Contents

1 Transmission Photoelasticity 1
1.1 Introduction 1
1.2 Physical Principle Used in Photoelasticity 1
1.3 Nature of Light 2
1.4 Polarization 3
1.5 Passage of Light Through Isotropic Media 4
1.6 Passage of Light Through a Crystalline Medium 5
1.7 Light Ellipse . . 6
1.8 Retardation Plates and Wave Plates 8
1.9 Stress-Optic Law 9
1.10 Plane Polariscope 10
1.10.1 Analysis by Trigonometric Resolution 13
1.11 Jones Calculus 14
1.11.1 Rotation Matrix 14
1.11.2 Retardation Matrix 15
1.11.3 Representation of a Retarder 16
1.11.4 Polarizer 17
1.11.5 Quarter-Wave Plate 17
1.12 Analysis of Plane Polariscope by Jones Calculus 17
1.13 Circular Polariscope 18
1.14 Use of White Light 20
1.15 Determination of Isoclinic and Isochromatic Fringe
Order at a Point 21
1.15.1 Ordering of Isoclinics 21
1.15.2 Ordering of Isochromatics 22
1.16 Tardy's Method of Compensation 22
1.17 Calibration of Photoelastic Model Materials 24
1.17.1 Stress Field in a Circular Disc Under Diametral
Compression 24
1.17.2 Conventional Method 25
1.17.3 Sampled Linear Least Squares Method 26
Need for a better methodology 26
Use of whole field data to evaluate material fringe
value 26
1.17.4 Theoretical Reconstruction of Fringe Patterns . . . . 28
1.18 Further Comments on Fringe Ordering 29
1.18.1 Properties of Isochromatic Fringe Field 30
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1.18.2 Properties of Isoclinic Fringe Field 31

1.18.3 Use of Fringe Field Properties to Identify
Fringe Ordering 32
1.18.4 Role of Principles of Solid Mechanics in
Fringe Ordering 33
1.19 Determination of the Sign of the Boundary Stresses . . . . 34
1.20 Resolving the Ambiguity on the Principal Stress Direction . . 34
1.21 Introduction to Three-Dimensional Photoelasticity and
Integrated Photoelasticity 35
1.21.1 Conventional Three-Dimensional Photoelasticity . . . 36
1.21.2 Principle of Optical Equivalence 37
1.22 Model to Prototype Relations 39
1.23 Closure 42
Exercises 42
References 46

2 Reflection Photoelasticity 47
2.1 Introduction 47
2.2 Reflection Polariscope 48
2.3 Stress and Strain-Optic Relations for Coatings 49
2.4 Coating and Specimen Stresses 50
2.5 Correction Factors for Photoelastic Coatings 52
2.6 Poisson's Ratio Mismatch 56
2.7 Coating Materials 57
2.8 Bonding the Coating 59
2.9 Selection of the Coating Thickness 60
2.10 Calibration of the Coating Material 62
2.11 Data Collection and Analysis 64
2.12 Application of Photoelastic Coatings 65
2.13 Closure 65
Exercises 65
References 66

3 Digital Image Processing 67

3.1 Introduction 67
3.2 Image Sampling and Quantization 67
3.2.1 Pictures as Functions 67
3.2.2 Uniform Sampling and Quantization 68
3.3 Video Standards 69
3.4 Image Sensors 71
3.5 Image Display 72
3.6 Image Perception • 73
3.7 Image Storage 74
 Contents XIII

