Accurate ex situ deformation measurement using an ultra-stable two-dimensional digital image correlation system Bing Pan,* Liping Yu, and Dafang Wu Institute of Solid Mechanics, Beihang University, Beijing 100191, China *Corresponding author: [email protected] Received 10 April 2014; revised 23 May 2014; accepted 24 May 2014; posted 29 May 2014 (Doc. ID 209613); published 27 June 2014
In situ planar deformation measurement using two-dimensional digital image correlation (2D-DIC) and a fixed camera has been fully investigated in many published works and widely used for various applications. However, in certain special cases (e.g., long-term deformation monitoring of engineering structures, or deformation measurement of a specimen subjected to uncommon loading), it is very difficult or impossible to carry out in situ deformation measurement using 2D-DIC, and necessitates ex situ 2D-DIC measurements for a repositioned specimen or using a relocated camera. To achieve accurate measurements, the error sources of ex situ 2D-DIC measurements should be identified and minimized, and the strain accuracy of ex situ 2D-DIC measurements should be quantified. In this work, the potential error sources of ex situ 2D-DIC measurements are first discussed in detail. Then, to mitigate the errors associated with these issues, an ultra-stable 2D-DIC system combining the idea of active imaging and a welldesigned bilateral telecentric lens is established, which is invariant to the potential variations in ambient lighting and the possible small out-of-plane motions in object surface and/or image plane. The established ultra-stable 2D-DIC system is first compared with the regular 2D-DIC setup in determining the surface strains of an unstrained sample. Then, ex situ residual stress measurement using the hole-drilling technique and the established 2D-DIC system was conducted and compared with the applied ones. The results of this work confirm that the accuracy of ex situ 2D-DIC deformation measurements using the proposed system is of high fidelity, and can be used for accurate deformation measurement in practical ex situ tests. © 2014 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (110.0110) Imaging systems; (150.3045) Industrial optical metrology. http://dx.doi.org/10.1364/AO.53.004216
1. Introduction
As a typical non-interferometric optical metrology noted for its simple principle and easy implementation, two-dimensional digital image correlation (2DDIC) [1–4] using a single digital camera has been widely used in various scientific research and engineering fields for in-plane displacement and strain measurements of planar objects. Over the last 30 years, this cost-effective, easy-to-use but practical 1559-128X/14/194216-12$15.00/0 © 2014 Optical Society of America 4216
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technique has undergone not only significant improvements in technique development, but also a burst of applications. In practical implementation of 2D-DIC for in-plane deformation measurement, the optical axis of the imaging lens should be aligned perpendicularly to the specimen surface before loading. Then, the random intensity distribution (also known as the speckle pattern) on the test specimen surface at different states is imaged onto the sensor target of a CCD/CMOS camera to form reference (or undeformed) and target (or deformed) images. By comparing the deformed images with the reference one using a well-established subset-based image-matching
algorithm [5,6], full-field displacements with subpixel accuracy can be determined. Also, full-field strain information can be further extracted by differentiating the obtained displacement fields [7]. In regular 2D-DIC measurements based on optically recorded images, it is a common sense and also an essential requirement that the imaging system (comprising a digital camera and an optical lens) and the specimen should be fixed and remain stationary during recording the surface images of the test specimen. It has been recognized [8–12] that, for 2D-DIC measurement using a common lens, even small out-of-plane motions between the camera and the specimen may induce additional large virtual displacements and strains, and corrupt the desired displacement and strain values induced by external loading. Clearly, this basic requirement ensures that the specimen surface and sensor target remain in their original planes, and thus avoid any small relatively out-of-plane rigid-body movements between these two planes. However, in certain special cases, it is very difficult or impossible to carry out in situ strain measurement using 2D-DIC. First, in some cases, for the purpose of protecting the measuring equipment, the 2D-DIC setup needs to be relocated to its original position to image the test sample at set intervals. For instance, during structural health monitoring of the most vulnerable regions of engineering structures (e.g., pipe welds in power generation industry) over a long period of time, and strain measurement for fatigue tests spread over several months, the 2D-DIC system cannot be left stationary at its original position for safety reasons [13,14]. Second, in other cases, it is difficult to arrange the 2D-DIC system to accommodate certain loading devices. One typical example is the determination of residual strain in a hole-drilling testing with the DIC method [15–17]. Generally, the test sample, whose residual stress is to be determined, is mounted at a sample stage, and a reference image is captured. Then the specimen is moved to a drilling machine, and a small shallow hole is drilled at the region of interest (ROI). Afterward, the specimen with released residual stress is moved back to original place to record a target image. In all these cases, small in-plane and out-of-plane rigid-body movements between the image plane and the object
plane unavoidably occur due to the repositioning of the imaging system or the specimen. Furthermore, considering the fact that image capture should be performed at different times after the repositioning of the imaging system or the specimen, the variation in ambient light may be too severe to be ignored, especially in outdoor field measurement [18]. Although in situ displacement and strain measurement using 2D-DIC has been fully studied, very little work has been dedicated to ex situ 2D-DIC measurement for a repositioned specimen or using a relocated camera. From the viewpoint of applications, the error sources of ex situ 2D-DIC measurements should be identified, and the strain accuracy of ex situ 2D-DIC measurements should be quantified. With these questions in mind, in the reminder of this work, the possible error sources of ex situ 2D-DIC measurements are first discussed in detail. Then, to minimize these potential errors induced by relocation of the camera or the specimen, an ultra-stable 2D-DIC system, which is invariant to the potential variations in ambient lighting and the possible small out-of-plane motions in object surface and/or image plane, is established. Afterward, ex situ 2D-DIC experiments including specimen repositioning tests and camera relocation tests were performed using a nondeformable sample. The surface strains of the sample were determined by the established ultra-stable 2DDIC system and a regular 2D-DIC setup. Finally, a real hole-drilling test was also carried out to further demonstrate the practicality of the proposed ultrastable 2D-DIC system for ex situ deformation measurements. 2. Ex Situ Deformation Measurement Using an Ultra-Stable 2D-DIC System A. Error Analysis of 2D-DIC Measurement due to Specimen or Camera Relocation
Figure 1 shows the schematic of a typical 2D-DIC setup using an optical imaging device for measuring surface deformation of planar objects. The test planar object surface should have a random speckle pattern, which deforms together with the specimen surface as a faithful carrier of deformation information. The fixed camera equipped with an optical imaging lens is aligned with its optical axis normal
Fig. 1. Schematic diagram of the 2D-DIC measurement using a regular optical imaging lens. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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to the specimen surface, imaging the planar specimen surface of different loading states onto its sensor plane. The digital images of the test object surface are saved and subsequently processed by wellestablished subset-based DIC algorithms to retrieve the surface deformation information. The common 2D-DIC system is able to deliver accurate displacement and strain measurements in which the specimen and camera are both tightly fixed during the testing period. However, for ex situ deformation measurement of a repositioned specimen or using a relocated camera, the position or the orientation of the test sample relative to the camera will change slightly due to unavoidable reposition errors, as schematically illustrated in Fig. 2. Generally, out-of-plane translation and rotation, and in-plane translation and rotation, may occur simultaneously after the reposition of the specimen or the camera. It should be pointed out first that, no matter what kind of reposition is conducted, the camera can be assumed to be fixed and the resulting change in imaging model can be depicted as a position and orientation change in the sample surface. It is known that in-plane translation and rotation of the specimen have no effect on the measured strains, and the displacements caused by in-plane motions can be
eliminated using a plane fitting approach as detailed in Appendix A. In other words, merely out-of-plane translation and rotation lead to an alteration in object distance ΔZ of a physical point on the test sample surface, which may vary at different positions. Theoretically, the displacement and strain errors due to out-of-plane translation and/or rotation have been thoroughly derived in Ref. [19]. To intuitively show the distributions of the errors and better understand the results in the next section, we simulated the theoretical displacement distribution according to the derived equations in Ref. [19], and then estimated the corresponding mean values and standard deviations. During the simulation, the ROI was set as 1000 × 1000 pixels, and the object distance Z and the image distance L were defined as 400 and 50 mm, respectively. Meanwhile, the magnification of the imaging system was assumed to be 100 pixel∕mm. Then, two cases, one in which the object surface has a small out-of-plane translation ΔZ 1 mm away from the camera and the other in which the object surface has a clockwise out-of-plane rotation θ 1° about the X axis, were simulated. As shown in Figs. 3 and 4, the simulated displacement fields for the two cases show regular distributions, as the equations indicated. Finally, the mean values
Fig. 3. Simulated (a) u and (b) v fields with a 1 mm out-of-plane translation away from the camera occurred on the specimen surface. 4218
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Fig. 4. Simulated (a) u and (b) v fields with a 1° out-of-plane rotation about the X axis occurred on the specimen surface.
