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Residual stress measurement using cross-slitting and ESPI An, Yuntao 2008

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RESIDUAL STRESS MEASUREMENT USING CROSS-SLITTING AND ESPI by YU1JTAO AN B.Sc., Beijing University of Technology, 2006 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2008 © Yuntao An, 2008 Abstract Residual stresses are “locked-in” within a material, and exist without any external loads. Such stresses are developed during most common manufacturing processes, for example welding, cold working and grinding. These “hidden” stresses can be quite large, and can have profound effects on engineering properties, notably fatigue life and dimensional stability. To obtain reliable and accurate residual stress measurements for uniform and non-uniform stress states, a novel and practical method using crossing-slitting and ESPI is presented here. Cross-slitting releases all three in-plane stress components and leaves nearby deformation areas intact. The ESPI (Electronic Speckle Pattern Interferometry) technique gives an attractive tool for practical use, because measurements provide a large quantity of useful data, require little initial setup and can be completed rapidly and at low per-measurement cost. A new ESPI setup consisting of shutter and double-mirror device is designed to achieve dual-axis measurements to balance the measurement sensitivities of all in-plane stress components. To evaluate data quality, a pixel quality control and correction procedure is also applied. This helps to locate bad data pixels and provides opportunities to correct them. The measurement results show that this procedure plays an important role for the success of residual stress evaluation. Based on the observed displacement data and finite element calculated calibration data, an inverse computation method is developed to recover the residual stresses in a material for both uniform and non-uniform cases. By combining cross-slitting and ESPI, more reliable results for the three in-plane residual stress components can be obtained. 11 Table of contents Abstract.ii Table of contents iii List of tables v List of figures vi Acknowledgements viii Chapter 1 — Introduction 1 1.1 Introduction to Residual Stresses 1 1.1.1 Motivation 1 1.1.2 Formation and Effects of Residual Stresses 1 1.2 Techniques and Challenges of Residual Stresses Measurement 3 1.2.1 Non-destructive Methods 3 1.2.2 Destructive Methods 4 1.3 Research Objective and Proposed Method 8 1.4 Overview 10 Chapter 2 — Solution of Residual Stresses Calculation for Cross-slitting Method 12 2.1 Introduction to Cross-slitting Method 12 2.2 Displacement and Model Relationship 13 2.2.1 Displacement and Residual Stress States 13 2.2.2 Displacement and Rigid Body Motions 14 2.2.3 Additional Motions from Air Turbulence 15 2.3 Residual Stress Calculation for Single Depth Case 15 2.3.1 Matrix Equation for Single Depth Case 16 2.3.2 Least Square Approximation 18 2.4 Residual Stresses Calculation for Incremental Depth Case 19 2.4.1 General Displacement and Model Relationship 19 2.4.2 Regularization Approach for Incremental Depth Case 22 2.5 Finite Element Calculation 24 2.6 Discussion 25 Chapter 3 — Electronic Speckle Pattern Interferometry (ESPI) 26 3.1 Fundamental of Laser Interference 26 3.2 Typical ESPI Setup 28 3.3 The Phase-shifting Method 30 3.4 The Unwrapping of the Phase Difference Change 32 111 Chapter 4— Modified ESPI Technique and Application 35 4.1 Limitation of Single-axis ESPI System 35 4.2 Dual-axis ESPI System 36 4.2.1 Dual-axis ESPI Setup 36 4.2.2 Shutter and Double-mirror Devices 38 4.2.3 Two Directional Measurements of Dual-axis ESPI System 38 4.3 ESPI Challenges and Improvements 40 4.3.1 Modulation during Phase-stepping 41 4.3.2 Speckle Size 42 4.3.3 ESPI Improvements 43 4.4 Pixel Quality Control and Correction Procedure 45 4.4.1 Pixel Quality Examination 45 Chapter 5 — Experimental Validation 49 5.1 Cross-slitting ESPI Experimental Setup 49 5.2 Measurement of a Known Stress Field 50 5.3 Experimental Measurement Result 53 5.3.1 Single Depth (Uniform Stress State) Measurement 54 5.3.2 Incremental Depth (Non-uniform Stress State) Measurements 59 5.4 Discussion 64 Chapter 6 — Conclusion 66 6.1 Contribution 66 6.2 Remaining Challenges 68 6.3 Future Work and Recommendations 69 References 71 iv List of tables Table 1 — Experimental equipments and parameters 50 Table 2 — Incremental stresses calculated from bending test 53 Table 3 — Measurement results from single depth case 54 V List of figures Figure 1 — Toughened glass .2 Figure 2 — Distorted cargo ramp 3 Figure 3 — Slitting method 4 Figure 4— Plane and side views of hole-drilling and ring-core method 5 Figure 5 — Existing ESPI arrangement 6 Figure 6— Fringe pattern by ESPI hole-drilling method 8 Figure 7 — Cross-slitting method 9 Figure 8 — Cross-slitting under in-plane stress state 12 Figure 9 — In-plane stress state decompositions 13 Figure 10— Three in-plane rigid body motions 14 Figure 11 — Uniform residual stress state 16 Figure 12 — Unit pulse functions used for residual stress calculation 20 Figure 13 — Physical interpretation of matrix coefficients for slitting method 21 Figure 14 — Detailed finite element mesh pattern for kernel matrix coefficients 25 Figure 15 — Interference of light waves 26 Figure 16 — Schematic arrangement of in-plane ESPI measurement 28 Figure 17 — Sensitivity vector 29 Figure 18 — Intensity change trend by phase stepping 31 Figure 19— Sets of images obtained by 5-step phase stepping 32 Figure 20— Phase map obtained from single-slitting 34 Figure 21 — Synthetic fringe patterns 35 vi Figure 22 — Newly designed dual-axis ESPI system 37 Figure 23 — Key components for new ESPI system 38 Figure 24 — Double-mirror assembly and sensitivity vectors 39 Figure 25 — High and low modulated signals 42 Figure 26 — Effect of speckle size 43 Figure 27 — Telecentric lens 44 Figure 28 — Evaluation of variation of pixels 47 Figure 29— Experimental setup for dual-axis ESPI measurement 49 Figure 30— Four-point bending experiment 51 Figure 31 — Four-point bending result 52 Figure 32 — Pixel quality contorl and correction for lower mirror fringe in cross-slitting case 56 Figure 33 — Fringe pattern and pixel quality image for single-slitting 57 Figure 34— Fringe pattern from dual-axis measurement 58 Figure 35 — Raw incremental fringe pattern using lower mirror 59 Figure 36 — Residual stress calculation from ESPI and cross-slitting measurement 61 Figure 37 — Residual stress a profile from ESPI/Cross-slitting measurement and bending test 63 Figure 38 — The wrongly unwrapped 4th incremental phase map without correcting the bad pixels 63 Figure 39 — o stress profiles from dual-axis measurement and single-axis measurement 64 vii Acknowledgements I sincerely thank my supervisor Dr. Gary Schajer for his guidance and support throughout my research work at UBC. I had a wonderful educational experience thanks to his wisdom, guidance and generosity. I also give my thanks to all the staffs in the department of Mechanical Engineering who gave me invaluable support throughout the duration of this research. I am truly grateful for my parents’ constant support throughout all my life. Their pure and kind personalities taught me the attitude towards life and inspired me to chase my dreams and never give up easily. Their constant encouragements always give me the confidence to achieve more accomplishments. VIII Chapter 1 — Introduction 1.1 Introduction to Residual Stresses 1.1.1 Motivation Residual stresses are stresses locked-in a material, and may exist without any external loads or thermal gradients. Most manufacturing operations, such as turning, grinding, heat treating, surface hardening and welding, can set up these stresses [1]. Residual stresses can be large, and can play a critical role on material behaviour, notably fatigue life and dimensional stability. Desired formation of residual stresses can enhance the mechanical performance of material, such as pre-stressed concrete, which performs much better under tensile loads. Undesired formation of residual stresses impairs strength and dimensional stability and leads to material wastage, and in extreme cases, catastrophic structure failure. Due to the critical role of residual stresses in engineering design, several destructive and non-destructive measurement methods have been developed. Further enhancements are yet needed to improve measurement stability, reduce noise sensitivity and enhance computational effectiveness. 1.1.2 Formation and Effects of Residual Stresses Residual stresses are self-equilibrated, that is, they have zero force and moment resultants. As an example, Figure 1(a) shows a typical residual stresses distribution profile for toughened glass. There are compressive stresses on the surface and tensile stresses in the middle region to satisfy the force and moment equilibrium. Figure 1(b) and (c) shows pictures of a shattered windshield glass. If only small cracks initiate on the near surface areas, the cracks will be prevented from opening by the surface compression. However, if the cracks happen to reach the deeper tensile areas, they will grow extensively as shown in Figure 1(c). (a) (b) (e) Figure 1 — Toughened glass: (a) Through- thickness stress profile, (b) shatteredpanel, (c) Detail ofshatteredpanel Residual stresses also substantially influence the dimensional accuracy of machined components. During any machining process, residual stresses will be redistributed and cause significant deformations. Figure 2 demonstrates a warped aircraft cargo ramp after machining. For specialized use in aerospace technology, all mechanical components must be very light. Therefore, substantial uncritical material must be removed from the billet. Relief and redistribution caused large deformation shown in Figure 2. 2 1.2 Techniques and Challenges of Residual Stresses Measurement Because of the major influence of residual stresses on failures, knowledge of residual stresses is crucial for the design of critical engineering structures. Both non-destructive methods and destructive methods have been explored for measuring residual stresses. 1.2.1 Non-destructive Methods The two primary non-destructive techniques are X-ray diffraction (XRD) and neutron powder diffraction (NPD) [2]. X.RD is capable of directly measuring the interplanar atomic spacing and from this quantity, the total stress on the metal can be obtained. However XRD can only evaluate stresses at the surface, maximum depth about 0.05mm, and furthermore could be only used with crystalline materials. NPD can measure residual stress depth of many millimetres but is generally constrained to measuring a volume no smaller than a cube 1 to 2 mm a side and requires the use of a nuclear reactor, which restricts the application of NPD greatly [3]. While the non-destructive methods have the I— - Figure 2— Distorted cargo ramp (D. Bowden, Boeing Company) 3 advantage of avoiding specimen damage, they have the disadvantage of lesser measurement accuracy when compared with destructive measurement methods. 