Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis

Laser speckle interferometry can be used to detect the locations of stress concentration of an object and deformation over a whole area to be measured through the shape information in fringe patterns (Cloud, 1998; Petzing & Tyrer, 1998). ESPI can obtain interferometric fringes by a subtraction process ( before after | | I I I   ) of the image data. Small displacement can be measured from such interferometric fringes corresponding to the order of the laser wavelength.


Application of ESPI to measurement of out-of-plane displacement in a spot welded canti-levered plate
Laser speckle interferometry can be used to detect the locations of stress concentration of an object and deformation over a whole area to be measured through the shape information in fringe patterns (Cloud, 1998;Petzing & Tyrer, 1998). ESPI can obtain interferometric fringes by a subtraction process ( before after || II I ) of the image data. Small displacement can be measured from such interferometric fringes corresponding to the order of the laser wavelength.
Several researches about in-plane displacement and vibration properties analyzed by ESPI have been reported and the application of ESPI is increasing (Rastogi, 2001). In this paper, the out-of-plane displacements of a partially spot welded canti-levered plate and of a normal canti-levered plate are measured and compared by application of 4-step phase shifting method to the analysis of fringe patterns in ESPI (Baek et al., 2002).

ESPI for measurement of out-of-plane displacement
Figure 1-1 shows the arrangement of the ESPI optical system for measuring out-of-plane displacement. The interferometric fringe patterns caused by phase differences between the specimen and the reference plane with a PZT (piezoelectric transducer) must be analyzed because the interferometric fringe patterns contain information about out-of-plane displacement of the specimen. The phase difference is made by an optical path difference. The out-of-plane displacement, w , caused by the optical path difference is as follows: In Eq. (1-1), w is z-directional (out-of-plane) displacement,  is the wavelength of the laser, and  represents the phase difference between before and after out-of-plane displacement.
Fig. 1-1. Block diagram of ESPI optical system for measuring out-of-plane displacement.

4-step phase shifting method in ESPI
The 4-step phase shifting method used in this paper is employed to move the reference plane with the PZT by /2  radians in each step and to obtain four fringe patterns with relative phase difference. The phase map is obtained from the four fringe patterns using the arc tangent function. The light intensity of the fringe pattern in ESPI, i I , is expressed as follows: (1-2) In the above equation, (, ) i Ixyis the measured light intensity, 0 I is the average intensity, (, ) mxy is the contrast, (, ) x y  is the phase difference, and  is the phase introduced by PZT. The phase map, where the magnitude and sign of displacement can be known, can be obtained as follow.
1 42 (, ) t a n 13 In Eq. (1-3), 1 I , 2 I , 3 I and 4 I are the light intensities at 0   , /2  ,  , 3/ 2  radians, respectively. Eq. (1-3) uses the four fringe patterns with different phases, so that this method is called the 4-step phase shifting method. The phase map obtained by this method has the phase between   radians and   radians due to the property of the arc tangent function.
Thus, the phase map has the discontinuous phase at every 2 radians, but this discontinuity can be eliminated by use of the phase unwrapping process and the continuous displacement of the specimen can be obtained (Ghiglia & Pritt, 1998).

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The speckle pattern contains lots of noise and the noise must be filtered out before the phase unwrapping process. The Gaussian blur process in a commercial image processing software package (Adobe Photoshop, Version 5.5) is used to eliminate the speckle noise.

Specimen and experimental set-up
The specimen used in this experiment is a 2 mm-thick canti-lever made of steel plate (E=200 GPa, 0.3   ). The shape and size of the specimen is shown in Fig. 1-2. Fig. 1-2 (a) is the normal canti-levered plate that is not spot-welded, and Fig. 1-2 (b) is the canti-levered plate that is spot-welded on the rear side.  Twyman-Green interferometry is used to measure the out-of-plane displacement. The phase shifting method is used to advance the precision degree of the speckle pattern in the measurement of out-of-plane displacement, and the phase shifting is performed by the PZT that is controlled with a personal computer.
The expression for the deflection curve for a canti-lever beam subjected to a concentrated load P , as shown in Fig. 1-2 In Eq. (1-4), E is the Young's elastic modulus, I is the moment of inertia, L is the distance from the fixed support to the loading point, and x is the distance from the fixed support to any arbitrary point.
www.intechopen.com Fig. 1-3 Optical systems for ESPI measurement of out-of-plane displacement. . Figure 1-5 shows the speckle fringe patterns of a spot welded cantilevered plate at difference phases for 0, / 2, , 3 / 2     . Those fringe patterns are obtained through a subtraction of before-displacement and after-displacement results of a specimen. The different phases in Figs. 1-4 and 1-5 are created by the PZT which controls the phases of fringe pattern by 0, / 2, , 3 / 2     , as mentioned before.