and standard deviations of these simulated displacement and strain fields were calculated, as shown in Table 1. Although the simulated displacement and strain fields are free of noise, their statistics can still be considered as a beneficial reference in the following analyses. As seen from Table 1, the errors caused by small out-of-plane translation or rotation are considerable and should be eliminated, especially when conducting high-accuracy deformation measurements. To minimize the effect of out-of-plane motions on 2D-DIC measurements, the following two possible measures were suggested in Ref. [8]: (1) use a welldesigned telecentric lens with large “effective” object distance, and (2) place the camera far away from the test object to approximate a telecentric lens. In addition, a newly developed error compensating technique using an unstrained compensating sample can also be used [18]. Other than unavoidable changes in object distance after specimen or camera reposition, the position and orientation of the sensor target (i.e., image plane) may also change slightly, due to the temperature variations caused by self-heating or ambient temperature. In addition, for ex situ DIC measurement, ambient light may change accordingly, especially when the measurements are performed in outdoor environments or at long intervals. In short, no matter what kind of repositioning is carried out, all these detrimental factors may more or less alter the imaging conditions assumed for the original 2D-DIC system, and lead to appreciable strain errors.
Table 1.
Mean Values and Standard Deviations of the Simulated u, v , εx , and εy Fields
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Out-of-plane 0 0.725 0 0.725 −2500 0 −2500 0 translation Out-of-plane 0 0.294 0.294 0.266 0 1000 −152 2000 rotation
B. Ex Situ Deformation Measurement Using an Ultra-Stable 2D-DIC System
In our recent work regarding high-accuracy 2D-DIC measurements, it is concluded that a well-designed bilateral telecentric lens is an essential optical component to realize high-accuracy strain measurement [12]. The bilateral telecentric lens is insensitive to small changes in both object distance and image distance, and can maintain constant magnification within its range of depth of focus. Also, the lens distortion of a high-quality bilateral telecentric lens is normally small enough to be neglected. These distinct merits of a bilateral telecentric lens enlighten us to build an ultra-stable 2D-DIC imaging system for ex situ strain measurement. Moreover, to deal with the serious light variation possibly occurring in outdoor measurement, the imaging system can be further improved by introducing the idea of active imaging [18,20], which incorporates an actively illuminated monochromatic light and an optical bandpass filter attached just before the bilateral telecentric lens. Shown in Fig. 5(a) is the schematic illustration of the proposed ultra-stable 2D-DIC system for accurate deformation measurement in an ex situ 2D-DIC experiment. The system comprises a high-resolution CCD camera (TGX50, Baumer, Inc., Switzerland), a high-quality bilateral telecentric lens (Xenoplan 1∶5, Schneider Optics, Inc., Germany), an optical bandpass filter with a center wavelength of 450 2 nm, and a monochromatic source emitting at 450–455 nm. The critical component in this 2DDIC system is the bilateral telecentric lens, as shown in Fig. 5(b). With the unique design of the aperture stop, the bilateral telecentric lens is not susceptible to small changes in the distance between the lens and the sensor plane. Therefore, the 2D-DIC system equipped with a bilateral telecentric lens performs significantly better in eliminating or avoiding the unfavorable effects of out-of-plane motions and outof-plane rotations compared with the use of a conventional lens. Moreover, according to our previous 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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Fig. 5. (a) Schematic diagram of the established ultra-stable 2D-DIC system. (b) Bilateral telecentric lens. (c) Transmission spectrum of the optical bandpass filter.
research [12], the 2D-DIC system using a bilateral telecentric lens presents very low lens distortion, which results in small errors in displacement and strain fields due to in-plane motions. With these distinct advantages of a bilateral telecentric lens, the errors due to the repositioning of a camera or specimen can be minimized. The 2D-DIC system using a bilateral telecentric lens is then considered to be a highly stable 2D-DIC system insensitive to any small out-of-plane motions occurring in the object surface and image plane. To compensate the illumination variations, improvements are also made to illumination and imaging in the 2D-DIC system. A tailor-made monochromatic blue LED lighting emitting at 450– 455 nm, instead of a common white light source, is adopted for illumination. Also, an optical bandpass filter, with a coupled center wavelength of 450 2 nm and a full-width at half-maximum value of approximately 32 nm, is mounted before the bilateral telecentric lens. Figure 5(c) shows the transmission efficiency curve of the bandpass filter. For the active illuminated monochromatic light, the bandpass filter shows a very high transmission efficiency exceeding 80%. As only a very limited portion of ambient light within the bandpass range of the filter can pass thorough the filter, the ambient lighting makes little and negligible contribution to the intensity of the recorded image compared with the actively illuminated monochromatic light. The system is therefore 4220
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capable of acquiring high-quality images with almost identical brightness and contrast even though serious changes have occurred in ambient lighting. 3. Experimental Verification Using an Unstrained Sample A. Experimental Details
To evaluate the effectiveness and accuracy of the established ultra-stable 2D-DIC system for ex situ deformation measurement, a series of camera reposition tests and specimen reposition tests were conducted in this work. The tested specimen is a flat glass plate with a premade random speckle pattern on its surface. As indicated in Fig. 6, the plate was first vertically fixed in a plate clamp and illuminated using the monochromatic source, and the imaging system, which was tightly fixed on a camera stage through four screws, was placed 268 mm in the front of the plate. In the following, three different experiments were designed. (1) Camera repositioning tests were conducted to investigate the errors due to unavoidable relocation errors. In this experiment, after getting a clear speckle pattern with sufficient contrast, a reference image was captured first. Then, the imaging system was moved from the fixed position by loosening the fixing screws and repositioned to the original position with an image recorded as a target image. The same process was repeated 20 times, and
Fig. 6. (a) Experimental setup for reposition tests. (b) Reference image captured using the established 2D-DIC system with the ROI and the corresponding calculation parameters defined.
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displacement data, a pointwise least squares [7] using a strain calculation window of 21 × 21 sets of displacement data (corresponds to a local region of 201 × 201 pixels) was employed to estimate the strain distributions. The two strain components (i.e., εx , εy ) at all calculation points are averaged to determine the average values of the ROI. B. Experimental Results
Figure 7 shows the mean values and standard deviations of the u and v displacements directly measured by the proposed ultra-stable 2D-DIC system for the three different experiments. For stationary tests, the mean displacements approach zero, and the standard deviations of the displacements are in a relatively low level, which agrees well with the practical deformation state. However, with respect to the camera or specimen relocation tests, it is clear that the repositioning of the camera or the specimen will induce artificial displacement and strain errors, which vary with each test and each practitioner. As a matter of fact, these errors are caused by the in-plane or out-of-plane translations and rotations of the test specimen relative to the camera. Since the in-plane translations and rotations exist in every test and make no contribution to the real deformation, we employed an approach based on the least squares method to cancel the in-plane translations and rotations from the directly measured displacements in
Standard deviation of displacements (pixel)
20 deformed images were recorded. (2) Specimen relocation tests were performed to investigate the strain errors caused by specimen reposition. The specimen was taken away from the plate clamp after a reference image was captured. Then the specimen was repositioned to record a target image for 20 times. (3) Regular static tests with fixed specimen and fixed camera were carried out to examine the strain accuracy of random noise and the DIC algorithm. In this test, 21 continuously captured images at a 10 s interval were used to study the uncertainty of 2D-DIC measurements, as an additional contrast to the above experiments. For comparison purposes, all three tests were also conducted using a common 2D-DIC system equipped with an ordinary industrial lens (Tec-55, Computar Optics, Inc.). For each test, 21 images including one reference image and 20 deformed images were recorded. Next, all the deformed images are compared with the reference image to extract the displacement and strains. During the correlation analysis, a rectangular region located in the middle of the reference image, containing 20,164 discrete points on a 142 × 142 grid, was chosen to be the ROI. Then, a 41 × 41 pixels subset size and a 10 pixels grid step were chosen as calculation parameters for each point of interest. The displacement components were first analyzed using the fast Newton–Raphson algorithm [5] we proposed recently. Based on the obtained discrete
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Fig. 7. (a) Average values and (b) standard deviations of measured displacement fields measured by the proposed ultra-stable 2D-DIC system in camera reposition, specimen reposition, and stationary tests. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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Fig. 8. (a) Average values and (b) standard deviations of measured strain fields measured by the proposed ultra-stable 2D-DIC system in camera reposition, specimen reposition, and stationary tests.