1.2.2 Destructive Methods Destructive methods, such as hole-drilling and slitting can provide relatively accurate and reliable results. The obvious disadvantage is the partial or complete damage of specimen. Even so, destructive methods are the most commonly used methods to measure residual stresses. Destructive methods involve cutting away some stressed material and measuring the resulting deformation in the adjacent material, typically displacements or strains [4]. The residual stresses existing in the removed material can then be calculated from the measured deformation. For all destructive methods, computational challenges arise because the calculated stresses exist in the removed material while the measurements are made in the remaining material. This characteristic is the major computational challenge. 1.2.2.1 Traditional Strain Gauge Destructive Method Traditional destructive methods involve using strain gauge to measure the deformations caused by the relief of residual stresses. surface strain gauge Figure 3 — Slitting method (Adaptedfrom Prime[3]) Figure 3 shows the stain gauge slitting method, which involves attaching one or more strain gauges to the specimen surface, and measuring the strain relaxation as a single narrow slot is cut. Because only one slot is produced on the workpiece, it relieves only back strain gauge 4 stresses perpendicular to the cutting edge and shear stresses in the cutting face. So the limitation of this method is that it can only indicate two of three in-plane stress components. Disregarding the directional limitation, the slitting method has some advantages. The slitting method could give results for the stresses that exist within the depth range of the slit. Practical considerations typically limit this range to about 90-95% of the specimen thickness [5]. If force and moment equilibrium are considered, this additional information could be used to estimate the residual stresses at all depths within the workpiece. Strain gauge Hole rosette core Figure 4—Plane and side views ofhole-drilling and ring-core method (Adaptedfrom Schafer [6]) The strain gauge hole-drilling method [7] is one of the most widely used techniques for measuring uniform residual stresses. In general, hole-drilling measurement is sensitive to all the in-plane stresses within the hole. It gives reliable results and creates only minor, often tolerable damage to the specimen. The ring core method is a variant of the hole drilling method. Instead of drilling a small hole, an annular groove is milled around the strain gage rosette. The strain gauge hole-drilling and ring-core method shown in Figure 4 5 have been well established for many years. Because of the advantages of low equipment cost, ease of use, moderate specimen damage, the strain gauge hole-drilling method has been standardized as an ASTM Standard Test Method [7]. However, the installation of strain gauges is very time consuming and for every measurement, it gives just 3 discrete readings, which are just sufficient to evaluate the three in-plane residual stresses. The hole-drilling and ring core methods are sensitive particularly to surface stresses with diminishing sensitivity to interior stresses far from the surface. The hole-drilling measurements are capable of resolving residual stresses down to about half the hole diameter and ring-core measurements can resolve stresses to about a quarter of the groove diameter [6], so they have only a limited capacity to evaluate stress .vs. depth profiles. 12.2.2 ESPI application for residual stresses measurement Mirror and piezo Figure 5— Existing ESPI arrangement (Adaptedfrom Stenzig and Ponslet [8]) Electronic Speckle Pattern Interferometry (ESPI) provides an alternative technique for measuring surface deformation (displacements). Figure 5 shows a typical arrangement, Object beam 6 the light from a coherent laser source is split into two parts using a beam splitter. One part of the beam illuminates the object, which is imaged by a CCD camera. The second part of the laser light passes through an optical fibre directly to the CCD. The two parts of the laser light interfere on the CCD surface to form a characteristic speckle pattern. The phase at each pixel of the CCD can be determined by taking multiple images of four or more phase angle steps controlled by the piezo stepper and mirror. The deformations of the object surface caused by hole-drilling, change the lengths of the illumination and object beams and hence the measured phases. These phase changes indicate the surface deformation. This technique provides accurate measurements in the order of a wavelength of light. Since the wavelength of visible light is small, the ESPI technique is specially suited to measuring very small displacements. Some researchers have combined the ESPI technique and the hole-drilling method to measure residual stresses. The ESPI method has the advantage of providing a very rich data set, over 300,000 pixels for a typical CCD camera. The ESPI technique also needs less surface preparation and avoids the time-consuming installation of strain gauges. A disadvantage is that the equipment cost is high. Figure 6 shows the contour interference fringe pattern of a surface obtained using the ESPI hole drilling method. A fringe pattern is obtained by plotting the wrapped phase angle shift (from - r to ) at each pixel due to surface deformation. This fringe pattern can be unwrapped and interpreted as surface displacements. The displacements can be used to calculate the released residual stresses that cause the deformation. 7 Small hole drilled Figure 6— Fringe pattern by ESPI hole-drilling method The ESPI hole-drilling technique also has some significant challenges. The signal-to- noise ratio of data obtained from ESPI measurements can be rather modest. Low quality pixels contain substantial noise content can impair results. In addition, the flow of chips from the cutting process damages the surface and causes the loss of ESPI data near the hole edge. However, the displacement data from these pixels contain highest deformation and are the most useful data for the calculation. Thus, it is important to seek ways to preserve these pixels. Furthermore conventional ESPI systems only have a single axis illumination, which means the method just gives one directional displacement measurement, which is less sensitive to perpendicular stresses. I 3 Research Objective and Proposed Method Both the slitting and ESPI methods have their particular advantages and disadvantages. Therefore there is a great interest to develop a new method which could combine the advantages of both methods and minimize the disadvantages. Furthermore, conventional residual stress measurement methods often assume that the residual stress field does not 8 vary with depth below the surface. However, for most of the industrial applications, the stress state is non-uniform. Thus, there is also a great interest to investigate the measurements of non-uniform stresses. The objective of this research is to develop a new residual stress measurement method that can give reliable and accurate measurement for uniform and non-uniform stress states. It involves using a new material removal technique, modifying existing the ESPI system, developing pixel quality control, and stabilizing residual stresses calculations. (a) z/L Figure 7— Cross-slitting method: (a) schematic drawing, (b) Picturefrom CCD camera To achieve the objective of this research, a cross slitting method is proposed. In contrast to the conventional single slitting method, two perpendicular slots are cut in the workpiece, releasing all three in-plane stress components. Cutting cross-slits has the advantage over cutting a similar size circular hole (shown by the red circle in Figure 7(b)) that it leaves many intact high-displacement pixels within the circular boundary. Compared with the hole drilling method, cross-slitting adds many measurable pixels within the circular boundary. Data in these pixels make the cross-slitting method more Vorkpece V (b) 9 sensitive to the residual stresses and enhances the accuracy of the residual stresses calculation. The ESPI measurements are also developed in this research. A shutter and double-mirror device is designed to create a dual-plane measurement. By doing this, displacement data in two independent directions can be obtained. This new ESPI system overcomes the shortcoming of the existing ESPI system that is predominantly sensitive to stresses in a single direction. A novel pixel quality control procedure is also applied in this research. This examines the displacement data pixel by pixel, interpolates the bad pixels and chooses only data from functional pixels for residual stresses calculation. The relative merits of cross slitting and ESPI are combined in this research to achieve the research objectives. 1.4 Overview The present work is organized as follows: Chapter 2 presents a discussion of residual stresses calculations for the cross-slitting method. The advantage of cross-slitting method, the relationship between residual stresses and surface displacements, the finite element calculations for calibration coefficients and the mathematical background of the inverse calculation are explained here. Chapter 3 provides a general explanation of ESPI technique. The principle and procedure of ESPI technique, phase stepping algorithm and phase map unwrapping are presented in this chapter. Chapter 4 introduces a modified dual-axis ESPI system, pixel quality control and correction procedure and improvements for ESPI technique in this research. 10 Chapter 5 reports the experimental result from single and cross slitting tests. A validation is made from the results from the proposed method in this research and a four-point bending test. Chapter 6 contains an overall assessment of the contributions and limitations of the research work presented. Recommendations for future work are also discussed. 11 Chapter 2— Solution of Residual Stresses Calculation for Cross-slitting Method 2.1 Introduction to Cross-slitting Method The established one-slot slitting method is limited being able to indicate only the stresses perpendicular to the slot. To overcome this limitation, a new cross-slitting technique is proposed here. tyx x Figure 8— Cross-slitting under in-plane stress state Figure 8 shows a specimen containing residual stresses. In the application of the cross- slitting method, two perpendicular slots are cut. Vertical slot 1 releases in-plane residual stress components o-, vi,, and horizontal slot 2 releases cr and. By making a pair of cross-slots, all three rn-plane stress components are relieved. In addition, cross-slitting gives access to measurements in the high deformation regions near the slot intersection. Unlike the same size hole or ring core, the four corner areas around the intersection of the two slots are still reachable after slitting. Data in these areas contain more deformation information and give more sensitive measurements txy 12 2.2 Displacement and Model Relationship Since the ESPI technique is applied in this research, the top surface displacements are measured. Displacement resulting from the release of residual stresses gives useful information to evaluate the locked-in stresses. However in practical use, even with carefully designed fixturing, it is still difficult to avoid rigid body motions and additional optical path changes caused by air turbulence. Thus, it is necessary to consider both displacement from stress relief and additional motion together to obtain accurate residual stress calculation. 