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In-Plane Stess Analysis Using Tensor Field Photoelasticity

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In-Plane Stess Analysis Using Tensor Field Photoelasticity IN-PLANE STESS ANALYSIS USING TENSOR FIELD PHOTOELASTICITY by Sui Gao B.A.Sc., University of Waterloo, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2011 © Sui Gao, 2011 Abstract Few experimental methods exist for evaluating all in-plane stress components in solid materials; because of the tensor nature of these quantities. Full field measurement of all three stress components is desirable, since plastic deformation or failure can result from any combination of the three. A new photoelastic stress measurement method is presented for evaluating all three in-plane stress components within a two-dimensional photoelastic material. The measurement method is based on the observation that the complex transmission factors that describe the optical phase changes due to stress-induced birefringence have a second order tensor character, similar to that of other tensor quantities such as stress and strain. The same transformation equations and Mohr’s circle construction can be applied to the rotation of optical axis. A Michelson type interferometer and phase shifting are used to quantify the phases of the complex transmission factors. Mohr’s circle calculation is applied to obtain the principal transmission factors and principal axis orientation. The principal stresses are then obtained from the principal transmission factors through the stresss optical relationship. The effectiveness of this technique is demonstrated by comparing the experimental and analytical results for a hollow circular ring under diametric compression. ii Table of Contents Abstract ........................................................................................................................................ ii Table of Contents ........................................................................................................................ iii List of Figures .............................................................................................................................. v Nomenclature ............................................................................................................................ viii Acknowledgements ...................................................................................................................... x Chapter 1 − Introduction .............................................................................................................. 1 1.1 Experimental Stress Analysis ...................................................................................... 1 1.1.1 Motivation ....................................................................................................... 2 1.2 In-Plane Stress Measurement Methods ........................................................................ 2 1.2.1 Strain Gauges ................................................................................................. 3 1.2.2 Thermoelasticity ............................................................................................. 4 1.2.3 Optical Displacement Measurement Methods ................................................ 5 1.2.4 Photoelasticity ................................................................................................. 6 1.2.4.1 Classical Photoelasticity .................................................................... 7 1.2.4.2 Interferometric Photoelasticity........................................................... 8 1.2.4.3 Photoelastic Stress Separation ........................................................... 9 1.3 Research Objective and Proposed Method ................................................................ 10 Chapter 2 − Theoretical Background ......................................................................................... 13 2.1 Introduction to Polarization of Light and Birefringence............................................ 13 2.2 In-Plane Stress ........................................................................................................... 15 2.3 Stress Optical Relationship ........................................................................................ 17 2.4 Tensor Field Character of Birefringence ................................................................... 22 2.5 Phase Stepping Interferometry ................................................................................... 26 2.6 Far Field Drift Compensation .................................................................................... 29 Chapter 3 − Modifications to Interferometric Photoelastic Measurement Method ................... 31 3.1 Introduction to Interferometric Photoelasticity .......................................................... 31 3.2 Proposed Experiment Configuration.......................................................................... 32 iii 3.2.1 Secondary Beam Expansion ......................................................................... 33 3.2.2 Optical Phase Detection ................................................................................ 36 3.2.3 Phase Shifting Implementation ..................................................................... 37 3.2.4 Polarization Control ...................................................................................... 41 3.3 Tuning and Adjustment........................................................................................................ 42 3.4 Experiment Challenges and Solution ................................................................................... 46 Chapter 4 − Experimental Validation ........................................................................................ 50 4.1 Experiment Overview ................................................................................................ 50 4.2 Photoelastic Test Specimen ....................................................................................... 50 4.3 Polarized Phase Measurement of Birefringence ........................................................ 53 4.4 Calculation of Principal Transmission Factors .......................................................... 55 4.5 Stress Optical Coefficient Evaluation ........................................................................ 56 4.6 Stress Distribution Comparison ................................................................................. 57 Chapter 5 − Discussions ............................................................................................................ 62 5.1 Comparison with Classical Photoelastic Method ...................................................... 62 5.2 Improvements to Current Measurement Method ....................................................... 64 5.3 Future Work and Applications ................................................................................... 66 Chapter 6 − Conclusions ............................................................................................................ 70 6.1 Research Summary and Contributions....................................................................... 70 6.2 Comparison with other in-plane stress measurement methods .................................. 71 References .................................................................................................................................. 74 iv List of Figures Figure 1 − Strain Gauge ............................................................................................................... 3 Figure 2 − Thermoelastic Experiment Setup ............................................................................... 4 Figure 3 − Digital Image Correlation........................................................................................... 5 Figure 4 − Photoelastic Induced Optical Birefringence ............................................................... 7 Figure 5 − Photoelastic Fringe Pattern in Polariscope ................................................................. 8 Figure 6 − Interferometric Photoelastic Optical Configuration ................................................... 9 Figure 7 − Horizontally Polarized Illumination of Birefringent Material ................................. 11 Figure 8 − Optical Tensor Axis Rotation ................................................................................... 12 Figure 9 − Electromagnetic Wave of Plane Polarized Light ..................................................... 13 Figure 10 − Electromagnetic Wave of Elliptically Polarized Light .......................................... 14 Figure 11 − Optical Transmission Through Wave Plate ........................................................... 15 Figure 12 − General State of Plane Stress ................................................................................. 16 Figure 13 − Observation Axis Rotation of Plane Stress ............................................................ 17 Figure 14 − Stress Optical Relationship .................................................................................... 18 Figure 15 − Plane Stress Poisson’s Effect ................................................................................. 19 Figure 16 − Polarization Measurement of Photoelatic Material ................................................ 23 Figure 17 − Mohr’s Circle Relationship of Complex Transmission Factors ............................. 24 Figure 18 − Superposition
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