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H. Aben, C. Guillemet

Photoelasticity of Glass

With 218 Figures

Springer-VerlagBerlin Heidelberg New YorkLondon Paris TokyoHong Kong Barcelona Budapest


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Dr. Hillar Aben

Instituteof

Cybernetics, 21, Akadeemia teeEE0026 Tallinn, Estonia

Dr.-Ing. Claude Guillemet

86, Rue Pierre Joigneaux, 92270 Bois-Colombes, France

ISBN 978-3-642-50073-2 ISBN 978-3-642-50071-8 (eBook)

DOI 10.1007/978-3-642-50071-8

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concemed,specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction onmicrofilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permittedonly underthe provisions ofthe German CopyrightLaw o fSepte mber9, 1965, in its current version, and permissionfor use must always be obtained from Springer-Verlag. Violations are liable for prosecution act under GerrnanCopyright Law.

Springer-Verlag Berlin Heidelberg 1993Softcover reprint of tbe hardcover 1 st edition 1993

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, evenin the absence of a specific statement, that such names are exempt from the relevant protective laws and regulationsand therefore free for general use.

Typesetting: Camera ready by author

61/3020 - 5 4 3 2 1 0 - Printed on acid -free paper


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Preface

Glass is th e oldest man-made material. Its invention about five thousand years ago shouldbe considered as one of the crucial events in th e history of mankind. Glass has given ma nthe possibility to have daylight in his protected living environment and to compensate th e

defects of his sight. Glass containers and tableware have played and still play an importantrole in man's everyday life. Glass elements in microscopes and telescopes have given us th epossibility to learn the secrets of micro- and macrocosm. Glass participates in th e mostsophisticated technologies: glass fibers have caused a revolution in telecommunication,glass is used as a material for many modern electronic devices. Although nowadays plasticsoften make a strong competition to glass, for many applications glass is still the bestmaterial due to its specific properties - its hardness, good transparency, resistance tochemicals, th e easiness to shape glass articles, feasibility to change the composition of th eglass in order to meet new specific demands, etc.

Two peculiarities of glass should be pointed out. Th e first is the fragility of glass - it

breaks easily due to tensile stresses. The second is th e fact that in every glass item thereexist residual stresses due to th e complicated technological process during which glass fromthe state of a viscous liquid at high temperature turns into solid sta te, while cooled down.Since during the cooling process the temperature field in th e glass is inhomogeneous,residual stresses appear which can considerably reduce or increase the strength of th earticle.

Therefore, measurement and control of the residual stresses is an inseparable part of th eglass technology. For residual stress measurement mostly photoelasticity has been used,beginning with 1816 when Sir David Brewster discovered th e photoelastic effect whileinvestigating pieces of glass in polarized light. Practically every glass plant is equippedwith photoelastic apparatus for residual stress estimation.

Since a lot of books have been written on photoelasticity, the quest ion may arise whyhave the authors decided to write a special book on th e photoelasticity of glass. Thereasons for that are the following.

First, most of the books on photoelasticity are primarily oriented to engineers whouse plastic models to investigate stresses in engineering constructions. Th e model material is usually optically highly sensitive. Therefore, in the polariscope one can observe alot of isochromatics (fringes) the interpretation of which is one of th e key problems inengineering photoelasticity. Since th e optical sensitivity of glass is very low, often one

cannot observe any fringes at all or there may appear only some of them. Therefore th emeasurement technique is different and mostly consists of pointwise measurements with


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vi Preface

a cornpensator. Somewhat different are also automatic measurements and separation of

the principal stresses.Secondly, dealing with three-dimensional stress fields most of the books on photoelasticity consider th e frozen stress and th e scattered light methods. The first cannot be used inthe case of glass in principle, while application of the second is connected with diflicultiesdue to the low intensity of light scattered by glass. The most suitable method for stressanalysis in glass articles of complicated shape is integrated photoelasticity, which usuallyis not dealt with in books on photoelasticity.

