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Improvements in electronic speckle pattern interferometry for residual stress measurements Cavusoglu, Mehmet Cagdas 2007

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IMPROVEMENTS IN ELECTRONIC SPECKLE PATTERN INTERFEROMETRY FOR RESIDUAL STRESS MEASUREMENTS by MEHMET CAGDAS CAVUSOGLU B.Sc , Middle East Technical University, 2005 A THESIS SUBMITTED IN PARTIAL F U L L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES (Mechanical Engineering) THE UNIVERSITY OF BRITISH C O L U M B I A August 2007 © Mehmet Cagdas Cavusoglu, 2007 Abstract Electronic Speckle Pattern Interferometry (ESPI) is an optical technique used for measuring surface displacements in the order of a wavelength of light by comparing interference patterns taken before and after surface deformation. Residual stress measurement is one of the applications where ESPI is useful. The technique is attractive because it provides very detailed information on deformation field and low per-measurement cost. However, ESPI data possess high noise content due to its high sensitivity to disturbances. In this research, factors that affect the quality of ESPI data were studied. The most important ones were found to be the specimen surface quality, illumination level and speckle size. The image quality was greatly improved by surface preparation. Good and faulty ,data were separated by evaluating their modulation level and identifying the saturated pixels. Mathematical methods were proposed to improve the data quality by either replacing the faulty data with good data or smoothing the data by filtering. Two common-path arrangements with single and double mirrors, which provide in-plane sensitive measurements, were designed to eliminate the separate and delicate optical paths. They improved the stability of ESPI measurements and greatly reduced the pixel drift that was a problem in the existing arrangement. The double mirror arrangement provided the measurement of full stress field. The single mirror method was confirmed by stress measurement, whereas the double mirror method could not be validated due to low data quality. Recommendations were made for an enhanced future design of this method. ii Table of Contents Abstract ii Table of Contents iii List of Tables v i List of Figures vii Acknowledgements ix Chapter 1 Introduction 1 1.1 Introduction to Electronic Speckle Pattern Interferometry 1 1.2 Residual Stresses 2 1.2.1 Formation and Effects of Residual Stresses 3 1.2.2 Measurement of Residual Stresses 6 1.3 Objective 7 1.4 Proposed Method 7 Chapter 2 Electronic Speckle Pattern Interferometry 9 2.1 ESPI Principles 9 2.2 ESPI Procedure for Measuring Displacements 12 2.3 Requirements 15 Chapter 3 Quality of ESPI Measurements 17 3.1 Parameters Affecting Measurement Quality 17 3.1.1 Phase Stepping Algorithm 17 iii 3.1.2 Modulation 20 Speckle Size 21 Illumination Quality 24 Surface Conditions 26 3.2 Mathematical Noise Reduction 28 3.2.1 Removing the Faulty Data 29 3.2.2 Sine-cosine Filtering 30 3.3 Discussion 32 Chapter 4 Improvements in ESPI Arrangement 34 4.1 Common-path Arrangements 34 4.1.1 Single Mirror Arrangement 34 4.1.2 Double Mirror Arrangement 39 4.2 Discussion 43 Chapter 5 Residual Stress Calculation 46 5.1 Forward Calculation 46 5.1.1 Data: Surface Displacements 46 5.1.2 Model: Stresses 46 5.1.3 Data and Model Relationship 47 5.2 Inverse Calculation 51 5.3 Calculation Stability 52 Chapter 6 Experimental Validation 59 6.1 Measurement of a Known Stress Field 59 iv 6.2 Discussion 66 Chapter 7 Conclusion 68 7.1 Contributions 68 7.2 Key Findings 70 7.2.1 Improving the measurement stability by eliminating separate light paths 70 7.2.2 The contribution of surface preparation to image quality 70 7.2.3 The importance of the quality of laser source and optical components 71 7.2.4 The importance of illumination level 71 7.2.5 The importance of the gap between the mirrors and object surface 72 7.3 Future Work and Recommendations 72 References 74 v List of Tables Table 5.1 - Norms of the column vectors and scaling factors for automatic scaling 54 Table 5.2 - Norms of the column vectors of combined kernel matrix for double mirror arrangement and scaling factors for automatic scaling 57 Table 5.3 - Condition numbers for single and double mirror arrangements 57 Table 6.1 - Statistics of the image sets 61 Table 6.2 - Calculated stresses using strain gages and single mirror ESPI 64 vi List of Figures Figure 1.1- Measurement points in: (a) ESPI fringe pattern, (b) strain gage rosette 2 Figure 1.2 - Residual stress profile in a windshield glass 4 Figure 1.3 - Effect of residual stresses in machining: (a) residual stress distribution before cutting, (b) distortion of the plate (Pingsha Dong, Battelle Labs.) 5 Figure 2.1 - Existing ESPI arrangement (Adapted from Stenzig and Ponslet [18]) 10 Figure 2.2 - Interference of light waves 11 Figure 2.3 - Sets of images obtained by phase stepping: (a) before surface deformation, (b) after surface deformation 13 Figure 2.4 - Intensity trend of a single pixel before and after deformation 14 Figure 3.1 - High and low modulated signals 21 Figure 3.2 - Arrangement for subjective speckle pattern (Adapted from Cloud [25]) 23 Figure 3.3 - Vertical fringes due to low illumination quality 26 Figure 3.4 - Diffuse and specular surfaces 27 Figure 3.5 - Phase data: (a) without surface preparation, (b) with surface preparation 28 Figure 3.6 - Phase maps: (a) raw data, (b) interpolated data 29 Figure 3.7 - Filtering the wrapped phase data (Adapted from Steinchen and Yang [28]) 30 Figure 3.8 - Phase maps: (a) raw data, (b) sine - cosine filtered data 31 Figure 4.1 - Common path arrangement with single mirror 35 Figure 4.2 - Illumination and reference beam vectors 36 vii Figure 4.3 - Synthetic fringe patterns: (a) under x-stress, (b) under y-stress 38 Figure 4.4 - Double mirror assembly and sensitivity vectors 39 Figure 4.5 - Double mirror arrangement 40 Figure 4.6 - Illumination and reference beam vectors in double mirror arrangement 41 Figure 4.7 - Addition of illumination and reflection vectors 41 Figure 4.8 - Fringe patterns generated using: (a) top mirror, (b) bottom mirror 43 Figure 4.9 - Used portions of the light: (a) without a beam splitter, (b) with a beam splitter 45 Figure 5.1 - Plane stress tensor 47 Figure 5.2 - Isotropic and shear stress states 48 Figure 6.1 - Diagram of the test specimen 60 Figure 6.2 - Image sets: (a) before hole drilling, (b) after hole drilling 62 Figure 6.3 - Phase maps: (a) with raw data, (b) with data cleaned by interpolation and sine-cosine filtering 63 Figure 6.4 - Badly functioning pixels 63 Figure 6.5 - Phase maps: (a) experimental, (b) theoretical, (c) residual 66 viii Acknowledgements I would like to express my sincere gratitude to my supervisor Dr. Gary Schajer, who has supported me throughout this thesis study with his wisdom and experience, and always showed me the right directions. Thank you for your encouragement and confidence in me. To my friends, thank you for making this Canadian adventure fun and helping me feel like I am home. Thank you for your patience toward me and for giving your precious time to edit this thesis. Finally, the biggest gratitude goes to my parents, who have always believed in me, who have provided moral and financial support even during the hardest times. You shared my dreams with me at the expense of missing your son for two years. Thank you for being proud of me. ix Chapter 1 - Introduction 1.1 Introduction to Electronic Speckle Pattern Interf erometry Electronic Speckle Pattern Interferometry (ESPI) is an optical technique that is used to measure very small surface displacements. ESPI determines the phase shift of coherent light waves to measure displacements. Since the wavelength of visible light is small (approximately 500 nanometers for green light), very small changes in the optical path length produce measurable changes in the intensity of an interference pattern. Thus, the method provides extremely accurate measurements in the order of a wavelength of light [1], and can be employed in applications such as residual stress measurements [2], vibration analysis [3], or material property identification [4], A CCD camera is used to collect the light and each pixel in the camera corresponds to a measurement point on the object surface. Hence, the technique measures the entire displacement field and provides a very large data set. A typical camera resolution is 480 x 640, which means the displacement measurement is made at more than three hundred thousand surface points. Figure 1.1a shows the contour map of a surface obtained using ESPI, also termed "fringe pattern". A fringe pattern is obtained by plotting the phase shift at each pixel due to surface deformation. Each line in this map shows the loci of constant phase difference, i.e., constant optical path difference. 1 ESPI technique brings an important challenge. The data obtained from ESPI measurements have high noise content. The grainy texture of the fringe pattern depicted in Figure 1.1a is caused by this noise. Since displacement measurement is based on the path length variations of light waves, the measurements are extremely sensitive to disturbances such as vibration and air current. These disturbances create undesirable variations in the light paths and result in inaccurate pixel readings. 1.2 Residual Stresses Residual stress measurement is an application of ESPI in which the stresses are calculated using surface displacements. These displacements occur due to the stress relief caused by the 2 removal of stressed material. Hole-drilling is a material removal technique used with ESPI in residual stress measurements. The richness of the ESPI data makes the method very attractive compared to typical three point strain gage measurements depicted in Figure 1.1b, because it provides much more detailed information on the deformation field. This method also eliminates the possible error that may occur with strain gages if the hole is not drilled in the center of the rosette accurately [5]. The non-contact nature of the measurement significantly decreases the time and effort spent for the installation of strain gages, providing a much lower per-measurement cost [2]. 1.2.1 Formation and Effects of Residual Stresses Residual stresses are the stresses that remain in an elastic solid body when all external loads and thermal gradients are removed [6, 7]. They are produced by the plastic deformation occurring during common processes such as welding, forging, heat treating, rolling, grinding and machining [6, 8, 9]. Undesired formation of residual stresses impairs the dimensional stability and leads to increase in machining cost. Performance of the material under thermal, mechanical and other kinds of loading depends on the state of residual stress induced during the manufacturing process [8]. 