Figures 9(a) and 9(b) show the mean values and standard deviations of the measured u and v displacements for the case of camera reposition, specimen reposition, and stationary tests using the regular 2D-DIC system. The corresponding mean values and standard deviations of strain fields are shown in Figs. 10(a) and 10(b), respectively. For stationary tests, since no external loads and movements were applied to specimen and the influence of illumination and temperature variations was small enough to be ignored, the errors can mostly be attributed to random noise and the 2D-DIC algorithm. The mean values of displacement fields and their corresponding standard deviations are all close to zero. The mean values of strain fields fluctuate around zero with an amplitude of 9 με, and the corresponding standard deviation slightly varies at a level of 60 με. All these approximately horizontal curves can be seen as the reference values with no errors except for small system errors and random errors. As shown in Fig. 9, the mean displacements of camera reposition tests [Fig. 9(a)] present no noticeable difference from the reference values, while the corresponding standard deviations [Fig. 9(b)] show a large volatility
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this work. The simple but effective approach is presented in Appendix A. After fitting the displacement fields using linear plane and removing these in-plane rigid-body displacements using the fitted coefficients, the mean values plotted in Fig. 7(a) are all right around zero. The standard deviations of the displacement fields exhibit extremely small variability compared with those of the static test. As shown in Fig. 7(b), the values are all less than 0.02 pixels. The corresponding mean values and standard deviations of strain fields are given in Fig. 8. Similarly, the results of stationary tests are chosen as references. Due to the advantages of insensitivity to out-of-plane motion and the extremely small distortion coefficient by using the Schneider bilateral telecentric lens, the induced strain errors are at a very low level, which can be reflected on the small mean strain errors shown in Fig. 8(a) and steady standard deviations shown in Fig. 8(b). For specimen reposition tests, the influence of reposition on strain fields is also limited. The mean values of strain fields range from −4 to 32 με, and standard deviations vary from 50 to 90 με. These relatively small errors can be ignored in real tests.
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Fig. 10. (a) Average values and (b) standard deviations of measured strain fields measured by the regular 2D-DIC system in camera reposition, specimen reposition, and stationary tests.
with four peaks from point A to D. Although the in-plane translations and rotations were removed, it is still a little difficult to distinguish the impact of out-of-plane motion and out-of-plane rotation on displacement fields merely from Fig. 9(b). By inspecting Table 1, it can be concluded that the mean value of the v-displacement field is sensitive to the outof-plane rotation. However, the mean displacements in camera reposition tests are all near zero, which indicates that the influence of out-of-plane rotation is negligible. Therefore, out-of-plane translation is a dominant error source in camera reposition tests. The highly variable mean values (ranging from −362 to 202 με) shown in Fig. 10(a) and small changes (ranging from 64 to 92 με) in standard deviations shown in Fig. 10(b) further demonstrate this inference. For specimen reposition tests, it can be seen that the induced mean strain errors vary between −82 and 65 με. This range is much smaller than that of camera reposition tests. The corresponding standard deviations of strain fields also have very small changes. These results indicate that the errors due to the repositioning of the specimen are smaller when compared with those of camera reposition tests. However, in real tests, the out-of-plane motion of the test specimen surface due to the imperfection of the loading device and Poisson’s ratio of a tested material is also a vital error source, which does not exist in these tests. In a word, all these results and above analyses show that the regular 2D-DIC system should not be directly used for accurate ex situ deformation measurements, as the unavoidable outof-plane motions, though they are quite small, can induce considerable mean strain errors. By investigating the influence of repositions of the specimen and the camera on the measured displacement and strain fields using the established ultra-stable 2D-DIC system and the regular 2D-DIC system, it is concluded that the proposed ultra-stable 2D-DIC system equipped with a bilateral telecentric lens can greatly mitigate the errors due to small out-of-plane motions and out-of-plane rotations, and these errors are randomly distributed in each
reposition test. As a result, the tendencies in Figs. 7–10 do not indicate regular distributions, but largely show the sensitivity of the two 2D-DIC systems to the repositioning of camera and specimen. In this view, the 2D-DIC system using a bilateral telecentric lens is highly stable, which is reflected on the extremely small changes in mean values and standard deviations of strain fields, and can be used as the practical 2D-DIC system for ex situ deformation measurement. 4. Application to Ex Situ Residual Stress Determination Using Hole-Drilling Technique A. Experimental Details
To investigate the practicality of the established ultra-stable 2D-DIC system, a validation test for ex situ residual stress measurement using the holedrilling technique was conducted. Generally, experimental validation for residual stress measurement needs to know the expected residual stress values, which will be compared with the measured ones. For this purpose it is necessary to fully release all stresses induced during the manufacturing of the specimen and artificially apply expected ones. A number of techniques have been utilized to fabricate the specimen with different residual stresses, such as shrinking fitting [15] and shot peening [16]. However, it is difficult and troublesome to accurately control the stress state and the stress values as expected. An alternate but more simple approach, which was also used to simulate the uniaxial residual stress state by other researchers [21,22], is available with mechanical loading. The experimental procedure is summarized as follows: (1) load the specimen to a known value of F, which is below the yield load; (2) acquire the reference image using the established 2D-DIC system; (3) remove the specimen to another location and drill a through hole in the center of the ROI; and (4) load the repositioned specimen with the load of F again, and then capture an image as the target image. It should be noted that a reposition process for the specimen is involved in 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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Fig. 11. (a) Experimental setup for validation test. (b) Reference image before drilling a through hole. (c) Target image after drilling a through hole of 3 mm in diameter.
the aforementioned approach, whereas the specimen in [21,22] were not removed during the whole tests. In present test, the specimen was a dog-bone-type aluminum specimen measuring 20 mm wide by 2 mm thick. The overall length of the specimen was 200 mm, and the distance between the two grips was approximately 110 mm. From the uniaxial tension tests, Young’s modulus and Poisson’s ratio were measured as 72.6 GPa and 0.33, respectively. A yield stress of 419 MPa was also determined from the tension test. Before loading, the specimen was subjected to a normalization annealing for the purpose of stress release, and prepared with a speckle pattern by applying white and black paints. Then, as shown in Fig. 11(a), the specimen was installed in the testing machine (UTM5105, Shenzhen Suns Technology Stock Co., Ltd., Guangdong Province, China) and imaged by a CCD camera (TGX50, Baumer, Inc., Switzerland) with a resolution of 2448 × 2050 pixels at 8-bit gray levels. Finally, the aforementioned experimental procedure was implemented with the applied load F equivalent to 7.5 kN, which indicates an axial stress of 258.26 MPa. The reference image before drilling a hole and the target image after drilling a through hole of 3 mm in diameter are shown in Figs. 11(b) and 11(c), respectively. By numerically correlating the reference image shown in Fig. 11(b) and the target image shown in Fig. 11(c) using a subset-based DIC algorithm, the induced displacement fields around the hole, which can only be attributed to stress relief, were obtained. During the correlation analysis, the ROI was selected in the reference image with the area of the hole carefully excluded. A 41 × 41 pixels subset size and a 5 pixels grid step were chosen as calculation parameters. As a consequence, the displacement data in x and y directions of 36,064 discrete points were calculated and then used to determine the residual stress. The approach to estimate the corresponding residual stress will be briefly introduced in the following. 4224
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B. Determination of the Residual Stress
Theoretically, radial and tangential displacement components, denoted as uri and uθi , induced by the stress relaxation produced by a through hole can be expressed in terms of residual stress components σ x , σ y , and τxy as [23] uri Ai Bi cos 2θi σ x Ai − Bi cos 2θi σ y 2Bi sin 2θi τxy uθi Ci sin 2θi σ x − Ci sin 2θi σ y − 2Ci cos 2θi τxy ;
(1)
where ri and θi are the polar coordinates of the ith point, and the corresponding coefficients Ai, Bi , and Ci have the following expressions: 3 r0 r0 r0 r0 r 1 ν ; 4 − 1 ν 0 Ai ; Bi 2E ri 2E ri ri 3 r r r ; (2) Ci − 0 21 − ν 0 1 ν 0 2E ri ri where r0 is the hole radius, ri the distance of the ith point from the center of the hole, ν is the Poisson ratio, and E is the Young modulus. Since the ultrastable 2D-DIC system with a telecentric depth of 10 mm is insensitive to the out-of-plane motions on the surface of the specimen, the effect of the out-ofplane displacement component (i.e., uzi ) caused by stress relaxation, along with out-of-plane motions due to the process of drilling and removing specimen, can be completely ignored. In other words, the inplane motions measured by the ultra-stable 2D-DIC system are free of the errors due to any out-of-plane motions on the specimen surface, and of high fidelity. The in-plane displacement components, Eq. (1), then can be projected to the x and y directions using a simple translation formula to finally obtain the following equations:
uxi u0 − ωyi uri cos θi − uθi sin θi u0 − ωyi α1i σ x α2i σ y α3i τxy uyi v0 ωxi uri sin θi uθi cos θi v0 ωxi β1i σ x β2i σ y β3i τxy ;
(3)
where uxi and uyi are the in-plane displacement components directly measured by DIC, u0 and v0 account for the in-plane rigid-body motion of the specimen, ω is the in-plane rotation, and the derived coefficients α1i , α2i , α3i , β1i , β2i , and β3i can be determined as α1i Ai Bi cos 2θi cos θi − Ci sin 2θi sin θi ; α2i Ai − Bi cos 2θi cos θi Ci sin 2θi sin θi ; α3i 2Bi sin 2θi cos θi 2Ci cos 2θi sin θi ; β1i Ai Bi cos 2 θi sin θi Ci sin 2θi cos θi ; β2i Ai − Bi cos 2θi sin θi − Ci sin 2θi cos θi ; β3i 2Bi sin 2θi sin θi − 2Ci cos 2θi cos θi :
(4)
For a specified ROI containing nn > 6 calculated points, Eq. (4) can be written compactly as XP U;
(5)
where X is a 2n × 6 matrix containing constants (i.e., 0 and 1), coordinates of n points, and values of the coefficients, P is a 6 × 1 vector including the stress components to be determined, and U is a 2n × 1 vector containing the measured displacement components in the x and y directions. The least squares solution for determining the desired stress component σ y in the parameter vector P can be obtained from P XT X−1 XT U: C.