Thirdly, due to the progress in glass industry it produces glass items with specific residual stress distribution which cannot be determined using classical photoelastic methods.For example, chemical strengthening of glass provokes extremely high stress gradient in athin layer near th e surface of the glass, in fibers and fiber preforms the residual stress distribution is discontinuous, etc. Innovation in glass technology presupposes also innovationin residual stress measurement methods.

To determine residual stresses in the aforementioned cases, during the last two decadesa number of new photoelastic methods have been developed which up to now haven'tbeen dealt with in monographic literature. Sometimes newly developed methods permitto quantitatively determine residual stresses where up to now only qualitative estimationof the latter was possible.

Thus, the present book is addressed to those who apply photoelasticity for investigatingstresses in glass. Its particular feature is that it contains description of a number of newrnethods which permit one to determine stresses also on the surface of glass even in thecase of high stress gradient, and in glass products of complicated shape an d of complicatedinternal structure.

Th e methods described are illustrated by numerous exarnples of stress investigation invarious glass articles. They demonstrate possibilities of the new methods and give alsoinformation about th e character of residual stresses in different glass products.

For reading th e book only a preliminary knowledge of mechanics of materials and ofpolarization optics is needed.

Many of the methods described in this book need specific apparatus and rather com

plicated software in order to calculate stresses by using the measurement data. Suchan apparatus and software have been elaborated under guidance of th e authors and aremarketed by several companies. Those who are interested in applying th e methods andalgorithms described in this book should not hesitate to contact the authors to obtaininformation about available apparatus and software.

This book is a summary of the work of many investigators whose publications are givenin the references. To a great extent th e book is based on research results obtained at theInstitute of Cybernetics of the Estonian Academy of Sciences and in the laboratories ofSaint-Gobain Recherche under the guidance of th e authors. Many of our colleagues in theaforementioned institutes supported our work on this book. The authors are particularly

indebted to Dr. J.Josepson, Mr. K.-J.Kell, Dr. A.Puro and Mr. A.Rumberg from th e Laboratory of Photoelasticity of th e Institute of Cybernetics, Estonian Academy of Sciences,to Mr. J.Prieur and Mr. S.Valladeau from Saint-Gobain Recherche, and to Dr. J.Kavkafrom the State Glass Research Institute of Hradek Kralove.


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Preface vii

The authors also wish to express their gratitude to the companies Saint-Gobain, Pilk

ington plc, Sharples Stress Engineers Ltd. and Photolastic Division of the Vishay Measurement Group as weil as to the Institute for Chemistry of Glass and Ceramic Materialsof the Czechoslovak Academy of Sciences who willingly provided product examples or original photos used in this book. The authors would like to acknowledge their indebtednessalso to McGraw-Hill International Book Company for the kind permission to reproducesome materials which previously appeared in H.Aben's book "lntegrated Photoelasticity".

C.Guillemet is indebted to his wife Eliane and to Miss E.Owens for their help in preparing his part of th e manuscript.

We are obliged to Mrs. E.Klement for the copy-editing, to Miss P.Veeber for preparingth e camera-ready manuscript and to Mr. V.Pihlo for drawing all the graphical represen

tations.Finaily we thank Springer-Verlag for the cooperation in preparing and publishing this

book.

Tailinn, AubervilliersNovember, 1992

Hillar AbenClaude Guillemet


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C o n t e n t s

Preface

Par t OneThe Basics of Photoelasticity and Glass

1 Basic Elasticity1.1 Elasticity1.2 Force and Stress1.3 Plane Stress . .1.4 Equations of Equilibrium1.5 Boundary Conditions1.6 Strain . . . . . . . . .1.7 Relations Between Stresses and Strains1.8 Plane Strain . . . . . . .1.9 Equations of Compatibility1.10 Stress Funct ion

2 Residual Stresses in Glass2.1 Int roduct ion . . . . . . . . . . . . . . . . . . . . . .2.2 Dependence of the Mechanical Strength on Residual Stresses2.3 Stresses Due to Indentations . . . . . . . . . . . . . .2.4 Residual Stresses Due to Thermal Annealing or Tempering