3 Residual stresses are self-equilibrating stresses. In equilibrium, the integral of the stress normal to a cross section of the specimen must be zero [6]. Figure 1.2 shows the cross section of a windshield glass, which is a type of toughened glass. There are compressive stresses on the surface introduced by a rapid cooling process and they prevent surface cracks from propagating. There are also tensile stresses at the midsection of the glass that provide the force and moment equilibrium throughout the thickness. i I Figure 1.2 - Residual stress profile in a windshield glass Residual stresses may be harmful or beneficial depending on their state and distribution in materials. For example, compressive residual stresses have a beneficial effect on the fatigue life, crack propagation and stress corrosion of materials, whereas tensile residual stresses reduce the performance [6]. Another effect of residual stress is the inaccuracy in machined components. This is often caused by the relief of residual stress, as stress relief produces deformation. Figure 1.3 demonstrates the distortion of a plate after machining. Figure 1.3a shows the residual bending stresses on the plate, where there are large compressions along the edges. Figure 1.3b shows that distortion occurs in the stressed plate after it has been cut into thin strips, while the stress free plate remains flat. The cutting operation separates the 4 sections of compression and tension allowing each strip to move freely. From the point of view of a manufacturer, this geometrical change is undesirable because it results in inaccurate product dimensions. Conversely, the change in geometry can be measured and used to determine the residual stresses that caused it. (b) Figure 1.3 -Effect of residual stresses in machining: (a) residual stress distribution before cutting, (b) distortion of the plate (Pingsha Dong, Battelle Labs.) 5 1.2.2 Measurement of Residual Stresses Residual stresses can be measured either by destructive methods through relieving the locked-up stress or by non-destructive methods [6, 7, 10]. Hole-drilling is the most commonly used destructive technique to obtain partial stress relaxation. It involves localized removal of stressed material by drilling a small hole, typically 1-4 mm in diameter, to a depth of approximately a diameter, and measurement of the deformation around the hole. The removal of stressed material within the hole redistributes the stresses in the surrounding material, resulting in surface deformation [5, 6]. Hole-drilling was first employed by Mathar in 1934. He used a mechanical extensometer to measure the displacements around a circular hole drilled through stressed plates [11]. This method did not provide a very stable measure of surface deformation. In 1950, Soete and Vancrombrugge introduced the use of strain gages, which significantly improved measurement accuracy and reliability [12]. However, strain gage measurements are very time consuming and expensive because they require meticulous installation and surface preparation. Also, the measurement accuracy is very sensitive to hole eccentricity [6]. Methods employing optical interferometry have been developed to measure surface displacements. In the mid 1980's, Antonov [13] and McDonach [14] showed that optical techniques could be used for surface displacement measurements around drilled holes. In the mid 90's, Nelson [15] proposed a method that uses holographic interferometry to measure the deformation map. Electronic Speckle Pattern Interferometry, which is the core of this 6 research work, is another optical technique commonly used by many researchers [2, 16, 17]. The richness of ESPI data is an important factor that makes the technique advantageous over strain gages that provide strain data from only three points. Ideally, ESPI is expected to provide higher measurement accuracy than strain gages due to the data averaging obtained from the rich data set. However, the high noise content of ESPI data diminishes this advantage and degrades the accuracy of measurements. 1.3 Objective The objective of this study is to improve the ESPI technique in order to obtain higher data quality for residual stress measurements. Hence, residual stresses can be calculated more accurately and the method can be used in practical applications with a higher level of confidence. 1.4 Proposed Method The factors affecting the quality of ESPI data are identified and their effects on the measurements are analyzed in Chapter 3. Also, a data analysis procedure is developed to assess the functionality of the data. The sources of non-ideal data are identified and methods are developed to reduce their effects. The existing ESPI technique is modified and two in-plane sensitive systems are developed in Chapter 4. With this modification, it is expected to obtain more useful data and a measurement setup that is more rigid and less susceptible to 7 disturbances. The proposed method is tested and validated for residual stress measurements in Chapter 6. 8 Chapter 2 - Electronic Speckle Pattern Interferometry 2.1 ESPI Principles The primary working principle of the ESPI technique is the interference of the coherent light waves and the measurement of the resultant light intensity occurring due to this interference. ESPI utilizes two laser beams as a requirement for interferometry and surface displacements are calculated using variations in the resultant light intensity. These intensity variations occur due to the phase shift of these laser waves with respect to each other as a result of the displacement of the surface points and consequently the change in the path length of the light. The phase shift information is directly related to surface displacements and is extracted from intensity data using a phase stepping procedure [ 1 8 , 1 9 ] . Figure 2.1 shows the arrangement used for ESPI measurements. In this arrangement, a laser source is used to provide a coherent beam that has a wavelength of 532 nm. This beam is split into two parts by a half-silvered mirror and transmitted through two different paths of fiber optic cables. One of these beams is pointed towards the specimen in order to illuminate its surface. This is the "illumination beam". Light reflected from the surface is then collected by a camera and an image of the surface is captured. The second beam, the "reference beam", is sent directly to the camera where it meets the illumination beam. These beams then combine on the camera surface giving a different intensity value at each pixel. The intensity 9 data are processed by a computer to create images from which calculations can be made [18, 19]. 4 RS232 j J F fl All E 1 iGRABlER \ 1. j i—^ |——^— • • ^  1 Compute Object Beam C C D Camera Illumination Beam Laser Source Reference beam Figure 2.1 - Existing ESPI arrangement (Adapted from Stenzig and Ponslet [18]) When the object surface deforms, the path length of the illumination beam changes, leading to a shift in its phase angle. Once this phase shift is determined, the displacement can be calculated in a straightforward manner. However, the camera used is only able to measure light intensity and cannot detect phase shift directly. The phase shift information is extracted by using a reference beam, whose phase angle is varied using "phase stepping" [18, 19]. The purpose of the piezoelectric actuator shown in Figure 2.1 is to create variations on the path length of the reference beam relative to that of the illumination beam. The piezoelectric 10 actuator expands and contracts depending on the voltage differences applied and moves the mirror in predetermined amounts. As the relative phase angle of the reference beam changes, the resultant intensity of the two beams varies due to the constructive and destructive interference shown in Figure 2 . 2 . If the beams are in phase, they interfere constructively and their amplitudes add up to give a bright pixel. If the beams are out of phase, they interfere destructively and their amplitudes cancel each other out creating a dark pixel. Any combination between these is also possible, creating different pixel shades. Using the phase stepping procedure before and after deformation, the phase shift of the illumination beam is calculated [18, 19]. / \ I l luminat ion Beam 7 \ + Reference Beam T L J Br ight Pixel Out of phase Reference Beam + \ Dark Pixel Figure 2.2 - Interference of light waves The ESPI measurements are only sensitive to the displacements occurring in the direction of the "sensitivity vector", labeled as k in Figure 2 .1. This vector bisects the object and 11 illumination vectors [18]. Therefore, the sensitivity vector can be changed by using a different optical arrangement according to the desired direction of displacement measurement. 2.2 ESPI Procedure for Measuring Displacements Two sets of four images are taken from the object surface before and after deformation. Within each set, the reference beam is stepped by a quarter of a wavelength, i.e., 90° phase angle, between consequent images. Figure 2.3 shows these two sets of images, which are called speckle patterns. Since these images depend on microscopic variations in the surface texture, they represent a random intensity distribution and each pixel is independent from the others. A pixel, however, is consistent with itself throughout the set and shows a sinusoidal intensity trend due to the phase stepping of the reference beam [18, 19]. Surface deformation is created using the hole-drilling method. A small hole is drilled into the stressed material, which causes a stress relief in the surrounding material due to the disturbance of the stress equilibrium [9, 2]. Single depth residual stress analysis assumes a uniform state of plane stress from the surface to the bottom of the hole because there are only in-plane stresses close to the free surface. Therefore, the displacements that occur after hole-drilling due to these in-plane stresses are mostly in-plane [18]. 12 Once the sets of images are obtained before and after deformation, the phase shift, and consequently the displacement at each pixel can be calculated. When the reference and object beams combine, the resultant intensity at a particular pixel follows the trigonometric relationship given in Equation 2.1 [1, 18, 20]. Image 1 Image 2 Image 3 Image 4 (b) Figure 2.3 - Sets of images obtained by phase stepping: (a) before surface deformation, (b) after surface deformation I = Iref+IobJ+2prefIobjcos(0-(p) (2.1) 13 Here, I r er and I 0 bj are the intensities of the reference and object beams, and (9-<p) is the phase difference between them. Furthermore, the angle cp is defined as the relative initial angle, whose variation in the course of measurements gives the phase shift. The pixel intensity I is measured, and the three unknowns are Iref, I0bj and <p. Therefore, at least 3 measurements are required for their solution. By using 4 different phase steps, 4 intensities are obtained for each pixel. Figure 2.4 shows the trend of these intensities Ii, I2, I3, I4 (before deformation), J | , J2, J3, J4 (after deformation) and the relative phase angles cp] and <p2. A 4-measurement algorithm is preferred over 3 in order to provide data averaging and decrease the sensitivity of the algorithm to the phase step errors [19, 20]. Pixel Intensity After hole, "J Before hole, "I 0 n/2 71 3TC/2 Step (9) Figure 2.4 - Intensity trend of a single pixel before and after deformation 14 By substituting the known variables in Equation 2.1, the relative phase angles can be calculated as given in Equations 2.2 and 2.3 [20, 21]. tanp hzl± (2.2) t a n ^ = ^ ^ - (2-3) The phase shift Q is found by subtracting (px and <p2, as shown in Equation 2.4. This, however, gives wrapped phase information because the inverse tangent function used to calculate (px and cp2 can only evaluate the angles in [-n, n]. Therefore, a phase unwrapping algorithm must be used in order to obtain the actual phase information [20, 21]. fi = - ? 