Results of experiment
As seen in Figs. 1-4 and 1-5, the change of fringe patterns is shown near the welded area of the spot welded specimen. To eliminate the noise of the high frequency component in the speckle, the Gaussian blur process is applied to the original image obtained in the experiment.   When Figs. 1-8 (a) and (b) are compared, the phase map of the normal canti-levered plate is uniform, but that of the spot welded canti-levered plate shows a phase reversal at the welded area.    Fig. 1-8. It shows that the result of ESPI is almost the same as that of the theoretical calculation for the normal canti-levered plate which is not spot welded. The maximum error of 0.076 m  occurs at approximately 7.9 mm from the fixed area of the canti-levered plate. However, in general, the measured displacement by the ESPI experiment is quite close to the theoretically expected displacement. Thus, it is proved that the physical out-of-plane displacement can be directly measured by the ESPI method.
www.intechopen.com  mm, 30 mm from the fixed area of the spot welded canti-levered plate, respectively.
As a reference, the other displacements, 1.006 m  , 3.219 m  , 5.432 m  , are estimated at 10mm, 20mm, 30mm from the fixed area of the normal canti-levered plate. Therefore, the displacement for the same load decreases as the canti-levered plate is reinforced by spotwelding, and the spot welded area that is not visible can be easily detected by use of speckle interferometry.

Conclusions and discussions
The 4-step phase shifting method applied to an ESPI experiment has been used for the measurement of out-of-plane displacement in the normal canti-levered plate and the spot welded canti-levered plate. The measured displacement of the normal canti-levered plate agreed to the theoretical value within 0.076 m  . That is, it is proved that the physical out-ofplane displacement can be directly measured and precise measurement with a nanometer resolution is possible. Also, a welded area that is not visible from the surface can be detected and a small out-of-plane displacement in the welded area can be measured. The distribution of displacement shows a slight non-linearity in the period of fringe, and therefore, further work is necessary for more precise measurement. Also, noise due to speckle has to be eliminated before the phase shifting method is applied. The Gaussian blur process is used to eliminate the noise in this work.
2. Measurement for in-plane displacement of tensile plates with throughthickness circular hole and partly through-thickness circular hole by use of speckle interferometry

Introduction
Speckle interferometry is an optical technique to measure displacement of a specimen by a coherent light from a laser. There are several phase extraction methods in speckle interferometry to measure the displacement, and phase shifting method (PSM) is one of the methods (Creath, 1988). In PSM, a known phase produced by piezoelectric transducer (PZT) is added into optical beam and resultant interference patterns are processed to get information about displacement of a specimen (Kim & Baek, 2006). In this paper, in-plane displacement of a steel plate with a partly through-thickness circular hole and a steel plate of a through-thickness circular hole is measured by simple optical system of speckle interferometry with PSM. Especially, the circular hole of steel plate with a partly through-thickness circular hole is not visible because the circular hole is cut on the rear side of the plate. This means that one cannot see any deformation or defect of the specimen.

Phase shifting method in speckle interferometry
Figure 2-1 is an optical system to measure in-plane displacement of specimens by use of PSM in speckle interferometry. A specimen is placed in a loading device and tensile load is applied to the specimen by the loading device in order to make in-plane displacement on the specimen. When two optical beams from a laser illuminate a specimen in speckle interferometry as shown Fig. 2-1, the optical beams make interference fringe patterns. The interference fringe patterns, i I , can be represented by Eq. (2-1).
where 0 I is the average intensity of fringe pattern and c I is the contrast of fringe pattern.
A known phase, i  , is added into one of the optical beams through controlling PZT.

