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Residual stress measurement of plates using shearography Walsh, Allan Frederick 2020

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Residual Stress Measurement of Plates using ShearographybyAllan Frederick WalshB.S. Mechanical Engineering, University of Maine, 2017A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of Applied ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)January 2020© Allan Frederick Walsh, 2020The following individuals certify that they have read, and recommend to the Faculty ofGraduate and Postdoctoral Studies for acceptance, the thesis entitled:Residual Stress Measurement of Plates using Shearographysubmitted by Allan Frederick Walsh in partial fulfillment of the requirements for thedegree of Master of Science in Mechanical EngineeringExamining CommitteeGary Schajer, Mechanical EngineeringSupervisorXiaoliang Jin, Mechanical EngineeringExaminerMauricio Ponga, Mechanical EngineeringExamineriiAbstractWood mills have used bandsaw blades for the primary breakdown of timber into lumberfor centuries. By reducing the width of the saw cutting edge, material waste can bereduced. Thin-kerf saw blades are susceptible to lateral forces during cutting. To combatthis behavior, saw filers introduce residual stresses into the blades. Classical methods formonitoring saw blade tension, such as the light-gap method, are subjective and analog.There exists a necessity to monitor the amount of residual stress induced into the bladesreliably, using a nondestructive test. A newly proposed method for stress evaluation,incremental shearography, offers the advantage of measuring transverse plate bendingbehavior at micron resolution. Shearography, a direct measure of surface slope, promises arobust, full-field measurement using low-cost components. By comparing measured slopesto those analytically found, a stress resultant of the tensioning process can be inferred.iiiLay SummaryMeasuring transverse deflection of tensioned bandsaw blades under bending using a lightinterference method. Contributions include novel methods of observing effects of residualstresses on transverse plate bending, modified shearographic approach for measuring largedeflections with micro scale resolution, and determining stress magnitudes of cold-rolledplates.ivPrefaceAll work presented henceforth was conducted under the supervision of Dr. Gary Schajerin the Renewable Resources Laboratory at the University of British Columbia. Dr. GarySchajer assisted with concept formulation and manuscript editing. This dissertation isoriginal, unpublished work by the author, A. F. Walsh.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiNomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Wood Processing in Lumber Mills . . . . . . . . . . . . . . . . . . . . . . . 11.2 Saw Filing and Stress Assessment . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Optical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Roll Tensioning Influence on Residual Stress . . . . . . . . . . . . . . . . . 72.1 Residual Stresses and their Influence . . . . . . . . . . . . . . . . . . . . . . 72.2 Imparting Residual Stresses: Roll-Tensioning . . . . . . . . . . . . . . . . . 102.3 Optimal Tensioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Evaluation of Residual Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1 Light Gap Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1.1 Quantifying the Light Gap Method . . . . . . . . . . . . . . . . . . . 153.1.2 Limitations of the Light Gap Method . . . . . . . . . . . . . . . . . 183.2 Derived Foschi Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.1 N th Magnitude Residual Stress: The Algorithm . . . . . . . . . . . . 213.2.2 Least-Squares Estimation . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Hole Drilling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.1 Computation of Residual Stress . . . . . . . . . . . . . . . . . . . . . 253.3.2 Limitations in Hole-Drilling . . . . . . . . . . . . . . . . . . . . . . . 263.4 A New Approach: Optical Methods . . . . . . . . . . . . . . . . . . . . . . . 274 Shearography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.1 Introduction to Interferometrics . . . . . . . . . . . . . . . . . . . . . . . . . 284.2 Shearography: The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31vi4.2.1 Introduction to Shearography . . . . . . . . . . . . . . . . . . . . . . 314.2.2 Shearing the Image: A Measure of Slope . . . . . . . . . . . . . . . . 344.3 Unwrapping the Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 Limitations of Classical Shearography . . . . . . . . . . . . . . . . . . . . . 374.4.1 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.4.2 Fringe Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4.3 Accurate Shearing Size . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4.4 Speckle Decorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5 Overcoming Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5.1 Incremental Method vs. Absolute Method . . . . . . . . . . . . . . . 404.5.2 Preserving the Phase Change . . . . . . . . . . . . . . . . . . . . . . 414.5.3 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.5.4 Optimized Optical Geometry . . . . . . . . . . . . . . . . . . . . . . 444.5.5 Analyzing Shearing Magnitude . . . . . . . . . . . . . . . . . . . . . 454.5.6 Dual Laser Configuration . . . . . . . . . . . . . . . . . . . . . . . . 454.5.7 Remedial Adjustments: Conclusions . . . . . . . . . . . . . . . . . . 465 Experimental Data and Results . . . . . . . . . . . . . . . . . . . . . . . . . 475.1 Procedure and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.1.1 Plate Rolling and Preparation . . . . . . . . . . . . . . . . . . . . . . 475.1.2 Shearographic Implementation . . . . . . . . . . . . . . . . . . . . . 485.1.3 Three-Point Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.2 Optical Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.1 Interferometric Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2.2 Bending Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3 Data Post Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.3.1 Filtering and Incremental Summation . . . . . . . . . . . . . . . . . 565.3.2 Data Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.3.3 Central Slope Extraction . . . . . . . . . . . . . . . . . . . . . . . . 565.4 Results: Plate Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.4.1 Measured Slopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.4.2 Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.4.3 Stressed Slopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.1 Highlights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.1.1 Robust Slope Measurement . . . . . . . . . . . . . . . . . . . . . . . 656.1.2 High Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.1.3 Incremental Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.2.1 Theoretical Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 667 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.1 Further Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.1.1 Multiple Roll Passes . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.1.2 Comprehensive FEA Investigation . . . . . . . . . . . . . . . . . . . 767.1.3 Industrial Prototype . . . . . . . . . . . . . . . . . . . . . . . . . . . 76References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80viiList of FiguresFigure 1.1 Saw mill production costs [1] . . . . . . . . . . . . . . . . . . . . . 2Figure 1.2 Two examples of cuts due to blade wandering. Adapted from P.F.Lister[2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Figure 1.3 Saw filer checking the light admittance of a pre-stressed bladeagainst a hand gauge. Adapted from P.F. Lister [2] . . . . . . . . . 3Figure 1.4 The saw filer making appropriate adjustments to stress profile us-ing a roll-tensioning machine. Adapted from P.F. Lister [2] . . . . 4Figure 1.5 An ESPI and Shearography phase map measurements for the samedeformation. Adapted from Zhu et. al. [3] . . . . . . . . . . . . . . 5Figure 2.1 Cold rolling of the bandsaw blade, central spreading of materiallongitudinally results in the lateral spreading of material . . . . . . 7Figure 2.2 Anticlastic deformation of a flat plate given a longitudinal bendingmoment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Figure 2.3 Synclastic deformation of a flat plate given a longitudinal bendingmoment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 2.4 An example of the effect of the seating profile on the bandmillwheel due to residual stresses . . . . . . . . . . . . . . . . . . . . . 10Figure 2.5 Side view of the roll-tensioning process. Adapted from P.F. Lister[2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 2.6 Detailed axial view of roll tensioning and the associated resultantstresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 3.1 The light-gap method applied to a plate containing centrally-rolledresidual stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 3.2 Detailed views of appropriate residual stress, over-tensioning, andunevenly distributed residual stress across the transverse of a blade 16Figure 3.3 The light-gap method according to Foschi, including a parabolicstress distribution and deflection δ. Adapted from R. O. Foschi [4] 17Figure 3.4 The components of transverse deflection summed together to pro-vide the combined deflection observed by the light-gap method . . 18Figure 3.5 The residual stress distributions given for various transverse de-flected shapes, noting that while two shapes may be similar, theresultant stress profile may be dissimilar. Adapted from R. O.Foschi [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Figure 3.6 Detailing the components of deflection numerically derived to rep-resent the components of slope . . . . . . . . . . . . . . . . . . . . 19Figure 3.7 Transverse slope due purely to induced residual stress . . . . . . . 20Figure 3.8 Stress distributions for varying nth order estimations: n = 1, n =5, n = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21viiiFigure 3.9 The assumed transverse stress distribution based on plate widthW and roller path width h . . . . . . . . . . . . . . . . . . . . . . . 22Figure 3.10 The least squares best fit model modifying the σ0 stress value,approaching the measured data for a stressed transverse slope . . . 24Figure 3.11 A before and after depiction of hole-drilling, outlining the effectof Poisson’s ratio caused by the introduction of the hole on thesurrounding gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 3.12 The strain gauge rosette used to determine residual stresses . . . . 26Figure 4.1 Detailing both constructive and destructive light wave interference 29Figure 4.2 ESPI configuration, noting a single light source with pathlengthinterference occurring on specimen’s surface . . . . . . . . . . . . . 29Figure 4.3 Shearography configuration, a single light source with interferenceoccurring directly before the CCD sensor . . . . . . . . . . . . . . 30Figure 4.4 Detailing how two overlapped images are slightly shifted due tothe angle of the shearing mirror . . . . . . . . . . . . . . . . . . . . 31Figure 4.5 The relative rotation of the surface as observed by the sensorthrough the shearing device . . . . . . . . . . . . . . . . . . . . . . 32Figure 4.6 Each intensity collected by shifting the phase-stepping mirror bya quarter of a wavelength . . . . . . . . . . . . . . . . . . . . . . . 33Figure 4.7 A pixel-wise operation calculating the phase distribution . . . . . . 34Figure 4.8 The difference between two phase distributions, taken before andafter the deformation is applied, resulting in a shearographic mea-surement of slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 4.9 The locations of each component related to the sensitivity of themeasurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 4.10 The wrapped measurement becoming unwrapped using the un-wrapping algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 4.11 A graphical representation of the unwrapped data versus the mea-sured wrapped data . . . . . . . . . . . . . . . . . . . . . . . . . . 37Figure 4.12 Raw data shown to be often fraught with random speckular noise,a centerline of data is plotted to emphasize the noise . . . . . . . . 38Figure 4.13 Fringes displayed as a function of cos2, noting the high fringe den-sity at the center which denotes the area of least rotation . . . . . 38Figure 4.14 While large shearings distances can be determined through vectormeasurement, small shearing distances cannot be as accuratelydetermined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 4.15 A literal representation of an absolute measurement, the final recordedphase distribution at 15 steps subtracted by the initial phase dis-tribution at step 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 4.16 The literal representation of the incremental calculation, whereeach smaller phase distribution changes are summed together . . . 41Figure 4.17 Incremental versus absolute measurements made for the same out-of-plane displacements . . . . . . . . . . . . . . . . . . . . . . . . . 42Figure 4.18 A 3D surface plot comparison of the raw data (colored) and incre-mentally filtered data (black) . . . . . . . . . . . . . . . . . . . . . 43Figure 4.19 The transverse slope comparison between the raw data (blue) andincrementally filtered data (black) . . . . . . . . . . . . . . . . . . 44ixFigure 4.20 An autocorrelation function applied to a sheared random specklepattern, the distance between the two peaks of correlation denotestwice the shearing distance . . . . . . . . . . . . . . . . . . . . . . 45Figure 4.21 Comparing the resultant intensity distributions when using a singleversus a dual laser illumination, the dual laser configuration on theright gives a more evenly illuminated region of interest . . . . . . . 46Figure 5.1 The physical experimentation setup . . . . . . . . . . . . . . . . . . 47Figure 5.2 For each roll pass, the plate was rotated 180° about the two prin-ciple axes, ensuring equal rolling was applied . . . . . . . . . . . . 48Figure 5.3 The rolling pressures applied to each plate, longitudinally . . . . . 48Figure 5.4 The shearographic setup used in the following experiments . . . . . 49Figure 5.5 A display of the program that controlled the acquisition of intensityvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 5.6 A simply supported plate . . . . . . . . . . . . . . . . . . . . . . . 51Figure 5.7 Experimentation procedure detailed in incremental steps . . . . . . 51Figure 5.8 The experimental shearographic setup, viewed from the top down . 