2.2.1 Displacement and Residual Stress States The in-plane stress state contains three stress components d = + + o, v as shown in Figure 9. Each of these stress state will cause displacements on the top surface. The superposition of the resultant displacements from the three stress states produce the final deformation map on the top surface of workpiece and only these displacements contribute to calculate the residual stresses inside the workpiece. - _ + + + ÷f Figure 9—In-plane stress state decompositions 13 2.2.2 Displacement and Rigid Body Motions In addition to the elastic deformations from stress relaxation, as mentioned, the measurement may also include arbitrary rigid-body motions caused by small relative movements [10]. These movements are caused by local temperature changes and bulk movements due to the slitting process [9]. Since the ESPI technique used here measures only the in-plane displacement components, the three rigid body motions of interest are: X and Y direction rigid body motion are represented as U and V. These are the whole workpiece bulk movement in X or Y direction. The X and Y displacement components due to bulk rotation are: z = —i’cv zy = xYI (1) (2) Where p’ is the counter-clockwise bulk rotation, and x, y are the coordinates of the pixel relative to the coordinate origin at the slot intersection. Y V Y Y U hi x (a) (b) (c) Figure 10— Three in-plane rigid body motions: (a) X direction translation (b) Y direction translation (c) in-plane rotation 14 2.2.3 Additional Motions from Air Turbulence ESPI is a very sensitive technique and is influenced by environmental factors. Air turbulence during an experiment alters optical path lengths and the measurement. Mathematically, air turbulence has the same effect as rigid body motions causing additional displacements besides the stress-released displacements. These displacements and rigid body motions could be considered together as additional motions as: d=o0+oy+a2x (3) where o, and a2, respectively, are the resultant translation and rotations from both rigid body motion and air turbulence. x, y are the coordinates of the pixel relative to the coordinate origin at the slot intersection. 2.3 Residual Stress Calculation for Single Depth Case For practical residual stress measurements, two types of stress states need to be examined. The first one is the single depth uniform stress case. This applies when the residual stress field doesn’t vary with depth below the surface or only near surface stress is of interest. The second type of measurement is non-uniform stress case where the residual stresses vary with depth. In this case, there is no longer a one-to-one relationship between the target stresses and the measured deformation. Instead, a measured deformation depends on the contributions of the various stresses contained in all parts of the removed material. Solutions for these two cases are discussed separately here. 15 2.3.1 Matrix Equation for Single Depth Case For the uniform stress state shown as Figure 11, the displacements caused by slot slitting equals the superposition of the displacements caused by each residual stress relief and additional motions. UniForn sr-ess stote Figure 11 — Ui4form residual stress state This could be expressed in a matrix form: Gm=d (4) Eq.(4) can be expanded as: o.x E 0x JyVxy’0O ‘0i (02 (04 (05 • . . • 00 E 0 0 Txy E • • • • • • 00 • • 0 0 0 • • • • • 0 0 0 • • (02 (0 G (1)5 (06 m Where G, the kernel matrix, represents the displacement relaxation by a unit uniform stress within the slitting depth and additional motions caused by rigid body motions or air 0 0 0 (n1--n2)x6 ———-a x d1 d1 [ d+ — d 16 turbulence. Model vector m includes three stress components normalized by Young’s modulus E and 6 additional motion terms since we take two directional displacement measurements for one stress component calculation (the additional motions for these two ESPI displacement can be different). Vector d is the measurement result from the ESPI experiment that includes two-directional displacements. The first three columns of the G matrix are the displacement relaxations corresponding to three unit uniform stress states o o o. The resulting top surface displacement maps are interpolated to match the CCD camera’s pixels. They are reformatted into vectors to fill the corresponding columns of kernel matrix G. Since a new ESPI setup used in this research gives top surface displacements in two directions. Thus all these displacements will be reformatted. The upper half of the G matrix represents the number of pixels used giving displacement in first direction and the lower half of G matrix represents the number of pixels used giving displacement in the second direction. Column 4,5,6 and 7,8,9 are the scaled additional motion terms for these two directional displacement measurements. Two major tasks need to be completed to calculate uniform stress and even non-uniform incremental stress profile. The first one is to build up the kernel matrix G and the second is to design an ESPI system that can achieve satisfactory displacement measurements. Finite element analysis (FEA) can be used to calculate displacement relaxation (the first three columns) from three unit uniform stress states, and additional displacement colunm in G matrix can be obtained from Eq. (l),(2),(3). The uniform stress solution could be obtained by Solving Eq. (4) 17 2.3.2 Least Square Approximation Eq. (4) is called an “inverse problem” because evaluation of the unknown quantity m requires a solution from right to the left. In a typical ESPI measurement, there are approximately hundreds of thousands of data points available from each measurement, but just 9 unknowns quantities to be determined ( three stress components and 6 additional motions). Thus, the problem is highly overdetermined. Meanwhile, inverse equations have a variety of associated challenges. They are typically ill-posed, causing small variances in data to be amplified in the solution [11]. The least-squares method is used here to find the best-fit solution. A least-squares model minimizes the misfit, or difference, between observed from direct measurement and predicted data [11]. data misfit=d0G.m (5) The minimization of the misfit norm is then represented by the least squares approximation: minG.m_d01j (6) Differentiating Eq. (6) with respect to m and setting it equal to zero provides the normal equations, the solution of this inverse problem [11]. GTG.m=GT.dobs (7) The model vector m is then determined by multiplying each side of Eq. (7) by the inverse of GTG as shown in Eq. (8). m = (GTG) GT . dobs (8) 18 2.4 Residual Stresses Calculation for Incremental Depth Case 2.4.1 General Displacement and Model Relationship The non-uniform stress state is much more complex than the uniform case because of the inverse relationship between the measured deformation and stress components. This relationship takes the form of an integral Eq. (9). d(h)=fg(H,h)’dH OHh (9) where d(h) is the released displacement when the slot is cut to the depth of h. The kernel function g(H, h) describes the displacement response due to a unit stress at depth H in a slit of depth h. This function depends on the geometry of the slots and the specimen. Eq. (9) shows the integral relation between released displacement and residual stresses. The measured displacement d(h) is the accumulated result of a combination of stresses o-(H) at all depth H within the slit depth. Eq. (9) is also an “inverse problem”: because the stresses o-(h) to be determined are within the integral, while the measured displacement data are outside. An effective solution method is to represent the stress profile in the form of a mathematical series çb (h) with unknown coefficients m1 [11]: (H)=mq51) (10) The stresses profile solution involves determining the unknown coefficients m Substituting Eq. (10) into Eq. (9) gives: d(h) =--m1G(H) (11) where: 19 G(h) = Jg(H,h)(H)dH OHh (12) In theory, any form of series with linearly independent term that spans the solution space (is able to represent any arbitrary stress profile) is acceptable [12]. Unit pulse functions are chosen here because of their conceptual and mathematical simplicity. 43(H) U — 45(H) H3H4 H5 Figure 12— Unit pulsefunctions usedfor residual sfress calculation (Adaptedfrom Schafer [4]) Five unit pulse functions are illustrated in Figure 12. The width of each pulse corresponds to the successive increment in material removal depth, which is not necessarily all equal. As an initial approach, equal increments are considered. After cutting each material increment, the top surface displacements are measured by the ESPI system. For unit pulses, the coefficients m. in Eq. (11) equal the stresses within jth material increment. The stresses within any given increment are assumed not to vary with depth. In this condition, the non-uniform stresses calculation is the extension of uniform residual stress case. Based on a five-increment model, the Eq. (11) can be rewritten in matrix form, analogous to the uniform stress state: I. UI I. ___... 1 1(H) 0 0 0 Hi H2 20 Gm=d (13) Figure 13 illustrates the composition of the the G matrix for the non-uniform stress incremental cutting case. 611 62t 622 111J 631 632 633 N1U 641 642 643 644 651 652 653 654 G55 Figure 13 — Physical interpretation ofmatrix coefficientsfor slitting method (Adapted from Schafer [13]) Physically, coefficient G represents the displacement relaxation by a unit uniform stress within increment “j” of a slot that is “i” increments deep. where, h1 h1 h_1 = J c(H, Ii, )dH = Jc(H, h1 )dH — f c(H, k)dH (14) h1_ 0 0 Extending Eq.(l 3) into a full matrix format, we get: G11 m1 d1 G21 G22 m2 d2 G31 G32 G33 in3 = d3 (15) G41 G42 G43 G m4 d4 G51 G52 G53 G54 G55 m5 d5 21 In Eq.( 15), taking consideration of additional motions after cutting each increment, matrix component G follows the format ofG in uniform stress state that contains the columns of displacement response to unit stress and scaled additional motion terms. The model vector rn includes three stress components in increment i and 6 additional motion terms. Vector d1 represents the real displacement map after each incremental slitting. The recovered stress vector [m1 m2 in3 rn4 m5] after the inverse calculation gives the stress components in each increment. 