2 - ^ , (2-4) The unwrapped phase is the projection of surface displacement, d, in the sensitivity vector direction (k ) as shown in Equation 2.5 [20, 21]. Q = kd (2.5) 2.3 Requirements There are certain conditions required to obtain useful ESPI measurements. The correlation between the pixels on the camera and the corresponding surface points is extremely important. A surface point that corresponds to a particular pixel before the deformation must 15 correspond to the same pixel after the deformation, so that the phase shift calculation can be done correctly. In order to avoid decorrelation, the optical setup and the specimen must be fixed on the optical table very rigidly, to eliminate relative motion. If the displacement of a surface point exceeds the pixel size during the measurement, it will move from one pixel to another and decorrelation will occur. Pixel decorrelation creates randomness in the data and degrades the measurement accuracy. The stresses introduced by the hole drilling process must not be significant compared to the stresses that are to be measured. In practice, this process creates some plastic deformation around the edge of the hole. Therefore, the data obtained from this region should not be included in the calculations [22]. 16 Chapter 3 - Quality of ESPI Measurements 3.1 Factors Affecting Measurement Quality The accuracy of measurements made by ESPI depends on the quality of data, which in turn is determined by various factors involved in the method. It is important to know and understand these in order to work with them efficiently and reduce any adverse effects. The most important factors affecting the data quality are: i. Phase stepping algorithm ii. Modulation a. Speckle size b. Illumination quality c. Surface conditions 3.1.1 Phase Stepping Algor i thm When two light waves have the same frequency and are polarized with their field vectors in the same plane, the resultant intensity occurring due to their interference can be expressed as I„(x,y) = a(x,y) + b(x,y)cos<p„(x,y) (3.1) where/,, is the intensity recorded by the pixel (x,y) at the nth frame. Also, a is the mean intensity, b is the amplitude and cp is the phase difference between two waves [1, 20, 21]. 17 When the phase of one of the beams is stepped by a known angle a, the interference equation becomes: I„{x,y) = a(x,y) + b(x,y)cos[tp(x,y) + an] (3.2) Here, a, b and cp are the three unknowns, which means at least three equations (three phase steps) are required to solve for them. In order to obtain a general solution for an N-step algorithm, a least squares method is used to calculate the unknowns. Minimizing the sum of the quadratic errors with respect to a, b and (p yields [20]: ur ^ E l n cos a„ )2 + (X I„ sin a„ J N and 12 (3.4) Y In sin an tan^(x,^) = = - (3.5) Z^1,, cosan When the four-step algorithm is used for phase stepping as mentioned earlier, Equations 3.3, 3.4 and 3.5 become: a = -i 2—* 1 (3-6) b = J[(f-h)2+(I2-h)2] ( 3 ? ) 18 tan<p = ^ - ^ L (3.8) W 3 Solutions can be derived for other algorithms with different number of steps and phase step angles by following the same procedure. The four-step algorithm can be modified by using the same step angle (0, 90°, 180°, 270°, 360°) and taking an additional frame (five steps). In this case, the phase difference can be calculated as [20]: tan«p = 7 ( 1 4 ~ ! l ) (3.9) 4 / , - / 2 - 6 / 3 - / 4 + 4 / 5 In theory, the five-step algorithm allows for better data averaging and improvement in error resistance. Huang and Yun [17] propose that the simulations for three different phase calculation algorithms show that the five step algorithm with accurate 90° phase step is less sensitive to the error in phase stepping angle. The five-step algorithm described above was implemented in the ESPI measurements in order to reduce the effect of phase stepping error. However, it was observed that the additional frame made no or insignificant improvement in the noise content. The phase step angle must be calibrated to the desired value before taking measurements. This angle can be calculated for each pixel by using the recorded intensities as given in Equation 3.10, where the angle is assumed to be constant throughout the phase stepping 19 process [20]. The calibration can then be done by controlling the piezo voltage until the desired value is reached. ( / 1 - / 7 ) + ( / 3 - / 4 ) cosa = —• = - — (3.10) 2 ( / 2 - / 3 ) 3.1.2 Modula t ion Equation 3.1 can be expressed as: I„(x,y) = a(x,y)[l + V(x,y) cos cp(x,y)] (3.11) where V(x,y)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 defined in Equation 3.1. > W ) = **4 (3-12) Following this definition, the intensity values recorded by pixels during the phase stepping process are used to calculate the modulation. When the four-step algorithm is used, modulation can be calculated as, J(I2 - 7 4 ) 2 +(/, - / 3 ) 2 V = XU i i L!_ (3.13) /, +/ 2 +/ 3 + / 4 where Ii, I2, I3 and I4 are the recorded intensities for a given pixel at sequential speckle patterns [20]. Figure 3.1 illustrates the intensity variations of high and low modulated pixels. Highly modulated pixels have high signal amplitudes, showing that phase stepping has a 20 significant contribution on intensity variations. Low modulated pixels, however, do not exhibit sizeable intensity variations and have low signal amplitudes. Thus, highly modulated pixels have higher signal-to-noise ratios and give more reliable information about the phase shift angle. In ideal conditions, all pixels are expected to give high modulations. In reality, however, the modulation level is affected by factors such as speckle size, illumination quality and surface conditions. Pixel Intensity / 1 , • • s~ High 14 Modulation i, ; Low Modulation i i 0 TC/2 7i 37T./2 Step (9) Figure 3.1 - High and low modulated signals Speckle Size When a diffuse object is illuminated by coherent light, such as laser, a grainy image structure is produced. This grainy light distribution, known as a speckle pattern, results from self-interference of numerous waves reflected from scattering centers on the surface of a diffuse object. The amplitudes and phases of these scattered waves are random variables [20, 23]. 21 Average speckle sizes can be calculated for two different types of speckle patterns. These are the objective and subjective speckle patterns. An objective speckle pattern is produced by the free space propagation of the scattered light waves. A speckle pattern formed at the image plane of a lens is termed the subjective speckle pattern, which is obtained from ESPI. It occurs as a result of the interference of waves from several scattering centers in the resolution element (area) of the lens. The average speckle size is calculated by CJS =— (3.14) D where A is the wavelength of light, D is the diameter of the aperture and b is the image distance [20, 21, 23]. In this case, the waves scattered from any single point of the object are focused to a corresponding point of the image as shown in Figure 3.2. The average speckle size can also be expressed by using the f-number (F#= f ID) as, as={\ + m)AF# (3.15) where m is the magnification (m = b/a) and / is the focal length of the lens. Since b is very small compared to a, magnification has a negligible effect on the speckle size. The f-number can be varied by adjusting the aperture in order to control the speckle size [20, 21, 23]. 22 coherent D 1 of phase -dark speckle waves in phase -bright speckle waves out a b Figure 3.2 - Arrangement for subjective speckle pattern (Adapted from Cloud [23]) The f-number should be adjusted such that the speckle size is greater than or equal to pixel size. This ensures that the phase changes produce maximum intensity variations on pixels, leading to high modulations. 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 the intensity change of individual speckles. Consequently, the signal modulation will decrease. When the f-number is increased to obtain larger speckles, the amount of light received by the camera may become insufficient, resulting in a low signal modulation. Hence, the optimum f-number should be found considering the available illumination level. The required f-number was calculated by using Equation 3.15 to enable proper adjustment of aperture. The CCD camera used has a pixel size of 5 um, which will also be the minimum 23 speckle size (crv m i n ) . Taking the wavelength of the light as 532 nm and setting m to zero, the smallest required f-number was calculated as 9.4. The camera aperture should be reduced as close to this number as possible. This reduces the received light intensity, which should not be allowed to drop below approximately 90, where the intensity scale is 0 to 255. Illumination Quality Illumination quality is one of the most important factors in ESPI because the phase data are directly related to pixel intensities. Illumination quality involves average light intensity and quality of the laser source and optical components. If the average light intensity received by the CCD camera is insufficient, the average modulation obtained from the image sets will be low due to the low signal-to-noise ratio. The average light intensity can be increased by using a larger camera aperture, which will result in a smaller speckle size. Modulation will drop if the average speckle size becomes smaller than the pixel size. In this case, using a higher powered laser source will produce a better solution if all safety precautions are taken. Another advantage of having a high intensity level is the possibility of using a smaller aperture without decreasing the average intensity under the desired level. Thereby, larger speckles can be obtained and the average modulation can be increased. If the light intensity received by a pixel is higher than the measurable intensity range, it will be saturated. These pixels do not yield any useful information and create noise 24 in the data. The average intensity should be adjusted such that a few pixels are saturated and the others see sufficient illumination. If the laser source used in the measurements is of low quality, the average intensity may fluctuate over time. This fluctuation may have a