52Figure 5.9 The National Instruments wiring configuration used for phase-stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 5.10 A top down image of the 3-point bending setup . . . . . . . . . . . 54Figure 5.11 The actuation stage, outlining the dimensions of the 3-point bending 55Figure 5.12 The actuator dimensions used to apply 3-point bending . . . . . . 55Figure 5.13 Isolating the measured transverse slope from the incrementally fil-tered summation, based on the center of actuation . . . . . . . . . 57Figure 5.14 The plate orientation . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 5.15 Measured slope visualized . . . . . . . . . . . . . . . . . . . . . . . 58Figure 5.16 Each individual transverse slope plotted, noting the increasingslope given the same bending deflection as a result of residual stresses 59Figure 5.17 A numerical integration of the slopes give a more physical repre-sentation of the transverse deflection, noting that plates withoutresidual stresses start out in anticlastic bending, but become moresynclastic as stress is introduced . . . . . . . . . . . . . . . . . . . 61Figure 5.18 The stressed components of the measured slopes, detailed with thecentral compressive stress magnitudes found for each . . . . . . . . 62Figure 5.19 The stress estimation versus the actuation distance for each plate,noting a reasonably stable estimation convergence as the deflectiongets larger for all plate rolling magnitudes . . . . . . . . . . . . . . 63Figure 5.20 The stress magnitudes determined for each plate plotted againstthe rolling pressures applied . . . . . . . . . . . . . . . . . . . . . . 64Figure 6.1 An unstressed, flat plate with an applied bending moment, result-ing in anticlastic bending behavior across the transverse width . . 67Figure 6.2 The transverse deflected shape and slope as generated by the the-oretical equations given the experimental plate properties and di-mensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 6.3 Given the fixed width and thickness, the longitudinal radius ofcurvature is varied in value to obtain a region of an appropriate ratio 69Figure 6.6 The transverse deflection and slope are plotted given the originalwidth and thickness, with a very small radius of curvature . . . . . 69xFigure 6.4 Given the fixed radius of curvature and width, the thickness of theplate is varied in value to obtain a region of an appropriate ratio . 70Figure 6.7 The transverse deflection and slope are plotted given the originalwidth and radius of curvature, with a reduced thickness value . . . 70Figure 6.5 Given the fixed radius of curvature and thickness, the width of theplate is varied in value to obtain a region of an appropriate ratio . 71Figure 6.8 The transverse deflection and slope are plotted given the originalradius of curvature and thickness, along with an increased width . 71Figure 6.9 The unpredictable stresses introduced by the various manufactur-ing processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 6.10 The whole-field FEA deflection with sensitivity in the vertical axis(y-direction) as a result of the actuation . . . . . . . . . . . . . . . 73Figure 6.11 The central deflection FEA result simulated along a single trans-verse path, the data displayed in the following figures relates tothis vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Figure 6.12 The FEA result for the central transverse deflection of the platecompared to the theoretically generated central deflection and thenumerically integrated central deflection from the measured data . 74Figure 6.13 The FEA result for the numerically derived slope compared to thetheoretically generated transverse slope and the measured trans-verse slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Figure 7.1 A figure detailing three rolls made across the width of the plate atvarying rolling pressures, a common characteristic of roll-tensioning 76xiNomenclatureB, b,W Transverse widthσc Compressive stressσt Tensile stressRL Longitudinal radius of curvatureRT Transverse radius of curvatureσx(y) Transverse stress profilet Thicknessw Transverse displacement profileδ Absolute transverse deflectionN,n Number of accuracy termsan Magnitude of each stress termy Transverse width vectorv Poisson’s ratiodwdy Transverse slopeE Modulus of elasticityα Geometric plate propertiesw0 Unstressed component of deflection∑nwn Stressed component of deflectionxiiγ1,2, A¯, B¯ Geometric components of unstressed deflectiondw0dy Unstressed component of slope∑ndwndy Stressed component of slope∂w∂y Measured transverse slopeσ0 Magnitude of central stressh Width of roll path Strain1,2,3 Principal strainsσx Stress in the x-directionσy Stress in the y-directionτx,y Shear stressa¯, b¯ Stress/strain sensitivity constantsδx,y Shearing directionIi Intensity captureI0 Average intensityγ Modulation termφ Phase angleλ Wavelength of light∆ Relative phase changeS Spatial orientation of illumination sourceO Spatial orientation of sensorP A point on the surfacexiiic Number of phase jumpsds Diameter of specklef f -number of camera lensda Peak distance between correlated images∆z Out-of-plane actuation distanceV Voltageβ Ratio determining nonlinear bending behaviorσ2 Secondary compressive stressxivAcknowledgmentsI would like to acknowledge my professor, Dr. Gary Schajer, for keeping me on the straightand narrow when I was struggling with my research. I’d also like to thank my labmatesfor their support and insight. Finally, I’d like to thank Bruce Lehmann at FP Innovationsfor taking the time to assist me with my experiments.xvDedicationDedicated to my loving parents;Eleanor and PeterxviChapter 1Introduction1.1 Wood Processing in Lumber MillsSawmills use bandsaw blades for the primary breakdown of timber into lumber. Theprocess of cutting timber produces a sizable amount of sawdust, which is considered rawmaterial waste. In BC alone, forest products account for 32 percent of commodity exports,making the business substantial in BC’s economy. Over 67 million cubic meters of timberwere processed in 2018[5]. Of the timber cut, approximately 10 percent of the raw materialis lost to sawdust during the initial stages of the sawmill [6]. This relatively high amountof material waste poses a severe economical burden on the lumber processing industry andreduces long-term profit margins of sawmills.To reduce the amount of material waste, ’thin-kerf’ band saws, or band saws withreduced thickness, have been implemented in sawmills. While full-kerf blades have athickness of 1/8”, thin-kerf blades have a thickness of 3/32” or less. This 25 percentreduction in kerf produces desirable advantages in wood processing, primarily a 30 percentsavings of the material when compared to a full-kerf blade[6], as well as a 25 percentreduction in energy. Thinner blades require less power to perform cuts [7]. In industry,raw material accounts for 75-80 percent of the total production costs, meaning that smallreductions in material waste can produce considerable increases in revenue [1]. Whilethere are advantages associated with reducing the kerf, there are also drawbacks.Reducing the sawblade kerf increases the risk of overheating, unstable vibrations, andwandering[8, 9]. Due to the minimalized thickness of the blade, the longitudinal stiffness isgreatly reduced, resulting in a tendency to succumb to lateral forces during cutting. Poordynamic behavior during operation causes the sawblade to deviate from the desired cutpath, resulting in ’snaking’ or wandering. These inaccurate cuts result in a gross materialwaste, which is highly undesirable.To counteract inaccurate cuts due to a reduction in blade stiffness, sawmills introduceresidual stresses into the blades during a process known as ’tensioning’. Tensioning isa mechanical process by which compressive stresses are induced longitudinally along theblade, typically by a compressive rolling machine. By introducing residual stresses via.1Figure 1.1: Saw mill production costs [1](a) Example 1 of wandering blade cut (b) Example 2 of wandering blade cutFigure 1.2: Two examples of cuts due to blade wandering. Adapted from P.F. Lister[2]longitudinal cold-rolling, the dynamic stability of the saw blade is greatly increased[10].Other methods of tensioning that have been investigated include shot peening, laser ten-sioning, and thermal tensioning[11, 12, 13]. However, roll tensioning is most commonlyused because of its quicker process and more uniform stress distribution. The rolling pro-cess includes feeding the bandsaw length-wise through two compressive, narrow crownedrollers that squeeze the middle section of the saw blade, spreading the surrounding ma-terial towards the free edges. Desired tensioning relies entirely on the amount of inducedresidual stress, as well as the stress profile. If there is too much tensioning the sawblade2will buckle, but if there is too little tensioning the blade will not have an adequate stiffnessappropriate for accurate cuts[14].1.2 Saw Filing and Stress AssessmentWhen assessing the induced residual stresses, there are several destructive testing (DT)and non-destructive testing (NDT) methods available. Of the destructive testing, residualstress hole drilling techniques are favorable because they provide accurate results, but notideal as they render the specimen damaged after holes are drilled[15]. The NDT methodsoffer the advantage of further use of the component after testing by only assessing thespecimen behavior within the elastic regime. Non destructive testing include methodswhich utilize hand gauges, strain gauges, surface probing, x-ray diffraction, and lightinterferometrics. Unfortunately, the latter four methods are only suitable for experimentaland laboratory use due to their complex and costly nature[16]. A simple, robust approachfor stress identification is necessary for day-to-day use in a sawmill.Figure 1.3: Saw filer checking the light admittance of a pre-stressed blade against a handgauge. Adapted from P.F. Lister [2]Historically, saw filing has been used to maintain saw blades since the 19th century. Thetrade requires years of practice and the adjustments made to each saw blade are highlysubjective to the individual saw filer. The saw filers do not base their adjustments onmechanical stresses, but utilizes hand gauges and bright light to assess the amount of lightadmittance between the saw blade and the gauge[17]. This qualitative measure is knownas the ’light-gap method’. The light-gap method assesses the transverse deflection of thetensioned blade under a longitudinal bending condition prescribed by the saw filer. Once3an area of stress concentration has been identified, the saw filer uses a roll tensioningmachine to correct the stress, making the blade once again suitable for use[14]. Thelight-gap method is mainly a measurement of the transverse deflection, while the trueindication of residual stress is in the transverse curvature. Using the light-gap method, oran empirically technical version using surface probing, the transverse deflection must benumerically derived twice to obtain curvature. Numerical derivation amplifies erroneousmeasurement data, causing large amounts of noise to propagate through the derived datasets.Figure 1.4: The saw filer making appropriate adjustments to stress profile using a roll-tensioning machine. Adapted from P.F. Lister [2]While the light-gap method has been a longstanding qualitative measurement to iden-tify the tensioned state of saw blades, the saw filers generally do their work based on anintuitive understanding of the relationship between the addition of stress and the trans-verse deflection. Foschi studied this relationship in 1975 and was able to determine amodel that related the transverse deflection to a state of residual stress[4]. Because Fos-chi’s equations related transverse deflection to residual stress, it has often been criticized,along with the light-gap method, as being a poor indicator of the saw’s true tensionedstate. The transverse deflection provides little indication of the specific stress magnitudeand distribution. While two transverse deflected shapes may be close to one another,the related stress distributions may be far apart due to the complex behavior of nonlinearplastic deformation [4]. There exists a need to quantitatively identify the stress magnitude4and distribution in a repeatable manner.1.3 Optical MethodsOf the NDT methods previously described, light interferometrics provide the greatestamount of resolution under loading conditions. The measurements associated with in-terferometrics are performed on the wavelength scale. These methods include electronicspeckle pattern interferometry (ESPI), holography, and shearography. ESPI and hologra-phy both measure in-plane surface deformations, while shearography measures the out-of-plane surface slope changes[18].