(6)
Experimental Results
Figures 12(a) and 12(b) show the directly measured displacement fields that were obtained for an applied
Fig. 12.
stress of 258.26 MPa, which is equal to 61.6% of the yield stress of the specimen. Note that the rigidbody motions and in-plane rotation were removed using the approach described in Appendix A. The u-displacement field shown in Fig. 12(a) reveals a symmetric distribution around the hole in the x direction, while the v-displacement field shows a symmetric distribution around the hole in the y direction with a higher amplitude. This is in good agreement with the theoretically simulated displacement distribution. The displacement components of all the calculated points measured by 2D-DIC were then submitted into Eq. (6) to determine the residual stress. With the least squares approach applied, the residual stress component σ y was found as 252.62 MPa, while the other two stress components (σ x and τxy ) were both close to 0. By comparing the residual stress obtained with the ultra-stable 2D-DIC system with the tensile stress measured by the load cell, the difference was found as 2.1%, indicating a good result has been obtained. 5. Conclusions and Future Work
Long-term monitoring of the deformation of engineering structures requires the repositioning of the camera. During the process of repositioning, position errors unavoidably occur and generally lead to considerable errors in the displacements and strains detected by the DIC method. In this work, an ultrastable 2D-DIC system using a well-designed highquality bilateral telecentric lens, bandpass filter imaging, and active monochromatic illumination is proposed for accurate ex situ displacement and strain measurements. The established imaging system is invariant to the potential variations in ambient lighting and the possible small out-of-plane motions in the object surface and/or image surface caused by reposition. Experimental results show that the ex situ 2D-DIC measurement using the proposed system is still of high fidelity, and can be used for accurate deformation measurement in ex situ tests. However, since a bilateral telecentric lens is limited to a fixed field of view and fixed working distance, it
Contour plots of (a) u- and (b) v-displacement fields directly measured by 2D-DIC. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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is not generally applicable. Thus future work should be focused on investigating effective ways for compensating these errors due a regular imaging system using unstrained objects or calibration targets [19,24–26]. Appendix A: Eliminating the In-Plane Rigid-Body Translations and In-Plane Rotations from the Original Displacement Data Using Least Squares Approach
In real tests, the directly detected displacement fields unavoidably include unwanted in-plane translations and rotations, which make no contribution to the deformation and can be completely removed using the least squares approach. Suppose xi and yi are the coordinates of the ith point among the n calculated points. Then, two planes [denoted as uxi ; yi and vxi ; yi ], which account for the in-plane translations and rotations, can be written as uxi ; yi u0 − ωyi ; vxi ; yi v0 ωxi ;
(A1)
where u0 and v0 are the in-plane rigid-body translations, and ω represents the in-plane rotations. Similar to the process described in Section 4.B, the corresponding coefficients u0 , v0 , and ω can be obtained using the least squares approach: 2u 3 x1
2
1
6 ux2 7 6 1 7 6 6 6 . 7 6 6 . 7 6 .. 6 . 7 6. 7 6 6 6u 7 6 6 xn 7 6 1 7 6 6 6u 7 60 6 y1 7 6 7 6 6 6 uy2 7 6 0 7 6 6 6 . 7 6. 6 . 7 6. 4 . 5 4. uyn
0
0
−y1
3
−y2 7 7 7 .. 7 . 7 72 u 3 7 0 0 −yn 76 7 7 4 v0 5 ; 1 x1 7 7 7 ω 1 x2 7 7 .. 7 .. 7 . 5 . 0 .. .
1
(A2)
xn
where uxi and vyi are the measured displacement components of the ith point. Then, by subtracting the displacements accounting for in-plane translations and rotations from uxi and vyi , the corrected displacements u0xi and v0yi can be obtained as u0xi uxi − uxi ; yi uxi − u0 ωyi ; u0yi uyi − vxi ; yi uyi − v0 − ωxi :
(A3)
This work was supported by the National Natural Science Foundation of China under grants 11172026, 11272032, and 11322220, the Program for New Century Excellent Talents in University under grant NCET-12-0023, the Science Fund of State Key Laboratory of Automotive Safety and Energy under grant KF14032, and the Beijing Nova Program under grant xx2014B034. 4226
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References 1. M. A. Sutton, J. J. Orteu, and H. W. Schreier, Image Correlation for Shape, Motion and Deformation Measurements (Springer, 2009). 2. B. Pan, K. M. Qian, H. M. Xie, and A. Asundi, “Twodimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20, 062001 (2009). 3. B. Pan, “Recent progress in digital image correlation,” Exp. Mech. 51, 1223–1235 (2011). 4. F. Hild and S. Roux, “Digital image correlation: from displacement measurement to identification of elastic properties—a review,” Strain 42, 69–80 (2006). 5. B. Pan and K. Li, “A fast digital image correlation method for deformation measurement,” Opt. Lasers Eng. 49, 841–847 (2011). 6. B. Pan, K. Li, and W. Tong, “Fast, robust and accurate digital image correlation calculation without redundant computations,” Exp. Mech. 53, 1277–1289 (2013). 7. B. Pan, A. Asundi, H. M. Xie, and J. X. Gao, “Digital image correlation using iterative least squares and pointwise least squares for displacement field and strain field measurements,” Opt. Lasers Eng. 47, 865–874 (2009). 8. M. A. Sutton, J. H. Yan, V. Tiwari, J. J. Orteu, and H. W. Schreier, “The effect of out-of-plane motion on 2D and 3D digital image correlation measurements,” Opt. Lasers Eng. 46, 746–757 (2008). 9. N. A. Hoult, W. A. Take, C. Lee, and M. Dutton, “Experimental accuracy of two dimensional strain measurements using digital image correlation,” Eng. Struct. 47, 718–726 (2013). 10. S. P. Ma, J. Z. Pang, and Q. W. Ma, “The systematic error in digital image correlation induced by self-heating of a digital camera,” Meas. Sci. Technol. 23, 025403 (2012). 11. Y. G. Wang and W. Tong, “A high resolution DIC technique for measuring small thermal expansion of film specimens,” Opt. Lasers Eng. 51, 30–33 (2013). 12. B. Pan, L. P. Yu, and D. F. Wu, “High-accuracy 2D digital image correlation measurements with bilateral telecentric lens: error analysis and experimental verification,” Exp. Mech. 53, 1719–1733 (2013). 13. G. Sposito, P. Ward, P. Cawley, P. B. Nagy, and C. Scruby, “A review of non-destructive techniques for the detection of creep damage in power plant steels,” NDT&E Int. 43, 555–567 (2010). 14. N. McCormick and J. Lord, “Digital image correlation,” Mater. Today 13(12), 52–54 (2010). 15. D. V. Nelson, A. Makino, and T. Schmidt, “Residual stress determination using hole drilling and 3D image correlation,” Exp. Mech. 46, 31–38 (2006). 16. J. D. Lord, D. Penn, and P. Whitehead, “The application of digital image correlation for measuring residual stress by incremental hole drilling,” Appl. Mech. Mater. 13, 65–73 (2008). 17. Z. X. Hu, H. M. Xie, J. Lu, and H. X. Wang, “Residual stresses measurement by using ring-core method and 3D digital image correlation technique,” Meas. Sci. Technol. 24, 085604 (2013). 18. B. Pan, D. F. Wu, and Y. Xia, “An active imaging digital image correlation method for deformation measurement insensitive to ambient light,” Opt. Laser Technol. 44, 204–209 (2012). 19. B. Pan, L. P. Yu, and D. F. Wu, “High-accuracy 2D digital image correlation measurements with low-cost imaging lenses: implementation of a generalized compensation method,” Meas. Sci. Technol 25, 025001 (2014). 20. B. Pan, D. F. Wu, Z. Y. Wang, and Y. Xia, “High-temperature digital image correlation for full-field deformation measurement at 1200°C,” Meas. Sci. Technol. 22, 015701 (2011). 21. M. R. Viotti and G. H. Kaufmann, “Accuracy and sensitivity of a hole drilling and digital speckle pattern interferometry combined technique to measure residual stresses,” Opt. Lasers Eng. 41, 297–305 (2004). 22. M. R. Viotti, A. E. Dolinko, E. G. Galizzi, and H. K. Guillermo, “A portable digital speckle pattern interferometry device to measure residual stresses using the hole drilling technique,” Opt. Lasers Eng. 44, 1052–1066 (2006).