(, )
x y  is the phase of fringe pattern caused by the in-plane displacement of a specimen which tensile load is applied to. When 0, /2  ,  , and 3/ 2  radians are used for i  , four fringe patterns are taken through CCD camera and stored in PC. When 1 I , 2 I , 3 I , and 4 I , represent the fringe patterns for the known phases of 0, /2  ,  , and 3/ 2  radians, respectively, the phase, (, ) x y  , is calculated by use of Eq. (2-2).
1 42 13 (, ) t a n II xy II is obtained with unwrapping algorithm of the wrapped phase (Ghiglia & Pritt, 1998). Then, the in-plane displacement of specimen, u , is obtained from the phase (, ) u x y  as follows (Rastogi, 2001) ; Schematic diagram of speckle interferometry with phase-shifting method for inplane displacement measurement (  is the angle between two incident lights onto the specimen).

Optical experiment
The specimens used in this experiment are a rectangular steel plate with a through-thickness circular hole at the center and a rectangular steel plate with a partly through-thickness circular hole at the center on rear side as in Fig. 2-2. Both of the rectangular steel plates have the same size that is 147 mm x 27.7 mm with thickness of 1.2 mm. The diameter of throughthickness circular hole is 12 mm. Also the diameter of partly through-thickness circular hole is 12 mm but it is uniformly cut by 0.8 mm on the rear side. Figure 2-3 is the picture of optical experiment system. Ar laser with wavelength of 515 nm is used in the optical experiment.
(a) (b) Fig. 2-2. Dimensions of specimens used in-plane displacement (unit : mm) (a) rectangular steel plate with through-thickness circular hole (b) rectangular steel plate with partly through-thickness circular hole.
At first, optical experiment is performed with the steel plate with through-thickness circular hole at the center. Four fringe patterns, 1 I , 2 I , 3 I , and 4 I , are taken in the experiment and they are stored in PC. However, the four fringe patterns have lots of speckle noises, so that they are processed by use of an image processing algorithm. In this work, Gaussian blur algorithm that is available commercially in Adobe Photoshop is used to process the fringe patterns. The processed four fringe patterns are shown in Fig. 2-4. The same experimental procedures are used to get four fringe patterns for the steel plate with partly throughthickness circular hole at the center on rear side. . The ANSYS discretization for the rectangular steel plate with a through-thickness circular hole at the center used 8 node quadrilateral elements. The ANSYS discretization for the rectangular steel plate with a partly through-thickness circular hole at the center used 10 node tetrahedral elements.
The results of the two specimens from ANYSYS are plotted also in Figs. 2-6 and 2-7. As shown in Figs. 2-6 and 2-7, the results of the optical experiments for speckle interferometry with 4-step phase shifting method agree with those of ANSYS.

Conclusions and discussions
In-plane displacements of two specimens are measured by simple optical system of speckle interferometry using 4-step phase shifting method; one is the rectangular steel plate with a through-thickness circular hole and the other is the rectangular steel plate with a partly through-thickness circular hole. The circular hole of the steel plate with a partly throughthickness circular hole is not visible because the circular hole is cut on the rear side of the plate, so that one cannot see any deformation or defect of the specimen. The fringe patterns acquired by optical experiment of speckle interferometry are processed by image processing algorithm of Gaussian blur in Adobe Photoshop and the in-plane displacements of the two specimens are obtained by the processed fringe patterns. Also the in-plane displacements of the two specimens are calculated by use of ANSYS.
The results of optical experiments are quite comparable to those of calculation with ANSYS. Based on the optical experiments, speckle interferometry can be applied to easily detect or measure defect or deformation in a specimen that is not visible.

Introduction
Stress raisers have been one of the main concerns when it comes to design analysis. Due to the complexities associated with it, numerous and continuing investigation are done to develop techniques to accurately measure stress concentration around the geometric boundaries. Several methods can be found in the literature ranging from FEM, the use of photoelastic-data, hybrid method and other various numerical and experimental procedures (Lekhnitskii;Tsai & Cheron, 1968;Kobayshi, 1993;Dally & Riley, 1991;Pilkey's, 2008).
In this paper, we present stress concentration measurement method using only xcomponent displacement data of selected points along straight lines away from the www.intechopen.com geometric discontinuity. In conjunction with our previous studies (Baek et al., 2000;Baek et al., 2006), the hybrid method employing the least-squares method integrated with Laurent series representation of the stress function was used to estimate reliable edge data around the circular hole in a tensile-loaded plate from a relatively few measured x-displacement data away from the boundary. Speckle interferometry has been explored and integrated with other methods for the optical measurement of in-plane and out-of-plane displacement in a material (Schwider, 1989;Steinzig & Takahashi, 2006). Different from the previous works, this study utilized considerably a few number of in-plane micro-scale x-displacement only measured by speckle interferometry using PSM/FT for hybrid stress analysis.
The present technique employs fairly general expressions for the stress functions, and traction-free conditions which are satisfied at the geometric discontinuity using conformal mapping and analytical continuation. The approach is illustrated using the x-displacement as input data obtained from phase-shifting method in speckle interferometry.