(a) ESPI in-plane deformation (displacement)phase map difference(b) Shearography out-of-plane relative rotation(slope) phase map differenceFigure 1.5: An ESPI and Shearography phase map measurements for the same deforma-tion. Adapted from Zhu et. al. [3]ESPI and holography require the use of two laser illumination fronts on the specimen’ssurface, resulting in a system prone to errors due to outside disturbances and low lasercoherence [16]. Shearography boasts a much more stable system through the use of aself-referencing light interference just prior to the sensor, allowing robust measurementsto be made under noisy conditions. Shearography, due to its self-referencing nature, doesnot require high laser coherence, so ordinary diodes can be used to the same effect. Thesecharacteristics make shearography ideal for analyses made in the sawmill environment.Shearographic measurements of saw blades, when compared to surface probing or thelight-gap method, offer a full-field analysis of the transverse slope. This brings the dataone numerical derivation closer to the curvature measurement, reducing the influence ofnoise due to numerical derivation.1.4 ObjectivesThe goal of this thesis is to implement the robust, high resolution measurement methodof shearography in conjunction with derived Foschi equations to determine the residualstress magnitude and profile induced by roll-tensioning in a bandsaw blade. This will beachieved by understanding the rolling process, adapting the shearography measurement5method to better suit larger range measurements (0.5mm of deflection), and deriving amodified Foschi model that can be used to compare the shearographic measurements to.For the residual stress distribution, a series of investigations are performed for insighton how the stresses are introduced, the dimensions of the rolling mechanism, and therolling pressure applied. In regards to the shearographic measurements, adjustments willbe made to the calculation of data such that the center of actuation is preserved andthe optical geometry is appropriate for long range measurements. The Foschi equationsrelating transverse surface deflection to longitudinal curvature will be derived to representslope, such that the measured data does not need to be numerically integrated or derivedto estimate the residual stresses.6Chapter 2Roll Tensioning Influence onResidual Stress2.1 Residual Stresses and their InfluenceResidual stresses are characterized by their ’locked-in’ nature. Once all external loadingshave been removed, the material remains in a state of residual stress. The residual stressesmust sum to zero in a self-equilibrating fashion [19]. For example, material that is sub-jected to a compressive residual stress must have an equal-and-opposite tensile residualstress throughout the volume. Almost all manufacturing processes introduce some degreeof residual stress into a finished product, particularly in casting and welding [20]. Resid-ual stresses can be induced in a number of fashions. The more notable sources of residualstresses are formed by plastic deformation, surface modification, and thermal gradients.Because the presence of residual stresses affect the lifespan of the manufactured product,it is important to understand their magnitude and distribution.Figure 2.1: Cold rolling of the bandsaw blade, central spreading of material longitudinallyresults in the lateral spreading of material7The contribution of residual stresses can be either detrimental or beneficial. In thecase of the tempered glass on any common smartphone screen, the compressive residualstresses create a crack-resistant boundary which protects tension sensitive areas of crackpropagation under impact or other external loads. These beneficial residual stresses createa bias in the material which increases the fatigue life of the component. In the case ofbandsaw blades, the stress redistribution increases the lateral stiffness of the blade whileunder external loads applied at either edge of the saw. This stress redistribution is achievedby introducing a compressive central stress, resulting in state of tensile stress along thefree edges. These effects are particularly beneficial for thin-kerf saw blades as it increasestheir stability. By stiffening thinner saw blades, their tendency to succumb to lateralforces during cutting is reduced, preventing snaking or wandering cuts [12, 14].In addition to increasing the lateral stability of bandsaw blades during operation, thetensioning process also allows the saw blade to seat correctly on the crowned mill wheels.Central stacking, or alignment, of the saw blade on the mill wheels is key to reducingslippage and maintaining consistent overhang of the cutting edge [21]. When flat, thin,unstressed plates are bent about their major longitudinal axis, their tendency is to bend’anticlastically’. Anticlastic behavior denotes a convex deflection about the transverse axiswhen the longitudinal axis is subjected to concave bending. This behavior is a result ofPoisson’s effect.Figure 2.2: Anticlastic deformation of a flat plate given a longitudinal bending momentDuring the cutting process, the blade is stretched around two large diameter wheelsand driven with a motor. As the section of unstressed blade wraps around the wheels,8Figure 2.3: Synclastic deformation of a flat plate given a longitudinal bending momentthe longitudinal axis becomes bent to the diameter of the wheels, causing anticlastic ’curl’to the free edges, similar to a hyperbolic paraboloid. Anticlastic behavior can cause theblade to shift laterally, and in extreme cases, become completely detached from the millwheels [22]. The introduction of residual stresses can counteract the anticlastic nature ofPoisson’s ratio if adequate pre-stress tensioning is applied. The addition of residual stressescreate a ’synclastic’ effect. Synclastic deformation denotes a same-sign curvature sharedbetween the longitudinal axis and transverse axis. Synclastic behavior is advantageous forthe crowned mill wheels due to the increased contact area.9Figure 2.4: An example of the effect of the seating profile on the bandmill wheel due toresidual stresses2.2 Imparting Residual Stresses: Roll-TensioningOf the methods available to saw filers used to impart pre-tensioning stresses, hammer-ing and roll-tensioning are the two most common methods. Hammer tensioning involvesstriking the surface of the saw blade, causing the steel to squeeze laterally in plane. Be-cause this method invokes highly localized adjustments, it is considered more of an artthan a universally repeatable method, as it takes a great amount of practice and skill toperfect. In comparison, roll-tensioning is much easier to control and produces results thatare much more consistent. Roll-tensioning has become an integral component of saw blademaintenance during the past 60 years, providing stable pre-stressed blades [23].10Figure 2.5: Side view of the roll-tensioning process. Adapted from P.F. Lister [2]During roll-tensioning, a section of the saw blade is fed longitudinally through twocompressive, crowned wheels. The rolling force is adjusted by the saw filer using a largelead screw, which changes the compressive forces between the two rollers. As the sawblade moves along between the wheels, a shallow groove is produced by the compressiverolling force. This groove is indicative of high compression along the path of the roll,causing the center of the saw blade to lengthen. As the center of the bandsaw blade islengthened, the free edges remain the same, causing them to ’tighten’ due to a low tensilestress equilibrium.11(a) Axial view of the rolling process(b) Resultant central compressive stress and tensile stresses due to central rollingFigure 2.6: Detailed axial view of roll tensioning and the associated resultant stressesThe factors that affect roll-tensioning include the rolling force, width of the rollingwheels, crown of the rolling wheels, plate thickness, roll path location with respect to theplate width, and number of rolls [24, 25, 26]. While multiple roll paths are typically usedin saw filing, only a central, highly compressive roll is considered in this dissertation.2.3 Optimal TensioningWhile extensive research on optimal circular saw blade tensioning has been investigated,there are few investigations regarding a benchmark in regards to bandsaw blades [14, 4].As mentioned previously, increasing the lateral stiffness and maintaining central stackingare the primary motivators to prescribing a stressed state within a blade. However, there12are few indicators of an ’optimally tensioned’ blade. In the case where the magnitude ofthe compressive residual stresses exceed 60 percent of the material yield strength, bucklingwill occur, rendering the blade useless [12]. Conversely, in cases of too little tensioning,the desired effects of the stress will not be noticeable and may lead to bandsaw cuttinginstability.The common criteria for assessing appropriate tensioning in pre-stressed bandsawblades come from a visual inspection. This method aims to match the synclastic deforma-tion of the transverse width with the crowned curvature of the mill wheel. The curvaturematch is a good indication of reliable stacking, but not of the stiffness and stability[17, 4].However, based on the principal stress distributions that give the synclastic deformationappropriate stacking, the stretching of the central material will always yield a state oftension towards the outer edges.13Chapter 3Evaluation of Residual Stress3.1 Light Gap MethodThe light-gap method is the most commonly used technique for assessing appropriatetensioning in bandsaw blades. As the bandsaw plate is bent with a longitudinal radius, ahand gauge is used to observe the transverse deflection profile. The term ’light-gap’ comesfrom the use of an illumination source on one side the gauge. As the blade is bent, theamount of light transmittance is observed between the gauge and the plate by the sawfiler.Figure 3.1: The light-gap method applied to a plate containing centrally-rolled residualstressesGauges have numbers inscribed on their surfaces corresponding to their radii in feet.14Depending on the bandmill crown and amount of roll-tensioning performed on the blade,different transverse radii of curvatures are desirable. The hand gauges will also show areasof localized stress including pockets (low points) and tight spots (high points). An ideallight-gap profile is one where the bandsaw blade adheres closely to the gauge, allowinglittle light through.The tensioning process results in synclastic bending behavior, which is directly relatedto the distribution and magnitude of residual stresses across the volume. As mentionedin chapter 2 this deflected shape, or light-gap, is dependent on the geometry of the blade,number of roll passes, pressure of roll passes, and geometry of the rollers [24, 25, 26].3.1.1 Quantifying the Light Gap MethodFoschi presented an analytical investigation relating the observed light-gap deflection toinduced residual stresses across the transverse width of the saw blade. All equationspresented in the following section pertain to y = ±B2 based on symmetry.If the centrally rolled stresses are considered symmetric, they can be approximatedusing a Fourier series,σx(y) ≈N∑n=1an cos(2npiyB)n = 1, 2, 3 . . . (3.1)where an is the term associated with the magnitude of stress and σx(y) is the stressdistribution,an =4B∫ B/20σx(y) cos(2npiyB)dy (3.2)The differential equation presented by Foschi relates transverse displacement w andlongitudinal curvature RL to the residual stress distribution σx(y),d4wdy4+12(1− v2)R2Lt2w =−12(1− v2)ERT t2σx(y) (3.3)where a particular solution to the differential equation is,wp =N∑n=1bn cos(2npiyB)n = 1, 2, 3 . . . (3.4)Originally bn is unknown, but can be solved using the particular solution of equation3.2, using free-end boundary conditions d2wdy2= vRL ,d3wdy3= 0 at y = ±B2 ,bn = −RLEan1 + 4[npiαB]4 (3.5)where α is associated with the plate geometric and material properties,15(a) A tensioned blade that adheres to the gauge due to rolling(b) A tensioned blade that has folding occurring centrally due to rolling(c) A tensioned blade that has a tight-spot occurring due to unevenly distributed stressFigure 3.2: Detailed views of appropriate residual stress, over-tensioning, and unevenlydistributed residual stress across the transverse of a bladeα = 4√3(1− v2)R2Lt2(3.6)The transverse deflection w(y) is expressed as the sum of the stress-free w0(y) and16Figure 3.3: The light-gap method according to Foschi, including a parabolic stress distri-bution and deflection δ. Adapted from R. O. Foschi [4]stressed wn(y) deflections,w(y) = w0(y) +∑nwn(y) (3.7)The first term associated with the stress-free deflection was determined by Nickola andConway[27],w0(y) = γ1 coshαy cosαy + γ2 sinhαy sinαy (3.8)where,γ1,2 =vRLα2sinh αB2 cosαB2 ∓ cosh αB2 sin αB2sinhαB + sinαB(3.9)The component of deflection related to residual stresses, wn(y), is given as,wn(y) =N∑n=1−RLEan1 + 4[ npiαB ]4[cos2npiyB+ (−1)n(2npiB)2w0(y)]n = 1, 2, 3 . . .(3.10)Substituting equation 3.10 into equation 3.7 we arrive at the relationship betweentransverse deflection and the influence of residual stress,w(y) = w0(y) +N∑n=1−RLEan1 + 4[ npiαB ]4[cos2npiyB+ (−1)n(2npiB)2 RLvw0(y)](3.11)Based on Poisson’s ratio, the term associated with stress-free deflection w0(y) will al-17ways produce an anticlastic behavior (concave-convex) due to transverse stress equilibriumunder bending conditions. Conversely, the stressed deflection wn(y) will produce highlysynclastic behavior (concave-concave). This synclastic behavior is a result of ’lengthening’the center of the blade, whilst the free-edges remain their original length. Based on thesummation of the two components of deflection, the measured deflection w(y) will be ini-tially anticlastic for a small stress distribution, but as the stress increases, the deflectionbehavior will become synclastic [4].Figure 3.4: The components of transverse deflection summed together to provide thecombined deflection observed by the light-gap method3.1.2 Limitations of the Light Gap MethodFoschi concludes in his investigation that while a transverse shape is uniquely related toa certain residual stress distribution, two different transverse shapes may be very close,but the corresponding stress distributions may be very dissimilar [4]. Looking back atequation 3.5, the deflected shape is inversely proportional to the 4th power of n, thus thehigher order terms produce a very small light-gap profile.The assumption is made that the stress distribution is parabolic, and by increasing thenth term accuracy, the parabolic shape becomes a high frequency, high amplitude solutionfor large values of an. This solution is virtually undetectable using the light-gap method.Foschi offers insight to overcoming this drawback by adjusting the tolerances by which thesaw filers use to approach the desired transverse profile.3.2 Derived Foschi EquationsA direct measurement of slope would omitt the tedious nature of measuring relative trans-verse displacements due to Poisson’s effect and would give a more direct indication of thestresses present. Taking the numerical derivation of Foschi’s transverse deflection compo-nents,dwdy=dw0dy+∑ndwndy(3.12)Based on the geometry of the plate, dw0dy can be inferred using relationships determinedby Nickola and Conway [27]. Since the quantity that is of most interest is the component18Figure 3.5: The residual stress distributions given for various transverse deflected shapes,noting that while two shapes may be similar, the resultant stress profile may be dissimilar.Adapted from R. O. Foschi [4]Figure 3.6: Detailing the components of deflection numerically derived to represent thecomponents of slopeof transverse slope due purely to stress, the following relationship is expressed as,∑ndwndy=dwdy− dw0dy(3.13)where the stress term,∑ndwndy is equal to the component of unstressed transverse slopedw0dy subtracted from the combined transverse slopedwdy .Substituting a measured transverse slope ∂w∂y into equation 3.13,19∑ndwndy=∂w∂y− dw0dy(3.14)Figure 3.7: Transverse slope due purely to induced residual stressa definition of measured transverse slope due purely to tensioning stresses can bedetermined. Given that dwody can be inferred and the slope quantity∂w∂y is measured, thecomponent of slope due to stress can be extracted and used as a comparative measureagainst equation 3.13.For this to be possible, the slope stress terms associated with∑ndwndy must be ex-panded,∑ndwndy=N∑n=1−RLEan1 + 4[ npiαB ]4[−2npiBsin2npiyB+ (−1)n(2npiB)2 dw0dy]n = 1, 2, 3 . . .