2.4.2 Regularization Approach for Incremental Depth Case Least square solution naturally reduces noise by averaging a large amount of data. However that is not enough sometimes, because modest noise content can still significantly impair the solution. Tikhonov regularization is commonly used to improve its stability of an inverse solution. This procedure effectively smoothes the stress solutions and diminishes the adverse effect of noise without significantly distorting the part of stress solution corresponding to the “true data” [11]. The Tikhonov method involves modif’ing Eq. (15) to penalize the local extreme values in the stress solution that occurs because of the presence of noise [11], (GTG+flWTw)m=GTd01 (16) ,6 is the regularization parameter that controls the level of regularization in the solution. Matrix W evaluates the chosen derivative of the stress solution that is to be penalized, Wsmaii , Wflag and Wsmooth are the operators that approximate the first and second derivatives, respectively, of the stress solution model. Each derivative operator is defmed in Eq. (17), (18) and (19). 22 1c11 = 1 (17) 1 0 —1 1 1 —1 1W =— (18)flat H —1 1 0 1 —2 1 1 1 —2 1 mooth (19) 1 —2 1 where H is the increment depth. For residual stress calculations, “smooth” (second derivative regularization) is a suitable choice because it does not significantly disturb force or moment equilibrium [11]. The regularization parameter fi is chosen according to Morozov’s Discrepancy Principle which requires that the data misfit IG m — d0 should equal the data noise level. Under the assumption that the standard deviations of all misfit errors are the same, the residual G d0sm should follow a Gaussian distribution with zero mean. According to 0 Morozov’s Discrepancy Principle (20) where N is the total number of measurement data points. ,8 is set in order to satisfy equation (20). Equations (16) and (20) are solved iteratively until convergence. 23 2.5 Finite Element Calculation For both uniform and non-uniform stress states, the displacement responses to unit shear or normal stress components need to be evaluated. These comprise the major part of the kernel matrix G. The finite element method provides a good opportunity to calculate these displacement responses. For finite element calculations, a model which is capable of dealing with three stress states o, o,,, r was developed. Program ANSYS 100 was used here for all the fmite element calculations reported. The dimensions of the workpiece used in this research are 4”x 4”, 0.5” deep. The slot is 1/16” wide, 0.25” deep and 0.5” long. With consideration of the symmetric geometry also the fact that cutting depth increases from one increment (uniform stresses state) to five increments, five one-quarter models with different cutting depth are built up. These models are capable of calculating displacement responses separately for all three stress states by just changing the loads and boundary conditions. By applying unit uniform stress in each increment correspondingly, all the components for kernel matrix coefficients can be obtained numerically. Also, all these models follow the same mesh pattern on the top surface as Figure 14(a). This provides convenience to interpolate and obtain kernel coefficients at each point on the top surface according to the pixels of the CCD camera in the ESPI measurement. 24 I1uIIIIN,,,’— lIIIII,I,,,,,,,, ••••••••u•i•••••u•• EØ :;::::::::•.iI•••P ; 3Z’ c • I— (a) Top view X (b) Side view (one increment case) Figure 14 — Detailedfinite element mesh patternfor kernel matrix coefficients calculation (Total mesh doubles the area shown in (a)) 2.6 Discussion In this chapter, the relationships between the displacement and residual stresses for both single and incremental depth cases have been discussed. The solution for residual stresses calculation has also been presented. Finite element models were built to calculate the displacement responses due to different stress states and kernel matrixes were built using the finite element calculation results. 25 Chapter 3 — Electronic Speckle Pattern Interferometry (ESPI) 3.1 Fundamental of Laser Interference The application of the ESPI technique relies on the interference phenomenon. When two laser light waves with same frequency or from the same source intersect, interference occurs. The resultant intensity at any point depends on whether they reinforce or cancel each other shown in Figure 15. If the beams are in phase, they interfere constructively and their amplitudes add to give a bright speckle. If the beams are out of phase, they interfere destructively and their amplitudes cancel each other out creating a dark speckle. Any case between these is possible. In phase / = r/ \ / / / Out of phase Figure 15— Interference oflight waves The complex amplitude at any point in the interference pattern is the sum of the complex amplitudes of these two waves, it could be written as: (21) where A1 = a1 exp(—i4) and A2 = a2 exp(—i) are the complex amplitudes and q, 2 are the phase angle of the two waves. The resultant intensity is therefore, 26 —I2 ‘2 = IA212 (22) = + J +2(J)”cosAçb where I and are the intensities due to each beam separately and Aq$ = —02 is the phase difference between them. If the two waves are derived from a common source, so that they have the same phase at the origin, then the phase difference Aq5 corresponds to an optical path difference Ap=(2I2r)Aq5 (23) In essence, ESPI technique uses the property of laser interference. A CCD camera is used to record the interference intensity before and after deformation happens. The phase difference AØ before deformation and phase difference Z\q5’ after the deformation could be obtained separately by applying a phase shifting method. The change of phase difference z between these two angles is: (24) According to the Eq. (23), the relative phase change corresponds to the total optical path length change EiP before and after deformation: = (2 / 2,z)tç (25) The total optic path length change zXP could be expressed as a projection of surface displacement: P=Kd (26) Where K is the sensitivity vector, d is the surface deformation. Substituting (26) into (25) gives, A, =(2r/2).K.d (27) 27 Equation (27) gives the basic idea of ESPI technique that is if the change of phase difference zç could be determined from the ESPI calculation, then the surface displacement can be obtained. 3.2 Typical ESPI Setup x I Object z Figure 16— Schematic arrangement of in-plane ESPI measurement (Adaptedfrom Steinchen [14]) The ESPI technique has been developed in many different arrangements depending on measurement needs. Figure 16 shows a diagram of a typical in-plane displacement measurement setup using laser light from a common source. In this arrangement, a laser source is used to provide coherent light. The beam emitted from the laser source passes through a beam splitter and is divided into two parts, one is as reference beam and another is illumination beam. A piezo with a mirror attached is placed on the path of the reference beam. After reflected by several mirrors, both reference beam and illumination Illumination Beam 28 beam reach the surface of workpiece and interfere. Light reflected from the object surface is then imaged by the CCD camera and the interference pattern is recorded. The ESPI technique is based on the concept that the surface displacements of interest changes the total optical path length linearly with phase angle. The phase angle before and after the deformation can be determined by recording the intensity changes from CCD camera and applying the phase-stepped algorithm. The measurement sensitivity obtained using different ESPI setup depends on the sensitivity vector. The sensitivity vector is defined as the bisector of the unit illumination and reference beam vectors given in Figure 17. Object Surface Figure 17— Sensitivity vector Here, k1 and k2 are the vectors for the reference beam and illumination beam. The resultant sensitivity vector is [14]: (28) When illumination angle with the surface normal is 0 with the surface normal, sensitivity vector can be expressed as: = -sin6T - coso] Ic2 sin0T - cosO] k Sensitivity Vector 29 k=k2-k1=2sin (29) Substitute Eq. (29) back to Eq. (27), the phase shift change is: = (2,r/2).k.d (2/2) . (2sin8T). (dT + dj +d2k) (30) =2(2sinO)d Eq. (30) shows that only the displacement component in sensitivity vector direction contributes to the phase change. 3.3 The Phase-shifting Method The major challenge for the ESPI technique is to determine the phase change of Aqi due to the deformation. According to Eq. (22), when the reference and illumination beams combine, the resultant intensity at any pixel in the CCD camera follows a trigonometric relationship. Eq. (22) can be rewritten as: I = A+ BcosAØ (31) where A is the mean intensity, B is the amplitude and AçS is the phase change of the interference. Aç5 can’t be calculated directly since there are three unknown quantities A, B, AqS, but just one known quantity I. To evaluate zqS, a phase shifting method is used here [15]. The basic idea of a phase shifting method is that the phase angle is shifted by a step angle fi several times. Images are recorded after each shift before and after deformation. Thus for each pixel, several intensities are used to evaluate the initial phase angle at each pixel point. 30 I, = A+ Bcos(zçS+/3) (32) a + u cos(fi) + vsin(fi) Where u = b cos(Ab), and v = —b sin(AØ), /3,, is a series of step angle used. A least square solution of Eq.(32) is: sin,8,, a sin,8,,cosJ3,, u = I,,cos,f3,, (33) sin218,, J,,sin/3,, This system is to be solved pointwise. Four and five step cases are used in this project. When four-step stepping is used (,8,, is 00,900,1800,2700) , by solving the equation (33), tanAØ=’2’4 (34) Pixel Intensity A1 Ii I I AqS1 ij1 Before slitting After slitting A2 o ,8 2,8 3,8 4,8 Figure 18— Intensity change trend by phase stepping Figure 18 shows the interference intensity change due to the phase shifting. Before and after slitting, two series of shifted images are taken, then the initial phase angle can be calculated using the least-square approach [15]. The intensity distribution could be rewrite as: 31 The four-step algorithm could be modified by using the same step angle (00,900,180, 2700,3600) and taking an additional frame. In this case, the phase difference can be calculated as [15]: tanziç= 4’2) (35) 411 —‘ —613 —14 + 415 The five-step algorithm consistently gives better results because by taking extra frames, it allows for better data averaging and improvement in error resistance. 3.4 The Unwrapping of the Phase Difference Change As discussed in the previous chapter, two sets of stepped images as shown in Fig 19 are needed to calculate the relative phase difference before and after deformation. Image 1 Image2 Image3 Image4 Image5 (b) Figure 19— Sets of images obtained by 5-step phase stepping: (a) before slitting (b)afler slitting Image 1 Image2 Image3 Image4 Images (a) 32 Using the Eq. (35), the phase difference before and after slitting can be calculated separately as: tanAq51= 7(14—12) (36) 411 ‘2 —613 —14 + 415 tanz = 7(J4 —J2) (3)2 4J1—J6J3+45 The phase difference change can then be obtained as: Aç=zb2—Aq (38) In Eq. (36) and Eq. (37), the resultant phase difference change lies in the range between —,r to ir. To better illustrate the resultant phase difference changes in the whole image, a fringe pattern with intensities p that vary over the range [0,1] can be formed. p=.