Holographic Moir´e Interferograms 68522.3 Introduction to Dynamic Holographic Interferometry 68922.4 Vibration Analysis 69322.5 Experimental Set up for Time Average Holography 69522.6 Investigation on Fracture Behavior of Turbine Blades Under Self-Exciting Modes 70022.7 Dynamic Holographic Interferometry. Impact Analysis. Wave Propagation 70822.8 Applications of Dynamic Holographic Interferometry 712References 721Index 723

“The book is highly recommended as a textbook in courses of experimental mechanics and can be used as a basis on which the researcher, the student and the practitioner can develop their ideas and promote research and applications of the experimental methods in engineering problems. The connection and interrelation of the various optical techniques is astonishing.”  (Wiley Experimental Techniques journal, 2012)