2.4.1 The Firs t Approaches .2.4.2 The Viscoelastic Theory . . . . . .2.4.3 The Structura l Theory . . . . . .2.4.4 Membrane Stresses and Form Stresses

2.4.5 Stress Redistribution by Cutt ing2.5 Stresses Due to Chemical Tempering

2.5.1 Stress Buildup . . . . . .2.5.2 Strengthening of Glass

2.6 Stresses Created in Glass by Radiations2.6.1 Corpuscular Radiat ion2.6.2 Electromagnetic Radiation

Thermal Effects . . . .Color Centers . . . . . .

2.7 Stresses Due to Heterogeneities

2.8 Stresses in Composite Glass Articles

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x Contents

2.8.1

2.8.22.8.32.8.4

Stresses in Glazes and Enamels

Stresses in Optical Fibers . . . . . . .Stresses in Glass-Metal and Glass-Ceramic SealsStresses Due to Inclusions

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3 Basic Photoelasticity 513.1 Polarized Light . . . . . . . . . . 51

3.1.1 Nature of Light . . . . . . 513.1.2 Natural and Polarized Light 513.1.3 Different Descriptions of Polarized Light 54

3.2 Artificial Double Refraction 553.3 Stress-Optic Law . . . . 563.4 Th e Plane Polariscope 573.5 The Circular Polariscope 583.6 Use of Double-Exposure Photography for the Elimination of the Isoclinics 603.7 Const ruction of Polariscopes 613.8 Measurement of Optical Retardation . 63

3.8.1 Color Matching . . . . . . . 633.8.2 Polariscope with a Tint Plate 633.8.3 The Babinet and Babinet-Soleil Compensators 653.8.4 Senarmont Method 663.8.5 The Azimuth Method . 67

4 Two-Dimensional Photoelasticity 694.1 General . . . . . . . . . . . 694.2 Stress Trajectories . . . . . . 694.3 Separat ion of Principal Stresses 71

4.3.1 Oblique Incidence Method 714.3.2 Shear Difference Method 724.3.3 Numerical Solution of the Compatibility Equation 734.3.4 Methods Based on Hooke's Law 73

4.4 Superposition of States of Stress

4.5 Determination of the Photoelastic Constant

5 The Scattered Light Method5.1 Introduct ion . . . . . . . . . . . . . . . . . . . . . .5.2 Scattering of Light . . . . . . . . . . . . . . . . . . .5.3 Th e Scattered Light Method with Polarized Incident Light5.4 The Scattered Light Method with Unpolarized Incident Light5.5 Using Interference of Coherent Scattered Light Beams

6 Integrated Photoelasticity6.1 Introduction . . . . . . . . . . . .6.2 Principle of Integrated Photoelasticity6.3 Basic Equations . . . . . . . . .6.4 Theory of Characteristic Directions6.5 Symmetrie Photoelastic Media

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6.6 The Case of Constant Principal Stress Axes . . . . . . . . . . .6.7 The Case of Weak Birefringence . . . . . . . . . . . . . . . . .6.8 Integrated Photoelastici ty as Optical Tomography of th e Stress Field6.9 Investigation of the General Three-Dimensional Sta te of Stress6.10 Axisymmetric State of Stress Due to External Loads

7 Photoelast ic Propert ies of Glass7.1 I n t r o d u c t i o n . . . . . . . . .

7.2 Discovery of th e Photoelastic Effect in Glass7.3 Influence of the Glass Composition7.4 Theories of th e Photoelastic Effect7.5 Influence of the Temperature and of the Thermal History7.6 Dependence of the Photoelastic Constant on Wavelength7.7 Anomalous Birefringence . . . . . . . . . . . . . . .