(l+costço) (39) The fringe pattern in Figure 20(a) illustrates the phase change distribution in the whole image. However the change of phase difference at each pixel is wrapped in the range of [—i, ]. To get the real phase change, all these data should be unwrapped. 33 Wrapped Phase Angle(radian) Unwrapped Phase Angle(radian) •1 4C 3L 30 20 10 — 1 H. I I : 0 100 200 360 400 500 606 700 0 100 203 200 400 500 70) Pixels Pixels (b) (c) Figure 20— Phase map obtainedform singi- slitting: (a) Fringe pattern (b)A sawtooth wrappedphase distribution (c)The unwrapped continuous phase distribution Figure 20(b) shows the raw wrapped phase distribution along the middle line of the fringe pattern shown in Figure 20(a). This sawtooth shape phase function shows the phase jumps of 2,r. After using a commonly used phase unwrapping algorithm [16], the phase angle can be unwrapped to the continuous phase distribution shown in Figure 20(c). These unwrapped real phase changes can then be directly related to the displacements in the sensitivity direction and used for the residual stress calculation. (a) 34 Chapter 4— Modified ESPI Technique and Application 4.1 Limitation of Single-axis ESPI System The Electronic Speckle Pattern Interferometry (ESPI) technique gives very sensitive (nanometer range) measurement of surface displacement. Figure 16 in Chapter 3 illustrates a commonly used single-axis in-plane displacement ESPI system. However, a typical limitation is that it measures surface displacement components in only one specific “sensitivity direction”. For example, the sensitivity direction in Figure 16 is along the x-axis, measurements are not sensitive to the displacements along the y direction. However, the associated displacement components are mostly controlled by the parallel in-plane stress components. Thus the single-axis ESPI displacement measurement has the preference for the same residual stresses in that direction. Sensitivity Vector Figure 21 — Syntheticfringe patterns: (a) under x-stress, (b) under y-stress Measurement sensitivity of single axis ESPI technique to x and y stresses was simulated mathematically using the code written in Matlab®. Theoretical stress fields were created ______ / (a) (b) 35 around the cross-slot using the data obtained by fmite element calculation and for both case, the sensitivity vectors lie in the x direction. In the first case, a stress state consisting of an x-stress of 80 MPa was created (o = 80 MPa , cr, =0 and v.., = 0). In the second case, the same stress state was created in the y-direction (cry = 0, o =80 MPa and = 0). Figure 21 shows the phase maps plotted for these cases. It can be seen that the single-axis illumination method produces approximately 6 fringes in response to the x stresses, whereas it produces only two fringes in response to y-stresses due to Poisson displacements. The number of fringes is directly proportional to the magnitudes of measured displacements. Since the y-stress causes only small displacements in this x-axis directional ESPI setup, the y-stress is much more noise prone than x-stress in the inverse calculation. 4.2 Dual-axis ESPI System 4.2.1 Dual-axis ESPI Setup A limitation of the current ESPI technique is that it measures only surface displacement components in a specific “sensitivity direction”. A novel dual-axis system is presented here that is capable of making ESPI measurement in two perpendicular directions that assure that all three stress components have similar calculation sensitivity. 36 Luble Mirror De.ice Double Mirror 1: Sensitivity Vector 1 2: Sensitivity Vector 2 Figure 22 — Newly designed dual-axis ESPI system Figure 22 shows the newly designed dual-axis ESPI system. To fulfill the purpose of making two directional measurements, a shutter and double-mirror device have been made. The coherent light from the laser is divided by the beam splitters and mirror on the shutter device into three switchable parts. The first part goes to the upper mirror of the double-mirror device and reflected to the workpiece surface. The second part beam goes to the lower mirror and also reflected to the workpiece. The third part as reference beam goes directly to the workpiece and could be stepped by the piezo attach behind the mirror. By controlling the shutter, the lower mirror and upper mirror could be used alternatively, and they give two different sensitivity directions and lead to two directional measurements. These two directional measurements give the three stress components with balanced sensitivity and make the residual stress calculation more reliable. Device 37 4.2.2 Shutter and Double-mirror Devices The key components of the newly designed ESPI setup are the shutter and double-mirror devices shown in Figure 23. Figure 23 — Key components for new ESPI system: (a) Shutter device (b) Double-mirror device Two beam splitters are fixed on the shutter devices for laser light division in Figure 23(a). A piezo is attached behind the mirror to step the reference beam. Three solenoids and a small control circuit are also mounted on this device to make the shutters switch the beams on and off as needed. Figure 23(b) shows the double mirror device. The lower and upper mirrors are placed at 45° and -45° with the horizontal axis and perpendicular to each other. This arrangement produces two in-plane sensitivity vectors that are perpendicular to each other and allows the measurement of the full stress field. 4.2.3 Two Directional Measurements of Dual-axis ESPI System Dual-axis measurement could be achieved by using a double-mirror and shutter devices. The dual-axis contains two sensitivity vectors that determine the direction of the measured displacements. (a) (b) 38 xz,k 6 LJppermirror2 DoubI\am Lower mirror 1 (a) (b) Lower mirror (c) Figure 24—Double-mirror assembly and sensitivity vectors: (a) Plane view ofDouble axis ESPI setup (b) Illumination and reference beam vectors in doubl- mirror arrangement (c) Addition of illumination and reflection vectorsfor lower mirror Sensitivity vectors can be derived using the diagram given in Figure 24, where lc shows the illumination direction (Illumination angle is 8 .), si and S2 show the normal directions of top and bottom minors, and lci and k2 show the directions of illumination beams reflected from these mirrors. All the vectors could be written as the following form: = (— sin 8)1 — (cos 8)k (40) Si 39 Normal directions of top and bottom mirrors are: — 1— 1- SI jl +J — 1-. 1- S2 Using Figure 24(c), illumination and reflection vectors in bottom mirror can be added as, k+k0=2[—(.si)siJ (41) that yields k0—k1=2(k0.SI)Si (42) Thus the resultant sensitivity vector kb for bottom mirror is ‘b =—sinOi—sinOj (43) Similarly, sensitivity vector Ic can be derived for the top mirror as: k=—sint9I+sin] (44) Substitute Eq. (43) and (44) back to Eq. (27), the phase shift change is: Açp =(2r/2).k.d (45) = (2,r/2).(.ñsinO).d Eq. (45) clearly shows for the double-axis case, the change of phase difference is the result of displacement in the sensitivity direction. By using the two mirrors, two directional displacements can be calculated and these displacement data can be used to recover the residual stresses in the workpiece. 4.3 ESPI Challenges and Improvements The accuracy of measurements made by the ESPI technique depends on the quality of data, which in turn is determined by various factors involved in this technique. To 40 achieve higher experiment performance, the adverse effects should be reduced. The most important factors affecting the data quality are the modulation of phase-stepping technique and speckle size. 4.3.1 Modulation during Phase-stepping The phase-stepping technique was discussed In Chapter 3. The quality of phase-stepping directly affects the calculation of the phase change. To examine the phase-stepping technique better, Eq. (31) can be rewritten as: I = A (1+ VcosAØ) (46) where V is the visibility, or modulation of the signal obtained from the interference. Hence, modulation is defined as the ratio of amplitude to mean intensity as: (47) Figure 25 illustrates the intensity variations of high and low modulated pixels. Highly modulated pixels have high signal amplitudes and show that phase stepping has a significant contribution to intensity variation. Low modulated pixels exhibit less stable variations and have very low signal amplitude. Highly modulated pixels have higher signal-to-noise ratio and give more reliable and accurate information about the phase shift angle. Under ideal conditions, all pixels are expected to give high modulation. However, in reality, the modulation level is affected by factors such as speckle size, illumination quality and surface conditions. 41 High Pixel Modulation Intensity Low Modulation 0 n/2 it 3it/2 Step (0) Figure 25—High and low modulated signals 4.3.2 Speckle Size When a diffUse surface is illuminated by coherent light, such as laser, a grainy image is produced. The grainy light distribution, known as a speckle pattern, results from self- interference of numerous waves reflected from scattering centers on the diffuse surface. The pixels on a CCD camera detect the intensity of the speckle. Since pixels define the spatial resolution of the measurements, it is necessary that the speckles are big enough to cover the pixel area. Figure 26 illustrates the effect of speckle size. If the average speckle size is smaller than the pixel size, several adjacent speckles will overlap on a single pixel. This will result in an intensity variation that is an average of intensity change of individual speckles that cause an unstable pixel reading. If the speckle size is greater than the pixel size, this will guarantee the correlation of speckles before and after deformation and allow each pixel to detect the right intensity variation during phase-stepping. Thus speckle size plays a critical role for the success of ESPI measurement. 