Photoelasticity of Glass

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1 H. Aben, C. GuiUemet Photoelasticity of Glass With 218 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

2 Contents Preface v Part One The Basics of Photoelasticity and Glass 1 1 Basic Elasticity Elasticity Force and Stress Plane Stress Equations of Equilibrium Boundary Conditions Strain Relations Between Stresses and Strahls Plane Strain Equations of Compatibility Stress Function 16 2 Residual Stresses in Glass Introduction Dependence of the Mechanical Strength on Residual Stresses Stresses Due to Indentations Residual Stresses Due to Thermal Annealing or Tempenng The First Approaches The Viscoelastic Theory The Structural Theory Membrane Stresses and Form Stresses Stress Redistribution by Cutting Stresses Due to Chemical Tempering Stress Buildup Strengthening of Glass Stresses Created in Glass by Radiations Corpuscular Radiation Electromagnetic Radiation 44 Thermal Effects 45 Color Centers Stresses Due to Heterogeneities Stresses in Composite Glass Articles 46 )

3 x Contents Stresses in Glazes and Enamels Stresses in Optical Fibers Stresses in Glass-Metal and Glass-Ceramic Seals Stresses Due to Inclusions 48 3 Basic Photoelasticity Polarized Light Nature of Light Natural and Polarized Light Different Descriptions of Polarized Light Artificial Double Refraction Stress-Optic Law The Plane Polariscope The Circular Polariscope > Use of Double-Exposure Photography for the Elimination of the Isoclinics Construction of Polariscopes Measurement of Optical Retardation Color Matching Polariscope with a Tint Plate The Babinet and Babinet-Soleil Compensators Senarmont Method The Azimuth Method 67 4 Two-Dimensional Photoelasticity General Stress Trajectories Separation of Principal Stresses Oblique Incidence Method Shear Difference Method Numerical Solution of the Compatibility Equation Methods Based on Hooke's Law Superposition of States of Stress Determination of the Photoelastic Constant 75 5 The Scattered Light Method Introduction Scattering of Light The Scattered Light Method with Polarized Incident Light The Scattered Light Method with Unpolarized Incident Light Using Interference of Coherent Scattered Light Beams 85 6 Integrated Photoelasticity Introduction Principle of Integrated Photoelasticity Basic Equations Theory of Characteristic Directions Symmetrie Photoelastic Media 92

4 Contents xi 6.6 The Case of Constant Principal Stress Axes The Case of Weak Birefringence Integrated Photoelasticity as Optical Tomography of the Stress Field Investigation of the General Three-Dimensional State of Stress Axisymmetric State of Stress Due to External Loads 99 7 Photoelastic Properties of Glass Introduction Discovery of the Photoelastic Effect in Glass Influence of the Glass Composition Theories of the Photoelastic Effect Influence of the Temperature and of the Thermal History Dependence of the Photoelastic Constant on Wavelength Anomalous Birefringence 115 Part Two Stress Analysis in Fiat Glass Thickness Stresses Different Kinds of Thickness Stresses Measurement of Thickness Stresses Using the Bending of the Light Rays Conventional Photoelasticity Membrane Stresses Introduction Uniaxial Membrane Stresses Edge Stresses Stresses Across a Ribbon Bidimensional Membrane Stresses Determination of the Total Stresses Introduction The Measurement of Surface Stresses Differential Refractometry The "Mirage" Methods 141 Observation of the Guided Waves Close to the Surface 141 The Case of Fiat Samples 142 The Case of Curved Samples 145 The Case of Stress Gradient Near the Surface 145 Observation of the Guided Waves at Infinity 146 Theory of the Differential Refractometry with Guided Waves Linear Index Profile 148 Determination of Stresses 149 An Example 150 Alternative Numerical Methods 152 Curved Surface 153 Thermally Tempered Glass 154

5 xii Contents 10.3 Measurement of Total Residual Stresses The Scattered Light Method 154 Spatial Modulation Method 154 Phase Modulation Method Magnetophotoelasticity 158 Part Three Stresses in Glass Articles of Complicated Shape Axisymmetric Glass Articles General Case of Axisymmetric Residual Stress Distribution Pecuharities of the Determination of the Residual Stress Determination of the Axial and Shear Stress Distributions Additional Tomographie Measurements Application of the Equilibrium and Boundary Conditions Stresses on the Externa! Surface Average Value of the Circumferential Stress Stresses in Long Cylinders Spherical Symmetry Stress Distribution in Spheres Quenching Stresses Around a Spherical Inclusion Bending of Light Rays Determination of the Components of the Dielectric Tensor Optimization of the Number of Terms in Stress Polynomials Experimental Technique Polariscopes Immersion Technique The Case of Mismatching Immersion Examples Quenched Long Cylinder An Article of Optical Glass High Voltage Insulator Closed Tube Two Bonded Tubes Containers and Other Thin-Walled Glassware Introduction Traditional Methods Determination of Stress in Cylindrical Part of the Container Axial Stress in an Arbitrary Section Determination of the Stresses Due to the Internal Pressure Sandwich Glassware Examples A Champagne Bottle A Beer Bottle TumblerN l TumblerN SaladBowl 212

6 Contents xiii Electric Lamp Ampule of a Fire Extinguisher System Optical Fibers and Fiber Preforms Introduction Axisymmetric Fibers and Fiber Preforms Refractive Index Profiles Determination of the Stress Distribution Application of the Method of Oblique Incidence Examples Fiber Preforms of Arbitrary Cross Section Determination of the Axial Stress Distribution Determination of Other Stress Components Internal Rotation of the Birefringence Axes in Polaiization-Holding Fibers Examples 225 Bibliography 229 Author Index 249 Subject Index 253

Fundamentals of Photoelasticity

Fundamentals of Photoelasticity Some Useful Definitions How Stress Is Calculated Principles of Photoelasticity Stress Measurement Techniques Strainoptic Technologies, Inc. Some Useful Definitions Residual