23. A. Makino and D. V. Nelson, “Residual-stress determination by single-axis holographic interferometry and hole drilling— Part I: Theory,” Exp. Mech. 34, 66–78 (1994). 24. M. Malesa and M. Kujawinska, “Modified two-dimensional digital image correlation method with capability of merging of data distributed in time,” Appl. Opt. 51, 8641–8655 (2012).
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In situ planar deformation measurement using two-dimensional digital image correlation (2D-DIC) and a fixed camera has been fully investigated in many published works and widely used for various applications. However, in certain special cases (e.g., long-term deformation monitoring of engineering structures, or deformation measurement of a specimen subjected to uncommon loading), it is very difficult or impossible to carry out in situ deformation measurement using 2D-DIC, and necessitates ex situ 2D-DIC measurements for a repositioned specimen or using a relocated camera. To achieve accurate measurements, the error sources of ex situ 2D-DIC measurements should be identified and minimized, and the strain accuracy of ex situ 2D-DIC measurements should be quantified. In this work, the potential error sources of ex situ 2D-DIC measurements are first discussed in detail. Then, to mitigate the errors associated with these issues, an ultra-stable 2D-DIC system combining the idea of active imaging and a welldesigned bilateral telecentric lens is established, which is invariant to the potential variations in ambient lighting and the possible small out-of-plane motions in object surface and/or image plane. The established ultra-stable 2D-DIC system is first compared with the regular 2D-DIC setup in determining the surface strains of an unstrained sample. Then, ex situ residual stress measurement using the hole-drilling technique and the established 2D-DIC system was conducted and compared with the applied ones. The results of this work confirm that the accuracy of ex situ 2D-DIC deformation measurements using the proposed system is of high fidelity, and can be used for accurate deformation measurement in practical ex situ tests. © 2014 Optical Society of America OCIS codes: (120.0120) Instrumentation, measurement, and metrology; (110.0110) Imaging systems; (150.3045) Industrial optical metrology. http://dx.doi.org/10.1364/AO.53.004216
1. Introduction
As a typical non-interferometric optical metrology noted for its simple principle and easy implementation, two-dimensional digital image correlation (2DDIC) [1–4] using a single digital camera has been widely used in various scientific research and engineering fields for in-plane displacement and strain measurements of planar objects. Over the last 30 years, this cost-effective, easy-to-use but practical 1559-128X/14/194216-12$15.00/0 © 2014 Optical Society of America 4216
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technique has undergone not only significant improvements in technique development, but also a burst of applications. In practical implementation of 2D-DIC for in-plane deformation measurement, the optical axis of the imaging lens should be aligned perpendicularly to the specimen surface before loading. Then, the random intensity distribution (also known as the speckle pattern) on the test specimen surface at different states is imaged onto the sensor target of a CCD/CMOS camera to form reference (or undeformed) and target (or deformed) images. By comparing the deformed images with the reference one using a well-established subset-based image-matching
algorithm [5,6], full-field displacements with subpixel accuracy can be determined. Also, full-field strain information can be further extracted by differentiating the obtained displacement fields [7]. In regular 2D-DIC measurements based on optically recorded images, it is a common sense and also an essential requirement that the imaging system (comprising a digital camera and an optical lens) and the specimen should be fixed and remain stationary during recording the surface images of the test specimen. It has been recognized [8–12] that, for 2D-DIC measurement using a common lens, even small out-of-plane motions between the camera and the specimen may induce additional large virtual displacements and strains, and corrupt the desired displacement and strain values induced by external loading. Clearly, this basic requirement ensures that the specimen surface and sensor target remain in their original planes, and thus avoid any small relatively out-of-plane rigid-body movements between these two planes. However, in certain special cases, it is very difficult or impossible to carry out in situ strain measurement using 2D-DIC. First, in some cases, for the purpose of protecting the measuring equipment, the 2D-DIC setup needs to be relocated to its original position to image the test sample at set intervals. For instance, during structural health monitoring of the most vulnerable regions of engineering structures (e.g., pipe welds in power generation industry) over a long period of time, and strain measurement for fatigue tests spread over several months, the 2D-DIC system cannot be left stationary at its original position for safety reasons [13,14]. Second, in other cases, it is difficult to arrange the 2D-DIC system to accommodate certain loading devices. One typical example is the determination of residual strain in a hole-drilling testing with the DIC method [15–17]. Generally, the test sample, whose residual stress is to be determined, is mounted at a sample stage, and a reference image is captured. Then the specimen is moved to a drilling machine, and a small shallow hole is drilled at the region of interest (ROI). Afterward, the specimen with released residual stress is moved back to original place to record a target image. In all these cases, small in-plane and out-of-plane rigid-body movements between the image plane and the object
plane unavoidably occur due to the repositioning of the imaging system or the specimen. Furthermore, considering the fact that image capture should be performed at different times after the repositioning of the imaging system or the specimen, the variation in ambient light may be too severe to be ignored, especially in outdoor field measurement [18]. Although in situ displacement and strain measurement using 2D-DIC has been fully studied, very little work has been dedicated to ex situ 2D-DIC measurement for a repositioned specimen or using a relocated camera. From the viewpoint of applications, the error sources of ex situ 2D-DIC measurements should be identified, and the strain accuracy of ex situ 2D-DIC measurements should be quantified. With these questions in mind, in the reminder of this work, the possible error sources of ex situ 2D-DIC measurements are first discussed in detail. Then, to minimize these potential errors induced by relocation of the camera or the specimen, an ultra-stable 2D-DIC system, which is invariant to the potential variations in ambient lighting and the possible small out-of-plane motions in object surface and/or image plane, is established. Afterward, ex situ 2D-DIC experiments including specimen repositioning tests and camera relocation tests were performed using a nondeformable sample. The surface strains of the sample were determined by the established ultra-stable 2DDIC system and a regular 2D-DIC setup. Finally, a real hole-drilling test was also carried out to further demonstrate the practicality of the proposed ultrastable 2D-DIC system for ex situ deformation measurements. 2. Ex Situ Deformation Measurement Using an Ultra-Stable 2D-DIC System A. Error Analysis of 2D-DIC Measurement due to Specimen or Camera Relocation
Figure 1 shows the schematic of a typical 2D-DIC setup using an optical imaging device for measuring surface deformation of planar objects. The test planar object surface should have a random speckle pattern, which deforms together with the specimen surface as a faithful carrier of deformation information. The fixed camera equipped with an optical imaging lens is aligned with its optical axis normal
Fig. 1. Schematic diagram of the 2D-DIC measurement using a regular optical imaging lens. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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(a)
(b) Specimen Sensor
v
v
Lens
Z
Repositioned camera
L
Repositioned specimen
Fig. 2. Schematic diagrams showing the effects of (a) camera reposition and (b) specimen reposition on the image displacements measurement by 2D-DIC.