P a r t TwoStress Analysis in Flat Glass

8 Thickness Stresses8.1 Different Kinds of Thickness Stresses8.2 Measurement of Thickness Stresses

8.2.1 Using the Bending of the Light Rays8.2.2 Conventional Photoelasticity

9 Membrane Stresses9.1 In t roduc t ion . .9.2 Uniaxial Membrane Stresses

9.2.1 Edge Stresses9.2.2 Stresses Across a Ribbon

9.3 Bidimensional Membrane Stresses

10 Determinat ion of the Total Stresses10.1 Introduct ion . . . . . . . . . . .

10.2 The Measurement of Surface Stresses10.2.1 Differential Refractometry . .10.2.2 Th e "Mirage" Methods

Observation of the Guided Waves Close to the SurfaceThe Case of Flat Sampies . . . . . . . . .The Case of Curved Sampies . . . . . . . .The Case of Stress Gradient Near the Surface

Observation of the Guided Waves at InfinityTheory of the Differential Refractometry with Guided WavesLinear Index Profile . .

Determination of StressesAn Example . . . . .Alternative Numerical MethodsCurved Surface . . . . .Thermally Tempered Glass . .

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xi i Contents

10.3 Measurement of Total Residual Stresses

10.3.1 The Scattered Light MethodSpatial Modulation MethodPhase Modulation Method

10.3.2 Magnetophotoelasticity

Pa r t ThreeStresses in Glass Articles of Complicated Shape

11 Axisymmet r ic Glass Articles11.1 General Case ofAxisymmetr ie Residual Stress Distribution

11.1.1 Peculiarities of the Determination of th e Residual Stress .

11.1.2 Determination of the Axial and Shear Stress Distributions11.1.3 Additional Tomographie Measurements . . . . .11.2 Applieation of the Equilibrium and Boundary Conditions11.3 Stresses on the External Surfaee . . . . .11.4 Average Value of the Cireumferential Stress11.5 Stresses in Long Cylinders . . . . .11.6 Spherical Symmetry . . . . .

11.6.1 Stress Distribution in Spheres11.6.2 Quenehing Stresses Around a Spherieal Inclusion

11.7 Bending of Light Rays . . . . . . . . . . . . . . . .11.8 Determination of the Components of the Dielectrie Tensor11.9 Optimization of the Number of Terms in Stress Polynomials11.10 Experimental Techuique

11.10.1 Polariscopes . . . . . . . . . . .11.10.2 Immersion Teehnique . . . . . . .11.10.3 The Case of Mismatching Immersion

11.11 Examples . . . . . . . . . . . .11.11.1 Quenehed Long Cylinder11.11.2 An Article of Optieal Glass11.11.3 High Voltage Insulator11.11.4 Closed Tube11.11.5 Two Bonded Tubes . .

12 Containers and Other Thin-Walled Glassware12.1 Introduct ion . . . . . . . . . . . . . . . .12.2 Tradit ional Methods . . . . . . . . . . . .12.3 Determination of Stress in Cylindrieal Part of th e Container12.4 Axial Stress in an Arbitrary Section . . . . . . . . . . .12.5 Determination of the Stresses Due to the Internal Pressure12.6 Sandwich Glassware12.7 Examples . . . . . . . . .

12.7.1 A Champagne Bottle12.7.2 A Beer Bot tle12.7.3 Tumbler N l12.7.4 Tumbler W212.7.5 Salad Bowl .

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12.7.6 Electric Lamp . . . . . . . . . . . 21312.7.7 Ampule of a Fire Extinguisher System 215

13 Optical Fibers and Fiber Preforms 216

13.1 Int roduct ion . . . . . . . . . . 21613.2 Axisymmetric Fibers and Fiber Preforms 217

13.2.1 Refractive Index Profiles 21713.2.2 Determination of the Stress Distribution 21713.2.3 Application of th e Method of Oblique Incidence 21813.2.4 Examples . . . . . . . . . . . . . . . . . 220

13.3 Fiber Preforms of Arbitrary Cross Section . . . . . . 22113.3.1 Determinat ion of th e Axial Stress Distribution 222

13.3.2 Determination of Other Stress Components . . 22313.3.3 Internal Rotation of the Birefringence Axes in Polarization-Holding

Fibers . . 22413.3.4 Examples

BibliographyA uthor IndexSubject Index

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