3.8 Some Basic Relationships and Mathematical Operations

Between Pixels 74
3.8.1 Neighbours of a Pixel 74
3.8.2 Arithmetic and Logic Operations 75
3.8.3 Neighbourhood Oriented Operations 75
3.9 Basic Steps in Image Processing . . . . 77
3.10 Typical Image Processing Systems for Digital Photoelasticity 77
3.11 Software Structure and Design 79
3.12 Image Acquisition 80
3.13 Tools for Image Understanding 82
3.13.1 Pseudo Colouring 82
3.13.2 Histogram 83
3.13.3 Two-Dimensional and Three-Dimensional
Intensity Plots 84
3.14 Filtering in Spatial Domain 86
3.14.1 Low Pass Spatial Filtering 88
3.14.2 Median Filtering 88
3.15 Image Enhancement 88
3.15.1 Contrast Stretching 89
3.15.2 Histogram Equalisation 90
3.16 Image Segmentation 91
3.16.1 Thresholding 91
Global thresholding 92
Semi thresholding 92
Dynamic thresholding 92
3.16.2 Edge Detection 94
Edge detection by convolution filters 94
Edge detection by non-convolution filters 95
Edge detection by thresholding 96
3.17 Morphological Filters 98
3.18 Further Discussions on Image Sensors 98
3.18.1 Operation of CCD Arrays 98
3.18.2 Interline Transfer CCD 100
3.18.3 Linearity and Dynamic Range 101
3.18.4 Sources of Noise 102
3.19 Digitisation of the Camera Video Signal 103
3.20 Resolution of an Image Processing System 103
3.21 Gamma Compensation 104
Exercises 104
References 105

4 Fringe Multiplication, Fringe Thinning and Fringe

Clustering 107

4.1 Introduction 107

4.2 Fringe Multiplication 108
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4.3 Half Fringe Photoelasticity (HFP) 110

4.4 DIP Methods for Fringe Thinning Ill
4.5 Algorithms Based on Considering the Fringe Patterns as
a Binary Image 112
4.6 Mask-Based Algorithms for Skeleton Extraction Using
Intensity Variation within a Fringe 113
4.7 Global Identification of Fringe Skeletons Based on
Intensity Variation 115
4.7.1 Edge Detection 115
4.7.2 Fringe Skeletonization 116
Row-Wise scanning algorithm 116
Algorithm for fringe skeleton extraction for arbitrarily
shaped fringes 117
4.7.3 Applications of the Algorithm to Actual Experimental
Conditions 120
4.8 Further Improvements on the Global Thinning Algorithm . . 123
4.9 Performance Evaluation of Various Fringe Thinning
Algorithms 126
4.9.1 Comparison of the Skeleton Extraction 127
Computer generated test images 127
Images recorded from actual experimental situations . 129
4.9.2 Comparison of the Computational Effort 131
4.10 Use of Tiling to Improve Information in Stress
Concentration Zones 131
4.11 Fringe Tracing Algorithm 133
4.12 Ordering of Fringes 135
4.13 Closure 136
Exercises 137
References 138

5 Phase Shifting, Polarization Stepping and Fourier

Transform Methods - 141

5.1 Introduction 141

5.2 Early Attempts for Automated Polariscopes 142
5.3 Phase Shifting in Photoelasticity 144
5.4 Intensity of Light Transmitted for a Generic Arrangement of a
Plane Polariscope 146
5.5 Intensity of Light Transmitted for a Generic Arrangement of a
Circular Polariscope 149
5.6 Evaluation of Fractional Fringe Order along an Isoclinic
Contour 152
5.7 Whole Field Evaluation of Photoelastic Data by Using a Plane
Polariscope 153
5.8 Whole Field Evaluation of Photoelastic Data by Using a
Circular Polariscope 155
 Contents XV

5.8.1 The Generic Procedure 155

5.8.2 Calculation and Representation of Whole Field Data . 159
5.8.3 Parameters Affecting the Generation of Phase Map and
its Solution 162
Influence of local oscillations of isoclinic parameter on
fractional retardation calculation 162
Importance of isoclinic parameter representing either a x
or <r2 direction over the domain 163
Ambiguity in experimentally evaluating the isoclinic
parameter 167
Interactive approach t o obtain a good phase map . . . 169
5.9 Error Sources and Methods to Minimise Their Influence . . • 172
5.9.1 Influence of Error in Measuring Intensities 174
5.9.2 Errors Due t o Mismatch o f Quarter-Wave Plates . . . 177
5.10 Evaluation of Isoclinic Value by Phase Shifting Technique - 181
5.10.1 Use of Two Loads to Get Continuous Isoclinic
Contours 181
5.10.2 Use of Multiple Wavelengths to Get Continuous
Isoclinic Contours 183
5.11 Polarization Stepping for Isoclinic Determination 185
5.12 Fourier Transform Methods for Photoelastic Data
Acquisition 188
5.12.1 Use of Carrier Fringes 188
5.12.2 Use o f Multiple Polarization Stepped Images . . . . 189
5.12.3 Use of Load Stepping 191
5.13 Comparative Evaluation of Phase Shifting, Polarization
Stepping and Fourier Transform Techniques 192
5.14 Closure 193
Exercises 193
References 194