to the specimen surface, imaging the planar specimen surface of different loading states onto its sensor plane. The digital images of the test object surface are saved and subsequently processed by wellestablished subset-based DIC algorithms to retrieve the surface deformation information. The common 2D-DIC system is able to deliver accurate displacement and strain measurements in which the specimen and camera are both tightly fixed during the testing period. However, for ex situ deformation measurement of a repositioned specimen or using a relocated camera, the position or the orientation of the test sample relative to the camera will change slightly due to unavoidable reposition errors, as schematically illustrated in Fig. 2. Generally, out-of-plane translation and rotation, and in-plane translation and rotation, may occur simultaneously after the reposition of the specimen or the camera. It should be pointed out first that, no matter what kind of reposition is conducted, the camera can be assumed to be fixed and the resulting change in imaging model can be depicted as a position and orientation change in the sample surface. It is known that in-plane translation and rotation of the specimen have no effect on the measured strains, and the displacements caused by in-plane motions can be
eliminated using a plane fitting approach as detailed in Appendix A. In other words, merely out-of-plane translation and rotation lead to an alteration in object distance ΔZ of a physical point on the test sample surface, which may vary at different positions. Theoretically, the displacement and strain errors due to out-of-plane translation and/or rotation have been thoroughly derived in Ref. [19]. To intuitively show the distributions of the errors and better understand the results in the next section, we simulated the theoretical displacement distribution according to the derived equations in Ref. [19], and then estimated the corresponding mean values and standard deviations. During the simulation, the ROI was set as 1000 × 1000 pixels, and the object distance Z and the image distance L were defined as 400 and 50 mm, respectively. Meanwhile, the magnification of the imaging system was assumed to be 100 pixel∕mm. Then, two cases, one in which the object surface has a small out-of-plane translation ΔZ 1 mm away from the camera and the other in which the object surface has a clockwise out-of-plane rotation θ 1° about the X axis, were simulated. As shown in Figs. 3 and 4, the simulated displacement fields for the two cases show regular distributions, as the equations indicated. Finally, the mean values
Fig. 3. Simulated (a) u and (b) v fields with a 1 mm out-of-plane translation away from the camera occurred on the specimen surface. 4218
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Fig. 4. Simulated (a) u and (b) v fields with a 1° out-of-plane rotation about the X axis occurred on the specimen surface.
and standard deviations of these simulated displacement and strain fields were calculated, as shown in Table 1. Although the simulated displacement and strain fields are free of noise, their statistics can still be considered as a beneficial reference in the following analyses. As seen from Table 1, the errors caused by small out-of-plane translation or rotation are considerable and should be eliminated, especially when conducting high-accuracy deformation measurements. To minimize the effect of out-of-plane motions on 2D-DIC measurements, the following two possible measures were suggested in Ref. [8]: (1) use a welldesigned telecentric lens with large “effective” object distance, and (2) place the camera far away from the test object to approximate a telecentric lens. In addition, a newly developed error compensating technique using an unstrained compensating sample can also be used [18]. Other than unavoidable changes in object distance after specimen or camera reposition, the position and orientation of the sensor target (i.e., image plane) may also change slightly, due to the temperature variations caused by self-heating or ambient temperature. In addition, for ex situ DIC measurement, ambient light may change accordingly, especially when the measurements are performed in outdoor environments or at long intervals. In short, no matter what kind of repositioning is carried out, all these detrimental factors may more or less alter the imaging conditions assumed for the original 2D-DIC system, and lead to appreciable strain errors.
Table 1.
Mean Values and Standard Deviations of the Simulated u, v , εx , and εy Fields
Simulations
u (pixel)
v (pixel)
εx με
εy με
Out-of-plane 0 0.725 0 0.725 −2500 0 −2500 0 translation Out-of-plane 0 0.294 0.294 0.266 0 1000 −152 2000 rotation
B. Ex Situ Deformation Measurement Using an Ultra-Stable 2D-DIC System
In our recent work regarding high-accuracy 2D-DIC measurements, it is concluded that a well-designed bilateral telecentric lens is an essential optical component to realize high-accuracy strain measurement [12]. The bilateral telecentric lens is insensitive to small changes in both object distance and image distance, and can maintain constant magnification within its range of depth of focus. Also, the lens distortion of a high-quality bilateral telecentric lens is normally small enough to be neglected. These distinct merits of a bilateral telecentric lens enlighten us to build an ultra-stable 2D-DIC imaging system for ex situ strain measurement. Moreover, to deal with the serious light variation possibly occurring in outdoor measurement, the imaging system can be further improved by introducing the idea of active imaging [18,20], which incorporates an actively illuminated monochromatic light and an optical bandpass filter attached just before the bilateral telecentric lens. Shown in Fig. 5(a) is the schematic illustration of the proposed ultra-stable 2D-DIC system for accurate deformation measurement in an ex situ 2D-DIC experiment. The system comprises a high-resolution CCD camera (TGX50, Baumer, Inc., Switzerland), a high-quality bilateral telecentric lens (Xenoplan 1∶5, Schneider Optics, Inc., Germany), an optical bandpass filter with a center wavelength of 450 2 nm, and a monochromatic source emitting at 450–455 nm. The critical component in this 2DDIC system is the bilateral telecentric lens, as shown in Fig. 5(b). With the unique design of the aperture stop, the bilateral telecentric lens is not susceptible to small changes in the distance between the lens and the sensor plane. Therefore, the 2D-DIC system equipped with a bilateral telecentric lens performs significantly better in eliminating or avoiding the unfavorable effects of out-of-plane motions and outof-plane rotations compared with the use of a conventional lens. Moreover, according to our previous 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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Fig. 5. (a) Schematic diagram of the established ultra-stable 2D-DIC system. (b) Bilateral telecentric lens. (c) Transmission spectrum of the optical bandpass filter.
research [12], the 2D-DIC system using a bilateral telecentric lens presents very low lens distortion, which results in small errors in displacement and strain fields due to in-plane motions. With these distinct advantages of a bilateral telecentric lens, the errors due to the repositioning of a camera or specimen can be minimized. The 2D-DIC system using a bilateral telecentric lens is then considered to be a highly stable 2D-DIC system insensitive to any small out-of-plane motions occurring in the object surface and image plane. To compensate the illumination variations, improvements are also made to illumination and imaging in the 2D-DIC system. A tailor-made monochromatic blue LED lighting emitting at 450– 455 nm, instead of a common white light source, is adopted for illumination. Also, an optical bandpass filter, with a coupled center wavelength of 450 2 nm and a full-width at half-maximum value of approximately 32 nm, is mounted before the bilateral telecentric lens. Figure 5(c) shows the transmission efficiency curve of the bandpass filter. For the active illuminated monochromatic light, the bandpass filter shows a very high transmission efficiency exceeding 80%. As only a very limited portion of ambient light within the bandpass range of the filter can pass thorough the filter, the ambient lighting makes little and negligible contribution to the intensity of the recorded image compared with the actively illuminated monochromatic light. The system is therefore 4220
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capable of acquiring high-quality images with almost identical brightness and contrast even though serious changes have occurred in ambient lighting. 3. Experimental Verification Using an Unstrained Sample A. Experimental Details
To evaluate the effectiveness and accuracy of the established ultra-stable 2D-DIC system for ex situ deformation measurement, a series of camera reposition tests and specimen reposition tests were conducted in this work. The tested specimen is a flat glass plate with a premade random speckle pattern on its surface. As indicated in Fig. 6, the plate was first vertically fixed in a plate clamp and illuminated using the monochromatic source, and the imaging system, which was tightly fixed on a camera stage through four screws, was placed 268 mm in the front of the plate. In the following, three different experiments were designed. (1) Camera repositioning tests were conducted to investigate the errors due to unavoidable relocation errors. In this experiment, after getting a clear speckle pattern with sufficient contrast, a reference image was captured first. Then, the imaging system was moved from the fixed position by loosening the fixing screws and repositioned to the original position with an image recorded as a target image. The same process was repeated 20 times, and
Fig. 6. (a) Experimental setup for reposition tests. (b) Reference image captured using the established 2D-DIC system with the ROI and the corresponding calculation parameters defined.