(3.15)The primary relationship between the stressed transverse profile and the distributionof stress is an, the magnitude of distribution. The stress profile is described as,σx(y) ≈N∑n=1an cos(2npiyB)n = 1, 2, 3 . . . (3.16)Therefore, assuming the stress distribution σx(y), the an terms can be numerically de-termined and substituted into equation 3.15. This substitution and comparison ultimatelyallows for the inverse solution of the component of slope due to stress.20Figure 3.8: Stress distributions for varying nth order estimations: n = 1, n = 5, n = 103.2.1 N th Magnitude Residual Stress: The AlgorithmIn order to determine the subsequent an terms, and by association, the stress distributionσx(y), an a-priori assumption must be made based on the residual stress profile createdby roll-tensioning.Given a single, central, highly compressive stress produced by roll-tensioning, a stressdistribution involving a compressive stress σ0, and lateral tensile stresses σt can be inferred21Figure 3.9: The assumed transverse stress distribution based on plate width W and rollerpath width hwhere h is equal to the width of the tensioning roller wheel and W is equal to thewidth of the plate. Due to equilibrium, the tensile stresses can be equated to,σt =−hW − hσ0 (3.17)Rearranging equation 3.16 into a Fourier cosine series,σx(y) =∑nan cos(2npiyW)(3.18)Solving for the stress magnitude term an,an =2W∫ W/2−W/2σx cos(2npiyW)dy (3.19)Inserting the a-priori stress distribution into equation 3.19 for a symmetrical pulse,an =4W∫ h/20σ0 cos(2npiyW)dy +4W∫ W/2h/2σt cos(2npiyW)dy (3.20)Evaluating and solving equation 3.20 the stress magnitude can be defined in terms of22the central compressive stress σ0 and lateral tensile stresses σt,an =2npiσ0 sin(npihW)− 2npiσt sin(npihW)(3.21)Finally, substituting equation 3.17 into equation 3.21 and simplifying,an =2npiWW − h sin(npihW)σ0 (3.22)Here, the stress magnitude term an that appears in equations 3.15 and 3.16 is governedby the geometries of the roll path width and plate width, as well as the central compressivestress magnitude σ0. This equation is suitable for theoretical slope generation because thestress term σ0 does not change in magnitude with respect to increasing nth term estimation.Equation 3.22 can finally be substituted into equation 3.15 resulting in,∑ndwndy=N∑n=1−2RLW sin(npi hW)Enpi(W − h)σ01 + 4[ npiαB ]4[−2npiBsin2npiyB+ (−1)n(2npiB)2 dw0dy](3.23)Using equations 3.13 and 3.23 with plate geometry and material constants, a transversestressed slope distribution∑ndwndy can be generated for a given central stress magnitudeσ0, allowing the comparison between the component of measured stressed slope∑ndwndy mand the generated stressed slope σ0∑ndwndy .In more direct terms, the generated stressed slope is multiplied by the stress term σ0to best-fit the component of measured slope data associated with residual stress,∑ndwndy m= σ0(∑ndwndy1σ0)(3.24)where σ0 is determined by a least-squares estimation algorithm.3.2.2 Least-Squares EstimationIn order to determine the central stress magnitude σ0 value that allows the generatedstressed slope to fit the measured stressed slope, a least squares fit is applied,σ0 =∑(∑ndwndy∑ndwndy m)∑(∑ndwndy) (3.25)The corresponding least-squares fit is determined by first setting the central stressdistribution magnitude σ0 = 1 MPa, generating a transverse stressed slope∑ndwndy andthen adjusting the value to fit the measured stressed slope∑ndwndy m.23Figure 3.10: The least squares best fit model modifying the σ0 stress value, approachingthe measured data for a stressed transverse slope3.3 Hole Drilling MethodHole-drilling is a residual stress measurement method that is widely used and well docu-mented with standardized procedures. It is a destructive testing method which involvesremoving a small amount of material on the surface of a specimen, causing a redistribu-tion of residual stresses around the hole. This stress redistribution results in localizeddeformations. Both strain gauging and optical methods are available for assessing thelocalized deformations, with strain gauging being the most accurate and reliable. Stan-dardized strain gauge geometry was established in 1966 by Rendler et. al. [28], along witha repeatable, reliable process which is the backbone of the current day standard: ASTMStandard Test Method E837 [29].Strain gauge rosettes used for residual stress analysis contain three radially locatedgauges, typically at 0°, −135°, and −270°. These gauges accurately ensure the measure-ment of in-plane stress components σx, σy, and τxy. As well, there are standard calibrationconstants that relate measured strain to residual stress for these proprietary gauges [30].The detailed surface preparation, gauge selection, and installation can be found in the24Figure 3.11: A before and after depiction of hole-drilling, outlining the effect of Poisson’sratio caused by the introduction of the hole on the surrounding gaugesASTM Standard Test Method E387.3.3.1 Computation of Residual StressResidual stresses in thin plates are assumed to be uniform with thickness. For a uniformdistribution, the computation is simple because there are only three unknown residualstress components being considered [31].The relationship between the measured strain at each radial location on the rosetteand in-plane residual stress becomes, =σx + σy2(1 + v)a¯E+σx − σy2b¯Ecos 2θ+ τxyb¯Esin 2θ (3.26)where σx, σy, and τxy are the three in-plane components. The angle between eachindividual gauge on the rosette is denoted by θ, and the calibration terms which definethe stress/strain sensitivity are given as the constants a¯ and b¯. Their numerical weightdepends on the hole diameter and depth. The in-plane components of strain can berearranged to represent stress, given the three measured strain components: 1, 2, 3.σx =E(1 + v)a¯3 + 12− Eb¯3 − 12(3.27)σy =E(1 + v)a¯3 + 12+Eb¯3 − 12(3.28)τxy =Eb¯3 − 22 + 12(3.29)Using equations 3.27,3.28,3.29, the material properties, calibration constants, andstrain indicator readings, the residual stresses can be found [32]. The primary stress25Figure 3.12: The strain gauge rosette used to determine residual stressesinterest of tensioned bandsaw blades is the magnitude of the transverse stress σx.3.3.2 Limitations in Hole-DrillingWhile the hole-drilling method is regarded as an accurate, repeatable measurement methodfor determining residual stresses in materials, it comes at a cost. From an economicalperspective, each measurement is expensive [32]. Tungsten carbide drill heads must bereplaced after each hole is drilled, as well as the single-use strain gauges. The standardizedstrain gauge rosettes must be installed correctly for a reliable measurement. Any inconsis-tencies with installation may lead to debonding. If the gauge, once removed, shows areasof debonding, the whole process must be started again with new experimental components.From an analytical standpoint, the information collected by the strain gauge onlypertains to a finite location; where the material was removed from the volume. Unlikefull-field measurement methods, hole-drilling only collects information from a single pointlocation. If the profile across the transverse width is desired, additional hole-drillingmeasurements must be made.Finally, from a safety standpoint, once the component is subjected to drilling it is26often unusable. In some cases the holes created can be repaired, which deems this a ’semi-destructive’ measurement [32]. However, in the case of bandsaw blade stress identification,once a small hole has been drilled through the material, the blade cannot be used foroperation again. Saw blades are subjected to high temperatures and large forces duringuse. The introduction of a through-surface hole, irregardless of size, could increase thelikelihood of premature failure.3.4 A New Approach: Optical MethodsBased on the limitations presented by the hole-drilling method, a non-destructive testingprocedure for assessing residual stresses in tensioned plates is desirable. Optical methodsoffer notable advantages when compared to both the light-gap and hole-drilling methods.Interferometric optical methods, in particular, include measurements that are full-field,of light wavelength sensitivity, and, depending on the optical configuration, can measurein-plane or out-of-plane deformations and strains [33].The optical interferometric method that is used in this investigation is known asshearography. Shearography is a form of electronic speckle pattern interferometry (ESPI).Instead of measuring in-plane deformations, as with classical ESPI, the optical arrange-ment measures out-of-plane relative slope changes. The first and second derivatives ofdeformation, slope and curvature, paint a much more sensitive and complete picture ofthe residual stresses.The background and implementation of the shearographic method specific to thisproject is described in the following chapter.27Chapter 4Shearography4.1 Introduction to InterferometricsLight interference methods offer measurements at extremely high resolutions, within frac-tions of a wavelength of light. Based on the optical configuration, sensitivity to out-of-plane or in-plane deformation data can be collected. Originally starting with holographyin 1971 [33], random speckle patterns displayed on a surface from an coherent illuminationsource contained useful information relating to surface roughness. This information is theconsequence of coherent light waves interfering with each other from the perspective of theobserver. The interference of two wave fronts can either be constructive or destructive,and by moving the sensor relative to the surface, interference can be observed. Modernadaptations of the original holographic method take the forms of ESPI and an adaptationof ESPI, shearography.ESPI is a method which has in-plane sensitivity through the use of two coherent wavefronts emitted from a single laser source. This method can be used to visualize andquantify small in-plane surface displacements. Based on the optical configuration usedwith ESPI, different levels of sensitivity are possible for problems including stress-strainanalysis, vibrational analysis, and other NDT investigations.Because of ESPI’s sensitive nature, it is prone to environmental disturbances such asvibrations, temperature gradients, or even air currents. ESPI collects the information be-tween two relative phase changes, once before deformation and once after deformation. Iflateral displacements perturb the speckle pattern away from its original position, the rel-ative phase differences will suffer from speckle decorrelation, failing to provide any usefulinformation. For this reason ESPI is primarily practical only for laboratory experimen-tal use, where outside factors such as temperature changes, air currents, humidity, andvibrations can be controlled or eliminated.28Figure 4.1: Detailing both constructive and destructive light wave interferenceFigure 4.2: ESPI configuration, noting a single light source with pathlength interferenceoccurring on specimen’s surface29An adaptation of ESPI, shearography, was first known by the name ’speckle shearinterferometry’. The same light physics that drive ESPI also apply to shearography, withsubtle differences. Instead of having two coherent wave fronts emitted from a single lightsource, a single wave front is displayed on the surface. This reflected illumination isthen observed by a shearographic interferometer containing a pair of laterally or verticallysheared images on the sensor plane. The two slightly transposed images interfere witheach other much like the two coherent wave fronts in ESPI. However, the informationrecorded by the phase difference is the relative rotational change of the surface before andafter deformation.Figure 4.3: Shearography configuration, a single light source with interference occurringdirectly before the CCD sensorIn shearography, the slope is measured, giving a first-order derivative of the out-of-plane surface displacement. Shearography offers a non-contact, non-destructive, whole-field measurement method. Additionally, shearography has the advantage of observingdirect slope changes due to surface deformations, as well as using self-referencing lightinterference. This self-referencing interference is advantageous as it is more robust tooutside disturbances. Shearography has a common optical path between the specimen,interferometer, and illumination source. This common optical path between the objectand sensor plane provides very stable measurements, even under harsh environmentalconditions.Shearography offers a powerful advantage to measuring the transverse surface slope ofbandsaw blades under three-point bending. By measuring the transverse slope directly, thedata can be compared directly to a theoretically generated transverse slope to determine30the magnitude of central compressive stress.4.2 Shearography: The Basics4.2.1 Introduction to ShearographyShearography was developed to overcome the limitations of classical holography. Thenotable advantages of shearography when compared to other interferometric methods in-clude (1) a single light illumination source, not requiring a reference beam, leading tosimplified, low-cost, and stable optical configurations and (2), a direct measurement offull-field surface slope, a first-order derivative of the out-of-plane surface displacement[34]. Shearography is a practical non-destructive measurement tool appropriate for a widerange of industrial testing applications.Figure 4.4: Detailing how two overlapped images are slightly shifted due to the angle ofthe shearing mirrorIn the optical configuration, a non-polarizing 45-degree beam splitting cube is placedin front of the camera sensor. Two adjacent mirrors are located next to the shearingcube, the first at a slight angle and the second attached to a piezoelectric element whichproduces small displacements for varying levels of phase interference. The shearing deviceproduces a pair of slightly transposed images of the specimen’s surface. Because of thisslight image shearing, one point on the image sensor is represented by two separate pointson the specimen surface. The direction of this slight transpose is known as the directionof image-shearing and is directly related to the sensitivity of the measurement.31Figure 4.5: The relative rotation of the surface as observed by the sensor through theshearing deviceThe object’s surface is illuminated with coherent laser light, so the two slightly shearedimages interfere with one another, creating a random speckle interference pattern. Asthe mirror attached to the piezoelectric device is modulated to and from the surface bycontrolled fractions of a wavelength, random interference can be collected as intensities andcalculated into a phase map. Sequential speckle interference is collected before and afterthe object undergoes deformation. The initial calculated phase map is then subtractedfrom the deformed phase map, producing a full-field