6 Phase Unwrapping and Optically Enhanced Tiling

in Digital Photoeiasticity 199
6.1 Introduction 199
6.2 Boundary Detection 200
6.3 Noise Removal in Phase Maps 201
6.4 Algorithm for Phase Unwrapping 202
6.5 Representation of the Unwrapped Phase 205
6.5.1 Three-Dimensional Plots 205
6.5.2 Total Fringe Order Viewing on the Image 206
6.6 Parameters Affecting Phase Unwrapping 207
6.6.1 Influence of the Selection of the Phase Unwrapping
Threshold 207
6.6.2 Influence of the Location of the Primary Seed Point. . 209
6.7 Use of Tiling Procedure for Phase Unwrapping 210
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6.8 Digital Magnification of High Fringe Density Zones . . . . 211

6.8.1 Replication 211
6.8.2 Linear Interpolation 212
6.8.3 Higher Order Interpolation 212
6.9 Optically Enhanced Tiling (OET) 212
6.10 Cementing of a Tile 213
6.11 OET Applied to a Circular Disc Under Diametral
Compression 215
6.12 OET Applied to a Ring Under Diametral Compression. . . . 217
6.13 Closure 219
Exercises 220
References 220

7 Colour Image Processing Techniques 221

7.1 Introduction 221

7.2 Colour Models 223
7.2.1 RGB Model 224
7.2.2 HSI Model 225
7.3 Colour Image Processing Systems 226
7.3.1 Hardware 226
Transmission Photoelasticity 227
Reflection Photoelasticity 227
7.3.2 Software 228
7.4 Typical Spectral Response of a Colour Camera 228
7.5 Intensity of Light Transmitted in White Light for
Various Polariscope Arrangements 230
7.6 Three Fringe Photoelasticity (TFP) 234
7.6.1 Calibration 235
7.6.2 Methodology 240
7.6.3 Application to the Problem of a Circular Disc Under
Diametral Compression 240
7.7 Green Image Plane as a Green Filter 243
7.8 Phase Shifting in Colour Domain 244
7.8.1 Transmission Photoelasticity 244
7.8.2 Reflection Photoelasticity 246
7.9 Spectral Content Analysis (SCA) 249
7.10 Digital Spectral Content Analysis (DSCA) 252
7.11 Hybrid Techniques 252
7.11.1 Polarization Stepping in Colour Domain 253
7.12 Tricolour Photoelastic Method 254
7.13 Closure 258
Exercises 261
References 261
 Contents XVII

8 Evaluation of Contact Stress Parameters and Fracture

Parameters 265

8.1 Introduction 265

8.2 Basic Data Required and its Digital Acquisition 266
8.2.1 Conversion of Pixel Co-ordinates to Model
Co-ordinates 266
8.2.2 Rotational Transformation 267
8.3 Stresses in Terms of Contact Length and Geometrical and
Elastic Properties of the Bodies in Contact 268
8.4 Evaluation of Contact Stress Parameters by
Least Squares Analysis 270
8.4.1 Validation for Hertzian and Non-Hertzian Contact . . 272
8.5 Developments in the Description of the Stress Field Equations
in the Neighbourhood of a Crack-tip 273
8.5.1 Mode-I Stress Field Equations 273
8.5.2 Mixed-Mode (Combination of Mode-I and Mode-II)
Stress Field Equations 276
8.5.3 Equivalence Between the Multi-Parameter Stress Field
Equations 278
8.6 Developments in SIF Evaluation Methodology 278
8.7 Evaluation of Mixed-Mode Stress Field Parameters Using Least
Squares Technique 281
8.8 Experimental Validation of the Methodology 284
8.8.1 Mode-I Loading 284
8.8.2 Mixed-Mode Loading 288
8.9 Contact Stress and Fracture Analysis of a Spur Gear . . . . 290
8.9.1 Loading Frame Design 290
8.9.2 Evaluation of Contact Parameters 292
Measurement of radius of curvature at the point
of contact 292
Experimental results 293
8.9.3 Evaluation of Fracture Parameters 294
8.10 Closure 298
Exercises 299
References 299