Mean value of displacements (pixel)
5
(a)
u (camera reposition test) v (camera reposition test) u (specimen reposition test) v (specimen reposition test) u (static test) v (static test)
4 3 2 1 0 -1 -2 -3 -4 -5
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displacement data, a pointwise least squares [7] using a strain calculation window of 21 × 21 sets of displacement data (corresponds to a local region of 201 × 201 pixels) was employed to estimate the strain distributions. The two strain components (i.e., εx , εy ) at all calculation points are averaged to determine the average values of the ROI. B. Experimental Results
Figure 7 shows the mean values and standard deviations of the u and v displacements directly measured by the proposed ultra-stable 2D-DIC system for the three different experiments. For stationary tests, the mean displacements approach zero, and the standard deviations of the displacements are in a relatively low level, which agrees well with the practical deformation state. However, with respect to the camera or specimen relocation tests, it is clear that the repositioning of the camera or the specimen will induce artificial displacement and strain errors, which vary with each test and each practitioner. As a matter of fact, these errors are caused by the in-plane or out-of-plane translations and rotations of the test specimen relative to the camera. Since the in-plane translations and rotations exist in every test and make no contribution to the real deformation, we employed an approach based on the least squares method to cancel the in-plane translations and rotations from the directly measured displacements in
Standard deviation of displacements (pixel)
20 deformed images were recorded. (2) Specimen relocation tests were performed to investigate the strain errors caused by specimen reposition. The specimen was taken away from the plate clamp after a reference image was captured. Then the specimen was repositioned to record a target image for 20 times. (3) Regular static tests with fixed specimen and fixed camera were carried out to examine the strain accuracy of random noise and the DIC algorithm. In this test, 21 continuously captured images at a 10 s interval were used to study the uncertainty of 2D-DIC measurements, as an additional contrast to the above experiments. For comparison purposes, all three tests were also conducted using a common 2D-DIC system equipped with an ordinary industrial lens (Tec-55, Computar Optics, Inc.). For each test, 21 images including one reference image and 20 deformed images were recorded. Next, all the deformed images are compared with the reference image to extract the displacement and strains. During the correlation analysis, a rectangular region located in the middle of the reference image, containing 20,164 discrete points on a 142 × 142 grid, was chosen to be the ROI. Then, a 41 × 41 pixels subset size and a 10 pixels grid step were chosen as calculation parameters for each point of interest. The displacement components were first analyzed using the fast Newton–Raphson algorithm [5] we proposed recently. Based on the obtained discrete
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0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
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Fig. 7. (a) Average values and (b) standard deviations of measured displacement fields measured by the proposed ultra-stable 2D-DIC system in camera reposition, specimen reposition, and stationary tests. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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Fig. 8. (a) Average values and (b) standard deviations of measured strain fields measured by the proposed ultra-stable 2D-DIC system in camera reposition, specimen reposition, and stationary tests.
Figures 9(a) and 9(b) show the mean values and standard deviations of the measured u and v displacements for the case of camera reposition, specimen reposition, and stationary tests using the regular 2D-DIC system. The corresponding mean values and standard deviations of strain fields are shown in Figs. 10(a) and 10(b), respectively. For stationary tests, since no external loads and movements were applied to specimen and the influence of illumination and temperature variations was small enough to be ignored, the errors can mostly be attributed to random noise and the 2D-DIC algorithm. The mean values of displacement fields and their corresponding standard deviations are all close to zero. The mean values of strain fields fluctuate around zero with an amplitude of 9 με, and the corresponding standard deviation slightly varies at a level of 60 με. All these approximately horizontal curves can be seen as the reference values with no errors except for small system errors and random errors. As shown in Fig. 9, the mean displacements of camera reposition tests [Fig. 9(a)] present no noticeable difference from the reference values, while the corresponding standard deviations [Fig. 9(b)] show a large volatility
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this work. The simple but effective approach is presented in Appendix A. After fitting the displacement fields using linear plane and removing these in-plane rigid-body displacements using the fitted coefficients, the mean values plotted in Fig. 7(a) are all right around zero. The standard deviations of the displacement fields exhibit extremely small variability compared with those of the static test. As shown in Fig. 7(b), the values are all less than 0.02 pixels. The corresponding mean values and standard deviations of strain fields are given in Fig. 8. Similarly, the results of stationary tests are chosen as references. Due to the advantages of insensitivity to out-of-plane motion and the extremely small distortion coefficient by using the Schneider bilateral telecentric lens, the induced strain errors are at a very low level, which can be reflected on the small mean strain errors shown in Fig. 8(a) and steady standard deviations shown in Fig. 8(b). For specimen reposition tests, the influence of reposition on strain fields is also limited. The mean values of strain fields range from −4 to 32 με, and standard deviations vary from 50 to 90 με. These relatively small errors can be ignored in real tests.
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Fig. 9. (a) Average values and (b) standard deviations of measured displacement fields measured by the regular 2D-DIC system in camera reposition, specimen reposition, and stationary tests. 4222
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Fig. 10. (a) Average values and (b) standard deviations of measured strain fields measured by the regular 2D-DIC system in camera reposition, specimen reposition, and stationary tests.
with four peaks from point A to D. Although the in-plane translations and rotations were removed, it is still a little difficult to distinguish the impact of out-of-plane motion and out-of-plane rotation on displacement fields merely from Fig. 9(b). By inspecting Table 1, it can be concluded that the mean value of the v-displacement field is sensitive to the outof-plane rotation. However, the mean displacements in camera reposition tests are all near zero, which indicates that the influence of out-of-plane rotation is negligible. Therefore, out-of-plane translation is a dominant error source in camera reposition tests. The highly variable mean values (ranging from −362 to 202 με) shown in Fig. 10(a) and small changes (ranging from 64 to 92 με) in standard deviations shown in Fig. 10(b) further demonstrate this inference. For specimen reposition tests, it can be seen that the induced mean strain errors vary between −82 and 65 με. This range is much smaller than that of camera reposition tests. The corresponding standard deviations of strain fields also have very small changes. These results indicate that the errors due to the repositioning of the specimen are smaller when compared with those of camera reposition tests. However, in real tests, the out-of-plane motion of the test specimen surface due to the imperfection of the loading device and Poisson’s ratio of a tested material is also a vital error source, which does not exist in these tests. In a word, all these results and above analyses show that the regular 2D-DIC system should not be directly used for accurate ex situ deformation measurements, as the unavoidable outof-plane motions, though they are quite small, can induce considerable mean strain errors. By investigating the influence of repositions of the specimen and the camera on the measured displacement and strain fields using the established ultra-stable 2D-DIC system and the regular 2D-DIC system, it is concluded that the proposed ultra-stable 2D-DIC system equipped with a bilateral telecentric lens can greatly mitigate the errors due to small out-of-plane motions and out-of-plane rotations, and these errors are randomly distributed in each
reposition test. As a result, the tendencies in Figs. 7–10 do not indicate regular distributions, but largely show the sensitivity of the two 2D-DIC systems to the repositioning of camera and specimen. In this view, the 2D-DIC system using a bilateral telecentric lens is highly stable, which is reflected on the extremely small changes in mean values and standard deviations of strain fields, and can be used as the practical 2D-DIC system for ex situ deformation measurement. 4. Application to Ex Situ Residual Stress Determination Using Hole-Drilling Technique A. Experimental Details
To investigate the practicality of the established ultra-stable 2D-DIC system, a validation test for ex situ residual stress measurement using the holedrilling technique was conducted. Generally, experimental validation for residual stress measurement needs to know the expected residual stress values, which will be compared with the measured ones. For this purpose it is necessary to fully release all stresses induced during the manufacturing of the specimen and artificially apply expected ones. A number of techniques have been utilized to fabricate the specimen with different residual stresses, such as shrinking fitting [15] and shot peening [16]. However, it is difficult and troublesome to accurately control the stress state and the stress values as expected. An alternate but more simple approach, which was also used to simulate the uniaxial residual stress state by other researchers [21,22], is available with mechanical loading. The experimental procedure is summarized as follows: (1) load the specimen to a known value of F, which is below the yield load; (2) acquire the reference image using the established 2D-DIC system; (3) remove the specimen to another location and drill a through hole in the center of the ROI; and (4) load the repositioned specimen with the load of F again, and then capture an image as the target image. It should be noted that a reposition process for the specimen is involved in 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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Fig. 11. (a) Experimental setup for validation test. (b) Reference image before drilling a through hole. (c) Target image after drilling a through hole of 3 mm in diameter.