significant effect on data quality, if its frequency is higher than the frequency of frame grabbing during phase stepping. In this case, there will be a random intensity change at each pixel due to the fluctuation, which will result in inaccuracy in the phase calculation. If the fluctuation is slow such that the light intensity is constant throughout an image set, it will not affect the phase calculation. The laser quality may also affect the smoothness of the light wavefront and a low quality source may lead to undesired diffraction patterns. Although the exact reason for this effect could not be identified, it was observed that these patterns lead to dark areas on the object surface and decrease the pixel modulation. A fringe pattern obtained during the optical tests that were done by using the single mirror method described in the following chapter is shown in Figure 3 .3. The first and second image sets were taken without drilling a hole, and hence, no fringes were expected. At first, the vertical fringes observed in this pattern were considered to indicate a rigid-body motion. However, after repeating the tests and plotting the modulation curve over the horizontal axis, the reason for these fringes was found to be the diffraction patterns on the laser wavefront. The modulation curve was plotted in Figure 3.3 by averaging the modulation in each column. This curve showed that the modulation decreases dramatically in dark areas. After replacing the laser source with a higher quality 2 5 one, these fringes were completely eliminated. The higher quality laser provided a wavefront that was much smoother and free of undesired diffraction patterns. Also, the stability of light intensity during the measurement period was better than that of the lower quality laser. Figure 3.3 - Vertical fringes due to low illumination quality Surface Conditions A diffuse surface, which is an uneven or granular surface that reflects an incident ray at a number of angles, is needed to create a speckle pattern. The opposite is a specular surface, in which the reflection is mirror-like. Light coming to a specular surface from a single incoming direction is reflected into a single outgoing direction [24], Figure 3.4 illustrates the reflection from these surfaces. If measurements are taken from a specular surface, light will not be scattered and speckle creation will be inhibited. Since the light ray goes in one direction, the camera will not receive sufficient illumination. This will result in dark images and consequently lower data 26 quality as explained in 3.1.4. Another factor important to illumination level is the surface color. If the surface is dark, light will not be reflected enough and a low illumination will be experienced. To obtain high quality images, the object surface must be diffuse and have a bright color. r ( I) _ (( i> Di f fuse surface Specular surface Figure 3.4 - Diffuse and specular surfaces In order to satisfy these requirements and improve the quality of speckle patterns, a surface preparation method was developed. The method involves using sandpaper to remove any surface defects and spraying a thin layer of white paint onto the object surface. The surface defects may be scratches or stains that will be visible on the images. Since paint grains form a rough texture, the paint layer creates a diffuse surface. Furthermore, the white color increases the illumination received by the camera. Figure 3.5a shows that a fringe pattern obtained without surface preparation is very noisy and the fringe contrast is low. Moreover, undesired surface features such as scratches are visible, 27 rendering some of the data useless. Figure 3.5b shows a fringe pattern taken after surface preparation. It can be seen that the fringe contrast is much better and the surface features are not visible. Thus, the data quality has improved greatly after using surface preparation. Figure 3.5 - Phase data: (a) without surface preparation, (b) with surface preparation 3.2 Mathematical Noise Reduction Two different methods were developed in order to reduce the effect of noise in the phase data. In the first one, noisy data points are marked and completely removed from the data set. Thereby, it is aimed to prevent the correct data points from the bad effects of faulty ones. Second one is a filtering technique that is suitable for cleaning data in the wrapped form. This technique aims to filter the data without doing a serious damage to the wrapped phase distribution. 28 3.2.1 Removing Faulty Data Smoothing is a method that averages the good and faulty data. However, smoothing distorts the good data as well as ameliorating the bad data. To avoid this, faulty data can be identified and rejected from the data set. The empty data spaces created in the wrapped phase data must be filled before unwrapping. This is done by finding the good data points in the neighborhood of empty ones, and interpolating them to fill the empty points. Thus, the faulty data points are replaced with better ones without damaging the good ones. Faulty data are separated from good data by identifying the low modulated and saturated pixels. By setting a threshold for acceptable modulation, low modulated pixels are distinguished and discarded from the data set. Also, the saturated pixels are rejected by setting a maximum intensity limit and removing the ones with higher intensities. The phase maps obtained before and after interpolation are shown in Figure 3.6. 29 3.2.2 Sine - Cosine Filtering To unwrap the wrapped phase data successfully, noise in the original data must be reduced. This is especially important since unwrapping algorithms are sensitive to noise. If the wrapped data are filtered directly, sharp peaks will be interpreted as noise and smoothed out by the filtering algorithm. Figure 3.7 depicts the theoretical, noisy and filtered wrapped phase distributions. The peaks must be kept because they contain the actual wrapped phase information that will be needed when unwrapping is to be done. Wrapped Phase Distribution 71 -71 -7t Theoretical Noisy Filtered Figure 3.7 - Filtering the wrapped phase data (adapted from Steinchen and Yang [25]) It was mentioned earlier that the phase data were calculated using the inverse tangent function shown in Equations 2.2 to 2.4. This function can only evaluate the angles between - 7 r and n , which is the reason for obtaining wrapped phase data. Before evaluating the phase shift using Equations 2.2 to 2.4, it can be separated into its sine and cosine components as shown in Equation 3.16. Since sine and cosine are continuous functions unlike tangent, sharp peaks observed in the wrapped phase data do not appear in them. Therefore, they may be filtered without losing a significant amount of good data and the wrapped phase image can be reconstructed from these filtered data [25]. 30 tariff , -<f>2) = sin(^| -<j>2) tan^, - tan^ 2 cos(^, - ^ 2 ) 1 + tan ^, tan <p2 (3.16) Using Equations 2.2 to 2.4, phase shift can be expressed as: (/ 1-/ 3XJ 2-y 4)-(/ 2-/ 4)(j 1-j 3) N tan(^, -<j)2) = tan(Q) (f-I^J.-J^-n-IVT-j.) D (3.17) Then, the sine and cosine components of the phase map will correspond to the normalized values of numerator and denominator: sin(Q) = A" J N 2 +D2 cos(Q) = D •yjN2 + D2 (3.18,3.19) Figure 3.8 shows the phase maps obtained before and after the application of sine - cosine filtering. It can be seen that the noise is reduced greatly and the fringe contrast is not lost. (a) (b) Figure 3.8 - Phase maps: (a) raw data, (b) sine - cosine filtered data 31 3.3 Discussion In order to investigate the effects of the factors discussed in this chapter, optical tests were done by using the single mirror method presented in the following chapter. During these tests, variation of noise content of data was observed by calculating the amount of low modulated and saturated pixels. The most important contribution to the data quality was obtained by surface preparation. It was found that a significant improvement in the overall data quality can be ensured by preparing the specimen surface before the measurements. The white paint sprayed onto the surface provides a diffuse surface that is free of defects. Moreover, it increases the light intensity received by the camera by providing a bright surface. Hence, the image quality improves with the use of surface preparation. Modulation is a parameter that provides a measure of the quality of data points. High modulated pixels are desired since they yield higher signal-to-noise ratios. One of the factors affecting the pixel modulation is the speckle size. The f-number must be chosen such that the average speckle size is greater than or equal to the pixel size. The required f-number for the camera was calculated as 9.4. However, a large f-number results in dark images because the amount of light received by the camera decreases. Thus, this choice is possible with a greater level of illumination. The laser quality was also found to be important in terms of obtaining a smooth light wavefront and higher data quality. 32 In theory, the phase stepping algorithm used in the measurements affects the data quality as well. By using an algorithm with a higher number of phase steps, data averaging can be achieved and the effect of phase stepping error can be reduced. However, during the optical tests, it was found that using different algorithms did not change the data quality significantly. Noisy data can be cleaned by filtering and removing the faulty data, or interpolation. Quality of the phase maps improves greatly when these methods are used. However, filtering causes distortion in the useful data as well as improving the faulty data. These methods should be used as a last resort after making every effort to obtain cleaner data from the measurements. 33 Chapter 4 - Improvements in ESPI Arrangement The ESPI arrangement shown in Figure 2.1 employs long separate fiber paths for illumination and reference beams that are extremely delicate and sensitive to external effects such as air currents and vibration [22]. These disturbances affect the separate paths differently and it becomes extremely difficult to keep them stable relative to each other. Instability in these paths results in drift in the interference pattern and causes noise in the measurements. This effect can be termed "pixel drift" due to the drift motion observed in live images. Pixel drift can be greatly reduced by modifying the ESPI arrangement. Using a common path for both illumination and reference beams eliminates the relative drift because both beams will experience the same effects that will cancel each other. 