42 Dark Speckle Bright Speckle Pixel Pixel - Large Size Speckle Figure 26— Effect ofspeckle size The average subjective speckle size in the image can be estimated in Eq. (48) [171: SbJ =2(l+M)2f (48) where 2 is the wave length, f is the “f-number” of a lens as the ratio of focal length to diameter, M is the magnification of the image. For the ESPI experiment, M >10 and 1 + M M. Thus the f-number makes the major contribution to the speckle size. The f number can be varied by adjusting the aperture size. However when f-number is increased to obtain larger speckles, the amount of light received by camera may become insufficient. Therefore, the use of large f-numbers needs to be accommodated by using larger exposure time. 4.3.3 ESPI Improvements To get better results, many improvements have been made on the ESPI experiments. All these improvements are designed to enlarge the ratio of speckle and pixel size without losing too much light. L_ —— -J 43 Illumination quality is one of the most important factors in ESPI measurements. If the laser source used in the measurements is of low quality, the average intensity may fluctuate over time and cause unstable intensity reading. A high intensity level allows possibility of using a smaller aperture, thereby giving larger speckles. The CDPS 532M laser is chosen in this research, which has 532nrn wavelength and output power of 50mW. This laser turned out to be veiy stable and have clean wavefronts. To increase the speckle-pixel size ratio, one way is to close the aperture to get large speckles and the other way is to use smaller pixels. The EC750 camera from Procilica is used here. This camera has smaller pixels size only 6pm x 6pm (typical size is 10pm x 10pm) and can transmit up to 64 frames per second. A program to adjust the exposure time for this camera was developed. The exposure time was changed from 20ms to 4Oms. This enabled the camera to collect more light and allows the use of a smaller aperture. Figure 27— Telecentric lens Lens choice also plays a critical role. After some comparative tests, a telecentric lens was chosen in this research. The term “telecentric” means that the chief rays that pass through the centre of the aperture stop are parallel to the optical axis in front of or behind the system. The advantage of this lens is that the size and shape of an image formed by such a lens is independent of the object’s distance or position in the field of view. In the newly 44 designed ESPI system, the CCD camera does not image the workpiece perpendicularly, so the use of this lens helps the depth of focus. By all these experimental improvements, the f-number can reach 28, which according to the equation (48) gives speckles size of about l4pm, more than sufficient to fit on the 6pm pixel size of the camera. 4.4 Pixel Quality Control and Correction Procedure By applying the phase-stepping technique, a phase map (finge pattern) showing the phase change A can be obtained, but the phase map yields only a sawtooth function showing the phase modulo 2.ir [14]. Phase unwrapping needs to be done to get continuous Açi’. However the phase unwrapping method can be performed only for a noiseless sawtooth image [14]. The general method to reduce the noise of raw phase map is filtering. Filter operation can be classified into two categories: those used for linear operation, i.e. mean filter, and for nonlinear operation, e.g. median filtering. However, the filtering just averages the noise to adjacent areas other than eliminating the undesirable noise. It can distort the good data as well as ameliorating the bad data. Thus, such filtering is of limited effectiveness. 4.4.1 Pixel Quality Examination To get a precise residual stress calculation result, the quality and accuracy of the phase map must be enhanced as much as possible. Each pixel of a raw phase map can be examined according to criteria: saturation, modulation and variation. 45 The intensities of light on pixels are reported as numerical values in the range of 0 to 255. However there is a limit for the light that could be received. When pixels received excessive light, the numerical values stop at 255. Such pixels gave an unrealistic reading. Using these faulty data could only cause noise to the phase unwrapping algorithm and damage the residual stress calculation. Thus all the saturated pixels must be culled from phase map. The modulation is another parameter that needs to be considered. It is directly related to signal-to-noise ratio. By solving Eq. (34), the modulation formula can be obtained as below: V (49) For the four-step case, = J(J — J)2 + (‘2 — I) (50) + ‘2 + 13+14 For the five-step case, = J(4] ‘2 —613 I 4J)2 +(7J 7J)2 (51) 2I + + 413 + 314 + 215 After many experimental attempts, the modulation of V=O. 1 was chosen to be a practical value to determine good or bad pixels. In practice, only pixels with modulation greater than 0.1, were used in the calculation. 46 • . , . . • . • . • • . • • • • 5x5block.. ••• • . . • • • • •.. • •• . . • . . • •I. • • • •I• • • • • . • . i • • ixel to be evaluated • • • . .1. . • • .1. . • • L ¶_ • • . • • • •. . . • • • • •.••••••••..•. •. • • S S • S S • • S S S S • S S •. S • • • •• S S •SS • S S •.• ••...•••.•.•. Figure 28—Evaluation ofvariation ofpixels Besides the saturation and modulation, the image variation of phase map is another parameter that is considered. Ideally, a phase map should be continuous and smooth. Thus, the value of phase change at each pixel should not vary greatly from the adjacent pixels. Figure 28 illustrates the evaluation of variation of pixels. A 5x5 block is used to evaluate the variation of phase change of the center pixel compared with the average of the adjacent 24 pixels. Var = ((25 sin(A) — ssum) / 24)2 + ((25 cos(A) — csum) /24)2 (52) where Aç is the phase difference change of the middle pixel, ssum is the superposition of all sinusoidal phase changes of all pixels in the block and csum is the superposition of all cos-sinusoidal phase changes of all pixels in the block. Theoretically, if the pixel is good, Var should approach to zero. In practice, 0.4 is a good limiting value for Var. The calculated phase changes from these pixels are believed and chosen to compose the real displacement measurement. In this research, a novel ESPI setup with dual-axis measurements and a pixel quality examination technique have been applied. All these efforts are aiming at improving the 47 ESPI measurement accuracy. The experiment validation will be presented in the next chapter. 48 Chapter 5— Experimental Validation 51 Cross-slitting ESPI Experimental Setup In this chapter, experimental validation of the ESPI technique for residual stress measurements is presented. Figure 29 shows the experimental setup and Table 1 lists the experimental equipments and parameters. The experimental setup follows the dual-axis ESPI design in Figure 22. Driller Figure 29— Experimental setupfor dual-axis ESPI measurement: (a) Overall setup, (b) Driller, double-mirror and workpiece, (c) Camera, shutter and laser Double Mirror (a) dimera (b) (c) 49 Equipments Specification Parameters Specification Camera Prosilica EC 750 CCD camera Cross-slot length 1/2” End mill 1/16” double flute end mill Cross-slot width 1/16” Moving device Daedal 3-axis translation table Incremental depth 1/20” Mirror device Two perpendicular mirrors Total cutting depth 1/4” Shutter device Beam splitters, piezo with F-number 28 mirror attached and_shutter lens Computar telecentric lens Illumination angle 450 Table 1 — Experimental equzments andparameters All the equipments are placed in a glass-covered experimental box. This greatly prevents the external air turbulence. An electric drill is fixed on a 3-axis translation table. A computer program was written to drive the table to complete the single-slitting or cross- slitting tasks. The test workpiece is a bent aluminum plate with dimension of 4 “x 14 “x 0.5”. The double mirror device is placed nest to the workpiece to create dual- axis measurements. Laser beams from a CDPS 532M laser are divided into three parts by a shutter device. A Prosilica EC 750 camera imaged the workpiece with f-number 28, therefore giving good depth of focus and large speckle size. 5.2 Measurement of a Known Stress Field To verif’ the residual stress measurements from the cross-slitting/ESPI technique, a test specimen with known residual stress field was created for use as a reference. Since the cross-slitting/ESPI technique is capable of evaluating both uniform and non-uniform stresses, a four-point bending test on a T6061 aluminium material plate was done to create a known stress vs. depth profile [18]. 50 Ff2 F/2 Workpiece Roller F12 Figure 30 — Four-point bending experiment Strain gauges were attached on the top and bottom surface of the workpiece. Four rollers were placed in the positions shown in Figure 30. Forces were exerted on the rollers by a Tinius-Olsen tensile testing machine. The four-point bent procedure gave a uniform bending moment in the middle section. Creation of known residual stresses involved loading the workpiece until a substantial amount of yielding occurred, and then unloading. As a result, substantial residual stresses are created within the workpiece. Eq. (53) determines the stress and strain curve in this test, 2sdM + 4Mds (53) bh2d where s is the average of the strain readings of top and bottom strain gauges. dM and d8 are the differentials of the applied moment and strain, that could calculated by four- point differential formula [18]. b and h are the width and height of the workpiece. The strain has a linear relationship with the depth of workpiece, thus the stress-strain curve can be interpreted into the stress distribution along the workpiece depth. Unloading follows a linear curve. By subtracting these two curves, the residual stresses remaining in the workpiece can be calculated. Gauge F/2 51 35c1 Unloading - —Loading kidth from the oenterimm; (a) Center line of the specimen Stress 145. Depth Profile I I I — I100 I I I s—fl \ P I I I 50 _/_ I I F 0. I I I 0 Ci) I I I -50 -----/ I-- -100 •/ -150 I I 0 1.27 2.54 3.81 5.08 6.35 7 Depth(mm) (b) Figure 31 — Four point bending test result: (a) Stress-height cun-’e for loading and unloading, (b) Residual stress distribution versus depth Figure 31 (b) shows the calculated residual stress distribution vs. depth. Because the cross-slitting/ESPI measurement gives five incremental stress components, this can also 52 be obtained by calculating the average stresses in these increments. This result can be compared with the measurements from the cross-slitting/ESPI technique. The incremental stress components o are listed in Table 2. This bending test mainly produced the residual stresses in the X direction. The stress components in Y direction and shear stresses were not measured but expected to be small. Thus o calculated from the bending test is used as the reference to judge the accuracy of measurements from the ESPI/cross-slitting method. Increment o (Mpa) 1 -73.8 2 -5.8 3 60.4 4 114.1 5 70.8 Table 2—Incremental stresses calculatedfrom bending test 5.3 Experimental Measurement Result By following the new ESPI setup, both single depth and incremental depth residual stress measurements were done. The results from single-slitting and double-slitting were compared. The procedure for the single depth experiment is to take a reference set of stepped images from each mirror before slitting and cutting the crossed slots on the pre stressed workpiece to the desired depth and length. After slitting, further sets of stepped fringe images were taken from each mirror. Using Eq. (35) and applying the unwrapping algorithm, the unwrapped phase map was obtained as data for the residual stress 53 calculation. The procedure for incremental depth experiment is the same as for a single depth experiment, but using the previous fringe set of images as the reference for the next increment. Thus, only the phase change due to cutting the new increment is used for incremental residual stresses’ calculation. In all experiments, the depth of a single increment is 0.05”, slot length is 0.5”, the end mill diameter is 1/16” and the number of increments is 5. 