the aforementioned approach, whereas the specimen in [21,22] were not removed during the whole tests. In present test, the specimen was a dog-bone-type aluminum specimen measuring 20 mm wide by 2 mm thick. The overall length of the specimen was 200 mm, and the distance between the two grips was approximately 110 mm. From the uniaxial tension tests, Young’s modulus and Poisson’s ratio were measured as 72.6 GPa and 0.33, respectively. A yield stress of 419 MPa was also determined from the tension test. Before loading, the specimen was subjected to a normalization annealing for the purpose of stress release, and prepared with a speckle pattern by applying white and black paints. Then, as shown in Fig. 11(a), the specimen was installed in the testing machine (UTM5105, Shenzhen Suns Technology Stock Co., Ltd., Guangdong Province, China) and imaged by a CCD camera (TGX50, Baumer, Inc., Switzerland) with a resolution of 2448 × 2050 pixels at 8-bit gray levels. Finally, the aforementioned experimental procedure was implemented with the applied load F equivalent to 7.5 kN, which indicates an axial stress of 258.26 MPa. The reference image before drilling a hole and the target image after drilling a through hole of 3 mm in diameter are shown in Figs. 11(b) and 11(c), respectively. By numerically correlating the reference image shown in Fig. 11(b) and the target image shown in Fig. 11(c) using a subset-based DIC algorithm, the induced displacement fields around the hole, which can only be attributed to stress relief, were obtained. During the correlation analysis, the ROI was selected in the reference image with the area of the hole carefully excluded. A 41 × 41 pixels subset size and a 5 pixels grid step were chosen as calculation parameters. As a consequence, the displacement data in x and y directions of 36,064 discrete points were calculated and then used to determine the residual stress. The approach to estimate the corresponding residual stress will be briefly introduced in the following. 4224
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B. Determination of the Residual Stress
Theoretically, radial and tangential displacement components, denoted as uri and uθi , induced by the stress relaxation produced by a through hole can be expressed in terms of residual stress components σ x , σ y , and τxy as [23] uri Ai Bi cos 2θi σ x Ai − Bi cos 2θi σ y 2Bi sin 2θi τxy uθi Ci sin 2θi σ x − Ci sin 2θi σ y − 2Ci cos 2θi τxy ;
(1)
where ri and θi are the polar coordinates of the ith point, and the corresponding coefficients Ai, Bi , and Ci have the following expressions: 3 r0 r0 r0 r0 r 1 ν ; 4 − 1 ν 0 Ai ; Bi 2E ri 2E ri ri 3 r r r ; (2) Ci − 0 21 − ν 0 1 ν 0 2E ri ri where r0 is the hole radius, ri the distance of the ith point from the center of the hole, ν is the Poisson ratio, and E is the Young modulus. Since the ultrastable 2D-DIC system with a telecentric depth of 10 mm is insensitive to the out-of-plane motions on the surface of the specimen, the effect of the out-ofplane displacement component (i.e., uzi ) caused by stress relaxation, along with out-of-plane motions due to the process of drilling and removing specimen, can be completely ignored. In other words, the inplane motions measured by the ultra-stable 2D-DIC system are free of the errors due to any out-of-plane motions on the specimen surface, and of high fidelity. The in-plane displacement components, Eq. (1), then can be projected to the x and y directions using a simple translation formula to finally obtain the following equations:
uxi u0 − ωyi uri cos θi − uθi sin θi u0 − ωyi α1i σ x α2i σ y α3i τxy uyi v0 ωxi uri sin θi uθi cos θi v0 ωxi β1i σ x β2i σ y β3i τxy ;
(3)
where uxi and uyi are the in-plane displacement components directly measured by DIC, u0 and v0 account for the in-plane rigid-body motion of the specimen, ω is the in-plane rotation, and the derived coefficients α1i , α2i , α3i , β1i , β2i , and β3i can be determined as α1i Ai Bi cos 2θi cos θi − Ci sin 2θi sin θi ; α2i Ai − Bi cos 2θi cos θi Ci sin 2θi sin θi ; α3i 2Bi sin 2θi cos θi 2Ci cos 2θi sin θi ; β1i Ai Bi cos 2 θi sin θi Ci sin 2θi cos θi ; β2i Ai − Bi cos 2θi sin θi − Ci sin 2θi cos θi ; β3i 2Bi sin 2θi sin θi − 2Ci cos 2θi cos θi :
(4)
For a specified ROI containing nn > 6 calculated points, Eq. (4) can be written compactly as XP U;
(5)
where X is a 2n × 6 matrix containing constants (i.e., 0 and 1), coordinates of n points, and values of the coefficients, P is a 6 × 1 vector including the stress components to be determined, and U is a 2n × 1 vector containing the measured displacement components in the x and y directions. The least squares solution for determining the desired stress component σ y in the parameter vector P can be obtained from P XT X−1 XT U: C.
(6)
Experimental Results
Figures 12(a) and 12(b) show the directly measured displacement fields that were obtained for an applied
Fig. 12.
stress of 258.26 MPa, which is equal to 61.6% of the yield stress of the specimen. Note that the rigidbody motions and in-plane rotation were removed using the approach described in Appendix A. The u-displacement field shown in Fig. 12(a) reveals a symmetric distribution around the hole in the x direction, while the v-displacement field shows a symmetric distribution around the hole in the y direction with a higher amplitude. This is in good agreement with the theoretically simulated displacement distribution. The displacement components of all the calculated points measured by 2D-DIC were then submitted into Eq. (6) to determine the residual stress. With the least squares approach applied, the residual stress component σ y was found as 252.62 MPa, while the other two stress components (σ x and τxy ) were both close to 0. By comparing the residual stress obtained with the ultra-stable 2D-DIC system with the tensile stress measured by the load cell, the difference was found as 2.1%, indicating a good result has been obtained. 5. Conclusions and Future Work
Long-term monitoring of the deformation of engineering structures requires the repositioning of the camera. During the process of repositioning, position errors unavoidably occur and generally lead to considerable errors in the displacements and strains detected by the DIC method. In this work, an ultrastable 2D-DIC system using a well-designed highquality bilateral telecentric lens, bandpass filter imaging, and active monochromatic illumination is proposed for accurate ex situ displacement and strain measurements. The established imaging system is invariant to the potential variations in ambient lighting and the possible small out-of-plane motions in the object surface and/or image surface caused by reposition. Experimental results show that the ex situ 2D-DIC measurement using the proposed system is still of high fidelity, and can be used for accurate deformation measurement in ex situ tests. However, since a bilateral telecentric lens is limited to a fixed field of view and fixed working distance, it
Contour plots of (a) u- and (b) v-displacement fields directly measured by 2D-DIC. 1 July 2014 / Vol. 53, No. 19 / APPLIED OPTICS
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is not generally applicable. Thus future work should be focused on investigating effective ways for compensating these errors due a regular imaging system using unstrained objects or calibration targets [19,24–26]. Appendix A: Eliminating the In-Plane Rigid-Body Translations and In-Plane Rotations from the Original Displacement Data Using Least Squares Approach
In real tests, the directly detected displacement fields unavoidably include unwanted in-plane translations and rotations, which make no contribution to the deformation and can be completely removed using the least squares approach. Suppose xi and yi are the coordinates of the ith point among the n calculated points. Then, two planes [denoted as uxi ; yi and vxi ; yi ], which account for the in-plane translations and rotations, can be written as uxi ; yi u0 − ωyi ; vxi ; yi v0 ωxi ;
(A1)
where u0 and v0 are the in-plane rigid-body translations, and ω represents the in-plane rotations. Similar to the process described in Section 4.B, the corresponding coefficients u0 , v0 , and ω can be obtained using the least squares approach: 2u 3 x1
2
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6 ux2 7 6 1 7 6 6 6 . 7 6 6 . 7 6 .. 6 . 7 6. 7 6 6 6u 7 6 6 xn 7 6 1 7 6 6 6u 7 60 6 y1 7 6 7 6 6 6 uy2 7 6 0 7 6 6 6 . 7 6. 6 . 7 6. 4 . 5 4. uyn
0
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−y2 7 7 7 .. 7 . 7 72 u 3 7 0 0 −yn 76 7 7 4 v0 5 ; 1 x1 7 7 7 ω 1 x2 7 7 .. 7 .. 7 . 5 . 0 .. .
1
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xn
where uxi and vyi are the measured displacement components of the ith point. Then, by subtracting the displacements accounting for in-plane translations and rotations from uxi and vyi , the corrected displacements u0xi and v0yi can be obtained as u0xi uxi − uxi ; yi uxi − u0 ωyi ; u0yi uyi − vxi ; yi uyi − v0 − ωxi :
(A3)
This work was supported by the National Natural Science Foundation of China under grants 11172026, 11272032, and 11322220, the Program for New Century Excellent Talents in University under grant NCET-12-0023, the Science Fund of State Key Laboratory of Automotive Safety and Energy under grant KF14032, and the Beijing Nova Program under grant xx2014B034. 4226
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