9 Stress Separation Techniques 303

9.1 Introduction 303

9.2 Oblique Incidence Method 304
9.2.1 Secondary Principal Stresses 304
9.2.2 The Methodology 305
9.3 Shear Difference Technique 307
9.3.1 Conventional Method 307
9.3.2 Improvement by Tesar 308
XVIII Contents

9.4 Survey of Numerical Methods 309

9.4.1 Integration of Compatibility Condition 309
Finite difference approach 309
9.4.2 Integration of Stress Difference Equations 312
9.4.3 Least Squares Method 314
9.4.4 Hybrid Techniques 316
9.4.5 Methods Using Only Isochromatic Data 317
9.5 Stress Separation by Combined Phase Shifting and FEM . . 318
9.5.1 Finite Element Formulation 319
9.5.2 Meaningful Discretization of the Domain 323
9.5.3 Plotting of Fringe Contours from FE Results . . . . 324
9.5.4 Influence of Error in Fringe Data 327
9.5.5 Application of the Technique to the Problem of Plate
with a Hole 330
9.6 Use of Integrated Photoelasticity Concepts for Stress
Separation 332
9.6.1 Least Squares Algorithm 333
9.6.2 Design of the Loading Frame 333
9.6.3 Application to the Problem of Disc under Diametral
Compression 334
9.7 Stress Separation in Three-Dimensional Photoelasticity . . . 337
9.8 Stress Separation in Reflection Photoelasticity 342
9.8 Closure 344
Exercises 344
References 345

10 Fusion of Digital Photoelasticity, Rapid Prototyping

and Rapid Tooling Technologies 347
10.1 Introduction 347
10.2 Difficulties in Conventional Three-Dimensional
Photoelasticity 348
10.3 Rapid Prototyping in Model Making 348
10.3.1 Software Issues in RP 349
10.3.2 Stereolithography Process 351
10.3.3 Solid Ground Curing 354
10.3.4 Fused Deposition Modelling 355
10.4 Direct Analysis of RP Models by Photoelastic Coatings . . . 356
10.4.1 Experimental Results 357
10.4.2 Analysis of the Results 358
Evaluation of Young's modulus by tensile test . . . . 358
Study on the seepage of the adhesive 359
Numerical simulation of fringe patterns 359
10.4.3 Recommendations 361
10.5 Direct Use of RP Models for Transmission Photoelastic
Analysis 361
 Contents XIX

10.6 Rapid Tooling for Model Making 363

10.6.1 Basic Steps in Rapid Tooling 363
10.6.2 Digital Photoelastic Characterisation of the Process . . 364
10.7 Closure 365
Exercises 366
References 366

11 Recent Developments and Future Trends 369

11.1 Introduction 369

11.2 Evaluation of Characteristic Parameters 369
11.2.1 Srinath and Keshavan's Method 370
11.2.2 Whole Field Determination of Characteristic
Parameters by Phase Shifting 371
Development of relevant equations 371
Experimental evaluation of characteristic parameters . 375
Whole field theoretical evaluation of characteristic
parameters • . 375
11.3 Tensorial Tomography 378
11.4 Developments in DIP Hardware 383
11.5 Developments in DIP Software 384
11.5.1 Development of a Device Independent Software . . . 385
Selection of software features 385
FRN_DAT software 385
An application 386
11.5.2 Future Possibility 387
11.6 Digital Dynamic Photoelasticity 388
11.6.1 Classification of High, Very-high and
Ultra-high-speed Photography 388
11.6.2 Classical Methods for High-speed Photography . . . 388
11.6.3 Digital Dynamic Recording 391
11.7 Application to Composites 395
11.7.1 Photo-Orthotropic Elasticity Theories 396
Stress-Optic law 396
Strain-Optic law 397
11.7.2 Calibration of Photo-Orthotropic Composites . . . . 398
11.7.3 Influence of Residual Birefringence 399
11.7.4 Separation of Stresses in Photo-Orthotropic
Elasticity 400
11.7.5 Application of Digital Photoelasticity to
Composites 400
11.8 Closure 401
Exercises 401
References 401
XX Contents

Index 405