measurement of the surface’s relativeslope change.For convenience, these intensities are collected at intervals of λ4 interference [35],I1 = 2I0[1 + γ cosφ] (4.1a)I2 = 2I0[1 + γ cos(φ+ 90°)] (4.1b)I3 = 2I0[1 + γ cos(φ+ 180°)] (4.1c)I4 = 2I0[1 + γ cos(φ+ 270°)] (4.1d)32Figure 4.6: Each intensity collected by shifting the phase-stepping mirror by a quarter ofa wavelengthwhere Ii is the intensity distribution of the random speckle pattern observed by thecamera sensor, γ is the speckle pattern amplitude of modulation, I0 is the average specklepattern intensity observed at each level of interference, and φ is the random phase angleto be determined.After an out-of-plane deformation is applied, a second set of intensities are collected,which contain a value ∆ associated with phase change as a result of the applied deforma-tion,I ‘1 = 2I0[1 + γ cos(φ+ ∆)] (4.2a)I ‘2 = 2I0[1 + γ cos(φ+ 90°+ ∆)] (4.2b)I ‘3 = 2I0[1 + γ cos(φ+ 180°+ ∆)] (4.2c)I ‘4 = 2I0[1 + γ cos(φ+ 270°+ ∆)] (4.2d)In order to determine the relative phase change, ∆, the intensity equations must berearranged to solve for their respective random phase angles,φ = tan−1(I4 − I2I1 − I3)(4.3)φ+ ∆ = tan−1(I ‘4 − I ‘2I ‘1 − I ‘3)(4.4)The phase change due to deformation, ∆, can then be determined by taking the dif-ference between the final phase distribution and original distribution,∆ = φ‘ − φ (4.5)where φ‘ = ∆+φ. The resultant phase change distribution ∆ is consequently boundedby ±pi by the arctangent function, and requires phase-jump ’unwrapping’ to provide acontinuous gradient.33Figure 4.7: A pixel-wise operation calculating the phase distributionFigure 4.8: The difference between two phase distributions, taken before and after thedeformation is applied, resulting in a shearographic measurement of slope4.2.2 Shearing the Image: A Measure of SlopeDigital shearography makes use of a single laser source for illuminating the test specimen.The light is reflected off the diffuse surface and is passed through the shearing device.The light is then split into two separated angular beams. These two non-parallel beamsof scattered light represent a single point on the objects surface, and are almost collinearon the image sensor plane, displaced by the shearing distance δx,y. The shearographicmeasurement can be collected with a relatively low resolution camera sensor, allowingfaster data acquisition. Interference of the two light paths will create an embedded specklepattern of the shearographic image, which will reveal relative surface rotation after thespecimen’s surface is deformed.The shearing direction on the sensor plane determines the sensitivity and magnitude ofthe observed relative rotation, and reveals interference fringe patterns accordingly. Con-sidering a single point P (x, y, z) on the camera sensor represented by P1(x, y, z) andP2(x + δx, y, z) on the specimen’s surface, given the shearing direction and magnitudeas δx. It can be explained that the relative phase change ∆ is related to the two respectivepoint displacements (δu, δv, δw) as follows,∆ =2piλ(Aδu + Bδv + Cδw) (4.6)34where (u+ δu, v + δv, w + δw) is the displacement vector between P1 and P2. A,B,Care sensitivity vectors dependent on the spatial orientation of the illumination sourceS(xs, ys, zx) and camera sensor O(xo, yo, zo),A =(x− x0)√x20 + y20 + z20+x− xs√x2s + y2s + z2s(4.7a)B =(y − y0)√x20 + y20 + z20+y − ys√x2s + y2s + z2s(4.7b)C =(z − z0)√x20 + y20 + z20+z − zs√x2s + y2s + z2s(4.7c)Figure 4.9: The locations of each component related to the sensitivity of the measurementWhen equation 4.6 is subjected to a small shearing distance δx, the relative displace-ment terms can be treated as partial derivatives of (u, v, w) with respect to the directionof the applied shear,∆ =2piλ[A∂u∂x+ B∂v∂x+ C∂w∂x]δx (4.8)As the incidence angle between the illumination source and the shearing device de-creases, the A and B terms tend to zero, and C tends to 2 [34], resulting in a simplifiedrelationship between relative phase change and numerical slope,35∆ =4piδxλ∂w∂x(4.9)Rearranging equation 4.9 to solve for numerical slope ∂w∂x ,∂w∂x=∆λ4piδx(4.10)By adjusting the shear magnitude vertically, the optical setup can be arranged tocollect slope information with sensitivity in the y-direction,∂w∂y=∆λ4piδy(4.11)4.3 Unwrapping the MeasurementMacy offers an unwrapping algorithm that determines every 2pi phase jump across therelative phase distribution change ∆ [36].Figure 4.10: The wrapped measurement becoming unwrapped using the unwrapping al-gorithmSince the arctangent is bounded by a range from ±pi, we consider a full phase as givenby,φ‘(xj , y) = φ(xj , y) + 2cpi (4.12)where c is the number of phase jumps.36Figure 4.11: A graphical representation of the unwrapped data versus the measuredwrapped data4.4 Limitations of Classical ShearographyWhile shearography offers a robust interferometric measurement, it is still hindered by lim-itations prone to many interferometric methods. These issues include but are not limitedto: noise in the phase distribution, necessity to unwrap, speckle pattern decorrelation,accurate shearing vector determination, and areas of high density phase-jumps. Theselimitations plague the effectiveness of quantitative shearographic measurements. In orderto remedy these limitations, each must be investigated and understood.4.4.1 NoiseNoise in interferometric measurements is inherent. Poor modulation of phase-steppingduring the intensity collection can cause poor pixel visibility γ of the phase distributionφ. While there is an underlying average speckle intensity distribution I0, the modulationterm γ must receive enough interference modulation during phase-stepping [37]. Dueto the ever present noise in interferometric measurements, filtering is often required toaverage the data, and can lead to a loss of information.37Figure 4.12: Raw data shown to be often fraught with random speckular noise, a centerlineof data is plotted to emphasize the noise4.4.2 Fringe DensityWhen the observed relative phase distribution change is large, phase-jumps will accumulatein areas of low rotation. If the phase-jump frequency exceeds ±pi per every 2 pixels,unwrapping is difficult [36]. The maximum phase gradient can be resolved by adjustingthe shearing magnitude or changing the magnitude of out-of-plane deformation, but thetwo must be resolved in tandem to keep the relative phase change within tangible limits.Figure 4.13: Fringes displayed as a function of cos2, noting the high fringe density at thecenter which denotes the area of least rotation4.4.3 Accurate Shearing SizeFor calibrating the shearogram from radians to quantitative slope in millimeters, the shear-ing distance δx,y is needed in equations 4.10, 4.11. Often, this magnitude is determinedthrough analog methods, such as pixel measurement using a line vector between the two38sheared images. For measurements of high sensitivity (i.e. large shearing distances), thisanalog method is applicable as the shearing distance does not need to be accurate to thesub-pixel level. However, if the setup is adjusted for a less sensitive measurement, theshearing distance will be small, and may be much harder to determine using analog mea-surement. For these small distances, sub-pixel shearing determination is necessary, as thesensitivity vector will rely heavily on its direction and magnitude.Figure 4.14: While large shearings distances can be determined through vector measure-ment, small shearing distances cannot be as accurately determined4.4.4 Speckle DecorrelationAs with all interferometric techniques, shearography suffers severely from rigid body mo-tion [38]. The distribution of Gaussian illumination, the airy disk size, dictates the diam-eter of an observed speckle on the sensor,ds = 1.22λf (4.13)where ds is the diameter of the speckle on the sensor, f is the f-number of the aperture,and λ is the wavelength of light.When rigid body motion exceeds ds orthogonal to the line of sight, the speckle patternwill decorrelate and the secondary phase map after deformation φ‘ cannot be resolved forthe phase distribution change. This decorrelation can also be tied to the sensitivity of theshearing distance δx,y. If the surface rotation of the specimen is high, and the shearingdistance is also large (δx,y > 20 pixels), decorrelation will occur in areas of high rotation.This limits the maximum observed rotation greatly, making long-distance measurements( 0.5 mm) virtually impossible given a high sensitivity.4.5 Overcoming LimitationsThe limitations described above can have detrimental effects on the shearographic mea-surement, but through carefully selected optical arrangements and calculation techniques,can be overcome. The out-of-plane deflection required for determining the residual stress39in flat plates need to be large. Shearography, a method with sensitivity in fractions of awavelength, is often limited to very small out-of-plane deflections. To increase the robust-ness of the measurement technique, allowing the capability to record large out-of-planedeflection gradients, and preserve the integrity of the measured data when filtered, a se-ries of adjustments to the classical shearography method are detailed in the followingsubsections.4.5.1 Incremental Method vs. Absolute MethodWhen calculating the absolute phase distribution change, a fringe pattern appears as aresult of a large relative phase change. This method is simple, requiring only an initial andfinal phase distribution. This calculation technique is known as the ’absolute method’,denoting the absolute difference in phase distributions before and after a deformation hasbeen applied,∆abs = φn − φ0 (4.14)where φn is phase distribution after a deformation has been applied and φ0 is theinitial, undeformed phase distribution. This method is quick and convenient, but requiresa large shearing distance and, consequently due to increased sensitivity, a small out-of-plane deformation. The absolute method will yield many phase-jumps bounded between±pi.Figure 4.15: A literal representation of an absolute measurement, the final recorded phasedistribution at 15 steps subtracted by the initial phase distribution at step 1For the purpose of measuring large out-of-plane deformations necessary in 3-point-bending, a secondary method, known as the ’incremental method’, is implemented,∆inc =N∑n−1φn+1 − φn (4.15)Instead of taking two absolute measurements, the incremental method records a seriesof small deformations, none of which exceed phase distribution differences greater than thebounds of the arctangent function ±pi. Once all the small incremental phase distributionchanges have been calculated, they are summed up, resulting in a final phase distributionwhich represents the continuous gradient.40An example of the method for five incremental distances is given as,∆5 = 5∆4 + 4∆3 + 3∆2 + 2∆1where each n+1∆n = φn+1 − φnDue to the nature of small relative rotations never exceeding ±pi, the laser speckle pat-tern will never decorrelate, so long as the relative rotations are small between increments.This introduces the possibility to observe large deformations in pseudo real-time. Un-like the absolute method, which requires unwrapping, the incremental method is a directmeasure of the surface slope change.Figure 4.16: The literal representation of the incremental calculation, where each smallerphase distribution changes are summed together4.5.2 Preserving the Phase ChangeWhile the incremental method provides the possibility of observing large deformations atwavelength precision, another attractive advantage is that the true gradient is summedtogether directly without producing any 2pi phase jumps.Since each increment does not exceed the bounds necessary for fringe propagation,a series of ∆n = φn+1 − φn ⊆ ±pi, are summed together, resulting in the true phasedistribution.41Figure 4.17: Incremental versus absolute measurements made for the same out-of-planedisplacementsBased on the slope profile in figure 4.17, the issue of unwrapping arises when deter-mining the zero-datum of the measurement, which cannot be preserved using the absolutemethod. In the incremental method, no phase-jumps larger than ±pi occur, leading to acontinuous phase with a definite zero-datum.4.5.3 FilteringNoise affects interferometric measurements even in the most ideal environments. Due toinadequate phase modulation, camera sensor capabilities, or outside disturbance, noisecan limit the effectiveness of these high sensitivity measurements. In speckle pattern in-terferometry, noise will arise when not every pixel on the camera sensor is representedaccurately during the phase-shifting intensity collection. These locations may not modu-late, and incorrectly represent the phase distributions. While there may be many singlepixels which do not modulate adequately, the majority, with proper phase-shifting, willdisplay the relative phase change due to deformation. A medium-pass filter can be ap-plied to the phase distribution differences to help ’smooth’ noisy pixels. In the case ofan absolute measurement, fringes are more well-defined once a medium-pass filter hasbeen applied and unwrapping can become more precise. However, in instances when highfringe densities accumulate in areas of low-rotation (center of actuation, end supports),42Figure 4.18: A 3D surface plot comparison of the raw data (colored) and incrementallyfiltered data (black)the medium-pass filter size is limited to the closest set of fringes.For example, if a cluster of dense phase-jumps appear with a periodicity of 3-pixels ata location on the phase distribution difference and a 7 × 7 medium-pass filter is appliedto smooth the data of random noise, the filter may compromise the area where the phase-jumps are too close together, smoothing them together.For the case of the incremental method, if a medium-pass filter is applied to eachincremental phase distribution change ∆n, the data will be more naturally preserved, asthe phase distribution change is so small that no phase-jumps are produced. Additionally,given that many small increments have been filtered individually and summed together,the filtering will take less effect on the final phase distribution change ∆inc than if the finalphase distribution change was determined using the absolute method.43Figure 4.19: The transverse slope comparison between the raw data (blue) and incremen-tally filtered data (black)4.5.4 Optimized Optical GeometryTo ensure no relative phase-jumps appear in the incrementally collected phase changedistributions, the sensitivity of the measurement must be reduced. Typically, shearingdistances ranging from 20 to 30 pixels are suitable for an absolute measurement [34], as asingle measurement is providing sensitive data for a small deformation.For the incremental summation, the goal is to keep each relative phase