4.1 Common Path Arrangements 4.1.1 Single Mir ro r Arrangement A new ESPI arrangement was designed and built to use a common light path for illumination and reference beams. Figure 4.1 shows a diagram of this arrangement. Here, the beam emitted from the laser source passes through a lens where it expands and propagates toward the object surface. A mirror attached to a piezo is placed on the path of this beam and set perpendicular to the object surface. A part of the expanding beam hits to the surface directly and acts as an illumination beam. The remaining part hits the mirror and is reflected toward the surface. When the mirror is stepped, this part of the beam functions as a reference beam. 34 Light reflected from the object surface is then imaged by the CCD camera and the interference pattern is recorded. I I Laser Source C C D Camera Figure 4.1 - Common path arrangement with single mirror Since the distance between the laser source and the object is large compared to the size of the illuminated area, beams falling onto the mirror and the surface can be assumed to be parallel. As a result of this, beams illuminating the object surface will be symmetrical with respect to the surface normal as shown in Figure 4.2. The error caused by the angle difference will not have a significant effect on calculations. In this case, any out-of-plane movement of the surface will change the path lengths and the phases of both beams by the same amount. 35 Consequently, the resultant intensity produced at the camera will stay constant. Thus, the measurement sensitivity obtained using this arrangement will always be in-plane. Object Surface z Figure 4.2 - Illumination and reference beam vectors This result can also be shown by calculating the sensitivity vector using the vectors given in Figure 4.2. Here, k\ and k2 are the vectors for illumination and reference beams. The sensitivity vector is [20]: k=k7-k, (4.1) When the laser beam makes an angle 6 with the surface normal, sensitivity vector can be expressed as: k=— [(sin 0)1 - (cos 0)k] - — [(- sin 6)1 - (cos 0)k] (4.2) X X k=—(2sm0)l (4.3) X 36 Then, the phase shift due to an in-plane displacement dx is [20]: n = —(2sm0)dx (4.4) A Besides being relatively stable, this arrangement is an improvement over the existing one with its in-plane measurement sensitivity. Since the surface displacements observed are mainly in-plane, this arrangement yields more useful displacement data. The sensitivity to in-plane displacements would be lower in the presence of an out-of-plane sensitivity component. Thus, in this case, the in-plane measurement provides higher measurement accuracy than that would be obtained by out-of-plane measurements. This arrangement has a disadvantage as well. Since the sensitivity vector is along the x-axis, measurements are not sensitive to the displacements along the y-direction. Stresses in the y-direction cause displacements mainly in the same direction. Only much smaller Poisson displacements occur in the x-direction. Thus, the measurement sensitivity to y-stresses is expected to be much less than for x-stresses. 37 rjx= 150 MPa CTy= 150 MPa Measurement sensitivity of the method to x and y stresses was simulated mathematically using the code written in Matlab®. Theoretical stress fields were created around the hole using the forward calculation procedure presented in Chapter 5 and resulting phase maps were plotted. In the first case, a stress state consisting of an x-stress of 150 MPa was created (err =150MPa ,a = 0 and r = 0). In the second case, the same stress state was created in the y-direction (crx =0, <jy =150MPa and z =0). Figure 4.3 shows the phase maps plotted for these cases. It can be seen that the method produces approximately three fringes in response to the x-stresses, whereas it produces only a fringe in response to y-stresses due to Poisson displacements. The number of fringes is directly proportional to the magnitudes of measured displacements and consequently to stresses that caused them. Hence, the accuracy of y-stresses measured using this method is about three times less than the accuracy of x-stresses. 38 4.1.2 Double Mirror Arrangement The single mirror arrangement was modified in order to be able to measure the full in-plane stress field. In this new arrangement, two stepping mirrors were used, each of which gives a different sensitivity direction. Figures 4.4 and 4.5 show a diagram of the mirror assembly and a picture of the arrangement respectively. The mirrors were set 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. Sensitivity vectors can be derived using the diagram given in Figure 4.6, where k] shows the illumination direction, nl and nh show the normal directions of top and bottom mirrors, and k2l and k2h show the directions of illumination beams reflected from these mirrors. Illumination direction is the same as in the single mirror arrangement, and given as: Object Surface Figure 4.4 - Double mirror assembly and sensitivity vectors 39 jfc, = — [ ( - s i n 6)1 -(cos 6)k] (4.5) Normal directions of top and bottom mirrors are: 1 r 1 -: Using Figure 4.7, illumination and reflection vectors in top mirror can be added as, k]+k2L=2[k,+(krn,)nt] (4.8) 40 that yields kt - k2t - kx = 2(k{ • nt )nt k, = ^[(-sm0)i+(sm0)j] A Similarly, sensitivity vector can be derived for the bottom mirror as: 2TT, kh=— [(sm0)i +(-sm0)j] A (4.9) (4.10) (4.11) y Top mirror Bottom mirror Figure 4.6 - Illumination and reference beam vectors in double mirror arrangement Mirror Figure 4. 7 - Addition of illumination and reflection vectors 41 Since the arrangement employs two phase stepping mirrors, two separate measurements must be made to obtain two data sets. The measurement steps are: 1. Take two sets of images using one mirror at a time. 2. Drill a hole and create surface displacements. 3. Repeat step 1. 4. Obtain two phase distribution maps (fringe patterns) corresponding to two measurements using the data collected before and after deformation. 5. Unwrap the phase distribution maps and obtain displacements. 6. Construct a vector composed of the displacement data obtained from both measurements. 7. Construct a matrix that relates the displacements to the residual stresses that causes them (construct the kernel matrix). 8. Solve the equation for the stress variables. (Steps 6, 7 and 8 will be explained in detail in Chapter 5.) Figure 4.8 shows the typical fringe patterns generated using the top and bottom mirrors separately. These fringe patterns are obtained from a specimen containing only x-stresses. Therefore, the fringes are aligned along an axis between the horizontal axis and the sensitivity axes. 42 (a) (b) Figure 4.8 - Fringe patterns generated using: (a) top mirror, (b) bottom mirror 4.2 Discussion The single mirror arrangement is advantageous over the existing ESPI arrangement by providing a common path for the illumination and reference beams that is more stable and insensitive to the disturbances. Pixel drift was greatly reduced and the improvement in the stability could easily be seen in the live images. Another feature of this arrangement is the in-plane measurement sensitivity. However, it provides high sensitivity only for the x-stresses and is relatively insensitive to y-stresses. The double mirror arrangement solves this problem by providing two measurements with sensitivity directions perpendicular to each other. This arrangement is able to detect the displacements in all directions and consequently reveal the full stress field. During the conduct of optical experiments done by using the single and double mirror methods, several important points were identified as being helpful to obtain high data quality. These are: 43 The amounts of light that directly reach the object surface and reflect from the mirrors must be approximately equal, so that the intensities of illumination and reference beam can be balanced. This can be achieved by pointing the laser beam toward the corner where the mirror and object planes intersect. The gap between the mirrors and the surface should not be too great. If this is not the case, the center portion of the light beam will uselessly illuminate the gap, leaving only the lower intensity light at the edges to illuminate the object. When double mirror assembly is used, the vertical distance between the mirrors should also be kept small in order to minimize the gap. In future, a beam splitter assembly can be used to separate the beam into parts and send them to the mirrors. Thus, the center portion of the beam can be used more effectively. Figure 4.9 illustrates this idea. The object and mirror assemblies must be fixed on the optical table rigidly. Otherwise, disturbances such as vibration and drilling forces will create small motions that will lead to pixel decorrelation. In order to reduce the affects of external vibration, a vibration damping mechanism must be incorporated in the optical table. Also, the measurement setup can be enclosed by a cover that prevents it from being affected by air current and the ambient light. The mirrors must be covered with a protective screen during hole-drilling because the chips produced by the drill may damage the mirror surface. The object surface must be 44 cleaned with a soft brush after hole-drilling in order to clean the chips that may be stuck on the surface. (b) Figure 4.9 - Used portions of the light: (a) without a beam splitter, (b) with a beam splitter 45 Chapter 5 - Residual Stress Calculations The ESPI technique provides a rich data set composed of surface displacements caused by local stress relief in the specimen. The original stresses can then be calculated by inverting the relationship between the displacements and stresses defined by the linear elasticity theory. The volume of displacement data is much higher than the number of stress components to be recovered. Therefore, the problem is highly overdetermined and a least squares solution is used to find the best fit to the data. 5.1 Forward Calculation 5.1.1 Data: Surface Displacements The detailed procedure for calculating the surface displacements was presented in Chapter 2. A displacement value D:j in the direction of the sensitivity vector is obtained for each pixel included in the calculation. For a video frame with 480 x 640 resolution, there are over 300,000 pixels available. 