Basic equations
In the absence of body forces and rigid body motion, the stresses and displacements under plane and rectilinear orthotropy can be written as (Gerhardt, 1984;Rhee & Rowlands, 2002).
The two complex stress functions 1 ()   and 2 ()   are related to each other by the conformal mapping and analytic continuation. For a traction-free physical boundary, the two functions within sub-region  of Fig. 3-1 can be written as Laurent expansions, respectively (Gerhardt, 1984;Baek & Rowlands, 2001)   11 2 2 2 () a n d ( ) The coefficients of Eq. (3-2) are kk k cai b   where k a and k b are real numbers. In addition to satisfying the traction-free conditions on the hole boundary , the stresses and displacements of Eqs. (3-1) and (3-2) associated with these stress functions ( 1 ) and ( 2 ) satisfy equilibrium and compatibility. The primes denote differentiation with respect to the argument. Complex material parameters µ l (l = 1, 2) are the two distinct roots of the characteristic Eq. (3-3) for an orthotropic material under plane stress (Lekhnitskii, Tsai & Cheron, 1968;Gerhardt, 1984)  ( where S ij are the elastic compliances. The material properties p l and q l are defined as The inverse of the mapping function  namely 1   , maps the geometry of interest from the physical z-plane into the -plane ( ll      ). For orthotropic materials, the conformal transformation from the unit circle in the -plane to the hole in the z-plane of radius R is shown in Fig. 3-1 and is given by where 1 i . The inverse of (3-5) is The branch of the square root of Eq. (3-6) is chosen so that  l 1 (l = 1, 2). Complex quantities B and C in Eq. (3-2) depend on material properties defined as (3-7)

Least-squares method
Combining Eqs. (3-2), (3-3) and (3-5) gives the following expressions for the displacements through regions 1  and 2  of Fig. 3-1. In matrix form, [] U is a rectangular coefficient matrix whose size depends on the number of terms k of the power series expansions of Eq. (3-1) and given by   Knowing {d} at various locations in Eq. (3-8) allows the best values of unknown coefficients {c} in a least square sense (Sanford, 1980 (3-10)

In-plane displacement by speckle interferometry
In speckle interferometry using phase-shifting method with Fourier transformation (PSM/FT) (Morimoto & Fujisawa, 1994), interference fringe pattern is obtained by subtracting the pattern from the pre-loading and post-loading conditions of the specimen. The intensity of the fringe pattern is calculated as where A(x,y) is the amplitude of the brightness in the pattern and B(x,y) is the average brightness. Piezoelectric transducer (PZT) can control an optical path length.  is the known phase which is added into one of the two optical beams by controlling the PZT. The known phase  covers the region from 0 to 2 radians at equal intervals.  (x,y) is the phase of fringe pattern caused by the in-plane displacement of a specimen.
A phase-shifting method using Fourier transform had been well-developed for the measurement of in-plane displacement by speckle interferometry (Morimoto & Fujisawa, 1994;Kim et al., 2005). The phase  can be calculated as , : ,t a n Re , :

Fx y xy
Fx y is the -directional Fourier transform of Eq. (3-11). In Eqs. (3-12) and (3-13), 0  is a fundamental frequency. Using the calculated phase,  , the displacement of a specimen can be obtained.