differencebounded by ±pi, therefore smaller shearing distance are required. Depending on the sizeof each incremental deformation, shearing distances range from 2 to 8 pixels. The necessaryshearing distance and incremental deformation distances are inversely proportional. Fora relatively large incremental deformation, the shearing distance must be reduced greatly.If the incremental deformation is small, the shearing distance can be increased.Because these measurements deal with such a small region of allowable pixel shearingdistances, it is important to ensure that the direction of sensitivity is entirely along the axisof the desired measurement, as slight offsets along the adjacent axis may cause directionaldrift in the summed data, as the offset shearing magnitude will be comparable to theintended shearing direction and magnitude. Particularly for measuring transverse slopes,44the longitudinal bending is much larger than the desired Poisson’s effect across the width.If the shearing magnitude is offset some small amount from the transverse direction, thescale of the longitudinal bending will interfere with the width-wise measurement.4.5.5 Analyzing Shearing MagnitudeUnderstanding the applied shearing shearing magnitude is important to any shearographicmeasurement, as it gives a basis weight to the calibration between the measured radiansand quantifiable units, such as millimeters. The nature of the incremental method requiresa small shearing distance for each phase distribution, and any variance in the shearingdirection and magnitude will give rise to inaccuracies in the incrementally summed data.Classically, shearing distances were determined using a pixel-to-pixel measurement ofan image captured in white light by the shearography device. While this is effective formeasurements made with large shearing distances (∼ 30 pixels) in the desired direction ofsensitivity, for small shearing distances (∼ 6 pixels) this method is not sufficient.Instead of an analog measurement method, a robust 2D autocorrelation function wasimplemented, which has a mathematical functionality described by the Wiener-Khinchintheorem [39]. This autocorrelation is able to discern the shearing magnitude and directiongiven one of the intensity images collected during data acquisition. While human eyes maynot be able to determine the amount of shearing imposed between two overlapping specklepatterns, the autocorrelation function is able to determine the amount of shearing downto the sub-pixel range. Additionally, the function can also determine the direction of theshearing magnitude, making accurate alignment possible for small shearing distances.Figure 4.20: An autocorrelation function applied to a sheared random speckle pattern,the distance between the two peaks of correlation denotes twice the shearing distance4.5.6 Dual Laser ConfigurationShearography is known for its robust behavior due to the self-referencing nature of thelight interference. Because interference occurs and is captured directly before the sensor,rather than on the object, multiple illumination sources of the same nominal wavelengthmay be used for illumination.To improve illumination and increase the field-of-view (FOV), two separate lasers of the45same nominal wavelength were used, giving a much more even light distribution across theplate. For the purposes of keeping the airy disk size the same as the width of a pixel on thesensor, the f -number had to be adjusted to f = 11, which corresponds to a small aperturewith low light admittance. With only a single laser, the light distribution observed by thecamera was highly Gaussian. The center of the illumination had saturated pixels, whichthe outer edges tapered off in intensity. The Gaussian light distribution is not ideal, as itgives rise to varying modulation while phase-stepping.Figure 4.21: Comparing the resultant intensity distributions when using a single versusa dual laser illumination, the dual laser configuration on the right gives a more evenlyilluminated region of interestThe dual laser configuration gives a more even distribution, while not affecting thephase-stepping modulation, as each beam has the same nominal wavelength. In shearog-raphy this is made possible by the self-referencing nature of the light interference occurringwithin the shearing configuration.4.5.7 Remedial Adjustments: ConclusionsUsing an incrementally filtered summation acquisition and post-processing schematic, thedata collected by the shearography device can be continuous over large out-of-plane deflec-tions. Coupled with an accurately determined shearing vector and adequate light distri-bution, these adjustments allow the shearography measurements to be used in conjunctionwith the residual stress determination from measured transverse slope.The experimental investigation surrounding the efficacy of these adjustments are de-scribed in the following chapter.46Chapter 5Experimental Data and Results5.1 Procedure and Methods5.1.1 Plate Rolling and PreparationSix individual plates were compressively rolled using a bandsaw tensioning machine, allwith the same central roll-path. Five of the plates were rolled using increasing roll loads.The rolling loads for the experiments corresponded to 1500 lbf., 2500 lbf., 3500 lbf., 4500lbf., and 5500 lbf. Each plate was rolled a total of three times longitudinally to induceenough residual stress to produce synclastic behavior under 3-point-bending. The finalplate was kept unstressed to check against the theoretically unstressed slope component.Figure 5.1: The physical experimentation setupThe rolling process involved performing a single pass roll at a predefined pressure47down the center of each plate, then rotating and flipping the plate orientation, performinga second single pass roll, and then orientating the plate back to the original position for athird and final single-pass roll. This process can be seen in the figure below,Figure 5.2: For each roll pass, the plate was rotated 180° about the two principle axes,ensuring equal rolling was appliedAfter the plates were rolled, a hammering technique was used to ’level’ the bladesections. The leveling process involved striking the plates with a hammer at areas oflocalized surface height irregularity. This leveling allowed the plates to lay flat, effectivelyevenly distributing the residual stresses that were produced by the roll-tensioning.Figure 5.3: The rolling pressures applied to each plate, longitudinallyFinally, a coating of flat-white paint was applied to create an optically diffuse surface.This diffusive surface allowed for a more repeatable intensity modulation when phase-stepping during the shearography measurement.5.1.2 Shearographic ImplementationThe bandsaw blade plate was aligned horizontally to the shearographic interferometer, sothat the sensitivity direction of interest was along the y-axis. For that reason, a y-shearingmagnitude was applied. Because the relative rotation along the horizontal axis is much48larger than the Poisson’s effect across the transverse width under bending conditions, theshearing direction had to be aligned accurately to 90° about the horizontal axis. This wasaccomplished using the autocorrelation function described in chapter 4.Figure 5.4: The shearographic setup used in the following experimentsThe shearography setup used a shearing distance of δy = 6.12 pixels. This smallshearing distance, coupled with the small incremental out-of-plane actuation distancesmade certain that no incremental phase differences exceeded the arctangent ±pi bounds.Additionally, a polarizing filter was placed in front of the camera lens to increase theamount of pixel modulation during phase-stepping. The polarizing filter would only allowreflected waves of light that were in the same orientation thereby maximizing interferencemodulation.49Figure 5.5: A display of the program that controlled the acquisition of intensity valuesThe entire system was controlled with a graphical-user-interface which was coded tocalibrate the phase-stepping angles, capture intensities using a camera trigger betweenphase-steps, and control the actuation distance along with the number of incrementalsteps. The program was also able to give data analysis on the quality of the measurement interms of phase-angle and pixel modulation. Once all the incremental steps were collected,the program would then save the data for post processing.5.1.3 Three-Point BendingThe three-point bending of a simply supported plate was crucial to the investigation of theexperiment. By simply supporting the plate and controlling the actuation distances, thelongitudinal radius of curvature could be controlled and determined through the theoreticalplate bending equations.50Figure 5.6: A simply supported plateThe initial phase-map was collected while the plate was flat (no deflection) on thebending mechanism, each plate was then actuated 100 steps at incremental sizes of 0.005mm, totaling a final out-of-plane deformation of 0.5 mm. After each actuation step, thefour intensities were captured and the incremental phase maps could be calculated, incre-mental differences between phase maps could be determined, and then summed together.The measurement schematic can be seen in the figure below,Figure 5.7: Experimentation procedure detailed in incremental steps5.2 Optical ApparatusThe optical apparatus consisted of a shearing device confined within an aluminum frame,clad with a 3D printed shell to reduce the interference of outside light, as well as anactuation stage which held the plates in 3-point bending using magnetic mounting feet.Specific details and dimensions are detailed below.515.2.1 Interferometric SetupThe interferometric setup used a camera, two laser sources, a shearing cube, a piezoelectriccontrol system, a phase-stepping mirror, and a shearing mirror,Figure 5.8: The experimental shearographic setup, viewed from the top downCameraThe camera used was an Applied Vision GE680 monochrome CCD camera with a resolu-tion of 640 x 480 pixels. The camera collected the intensities at an average frame rate of13 frames-per-second with an exposure time of 0.075 seconds. The pixel sizes were 7.4 x7.4 µm. To optimize the airy disk speckle diameter to one pixel, the aperture was set tof = 11. Before any images were collected, the camera was allowed to warm for 10 minutesto full operating temperature at around 38°C. All intensities were captured using a Mono8format through a USB interface.Illumination SourcesTwo 50 mW red diode lasers were used for surface illumination. The nominal wavelengthof both lasers was 632 nm. Both lasers were aligned with the camera sensor in the verticalplane and angled towards the specimen. No additional lenses were used for collimation ordivergence because the open diodes were sufficient enough for adequate illumination. Anyinfluence of temperature gradients were considered negligible as the lasers were allowed towarm to operating temperature before any measurements were made.52Shearing DeviceThe shearing device, contained within an aluminum frame, consisted of a non-polarizingbeam-splitter, a phase-stepping first surface mirror, an angled first-surface mirror (y-shearing), a polarizing filter, and a 24v piezoelectric control system. Both mirrors weremounted on angle-adjustable mounts, which allowed simple alignment of the shearingdirection.The phase-stepping piezoelectric devices were run by a National Instruments NI USB-6009 multifunction I/O DAQ card, which took a digital command and fed a corresponding1.1v to 5v analog source into an amplifier capable of 24v output. The amplifier was at-tached to the piezoelectric devices connected in series which were mounted to the backof the phase-stepping mirror. Suitable voltages for angular intensity captures were deter-mined using an interference voltage scan calibration.Figure 5.9: The National Instruments wiring configuration used for phase-steppingThe beam-splitter and adjacent mirrors were aligned so that each mirror had thesame nominal distance from the respective faces on the beam-splitter. This alignmentwas crucial because different distances cause magnification effects between the two slighttransposed images. Each mirror was placed 5 mm from the cube. While the shearingmirror had a slight tilt to it, this tilt did not cause noticeable magnification effects as theangle was << 1°.535.2.2 Bending MechanismThe bending mechanism was assembled using aluminum stock, a high resolution linearactuator, high powered magnets, and 3D printed mounting feet. The assembly mimickedsimply supported conditions of 3-point-bending.Figure 5.10: A top down image of the 3-point bending setupActuation StageThe actuation stage was mounted to an optical table. It was long enough to span thelength of each bandsaw blade plate. By using strong magnetic mounting feet, the platecould be actuated from behind without the risk of becoming detached from the bendingstage.ActuatorThe actuator used in these experiments was a Newport CMA25PP, a micrometer drivensystem capable of 15µm bi-directional steps, as well as sub-micron incremental motion.The actuator was driven with a Newport Universal Motion Controller/Driver ESP100.This controller allowed interfacing between the GUI command window and the actuator.With a maximum stroke of 25 mm, the Newport actuator proved very useful for small,repeatable, incremental steps necessary for the long-range measurement technique.54Figure 5.11: The actuation stage, outlining the dimensions of the 3-point bendingFigure 5.12: The actuator dimensions used to apply 3-point bending555.3 Data Post Processing5.3.1 Filtering and Incremental SummationOnce each incremental phase-map had been recorded, the incremental phase differenceswere calculated and filtered using a 9 × 9 medium-pass filter. The bandwidth of thefiltering method was decided subjectively, the 9 × 9 medium-pass filter gave desirablysmooth gradients and limited the time required for post-processing. After filtering, eachincremental phase distribution change was summed to give a continuous gradient of thetransverse slope.The summed slopes could then be viewed as the continuous incremental summation,or with a fringe representation by applying a 2pi wrapping algorithm. The phase-jumps,while not particularly useful for quantifiable data, gave a nice representation qualitatively.5.3.2 Data CalibrationAfter obtaining the continuous gradient of transverse slope, the shearing size was deter-mined by the autocorrelation function to be 6.12 pixels, which was then used to calibratethe data using the equations described in chapter 4. The pixels to mm calibration termwas 0.4507 mm/p, which was applied to the shearing magnitude within the calibrationcalculation.