5.3.1 Single Depth (Uniform Stress State) Measurement Four cases of single depth measurements are discussed here: Cross-slitting measurement with and without the pixel quality control and correction procedure; Single-slitting measurement; Cross-slitting using only one direction data. The measurement results are tabulated in Table 3. Measurements O (Mpa) (Mpa) V,, (Mpa) Single-slitting (corrected data) -78.3 -80.2 0.8 Cross-slitting (lower mirror) -78.1 -15.2 3.4 Cross-slitting (upper mirror) -81.6 -16.9 -9.4 Cross-slitting (raw data) -48.1 -14.5 -1.9 Cross-slitting (corrected data) -74.5 -11.2 -1.0 Bending test -73.8 0 0 Table 3 — Measurement resultsfrom single depth case Based on the single depth measurement results, the improvements from ESPI/cross slitting method are discussed in the following sections. 54 5.3.1.1 The Effect of Pixel Quality Control and Correction Procedure The big improvement for the ESPI technique in this research is the pixel quality control and correction procedure. By examining the saturation, modulation and variation of the fringe data, the image quality can be visualized as Figure 32. Figure 32 (a) shows the image quality for the fringe pattern. White, red, green and blue colours are assigned to the pixels in the image. The pixels in red are saturated. Pixels in green mean the modulations are lower than acceptable level. Blue indicates pixels with high variation. The uncoloured pixels are well functioning pixels. The points enclosed by the cross-slot boundary were completely decorrelated because the surface in that portion was removed by slitting. The blue area adjacent to the cross-slot boundary was damaged possibly by the chips produced during slitting. All the data in the blue areas are useless. In practice, only data within the four pink boxes were used to recover the residual stresses. Even so, in these pink boxes, there are still many low quality pixels. This causes the grainy fringe pattern and the disturbance during the unwrapping in Figure 32(b). To improve the accuracy of the data and stabilize the unwrapping, bad pixels were corrected by interpolating between the adjacent good pixels. After the interpolation, the data noise is greatly reduced. Figure 3 2(c) shows the result after the correction: the fringe pattern becomes more distinct and the unwrapped phase map become more consistent and cleaner. 55 Data areas used / for calculation Figure 32— Pixel quality examination and correctionfor lower mirrorfringe in cross- slitting case: (a) Pixel quality image, (b) Rawfringe pattern and unwrappedphase map (c) Fringe pattern and unwrappedphase map after correction Table 3 lists the result for cross-slitting using corrected phase change data and raw phase change data. The pixel quality control and correction procedure greatly enhances the (a) (b) (c) 56 calculated single depth cr stress component from 48. iMpa to 74.5Mpa, which closely agrees with the stress value 73.8Mpa from the bending test. 5.3.1.2 Single-slitting versus Cross-slitting Both single-slitting and cross-slitting experiments were done in this research. Figure 33 shows the fringe pattern and pixel quality image for the single-slitting case. Figure 33 — Fringe pattern andpixel quality imagefor single-slitting: (a) Fringe pattern after correcting badpixels (b) Pixel quality image using lower mirror For the single-slitting case, the corrected data in those two pink rectangles were used to recover the residual stresses in the single depth case. As shown in Table 3, the calculated uniform stresses o for both single-slitting and cross-slitting agreed well and they converged to the measurement from the bending test. The large single-slitting result o (- 80.2Mpa) is obviously wrong. This is because single-slitting only releases the stresses perpendicular to the cutting face and the displacement from single-slitting has little capability to evaluate the stress component it hardly released. Thus, cross-slitting is more capable of recovering all the in-plane stress components. (a) (b) 57 5.3.1.3 Dual-axis Measurement versus Single-axis Measurement Another big improvement for ESPI/cross-slitting method is that a dual-axis measurement is achieved. Dual-axis measurements using two orthogonal displacement measurements rather than one directional displacement data makes the residual stress measurement more robust. Figure 34 shows the Fringe patterns measured in two sensitivity directions Sensitivity direction Figure 34— Fringe patternfrom dual-axis measurement: (a) Fringe pattern using Lower mirror (b) Pixel quality image using upper mirror The phase maps for this two cases favour the stress components in two sensitivity directions. Using only one of them gives a biased estimation of all in-plane stresses. By considering these two together, all the in-plane stress components can be balanced and. Table 3 lists the results of using these two directional data separately and together. It can be shown that by using the two mirrors together, all the three calculated stress components reproduce the expected values from the bending test better than by using just one of them. For example, using the lower mirror alone, o is 78.1 Mpa, and using upper mirror alone, o is 81 .6Mpa, but using both mirrors together, o- is 74.5Mpa much closer to the bending test. F Sensitivity direction (a) (b) 58 5.3.2 Incremental Depth (Non-uniform Stress State) Measurements The purpose of incremental depth measurements is to identify the residual stress profile vs. depth. The procedure follows that fringe images used for identifying uniform stresses at one depth increment are used as the reference images for the next increment. This procedure reduces the time interval between ESPI measurements and improves the measurement quality. Using least square method and solving Eq. (15), the incremental stress profile can be obtained. For the five-incremental cutting case, five fringe patterns can be obtained for each direction measurement. Figure 35 shows the five fringe patterns using lower mirror measurements. (d) (e) Figure 35—Raw incrementalfringe pattern using lower mirror: (a) Fringe pattern by cutting increment 1, (b) Fringe pattern by cutting increment 2, (c) Fringe pattern by cutting increment 3, (d) Fringe pattern by cutting increment 4, (e) Fringe pattern by cutting increment 5 (a) (b) (c) 59 Each of these fringes shows the additional displacement relaxation or phase change by cutting one more increment. The additional fringes observed after cutting the first three slot depth increments follows the expected pattern. The largest change occurs after the first depth increment, with progressively smaller changes after the second and third increments. Thus, the fringe pattern in Figure 35 shows many fringes in image (a), a moderate number in image (b) and few in image (c). Figure 35 (d) and (e) show that large rigid —body motions occurred during the cutting of the fourth and fifth slot depths increments, thereby creating many fringes that are observed. The reason for these rigid- body motions is not yet determined, but they occurred consistently with repeated measurements. While this behaviour is not desirable, neither is it destructive. The calculation method for evaluating residual stresses takes such rigid-body motions into account and eliminates their effects from the calculations. 60 --- -—-— Normal Stress X ,‘ \ —e—— Normal Stress Y L 1 2 3 4 5 6 Depth(mm) (a) 200. _______________________ 4 Bending Test I —— ESPI and Cross-slitting ‘ \150H -150 0 1 2 3 4 5 6 7 Depth(mm) (b) Figure 36— Residual stress calculationfrom ESPI and cross-slitting measurement: (a) Incremental residual stress profilefor all theses stress components, (b) Comparison of between ESPI cross-slitting measurement and Bending test Figure 36 shows the calculated stress profile from ESPI cross-slitting measurement. It can be seen that the profiles of all three stress components follow the expected trends. This 61 can be verified especially in Figure 36(b). The a stress profile from ESPI and cross- slitting agrees with the bending test result very closely. For the first three increments and last increment, the stress prediction almost falls on the stress curve from bending test. For the forth increment, there is an obvious discrepancy that needs to be further examined. These ESPI and cross-slitting measurements had been repeated for 4 times. All the measurements gave similar results and demonstrated the consistency of this measurement. According to Figure 36, ESPI and cross-slitting measurement gained success to predict trustable stress profile. 5.3.2.1 The Effect of Pixel Quality Control and Correction Procedure For incremental stress calculation, the pixel quality control and correction procedures play a particularly important role. This is because any defect in the phase map will cause errors that accumulate to the calculated stresses at greater depths. Figure 37 shows the ESPI cross-slitting measurement o profile from using and not using Pixel Quality Control. 62 200 150 100 - 1: i zzL Bending Test ESPI without Pixel Quality Control —— ESPI with Pixel Quality Control 0 1 2 3 4 5 6 7 Depth(mm) Figure 37— Residual stress o profilefrom ESPI/cross-slitting measurement and bending test Without using pixel quality control and correction procedure, the stress profile had more fluctuations and did not agree well with the one obtained from bending test. This is because without using pixel quality control and correction procedure, some of the phase maps as Figure 38 are not unwrapped correctly, causing faulty stresses calculations. Figure 38— The wrongly unwrapped 4th incremental phase map without correcting the badpixels -50 -100 63 5.3.2.2 Dual-axis Measurement versus Single-axis Measurement Both dual-axis and single-axis measurements has been done in this research. Figure 39 shows the o-, stress profile from dual-axis measurement and single-axis measurement. — Bending Test 200 L —e—— ESPI using Lower Mirror - - - —*-—- ESPI using Upper Mirror 150L_ —— ESPlusingTwoMirrors 100 ----- - Depth(mm) Figure 39— o, stress profilesfrom dual-axis measurement and single-axis measurement For these measurements, both the single-axis and dual-axis measurements gave similar results. However dual-axis measurements are preferred because they provide superior error correction when the data from a single-axis are faulty or of low quality. 