5.1.2 Model: Stresses Plane stress can be assumed when considering the stress state since hole drilling is used to calculate the residual stresses that exist close to the free surface. The model parameters are the two in-plane normal stresses (cr x, ay) and the in-plane shear stress (rV ).) as shown in 46 Figure 5.1. Also, in-plane rigid body motions can be included in the model in order to separate them from the displacement data. These are the translations in x and y directions (wx and wv) and the rotation with respect to z axis (w,). xy CT, xy C T = C T r yx 0 xy C T 0 0 0 0 Figure 5.1 - Plane stress tensor 5.1.3 Data and M o d e l Relationship When hole drilling is applied on the specimen, the surface around the hole deforms in three dimensions in response to the residual stresses. Therefore, for each surface point, there are axial, radial and circumferential displacement components. In order to improve numerical conditioning, the stress components crx, av and rxy are transformed into the stress variables given in Equations 5.1-5.3. Here, P represents the isotropic stress and Q and T represent the shear stress states as shown in Figure 5.2 [2, 26]. P = {ax+cry)/2, Q = (ax-cjy)l2, T = zxy (5.1,5.2,5.3) 47 P Q T Figure 5.2 - Isotropic and shear stress states Each displacement is related to these stress variables via basis functions that are determined from linear elasticity theory. These basis functions give the displacement responses when the stresses are considered to act separately on a plate weakened by a circular hole [27]. The displacement at each point can be calculated as a response to the superposition of these stresses (P, Q and T) by using the basis functions. Since the proposed ESPI arrangements are only sensitive to the in-plane displacements, the axial displacement components need not be considered. When the isotropic loading, P, is acting alone on an isotropic specimen, the deformations are [2,27]: Ur(r,e) = ur(r\ Ue(r,0) = O, ur(r) = r(\ + v)(a/rf (5.4,5.5,5.6) 48 In equations 5,4-5.6, r and 9 indicate the position of the pixel in cylindrical coordinates, Ur(r,9) and Ue(r,6) are the radial and circumferential displacements due to P, and v is the Poisson's ratio of the specimen. In order to express the displacement in dimensionless form, the loading is normalized with respect to Young's modulus (E). When the shear loading at 45° to the x-y axes, Q, is acting alone, the deformations are [2, 27], Vr(r,9) = ur(r)cos29, V9(r,9) = -os(r)sm29, (5.7,5.8) or(r) = r[4(a/r) 2 -(1 + v)(aIr)4], ve{r) = r[2(l-v)(a/rf + (1 + v)(aIr)4] (5.9,5.10) where Vr(r,9) and Vg(r,9) are the radial and circumferential displacements due to Q. When the shear loading in the axial directions, T, is acting alone, the deformations are [2, 27], V;(r,9) = ur(r)sm29, V'g{r,9) = ve(r)cos29, (5.11,5.12) where the superscript is added to indicate that the deformations for T loading are the same as for Q, rotated by 45°. In addition to the surface deformations, terms that account for the rigid body motions can be included in the calculations. The rigid body motions in the x and y axes are, Wx(r,9) = wx+w.-sm9 (5.13) a and 49 Wv (r,0) = wy + w, — cos 0 (5.14) where wx, wy and w, respectively, are the normalized amplitudes of the rigid body translations and the rotation around the z axis at r = a, the hole radius. Combining these cases, the relationship between the stresses and the displacements can be written in Cartesian coordinates as [2]: d(r,0) = ^ {Ur (r,6)cos0 i + Ur (r,0) sin6 j) + % E E Vr(r,0)cos0i-Vd(r,0)sm0 i + Vr(r,0)sin0 j + V0(r,6)cos0 j rV'r(r,6)cos0 i - V'g(r,0)sin0 i + V;Xr,0)sm0 j + V'g{r,0)cos0 j f + v wx + w, — sin 0 a ( i + (5.15) W + W —COS0 v 'a j Then, the measured displacement is [2], L\r,0) = d(r,0)-k= — {2(kx cos0 + ky sin#L (r)} 2E Q f cos6* - ky sin6*) + (kx cos36> + ky sin36>)] vr (r) IE |+ [(kx cos0 - ky sm0)- (kx cos36? + ky sm30)} ve(r) j T \ \jcx sin0 + ky cos0)+(kx sin36>- ky cos36>)] vr (r) 2E\+\kx sm6 + kv cos0)-(kx sin30-fc cos3i9)]^(r)j + + + wxkx + w k + wv r r — sin 0kx + — cos0k a (5.16) where k - kx i + kv j is the sensitivity vector. The forward calculation can be set up such that the displacement data for all measurement points can be obtained by applying a known stress field and rigid body motions. Equations 50 5.17 and 5.18 show the matrix representation of the forward calculation. Here, G is the kernel matrix that contains the basis functions given in Equation 5.16, D is a vector containing measured displacements and S is a vector containing the stress variables and rigid body terms. Gu Gu G l 3 G 1 4 G l 5 G 1 6 pn\ Gn7 GnZ G„4 GnS Gn6 GS = D (5.18) When the single mirror arrangement is used, sensitivity in the y-direction will be lost and kv will be zero. In this case, the rigid body term w and the corresponding basis function must be omitted from the calculation and the kernel matrix will be left with five columns. Otherwise, the matrix will be rank deficient and the inversion will be unstable. 5.2 Inverse Calculation It can be seen in Equation 5.18 that the stress calculation is an inverse problem since the stresses, that are convoluted with the basis functions, are to be recovered using the observed displacements. Moreover, there are approximately 300,000 data points used in order to p 2E Q D \ 2E T = i • r 2E Dn w. w y w. (5.17) 51 calculate 6 quantities. Thus, the problem is highly overdetermined and a least squares method, shown in Equations 5.19, is used to find the best-fit solution. G'GS = GTD (5.19) S = [GrG]-]GrD (5.20) The in-plane stresses can then be determined as v ^ P + Q, <rv=P-Q, TXV=T (5.21) 5.3 Calculation Stability Stability of the least squares calculation can be determined by evaluating the condition number of G' G . The condition number is a measure of sensitivity of a matrix to numerical operations. A matrix with a high condition number is said to be ill-conditioned. If the matrix G'G is ill-conditioned, noise in the displacement data will be amplified through inversion and lead to much larger errors in the recovered stresses. Thus, the condition number of this matrix must be kept as small as possible in order to minimize the sensitivity to noise. If any of the columns of the kernel matrix (G) have elements with magnitudes much smaller than the elements in other columns, this matrix may become numerically ill-conditioned. This occurs because these small numbers are rounded to zero in numerical calculations, rendering the matrix rank deficient. Thus, scaling can be done on the column elements of this matrix in order to achieve a better condition number and consequently more stable solutions. 52 The kernel matrix was constructed for the single mirror arrangement using the sensitivity vector given in Equation 4.5. Poisson's ratio was assumed to be 0.33 for aluminum and the hole radius (a) and illumination angle (0) were taken as 0.8 mm and 45° respectively. The measurement area was taken as 51 x 51 pixels to limit the data size so that a Singular Value Decomposition (SVD) could be done. The condition number of G'G was calculated as 4.9 • 10 7 . Since G1 G has a high condition number, this matrix is very ill-conditioned without scaling. In order to compare the orders of magnitudes of the basis functions, the norms of each column in G matrix were calculated and are given in Table 5.1. It can be seen that there is an order difference between the norms of first three columns and the last two. This problem was overcome by writing an algorithm that scales the norms of the column vectors automatically by taking the norm of the first column as reference. For this case, the scaling factors were calculated as given in Table 5.1. After scaling, the condition number of G1 G was calculated as 39. Thus, scaling the kernel matrix greatly improved the condition of this matrix by decreasing the condition number from 4.9 -107 to 39. Hence, more stable solutions and more accurate results are expected to be achieved by using the scaled kernel matrix. 53 G1(P) G2 (Q) G3 (T) G4(wx) G5(w,) Norm 0.04 0.08 0.08 48 76 Scaling Factor 1 0.5000 0.5000 0.0008 0.0005 Table 5.1 - Norms of the column vectors and scaling factors for automatic scaling Secondly, the condition of the double mirror arrangement was analyzed. Kernel matrices for top and bottom mirrors (G t and Gh) were constructed separately using the sensitivity vectors given in Equations 4.12 and 4.13. Although all six columns are used in these matrices, they have a rank of 5, because the sensitivity vectors produced by the mirrors are sensitive to the displacements in only one principal axis. Hence, inverting these matrices separately leads to unstable solutions because of their rank deficiency. The kernel matrices G, and Gh were decomposed into their eigenvectors using SVD in Matlab® as [U,A,V] = SVD(G), where U and V are the matrices containing data and model eigenvectors, and A is the diagonal matrix containing the singular values. The model eigenvectors for the two mirrors are shown in Equations 5.22 and 5.23. 54 V, = -0.0004 -0.0000 -0.0000 0.3565 -0.9343 0.0000 0.0000 0.0000 -1.0000 -0.0000 0.0000 0.0000 0.0008 0.0000 0.0000 -0.9343 -0.3565 0.0000 0.0000 -0.7071 0.0000 -0.0000 0.0000 0.7071 -0.0000 0.7071 0.0000 0.0000 0.0000 0.7071 1.0000 0.0000 -0.0000 0.0009 -0.0001 0.0000 -0.0004 0.0000 -0.0000 0.3565 0.9343 0.0000 -0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 -0.0008 -0.0000 -0.0000 0.9343 -0.3565 -0.0000 0.0000 -0.7071 0.0000 0.0000 0.0000 -0.7071 0.0000 -0.7071 -0.0000 -0.0000 -0.0000 0.7071 -1.0000 -0.0000 0.0000 -0.0009 -0.0001 -0.0000 (5.22) (5.23) Each row element in the columns of these matrices corresponds to a model parameter in the order P, Q, T, wx, wy and wz. In the ideal case where all the basis functions are completely different from each other, each of the eigenvectors will contain a unity element and the others will be zero. Thus, the model parameters will be easily separable and the matrix will be well-conditioned. Here, it can be seen that these matrices are close to the ideal case, except for the second and sixth columns. The 0.7071 values occur because the sensitivity directions are ±45° to x and y axes. If the coordinate axes were rotated by 45°, unit numbers could be seen in each of the matrices and the kernel matrices would be left with five columns. Alternatively, G, and Gh can be combined to construct a larger kernel matrix and stresses can be calculated as given in Equation 5.24, where Dt and Db are the displacement vectors 55 obtained using the top and bottom mirrors separately. Since this combined matrix is sensitive to the displacements in both principal directions, it has a full rank of 6 unlike the one for single mirror. The condition number of this combined matrix Gth'Glh was calculated as 3.9 • 10 6, which shows that the kernel matrix for this arrangement is also highly i l l -conditioned. G,, G,-, G,-, G<A G,c G, G b \ G b 2 J i 3 ^ 1 4 Gb3 G, IS w ) 6 b4 Gb5 Gh6 2E Q 2E T 2E w. (5.24) Using SVD, the matrix containing the eigenvectors was calculated similarly and is shown in Equation 5.25. It can be seen from this matrix that the values corresponding to wx and w are fairly distinct and the matrix is very close to the ideal case. 0 0 0 0 0 -1.0000 -0.0000 -0.0000 -0.0000 -1.0000 -0.0000 0 -0.0008 0.0000 -0.0000 -0.0000 1.0000 0 0.0000 -0.9497 0.3133 0.0000 0.0000 0 -0.0000 -0.3133 -0.9497 0.0000 0.0000 0 -1.0000 0.0000 -0.0000 0.0000 -0.0008 0 (5.25) 56 The combined matrix can be scaled similarly to decrease its condition number. The norms of the column vectors and the scaling factors are: Gl (P) G2 (Q) G3 (T) G4(wx) G5(wy) G6 (wz) Norm 0.04 0.08 0.08 34 34 76 Scaling Factors 1 0.5000 0.5000 0.0011 0.0011 0.0005 Table 5.2 - Norms of the column vectors of combined kernel matrix for double mirror arrangement and scaling factors for automatic scaling After scaling, the condition number for Gth' Glh was calculated as 9. The condition numbers for different cases are presented in Table 5.3. Arrangement Condition Number Single Mirror 4.9-107 Single Mirror (scaled) 39 Double Mirror (combined) 3.9-106 Double Mirror (combined and scaled) 9 Table 5.3 - Condition numbers for single and double mirror arrangements 57 These numbers show that a well-conditioned matrix has been obtained by using the double mirror arrangement and scaling the kernel matrix. The single mirror arrangement is sensitive to the displacements in only one principal direction, giving a matrix with a lower rank. The double mirror arrangement provides a better conditioned matrix because it is sensitive to the displacements in both principal directions. Thus, all the stress components and rigid body motions can be included in the kernel matrix and identified using this arrangement. It can be concluded that the double mirror arrangement provides better calculation stability than the single mirror arrangement. 