∂w∂y i=∆iλ4piδy=0.000 632 mm× ∆i4pi× 0.4507 mm/p× 6.12 p =mmmm(5.1)where i represents each incremental summation.Likewise, the pixel width and height of the measurements were also calibrated torepresent their respective sizes in millimeters using an interpolation function.5.3.3 Central Slope ExtractionPoisson’s effect is most influential at the location of bending across the transverse width.Therefore, the center of the shearographic measurement was isolated and used for com-parison rather than the whole data field,56Figure 5.13: Isolating the measured transverse slope from the incrementally filtered sum-mation, based on the center of actuationOnce the central slope was isolated, the theoretically unstressed slope component wassubtracted, resulting in the measured transverse slope component due entirely to residualstresses.∑ndwndy=∂w∂y− dw0dyUsing the equations determined by Conway and Nickola, the theoretically unstressedcomponent of slope could be generated using plate geometry and material properties [4, 27].Finally, the least-squares best fit model was applied to the measured transverse stressedslope component to determine the magnitude of compressive stress due to cold-rolling.5.4 Results: Plate ComparisonEach plate was deformed incremental distances of 0.005 mm for 100 steps, totaling amaximum deflection of 0.5 mm. The results from each plate are displayed below. Theorientation of the data results are given by,57Figure 5.14: The plate orientationFigure 5.15: Measured slope visualizedResults presented are measured stressed slope, the components of transverse sloperelated to stress-free deflection, magnitudes of central residual stress σ0, and measurementstability over the incremental actuation range.585.4.1 Measured SlopesThe measured slopes were taken directly from the center of each incrementally filteredsummation, shown in the following figure: ∂w∂yFigure 5.16: Each individual transverse slope plotted, noting the increasing slope giventhe same bending deflection as a result of residual stressesThe measured transverse slopes, taken from the actuation center of the shearographymeasurement, demonstrate how the magnitude of transverse slope changes with differentmagnitudes of residual stress. From figure 5.16 it is evident that as the roll-tensioningload is increased, the effect of the stressed component of slope gets larger, out-weighingthe unstressed component of transverse slope.For the 0 lbf. and 1500 lbf. plates, there is little difference between their respectiveslopes, indicating that the roll-tensioning had little effect on the anticlastic behavior.This may be due to a lack of plastic deformation during the rolling process, as plasticdeformation along the roll path leads to the residual stresses.At 2500 lbf. and above, the effects of roll-tensioning become more pronounced, re-sulting in increased magnitudes of the transverse slopes. It should also be noted that thequality of measurement improves for larger magnitudes of slope distributions. It can alsobe noted that for larger magnitudes of transverse slope, the data becomes smoother. Forlow stress plates, Poisson’s effect produces little slope action under bending conditions.As the stress increases, a larger transverse slope is produced, and the signal obtained bythe shearographic measurement is larger, leading to more well defined data.Finally, the theoretically unstressed transverse slope is observed to follow a linear59transverse slope. This contradicts the measurement data, as the measured unstressed datashows nonlinearity, and appears to have influence of residual stresses. Therefore, whencompared with the least-squares fit, the unstressed plate appears to have some magnitudeof residual stress. A further discussion on this conflict is discussed in chapter 6.605.4.2 DeflectionsThe respective deflections were calculated using a cumulative trapezoidal integration: w =∫∂w∂y dyFigure 5.17: A numerical integration of the slopes give a more physical representation ofthe transverse deflection, noting that plates without residual stresses start out in anticlasticbending, but become more synclastic as stress is introducedThe deflection data provides a more physical meaning to the slope data from the light-gap method perspective. As the rolling pressures increase, the plates transition from ananticlastic behavior to a synclastic behavior, increasing in magnitude. All measured datasets show nonlinear transverse deflection behavior, as expected when considering Poisson’seffect on simply supported bending.The resolution of the measurement provides accurate transverse deflection detail downto the micron scale. This is a distinct advantage of light interference measurement methodsand would be extremely hard to replicate using surface probing. Additionally, the datapresented accounts for a vector of values taken from a whole-field measurement, givingmany data points in a single incremental measurement.The theoretically unstressed deflection is represented as a ’pseudo-cylindrical’ shapewith no nonlinearity. This is to be expected when numerically integrating the linear slopein figure 5.16. Based on the measurement data, it is observed that even at these smalldeflections, the resultant transverse slope and deflection are nonlinear.615.4.3 Stressed SlopesThe stressed slope components were determined by subtracting the component of un-stressed slope from each measured slope:∑ dwndy =∂w∂y − dw0dyFigure 5.18: The stressed components of the measured slopes, detailed with the centralcompressive stress magnitudes found for eachThe transverse stressed slope is the component of slope which influences the transversebending characteristics. By removing the inferred unstressed theoretical slope from themeasurement data, the stressed transverse slope component can be interpreted in a moreintuitive fashion.As stress is added to the plates, the component of transverse slope due to stressincreases from an initially ’unstressed’ state. The addition of stress appears to rotate andincrease the magnitude of the transverse stressed slope component in a nonlinear manner.The measured unstressed plate had a compressive stress determined to be σ0 = 90 MPa,much larger than the expected zero-stress state. From there, the compressive stress mag-nitudes associated with each plate increase as expected, reaching σ0 = 338 MPa for the5500 lbf. plate.62Stress Magnitude vs. Incremental DistanceTo check the stability of the measurement, the stress at each incremental actuation distancewas calculated,Figure 5.19: The stress estimation versus the actuation distance for each plate, noting areasonably stable estimation convergence as the deflection gets larger for all plate rollingmagnitudesFigure 5.19 shows that even for low incremental distances, the best-fit stress magni-tudes were found to converge quite quickly and were stable throughout the incrementaldistances. This is a promising realization, as shorter actuation distances could be used toprovide the same results as longer, more lengthy measurements.This is also indicative of consistent slope measurements between incremental distances,one of the advantages of the incremental summation method. The relative slope changebetween two increments should be consistent throughout the entire measurement, regard-less of the absolute actuation distance. This consistency can be inferred from the resultsin figure 5.19.63Estimated Stress vs. Rolling PressureFigure 5.20: The stress magnitudes determined for each plate plotted against the rollingpressures appliedThe determined residual stress magnitude versus the rolling pressures shows the transitionfrom anticlastic to synclastic bending behavior between 2500 lbf. and 3500 lbf. At thelower rolling pressures, the stress magnitude remains nearly constant, but then experiencesa large, linear increase after 3500 lbf.64Chapter 6Discussion6.1 HighlightsThe shearographic residual stress investigation has many advantageous features when com-pared to the classical light-gap method. Most notably a robust, inexpensive configurationwith a full-field wavelength-resolution measurement, the method proved practical whendetermining residual stresses that result from roll-tensioning.6.1.1 Robust Slope MeasurementThe system, consisting of two inexpensive laser diodes, a camera, a beam splitting cube,and two adjacent mirrors, provided excellent slope measurement results. By making anincremental measurement, relatively long distance deformations were able to be observed.The measured slope data could then be related back to the component of slope relatedto stress, thereby determining the magnitude of residual stress. The setup provided fast,repeatable measurements and could quickly converge on a stress magnitude over a largeseries of incremental deflections.Influence of Residual StressThe influence of residual stresses could be seen in the results. As the rolling pressure wasincreased per plate, the effects of the stress increased the behavior of synclastic deforma-tion. It was observed for the 1500 lbf. rolling pressure, there was little difference betweenthe unstressed plate and the rolled plate. However, for higher rolling pressures, the effectsof stress become more evident, leading to greater transverse slopes and, consequently,greater synclastic deformations. Additionally, a roll-pressure gradient could be seen forincreasing rolling loads. Between 2500 lbf. and 3500 lbf. there is a shift between anticlas-tic behavior and synclastic behavior. This implies the idea that there may be a thresholdof rolling pressure which, when paired with the geometry and material properties of plate,result in a synclastic behavior.656.1.2 High ResolutionWhen the measured slope data was numerically integrated to represent displacement, thetransverse deflection ranged [−0.017 mm, 0.013 mm]. Integration results in an inherentsmoothing of data, therefore the resultant deflection data was clear and well represented.Even with a modern micrometer probe, the deflection data may not have these smoothcharacteristics under such a small measurement range.Importantly, the deflection of interest for the Foschi light-gap analysis was the relativedeflection, not the absolute deflection. The shearographic method records a measurementof relative surface rotation, and once integrated, results in a relative surface deflection.With a standard probing method, the transverse surface would be displaced in 3-pointbending as both a result of Poisson’s effect and the longitudinal bending, requiring thezero-datum to be reestablished. This introduces uncertainties when compared to a direct,relative slope-change measurement.6.1.3 Incremental FilteringThe incremental filtering method was essential for successfully measuring long-range out-of-plane slopes. By collecting many small slopes in between actuation distances and sum-ming them in sequence, the resultant full-field measurement could be seen as a ’real-time’display. This real-time display could be monitored for repetition between incrementalmeasurements, as well as give a series of central compressive stress terms over the entirelength of the measurement.The incremental filtering also provided a well-preserved data set. Because many smalltransverse slopes of the same nominal magnitude were observed repetitively, the filteringwould only adjust small magnitudes of each increment. When compared to the overallsummation of data, the filtering did not cause any compensation issues.6.2 LimitationsWhile the measurement technique was effective, there were still limitations. One of themain limitations was comparing the unstressed measured slope to the unstressed theoret-ical slope. The details surrounding this issue are outlined in the following subsections.6.2.1 Theoretical LimitationsWhen generating the theoretically unstressed deflection using Nickola and Conway’s equa-tions [27], the nonlinearity of the central transverse deflection disappears under very smallbending conditions. It can be seen in the unstressed plate measurement that this is not thecase in reality. Even with no known residual stresses, the transverse slope and integrateddeflection show nonlinear bending behaviors, contradictory to the theoretically generatedprofiles. This poses a discrepancy when applying the least-squares best fit model to themeasured data. If the unstressed plate has a measured transverse slope profile different66from that generated by the theoretical equations, the least-squares best fit will find a stressvalue larger than the realistic magnitude.Figure 6.1: An unstressed, flat plate with an applied bending moment, resulting in anti-clastic bending behavior across the transverse widthThe problem has been investigated by Bellow in 1965 [40]. Bellow describes the trans-verse deflection w under 3-point-bending conditions as non-dimensional,wt= A¯ coshαy cosαy + B¯ sinhαy sinαy (6.1)where,α = 4√3(1− v2)R2Lt2(6.2)A¯ =−vα2RLtsinh(αB/2) cos(αB/2)− cosh(αB/2) sin(αB/2)sinh(αB) + sin(αB)(6.3)B¯ =−vα2RLtsinh(αB/2) cos(αB/2) + cosh(αB/2) sin(αB/2)sinh(αB) + sin(αB)(6.4)and B is the the transverse width of the plate, t is the thickness of the plate, w is thetransverse deflection of the unstressed plate, y ranges from 0 mm at the center of the widthto ±B2 at the outer free edges. The additional terms associated with α are v Poisson’sratio and RL the radius of longitudinal curvature.It should be noted that Bellow’s equations are exactly that of Nickola and Conway’spresented in Chapter 3, part of Foschi’s adaptation. The terms A¯, B¯ correspond to γ1,2,and describe the same transverse deflection for an unstressed plate in 3-point-bending.Bellow notes in his investigation that αB/2 is directly proportional to the dimensionless67ratio β,β =√B2/RLt (6.5)This ratio β is considered the ’ideal ratio’ in terms of theoretical transverse deflec-tion. When solving equation 6.1 numerically, it is shown that if β < 1.26, the transversedeflection is described by an arc of radius RL/v, a problem discussed by Saint-Venant [41],Figure 6.2: The transverse deflected shape and slope as generated by the theoreticalequations given the experimental plate properties and dimensionsIf, however, the ratio exceeds β > 1.26, the transverse deflection becomes nonlinear.This is the expected and observed behavior as shown by the experimental data presentedfor the unstressed plate in chapter 5.For the experimental investigation presented in this thesis, the values associated withthe width, radius of longitudinal curvature, and thickness are 130 mm, 51 600 mm, and1.6 mm, respectively. This yields a ratio of,β =√(130 mm)251 600 mm× 1.65 mm = 0.46consequently falling into the region of a pseudo-cylindrical arc described by RL/v. Thistheoretical evaluation of the unstressed deflection component is not suitable for accuratetransverse slope estimation. Because the interferometric measurement has such a high res-olution, small slope