Optical experiments
Speckle interferometry experiment was performed with setup as shown in Fig. 3-2 to acquire the needed data. In Fig. 3-2, LA is a laser, PA is a pin-hole assembly, CL is a collimating lens, BS is a non-polarizing beam splitter, MR1 and MR2 are mirrors. SL is a specimen installed in a tensile loading device, CCD is a CCD camera, and PC is a personal computer. PC controls the movement of PZT through the control board CNT . ) which was subjected to tensile load during the procedure. The test specimen of Fig. 3 had been used for experiment for the development and application of phase-shifting method in speckle interferometry. The accuracy and reliability of the said in-plane displacement measuring method had been established (Baek et al., 2008). For this reason, it is a useful tool for the hybrid stress analysis presented herein Baek & Rowlands, 2001). In this study, phaseshifting method using Fourier transform was utilized.  where u is the longitudinal displacement,  is the wavelength of light from a laser and u  is the phase obtained from longitudinal displacement. The laser used in the experiment is He-Ne laser and  is 633 nm. The angle between incident light and vertical line to the specimen,  as shown in Fig. 3-2, is around o 26 . The specimen is installed in a loading device that applies tensile load to the specimen in order to make in-plane displacement. Fringe patterns of (, : ) Ixy  in Eq. (3-11) are taken through CCD camera in Fig. 3-2 The accuracy of the least-squares method for calculating displacement components {,} T u  as presented in the preceding section was related with the input data acquired through experiment by phase-shifting method in speckle interferometry (Baek & Rudolphi, 2010).
Percent errors for each data was calculated and shown in Table 3-1. As can be observed, the level of precision of the hybrid method is too close with the original input data as indicated by the % Errors. Thus, the method is significantly accurate and reliable for calculating such parameter.
The simplicity of the test specimen being isotropic, containing symmetrically-shaped discontinuity facilitates the ease of reliable verification. Finite element anlysis was done to establish a benchmark for comparison purposes. Figure 3-7 shows the ABAQUS discretization of the quarter steel plate. In the vicinity of the circular hole, the ABAQUS model of Fig. 3-7 utilizes elements on the edge of the hole as small as o 1.5 by 0.013r, where r is the radius of the circular hole. Tangential stress around the quarter hole was determined and compared as shown in Fig. 3-8. Different values of m term in the complex stress functions was tested to see its effect, it came out that at point of high stress concentration ( o 90   in Fig. 3-8). Well-comparable results were attained at decreasing value of m with the best value equal to 1 -accurate by less than one percent error. Figure 3-9 reveals the preceding observation and shows normalized tangential stress at the edge of the quarter hole in a steel plate with 1 m  in the complex stress function.

Conclusions and discussions
A reliable hybrid method for characterizing stress around the circular hole in a tensileloaded steel plate is presented. The method utilized only few micro-scale x-displacement data measured by a well-established optical technique, speckle interferometry, in conjunction with phase shifting method using Fourier transform to calculate stress components and eventually stress concentration at o 90   . The use of few input data may reduce experiment time and relatively increase data processing speed.
Different values of m term in the complex stress functions were tested to see its effect. In the comparison between input x-displacement data and calculated data the hybrid method is effective with an error below 1% in all values of m. On the other hand, it came out that at point of high stress concentration ( o 90   ), well-comparable results were attained at decreasing value of m. The best value of m is consistently known to be equal to 1 of which results are accurate by less than one percent error. Results showed that the method is accurate and reliable as compared with the widely-used FEM software, ABAQUS. This book provides the most recent studies on interferometry and its applications in science and technology. It is an outline of theoretical and experimental aspects of interferometry and their applications. The book is divided in two sections. The first one is an overview of different interferometry techniques and their general applications, while the second section is devoted to more specific interferometry applications comprising from interferometry for magnetic fusion plasmas to interferometry in wireless networks. The book is an excellent reference of current interferometry applications in science and technology. It offers the opportunity to increase our knowledge about interferometry and encourage researchers in development of new applications.

How to reference
In order to correctly reference this scholarly work, feel free to copy and paste the following: Tae Hyun Baek and Myung Soo Kim (2012). Speckle Interferometry for Displacement Measurement and Hybrid Stress Analysis, Interferometry -Research and Applications in Science and Technology, Dr Ivan Padron (Ed.), ISBN: 978-953-51-0403-2, InTech, Available from: http://www.intechopen.com/books/interferometryresearch-and-applications-in-science-and-technology/experimental-and-hybrid-displacement-measurementby-use-of-speckle-interferometry