58 Chapter 6 - Experimental Validat ion The ESPI apparatus was designed and built, and the calculation procedure was formulated as described in Chapters 4 and 5. In this chapter, experimental validation of the ESPI technique on stress measurements will be presented. The stress field measured by ESPI will be compared to the one found by strain gages. Since the physical arrangements and the calculations were designed for measuring in-plane stresses, an in-plane stress state was introduced to the measurement specimen. 6.1 Measurement of a Known Stress Field A bent plate assembly was used to create a test specimen with known stress state as illustrated in Figure 6.1. A thin aluminum plate with a thickness of 1.25 mm was fixed on a steel block that has a curved surface with a radius of 508 mm. The block also has a flat surface that is used to clamp the assembly on the optical table rigidly. With this arrangement, in-plane stresses were introduced to the curved specimen. The surface of the specimen was prepared by spraying a thin layer of white paint onto its surface as described in Chapter 3. The single mirror arrangement was set up and adjustments were made as discussed in Chapter 4. The direction of the laser beam was varied until the intensities of reference and illumination beams became equal. It was found that these intensities were equal when the laser beam was pointed toward the intersection of the mirror and object surface planes, with 59 an angle of 45° with the surface normal. A mean intensity of 94 was reached, where the measurable intensity range obtained from the camera is 0 to 255. Drilling direction • / Plate thickness: 1.25 mm Top View *• Bent plate on the curved surface Flat surface for clamping the assembly Figure 6.1 - Diagram of the test specimen Before making the actual measurement, several image sets were taken with and without hole-drilling, in order to check the rigidity of the specimen, observe the noise content of the images and obtain statistical information such as modulation and the percentage of saturated pixels. During these tests, the highest average modulation that could be reached was 0.30. A saturation of 2 % was found to be satisfactory in terms of the noise content of the images. The camera aperture was reduced further to decrease the light intensity and to reduce the 125 mm Front View 60 saturation, which, however, resulted in dark images. The stepping angle was calibrated to 90°. Various fringe patterns were obtained by using drill sizes 3/64", 1/16" and 7/64". It was found from these tests that a hole with a diameter of 7/64" (~ 2.8 mm) produced a visible fringe map that provided sufficient data for stress calculation. Once the drill location was fixed, the direction and the zoom of the camera were adjusted so that the hole is placed at the center of the measurement area. The stress measurement was initiated by taking the first set of images as described in Chapter 2 and storing them in the computer. Then, a hole with a diameter of 2.8 mm and a depth of 0.6 mm was drilled on the specimen surface. The hole depth was limited with this value in order not to cross the neutral axis and penetrate into the compressive stress zone. The second image set was taken after the surface deformation due to hole-drilling. Statistics of these sets are given in Table 6.1 and the speckle images are shown in Figure 6.2. Mean Saturated Low Modulated Rms Step Intensity Pixels (%) Pixels (%) Modulation Angle (deg.) First Set 94 2 4 0.29 89 Second Set 94 2 5 0.29 89 Table 6.1 - Statistics of the image sets 61 (a) (b) Figure 6.2 - Image sets: (a) before hole drilling, (b) after hole drilling Using the image sets, the raw phase map was plotted as shown in Figure 6.3a. It can be seen that the phase map has a grainy structure, indicating the high noise content. The noisy data were cleaned by using the interpolating algorithm and the sine-cosine filtering described in Chapter 3. Figure 6.3b shows the phase map obtained after these operations. The phase map was then unwrapped using an unwrapping algorithm incorporated in the software and the displacement data were obtained. However, some of the data points obtained from the measurement cannot be used to calculate the stresses because they do not provide any useful information. In order to determine the region that is to be included in the calculation, a map showing the badly functioning pixels was plotted and shown in Figure 6.4. 62 In this figure, the gray color represents the points that are low modulated or saturated. This map does not show the noise, but is used for identifying the good and faulty data points. Figure 6.4 - Badly functioning pixels The points enclosed by the hole boundary were completely decorrelated because the surface in that portion was removed by hole drilling. The area adjacent to the hole boundary was 63 damaged possibly by the chips produced during hole-drilling. It can also be seen that there are regions of decorrelated points on the left and right edges of the measurement area. The data obtained from these regions were rejected by choosing the inner and outer integration radii as r, / a = 1.35 and r2la = 2.87. The stresses were calculated using the displacement data and the calculation procedure described in Chapter 5. Table 6.2 presents the calculated stresses along with the stresses found by strain gage measurements. The results show that the proposed ESPI method gives stress values in agreement with the expected values. ESPI results indicate that some rigid body motions occurred during the measurement. This can also be seen from Figure 6.3. The areas far away from the hole are expected to be white since the displacements at these regions are zero. However, the phase map shows dark areas in these regions indicating rigid body motions. Method (MPa) (MPa) (MPa) x-translation (Um) y-translation (jam) z-rotation (um) Principal Angle (deg.) Strain Gage 77 21 4 - - - 4.1 Single Mirror ESPI 71 22 5 -0.146 - 0.003 5.8 Table 6.2 - Calculated stresses using strain gages and single mirror ESPI 64 In order to validate the ESPI results, the calculated stresses and rigid body motions were used in forward calculation to create theoretical phase data. These forward calculated data were then subtracted from the experimentally obtained ones to plot the residual phase map. Figure 6.5 shows the experimental, theoretical and residual phase maps. In the residual phase map, white points represent zero or very small residual, showing that the data were reproduced well. Hence, it can be seen from these maps that the theoretical calculation produces a good approximation of the experimentally obtained data in Figure 6.3, providing a further confirmation of the proposed method. The residual phase map in Figure 6.5c shows dark areas around the hole that occurred due to decorrelation as explained before. These areas were not included in the stress calculation. Also, the black points scattered on the white background represent the noise in the experimental data. Using the data within the integration area of the residual phase map, the root-mean-square (rms) of the noise was calculated as 0.77 rad. The rms of the theoretical phase map that is free• of noise was calculated as 2.99 rad, giving a signal-to-noise ratio of 3.88. 65 (a) (b) (c) Figure 6.5 - Phase maps: (a) experimental, (b) theoretical, (c) residual 6.2 Discussion The signal-to-noise ratio indicates that the experimental data possess high noise content. However, the method provides a great tolerance for noise by using a large number of data points. Since the residual phase map has a mostly random noise distribution without substantial structure, the effect of this noise was overcome by averaging the data. In theory, the effect of the noise is inversely proportional to the square root of the number of data 66 points. Since there are over 100,000 data points used in the stress calculation, the effect of the noise should be about 300 times smaller than that could be obtained by using one data point. Thus, if the noise is random with zero mean, the effective signal-to-noise ratio would be approximately 1000. However, this number is unrealistic and the actual number is likely much lower. This is because a modest amount of structure remains in the computed residual. Also, this number only takes the optical error into account and does not include the error caused by the machining stresses. The single mirror measurement produced realistic results for the stresses created by bending a thin plate because the principal stress direction was very close to the x-axes. The principal angle was calculated as 5.8°. If the specimen contained a large y-stress relative to the x-stress, the stress state could not be identified accurately with this arrangement. In that case, the double mirror arrangement must be used to measure the stresses. However, the double mirror arrangement could not be verified for stress measurements due to the low data quality obtained from the optical tests. The noise content of the images obtained using this arrangement was found to be very high, and consequently the modulation values were very low. The portion of the light that was reflected by the mirrors was too small, and the balance between the intensities of reference and illumination beams could not be reached. This occurred because the double mirror arrangement uses the light at the edges of the beam. The major part of the light at the centre of the beam is effectively lost. The possible solutions to overcome these problems and obtain high quality double mirror measurements are explained in the next chapter. 