changes are measured accurately but the maximum deflection rangeis limited to very small distances. Small actuation distances in the 3-point-bending con-figuration result in very large radii of curvature, all of which are theoretically determinedas being pseudo-cylindrical by the Bellow and Nickola equations.By adjusting either the radius of longitudinal curvature RL, the transverse width ofthe plate B, or the thickness of the plate t, the ideal ratio region can be determined for aplate-by-plate case. By fixing two of the values, the third can be plotted against the ratio.Therefore, this phenomenon can be seen as each of the variables are fixed to a valuewhich fits within the appropriate ratio range. Once these values are adjusted, the trans-68Figure 6.3: Given the fixed width and thickness, the longitudinal radius of curvature isvaried in value to obtain a region of an appropriate ratioverse deflection and slope profiles become nonlinear.Figure 6.6: The transverse deflection and slope are plotted given the original width andthickness, with a very small radius of curvature69Figure 6.4: Given the fixed radius of curvature and width, the thickness of the plate isvaried in value to obtain a region of an appropriate ratioFigure 6.7: The transverse deflection and slope are plotted given the original width andradius of curvature, with a reduced thickness value70Figure 6.5: Given the fixed radius of curvature and thickness, the width of the plate isvaried in value to obtain a region of an appropriate ratioFigure 6.8: The transverse deflection and slope are plotted given the original radius ofcurvature and thickness, along with an increased widthAs these adjusted values are physically unfeasible, the theoretical deflection that isgenerated will always have pseudo-cylindrical profile, leading to a linear slope. For thisreason, the theoretical unstressed slope generation is a limiting factor when analyzing71the central compressive stress magnitudes, as it does not accurately estimate the trueunstressed slope profile seen in the measurements. For accurate estimation, the deflectionwould need to increase, the thickness would need to decrease, the plate would need tohave a very large width, or, more easily, the unstressed plate slope data could be taken asthe zero-slope profile.Unknown Manufacturing StressesA secondary limitation is an unknown residual stress profile due to manufacturing pro-cesses. To produce sheet steel (as is used for bandsaw blades), the metal undergoes aseries of melting and forming procedures before it is shipped to a mill. All stages of man-ufacturing introduce some unknown amount of residual stress to a component, and it’sthis uncertainty which makes it difficult to predict the stress due to roll-tensioning.Figure 6.9: The unpredictable stresses introduced by the various manufacturing processesThe production of flat sheet steel involves melting the steel down, pouring the moltensteel into a rectangular mold where it is cooled into a block, chemically cleaned, pressedthrough a rolling machine for many passes, and finally annealed using heat. All of thesesteps could introduce any number of unpredictable residual stresses. For this reason,offsets from the theoretically generated unstressed profile could be due to residual stressesintroduced before the plate has undergone tensioning.FEA ComparisonFinally, to check the limitations of the theoretical equations against a finite element anal-ysis (FEA), a plate in 3-point-bending was simulated. The dimensions and propertieswere kept equal to those of the bandsaw plate sections measured in the experiments. Ifthe theoretical equations presented by Bellows and Nickola were accurate, then the FEAcomparison would yield a cylindrical transverse deflection and a linear slope.For the FEA simulation, the plate was initially flat between the two mounting feetidentical to those used in the experiments. The plate was then actuated a distance of ∆z =0.5 mm, and the central deflection across the transverse width was recorded. The analysiswas predefined as nonlinear deformation so that Poisson’s effect would be produced. Thetransverse FEA deflection data was then moved into a program which would calculatethe numerical derivative and compare the transverse slope profile to those made by themeasurement on the unstressed plate.72Figure 6.10: The whole-field FEA deflection with sensitivity in the vertical axis (y-direction) as a result of the actuationFigure 6.11: The central deflection FEA result simulated along a single transverse path,the data displayed in the following figures relates to this vectorThe FEA results show a nonlinear deflection at the center of the plate, which agreeswith the measured data. However, there is a magnitude offset between the FEA analysisand the measured data. In fact, the FEA, while nonlinear, coincides with the magnitudesgenerated by the theoretical equations. This is indicative of some residual stresses withinthe plate from an unknown source before rolling is applied.73Figure 6.12: The FEA result for the central transverse deflection of the plate comparedto the theoretically generated central deflection and the numerically integrated centraldeflection from the measured dataFigure 6.13: The FEA result for the numerically derived slope compared to the theoreti-cally generated transverse slope and the measured transverse slope74Chapter 7ConclusionIn this thesis, the necessary methods and procedures for measuring transverse slope oftensioned flat plates were presented. By modifying the shearographic calculation methodto an incrementally filtered summation, and tuning the optical arrangement, the full-fieldcontinuous slopes were measured over a deflection distance of 0.5 mm. From the measuredslope data, a best-fit model was able to determine the magnitude of the residual stress usingan a-priori assumption of the stress distribution profile. The magnitudes of the measuredslope profiles were seen to be directly affected by the addition of residual stressed impartedduring the roll-tensioning process. By increasing the rolling pressure, the profile of thetransverse slope was seen to tend from anticlastic behavior to highly synclastic behavior.These same results can be seen when taking the numerical integration of the measuredslope data.It may also be determined that for a small longitudinal deflection distance, the classicaltheoretical equations describing anticlastic plate bending are not accurate. This resultcan be seen when comparing the theoretical unstressed transverse deflection profile tothe finite element analysis results, and the measured transverse slope for the unstressedplate. Further investigations into the anticlastic bending behavior of plates at small radiiof longitudinal curvature should be analyzed for a more inclusive model.In conclusion, the shearographic method offers wavelength level precision even whenmeasuring macro level deflections. Furthermore, the magnitudes of the induced stressprofiles are able to be determined by checking the measured slope data against theory.7.1 Further Works7.1.1 Multiple Roll PassesAs many saw filers use more than a single pass roll to appropriately tension a bandsawblade, a further investigation into a series of rolls of varying rolling pressure would beadvantageous. The suggestion would be to further expand the Fourier cosine series de-scribed in chapter 3 to include more compressive pulses at different locations across theplate width. The assumption would still need to be symmetrical, as asymmetry wouldn’t75allow the stress profile to be determined. This modification would further allow complexstress profiles to be analyzed given the number of rolls, locations, rolling pressures.Figure 7.1: A figure detailing three rolls made across the width of the plate at varyingrolling pressures, a common characteristic of roll-tensioning7.1.2 Comprehensive FEA InvestigationA more complete FEA investigation should be performed on the plate deflection after therolling has been applied. The investigation would give more insight into the accuracy ofthe theoretical equations, as well as give a numerical, repeatable check against the slopesmeasured using the shearographic method.7.1.3 Industrial PrototypeWhile the prototype created for the experiments conducted in this thesis was appropriatefor laboratory investigations, a more robust prototype would need to be constructed formill use.With rigidity and repeatability in mind, the prototype would need a stiff bandsawfeeding mechanism which would move sections of the plate longitudinally into the field76of view. The measurement device would need to be housed in a stable, closed unit withpredefined distances and component orientations. Vibration damping would be necessary.77References[1] M. Higgs, “Economic advantages of saw management,” Forest Industries, vol. 166,no. 5, pp. 17–19, 1989.[2] P. Lister, “The Effectiveness of the Light-Gap Method for Monitoring Saw Tension-ing,” Master’s thesis, University of British Columbia, Vancouver, Canada, 1989.[3] L. Zhu, Y. Wang, N. Xu, S. Wu, M. Dong, and L. Yang, “Real-time monitoring ofphase maps of digital shearography,” Optical Engineering, vol. 52, no. 10, pp. 1 – 9,2013.[4] R. O. Foschi, “The light-gap technique as a tool for measuring residual stresses inbandsaw blades,” Wood Science and Technology, vol. 9, no. 4, pp. 243–255, 1975.[5] E. Duncan, “2019 fii key forest sector data and stats,” Forestry Innovation Investment,2019.[6] H. Hallock, “A mathematical analysis of the effect of kerf width on lumber yield fromsmall logs,” USDA Forest Service, vol. 2254, 1962.[7] “Circular saw blades.” http://circularsawblade.net/thin-kerf.[8] H. Sugihara, “Influence of temperature on the stability of saw blades,” IUFRO Proc,vol. 3, no. 41, 1965.[9] H. Sugihara, “Forces acting on a bandsaw,” Jap. Wood. Ind., vol. 8, no. 5, pp. 38–45,1953.[10] I. I. Trubnikov, “Stresses in bandsaw blades subjected to surface tensioning,” Lens.Zh., vol. 8, pp. 104–108, 1965a.[11] J. Umetsu, M. Noguchi, and I. Matsumoto, “A novel technique for tensioning circularsaws using shot peening,” Mokuzai Gakkaishi = Journal of the Japan Wood ResearchSociety, vol. 40, no. 1, pp. 14–19, 1994.[12] A. G. Ulsoy, C. D. Mote, and R. Szymni, “Principal developments in band sawvibration and stability research,” Holz als Roh- und Werkstoff, vol. 36, no. 7, pp. 273–280, 1978.78[13] H. Huber, “Straightening and tensioning of circular saws by means of laser,” Proceed-ings of the Tength International Wood Machining Seminar, 1991.[14] G. S. Schajer, “Understanding saw tensioning,” Holz als Roh- und Werkstoff, vol. 42,no. 11, p. 425–430, 1984.[15] G. S. Schajer, “Measurement of non-uniform residual stresses using the hole-drillingmethod,” vol. 110, no. 4.[16] A. Miks and J. Novak, “Interferometric method for deformation measurement of struc-tures in industry,” in Photonics, Devices, and Systems II (M. Hrabovsky, D. Sender-akova, and P. Tomanek, eds.), vol. 5036, pp. 20–24, International Society for Opticsand Photonics, SPIE, 2003.[17] D. Dugale, “Theory of circular saw tensioning,” International Journal of ProductionResearch, vol. 4, no. 3, pp. 237–248, 1965.[18] P. Hariharan and P. Hariharan, “13 - holographic and speckle interferometry,” inBasics of Interferometry (Second Edition) (P. Hariharan and P. Hariharan, eds.),pp. 111 – 120, Burlington: Academic Press, second edition ed., 2007.[19] G. S. Schajer, Practical residual stress measurement methods. Wiley-Blackwell, 2013.[20] B. Liscic, H. M. Tensi, and W. Luty, Theory and technology of quenching. Springer-Verlag, 1992.[21] B. F. Lehmann and S. G. Hutton, “The mechanics of bandsaw cutting,” Holz als Roh-und Werkstoff, vol. 54, no. 6, p. 423–428, 1996.[22] H. Sugihara, “Theory on running stability of bandsaw blades,” Proc. Wood Mach.Sem. Un. Cal., pp. 99–110, 1977.[23] E. Barz, “Comparative investigations on tensioning circular saw blades with machinesand with hammers.,” Holz Roh- u. Werkstoff., vol. 21, no. 4, pp. 135–44, 1963.[24] T. Aoyama, “Tensioning of bandsaw blades by rolls, i. calculation of crown. ii. calcu-lation of tension,” Jap. Wood Res. Soc., vol. 16, no. 8, pp. 370–381, 1970.[25] T. Aoyama, “Tensioning of bandsaw blades by rolls, iii. effect of saw blade thickness.iv. effect of saw blade width,” Jap. Wood Res. Soc., vol. 17, no. 5, pp. 188–202, 1971.[26] T. Aoyama, “Tensioning of bandsaw blades by rolls, v. the effects of rolling repetition,saw blade hardness, and rolls with different sectional diameters,” Jap. Wood Res. Soc.,vol. 20, no. 11, pp. 523–527, 1974.[27] H. D. Conway and W. E. Nickola, “Anticlastic action of flat sheets in bending,” Exp.Mech., vol. 5, no. 4, pp. 115–119, 1965.79[28] N. J. Rendler and I. Vigness, “Hole-drilling strain-gage method of measuring residualstresses,” Exp. Mech., vol. 6, no. 12, pp. 577–586, 1966.[29] ASTM, Determining Residual Stresses by the Hole-Drilling Strain-Gage Method.Standard Test Method E837-08. American Society for Testing and Materials, WestConshohocken, PA, 2008.[30] G. S. Schajer, “Application of finite element calculations to residual stress measure-ments,” Journal of Engineering Materials and Technology, vol. 103, no. 2, pp. 157–163, 1981.[31] G. S. Schajer and L. Yang, “Residual-stress measurement in othotropic materialsusing the hole-drilling method,” Experimental Mechanics, vol. 34, no. 4, pp. 217–236,1994.[32] G. S. Schajer, Practical Residual Stress Measurement Methods. Wiley, 2013.[33] D. Gabor, “Holography, past, present and future,” SPIE Proceedings, Developmentsin Holography, vol. 25, pp. 129–136, 1971.[34] Y. Hung and H. Ho, “Shearography: An optical measurement technique and appli-cations,” Materials Science and Engineering, vol. 49, pp. 61–87, 2005.[35] Y. Hung, K. W. Long, and J. W. Wang, “Measurement of residual stress by phaseshift shearography,” Optical Lasers Engineering, vol. 27, pp. 61–73, 1997.[36] W. M. Macy, “Two-dimensional fringe-pattern analysis,” Applied Optics, vol. 22,no. 23, pp. 3998–3901, 1983.[37] M. O. Petersen, “Decorrelation and fringe visibility: on the limiting behaviour of var-ious electronic speckle pattern correlation interferometers,” J. Opt. Soc. Am., vol. 8,no. 7, pp. 1082–1089, 1991.[38] H. T. Yura, S. G. Hanson, and T. P. Grum, “Speckle statistics and interferometricdecorrelation effects in complex abcd optical systems,” J. Opt. Soc. Am., vol. 10,no. 2, pp. 316–323, 1993.[39] N. Wiener, “Generalized harmonic analysis,” Acta Mathematica, vol. 55, pp. 117–258,1930.[40] D. G. Bellow, G. Ford, and J. Kennedy, “Anticlastic behavior of flat plates,” Exp.Mech., vol. 5, pp. 227–232, 1965.[41] L. M. H. Navier, “Resume des lecons, troisieme edition avs des notes et des appendiciespar m. barre de saint-venant,” Paris, pp. 32–36, 1864.80

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