5.4 Discussion The ESPI/cross-slitting method gives very good results for single depth (uniform stress) and incremental depth (non-uniform stress) measurements. Comparing with the result from a bending test, in the single depth case, the difference is about 0.9%. For the incremental depth case, the stress profile also closely follows the trend of stress profile obtained from bending test and only loses some accuracy in the 4th increment. The cross slitting technique helps to release all the in-plane stress components while single-slitting 64 only releases perpendicular stress components. The pixel quality control and correction procedures are effective at identifying and correcting the bad pixels. This stabilizes the unwrapping of the phase maps, greatly enhancing the data accuracy and stabilizing the residual stress calculation. Dual-axis measurements provide more data and balance the sensitivity to all three in-plane stress components. The reason for a large error of the 4th increment stress component needs to be examined further to achieve better stress profile calculations. 65 Chapter 6— Conclusion 6.1 Contribution Six major parts of work have been defined and achieved in this research work: 1. A novel cross-slitting method was proposed and executed. Cross-slitting releases all three in-plane stress components. This rectifies the disadvantage of the conventional single-slitting method, which indicates only the perpendicular stress components. Cross- slitting also leaves intact many high-displacement pixels near the cutting edge. Data from these pixels make the cross-slitting method more sensitive to the residual stresses and enhance the accuracy of the residual stresses calculation. An electric drill was fixed on a 3-axis working table programmed to move precisely to control the cutting length and direction. 2. A new dual-axis ESPI measurement setup was designed. This was achieved by introducing double-mirror and shutter devices. They created dual-plane illumination with perpendicular sensitivity vectors. This overcomes the limitation of the commonly used single-axis ESPI measurement that has a bias for the residual stresses in the sensitivity direction. Dual-axis measurements ensure that all three in-plane stress components have similar calculation sensitivities. 3. Many optical improvements have been made. A more powerful laser and a higher quality CCD camera with smaller pixel size were chosen. A new camera driver was written to change the exposure time and a telecentric lens was used to get a greater depth of focus. All these efforts led to an increased in the f-number to 28, which in turn 66 increases the speckle to 12pm, more than sufficient to fit on the 6pm pixel size of the camera. 4. A novel pixel quality control and correction procedure was successfully introduced. The procedure involves examining the displacement data pixel by pixel and verifying the saturation, modulation and variation are within acceptable ranges. All bad pixels were interpolated and recovered from adjacent functional pixels. This stabilizes the unwrapping algorithm and plays a very important role in achieving accurate stress calculation results. 5. The relationship between residual stresses and surface displacements has been examined for both single depth (uniform stress state) and incremental depth (stress profile) cases. The algorithm has been described in Chapter 2. The kernel matrices for both cases were built up from finite element calculations. Single depth and incremental finite element models were built in ANSYS 10.0. All the displacement data from the FE calculations were interpolated to compose the components in the kernel matrix. 6. In ESPI measurements, there are hundreds of thousands of data points available from each measurement, but just 9 unknowns quantities to be determined (three stress components and 6 additional motions). Thus, the problem is highly overdetermined. A least squares solution was used to find the best fit solution to the data in the calculation of residual stresses. This method uses most of the available data and therefore retains the averaging advantages. The residual measurement results demonstrate that for both single depth and incremental depth cases, the least squares solution both gives good measurement results. 67 The relative merits of cross slitting and ESPI are combined in this research. All these efforts lead successively to fulfill the objective of this research that is to obtain reliable and accurate measurement for uniform and non-uniform stress states. 6.2 Remaining Challenges Satisfactory measurement results have been obtained from this research. However there are still some challenges. 1. The quality of the interference patterns is not ideal. By examining the image quality, it can be found that the illumination intensity distribution on the whole field of the image is not even everywhere. The light is concentrated in the middle causing some of the central pixels to be saturated. Loss of these pixels is significant because their potential data content is high. In addition, the intensities from reference and illumination beams are not identical. If the intensities of these two beams could be balanced well, the modulation can become a little greater. 2. For the single depth uniform stress case or the first increment cutting of the incremental depth case, a large surface area near the slots was lost because the chips flying out of the cut tended to scratch the surface. Therefore, many useful pixels with high deformation around the cutting edge of the slots were lost. If these lost pixels can be saved, more useful data can be acquired and that can improve the accuracy and stability of the residual stress calculation. 3. The stress profile obtained from ESPI and cross-slitting measurements agrees with the measurement from bending test well. However, the solution presently does not include regularization capabilities. Regularization would reduce the noise content in the stress 68 profile measurements. Further mathematical work is needed to include regularization in the residual stress calculation. 4. At present, the residual stress calculation is limited to specific incremental depths, specific end mill size and specific workpiece dimension. This is not flexible for industrial use. Adaptability and convenience should be further extended in the future. 6.3 Future Work and Recommendations The cross-slitting and dual-axis ESPI measurements were successful in identifying the uniform and non-uniform residual stresses. Following the path of this research, more work can be done to make the measurement more stable and accurate. The laser beam used for illumination could be expanded so that it illuminates a larger area. This could make the light intensity more even in the central area used for imaging, and thus reduce the saturated pixels. Beam splitters with slightly different transmission and reflection ratios could be used to balance the intensity of the illumination and reference beams. A suction device could be designed to suck the chips created during cutting and prevent these chips scratching the surface of the workpiece. If this is successful, the image quality could be much improved and the number of high variation pixels could be greatly reduced. The data area used for calculations can then reach closer to the slot boundary, thereby facilitating more accurate measurement results. To get better profile prediction, a regularization technique should be introduced to stabilize the least squares calculation and reduce noise content. This involves estimating the standard deviation of the error in each of the incremental phase maps and choosing an 69 appropriate level of regularization to reduce noise content without distorting the true stress solution. A database should be developed from a family of finite element calculations which are more adaptable for practical use, e.g. different diameters of cutters, different depth of increment and different dimensions of workpieces. This constitutes the next stage of work regarding the practical use for industry application. 70 References [1] Noyan I. C., Cohen J.B. “Residual Stresses in Materials.” American Scientist. Vol. 79, No. 2, PP. 142-154, 1991. [2] Lu, Lian “Handbook of Measurements of Residual Stresses”. Fairmont Press, INC. Lilburn, GA. 1996. [3] Prime, M. B. “Residual Stress Measurement by Successive Extension of a A Slot: The Crack Compliance Method,” App. Mech. Vol. 51, pp. 375-381, 2006. [4] Schajer, G. S. “Use of Inverse Solutions for Residual Stress Measurements.” Journal of Engineering Material and Technology. Vol. 128, pp. 375-38 1, 2006. [5] Schajer, G. S. “Residual Stress Solution Extrapolation for the Slitting Method Using Equilibrium Constraints,” Journal of Engineering Materials and Technology-Transactions of the American Society of Mechanical Engineers. Vol.129, pp. 227-232, 2007. [6]Schajer, G. S. “Residual Stresses: Measurement by Destructive Testing.” Encyclopedia of Materials: Science and Technology, Elsevier, pp. 8152—8158, 2001. [7] ASTM. “Determining Residual Stresses by the Hole-Driling Strain-Gage Method.” ASTM Standard Test Method E837-08. American Society for Testing and Materials, West Conshohocken, PA. 2008. [8] Steinzig M, Ponslet E. “Residual Stress Measurement using the Hole Drilling Method and Laser Speckle Interferometry part I”. Experimental Techniques. Vol. 27, No. 3, pp. 43-46, 2003. 71 [9] Schajer GS, Steinzig M. “Full-field Calculation of Hole-drilling Residual Stresses from Electronic Speckle Pattern Interferometry data”. Experimental Mechanics. Vol. 45, pp. 526-532, 2005. [10] Ham S., Lee J., Park S. “Vibration Analysis of Gyro Sensors by using ESPI Technique”. Proceedings of the SPIE - The International Society for Optical Engineering. Vol 58, No. 1, pp. 220-225, 2005. [11] Scales, J., Smith, M., Treitel, S. “Introductory Geophysical Inverse Theory”. Samizdat Press. Center for Wave Phenomena. Dept. Geophysics, Colorado School of Mines”, 2001. [12] Schajer GS, Prime, MB. “Residual Stress Solution Extrapolation for the Slitting Method Using Equilibrium Constraints”. Journal of Engineering Materials and Technology. Vol. 129, pp. 227-229, 2007. [13] Schajer GS. “Measurement of Non-uniform Residual-stresses using the Hole- Drilling Method .1. Stress Calculation Procedures”. Journal of Engineering Materials and Technology-Transactions of the Asme. Vol. 110, pp. 338-343, 1988. [14] Steinchen W, Yang L. “Digital Shearography: Theory and Application of Digital Speckle Pattern Shearing Interferometry”. SPIE - The International Society for Optical Engineering. Bellingham, WA. 2003. [15] Sirohi RS, Chau FS. “Optical Methods of Measurement: Wholefield Techniques”. Marcel Dekker, Inc. New York, NY. 1999. [1 6]Kadono H, Takei H. “A Noise-immune Method of Phase Unwrapping in Speckle Interferometry”. Optics and Lasers in Engineering. Vol. 26, pp. 151-164, 1997. 72 [17] Cloud G. “Optical Methods in Experimental Mechanics. Part 27: Speckle Size Estimate”. Experimental Techniques. Vol. 31, No. 2, pp. 19-21, 2007. [18] Mayville RA, Finnie I. “Uniaxial Stress-Strain Curves from a Bending Test”. Experimental Mechanics, pp. 197-201, 1981. 73

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