67 Chapter 7 - Conclusion 7.1 Contributions A data analysis method was developed to assess the quality of ESPI data. Average modulation and saturation were found to be the quantities that give a measure of data quality. Functionality of each data point is evaluated by calculating their modulation values. Saturated pixels are marked because they do not produce any useful information. Great improvement in data quality was obtained by painting the specimen surface before the measurement and using data interpolation and smoothing. The improvement in the quality of phase maps was verified by comparison with the raw data. It was found during the optical tests that the illumination level is one of the important factors as it directly affects the pixel modulation. High illumination level allows the reduction of the camera aperture, by which the average speckle size can be increased. Higher modulations are expected when the average speckle size is greater than or equal to the pixel size. For the particular camera used (Matrox, black & white, 480 x 640 resolution camera), the required f-number was calculated as 9.4. The proposed common-path arrangements reduced the pixel drift greatly provided more stable measurements. The improved stability could be observed in live images. These 68 arrangements provide more useful data due to their in-plane measurement sensitivity. It was found that the accuracy of y-stresses measured using the single mirror arrangement is about 3 times lower than the accuracy of x-stresses because the measurement is only sensitive to x-displacements. The double mirror arrangement allows the measurement of full in-plane stress field by providing two orthogonal sensitivity directions. A least squares solution is used to find the best fit to the data in the calculation of residual stresses. The kernel matrices used in the calculations were found to be ill-conditioned when rigid body terms are included due to the size difference between the basis functions. The columns of the kernel matrices for both arrangements were scaled automatically to decrease their condition number and improve the calculation stability. The condition number of the kernel matrix used in double mirror measurements was low and its stability was slightly better than that of single mirror measurements. The proposed single mirror ESPI method was verified by measuring the stresses on a bent plate and comparing them to the ones found by strain gage measurements. The results obtained from these two methods were in agreement within 6 MPa. The residual phase map also showed that the calculated stresses reproduced the experimental data successfully. The double mirror arrangement could not be tested due to the low data quality obtained from the measurements. For this case, the illumination level was found to be low, which lead to low modulation values. This issue must be resolved by a future re-design, by which a higher 69 illumination level can be achieved. Initial ideas for improving this method are presented in the Future Work and Recommendations section. 7.2 Key Findings 7.2.1 Improving the measurement stability by eliminating separate light paths The existing ESPI technique uses separate and extremely delicate fiber paths to carry the illumination and reference beams. During the optical tests, it was found that these separate paths are being affected by disturbances such as vibration and air current and this resulted in pixel drift. The proposed common-path arrangements provided more stable measurements and reduced the pixel drift greatly. The rigidity of the measurement setup was also improved by removing the fiber couplings. 7.2.2 The contribution of surface preparation to image quality It was found that spraying a thin layer of white paint on the specimen surface increases the image quality greatly. This paint layer produces a diffuse surface that scatters the light in many different directions. The white color increases the amount of light that is being reflected from the surface. These two features of the surface preparation allow the camera to receive sufficient illumination and result in higher modulation. Also, with this technique, undesired surface features such as scratches are eliminated. 70 7.2.3 The importance of the quality of laser source and optical components If the laser source is of low quality, there may be fluctuations in the light intensity that may affect the measurement stability depending on the frequency of variation. Also, some undesired diffraction patterns may be seen on the laser wavefront that are caused by the laser source and the imperfections on the optical components such as lenses. These patterns lead to dark areas on the images and consequently decrease the modulation of the pixels in these areas. 7.2.4 The importance of illumination level The laser source used in the measurements must have sufficient power so that the illumination level is high. The required laser power depends on the arrangement and the components used, such as the distance between the laser source and the object, or the focal length of the lens. For the particular arrangement used in this research work, a laser power of 100-150 W was found to be sufficient. It was found that the images with high average intensities give high average modulations. The light intensity received by the camera must be adjusted such that most of the pixels have high intensities and few of them are saturated. Having a high illumination level also allows the adjustment of the camera aperture. Larger speckle sizes can be obtained by decreasing the aperture. However, this leads to dark images if the illumination is not sufficient. Hence, a high illumination level provides higher quality data. 71 7.2.5 The importance of the gap between the mirrors and object surface It was found that the highest modulation could be reached when the intensities of illumination and reference beams illuminating a surface point are approximately equal. This was achieved by pointing the laser beam toward the corner where the mirror and object planes intersect. The gap between the mirror and the object surface must be decreased in order to make use of the center portion of the laser beam efficiently. When using double mirror arrangement, this issue becomes more important. Since there are two mirrors, the area that is to be illuminated increases and the gap between the mirrors and the object surface becomes larger. As a result, a significant amount of light cannot be used. 7.3 Future Work and Recommendations The design of the double mirror arrangement can be improved in order to enhance the data quality and make reliable stress measurements. One possible modification is to use a smaller drill assembly and place the mirrors closer to each other and to the drilling location so that a higher intensity on the object surface can be achieved. This will also allow flexibility on drilling location because the horizontal movement of the drill will not be restricted as much. A lens with a shorter focal length can also be used to expand the laser beam so that a larger area can be illuminated. However, expanding the beam further will result in a decrease in the intensity, which can be overcome by using a more powerful laser. If a high powered laser is used, all precautions must be taken to ensure safety. 72 The most effective solution would be to design a beam splitter assembly, as described in Section 4.2, to divide the beam into two or three parts so that both the mirrors and the object surface receive sufficient light. This will prevent the light beam from illuminating the gap between the mirrors and the surface. Hence, the high intensity center portion of the light beam can be used effectively. 73 References [I] Hariharan, P., "Basics of Interferometry", Elsevier Inc., 2007 [2] Schajer GS, Steinzig M . "Full-field calculation of hole-drilling residual stresses from electronic speckle pattern interferometry data". Exp Mech. 2005, 45:526-532. [3] 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, v 5852, n 1, 2005,p 220-2 [4] Shareef S, Schmitt DR. "Point load determination of static elastic moduli using laser speckle interferometry". Optics and Lasers in Engineering. 2004; 42:511-527. [5] Nelson, D. V. , "Effects of residual stress on fatigue crack propagation". Residual Stress Effects in Fatigue, ASTM STP 776, American Society for Testing and Materials, 1982, pp. 172-194. [6] Lu, Lian "Handbook of Measurements of Residual Stresses". Fairmont Press, INC. Lilburn, GA. 1996. [7] Heindlhofer, K. "Evaluation of Residual Stress". McGraw Hill Book Company, INC. N Y , N Y . 1948. [8] Almen, J., Black, P., "Residual Stresses and Fatigue in Metals". McGraw Hill Book Company, INC. N Y . 1963. [9] Schajer, G. S., 2001, "Residual stresses: Measurement by destructive testing," Encyclopedia of Materials: Science and Technology, Elsevier, pp. 8152-8158. [10] Ruud, C O . , "Nondestructive and semidestructive methods for residual stress measurement". Residual Stress Effects in Fatigue, ASTM STP 776, American Society for Testing and Materials, 1982, pp. 3-5. [II] Mathar, J. 1934. "Determination of initial stresses by measuring deformations around drilled holes". ASME Transactions, v 56, n 4, Apr, 1934, p 249-254. [12] Soete, W. and Vancrombrugge, R., "An industrial method for the determination of residual stresses". Proceedings ofSESA, 8 (1), pp. 17-28 (1950). [13] Antonov, A. A. , "Inspecting the level of residual stresses in welded joints by laser interferometry," Weld. Prod. Vol . 30, pp 29-31 (1983). 74 [14] McDonach, A. , McKelvie, J., MacKenzie, P.M., and Walker, C.A., "Improved moire interferometry and applications in fracture mechanics, residual stress and damaged composites," EXPERIMENTAL TECHNIQUES, Vol . 7, pp 20-24 (1983). [15] Makino A, Nelson D. "Residual-stress determination by single-axis holographic-interferometry and hole-drilling. 1. Theory" Exp Mech. 1994, 34:66-78. [16] Diaz FV, Kaufmann GH, Moller O. "Residual stress determination using blind-hole drilling and digital speckle pattern interferometry with automated data processing". Exp Mech. 2001, 41:319-323. [17] Huang MJ , Yun BS. "Self-marking phase-stepping electronic speckle pattern interferometry (ESPI) for determining a phase map with least residues". Opt Laser Technol. 2007, 39:136-148. [18] Steinzig M , Ponslet E. "Residual stress measurement using the hole drilling method and laser speckle interferometry part I". Exp. Tech. v 27, n 3, May/June, 2003, p 43-46. [19] Alava R. "Deep hole residual stress measurements using Electronic Speckle Pattern Interferometry". University of British Columbia, 2005. [20] Sirohi RS, Chau FS. "Optical Methods of Measurement: Wholefield Techniques". Marcel Dekker, Inc. New York, N Y . 1999. [21] Williams D . C , "Optical Methods in Engineering Metrology". Chapman & Hall. London. 1993. [22] Ponslet E., Stenzig M . , "Residual stress measurement using the hole-drilling method and laser speckle interferometry Part III: Analysis technique". Exp. Tech. v 27, n 5, Sept.-Oct. 2003, p 45-48. [23] Cloud G. "Optical methods in experimental mechanics. Part 26: Subjective speckle". Exp. Tech. v 31, n 2, March/April, 2007, p 17-19. [24] Pedrotti F.L., Pedrotti L.S., "Introduction to Optics". Prentice-Hall, Inc., Englewood Cliffs, New Jersey. 1987. [25] 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. 75 [26] 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. 1988, 110:338-343. [27] Muskhelishvili, N.I. "Some Basic Problems in the Mathematical Theory of Elasticity". Noorhoff. 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