Microscopic strain mapping using scanning electron ...

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Microscopic strain mapping using scanning electron microscopy topography image correlation at large strain J Kang1 *, M Jain2 , D S Wilkinson1 , and J D Embury1 1 Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario, Canada 2 Department of Mechanical Engineering, McMaster University, Hamilton, Ontario, Canada The manuscript was received on 1 October 2004 and was accepted after revision for publication on 22 April 2005. DOI: 10.1243/030932405X16151

Abstract: Measuring the distribution of local strain at the microscopic level is a challenging problem, especially for materials subjected to large overall strain. In the present study, a novel microscopic strain mapping technique has been developed based on the analysis of surface topography using digital image correlation (DIC) software. The input is a series of scanning electron microscopy (SEM) images. The method uses topographic features (such as surface slip traces) found in these images as the input. A commercially available optical strain measurement system (ARAMIS1 , which is a trade name of the equipment from GOM mbH, Braunschweig, Germany) that utilizes the DIC methodology is used for this purpose. It was found that the best results were obtained using an incremental approach in which DIC is used to map the local strain increments following a modest amount of macroscopic deformation. This is essential when using topographic features such as slip traces that are not static. The accuracy and scale of the measurements are affected by image and facet size. The method has been validated, based on in situ deformation of an aluminium alloy within an SEM, using strains measured independently by means of surface indents. The results clearly reveal the details of the local shear on a sub-grain-size scale and the evolution of shear bands within the necking area, leading to local strains that exceed the average strain by a factor of 2.3. Keywords: digital image correlation, scanning electron microscopy (SEM), in situ tensile test, strain mapping

1

INTRODUCTION

Full-field measurement of the microscopic strain distribution in materials that have been subjected to a large amount of macroscopic deformation has been of interest for many years. Several methods have been previously utilized. Of these, the fiducial grid method, which uses fine cross-line grids to measure the local deformation, is the most straightforward. A key issue in this method is that the pattern must not disturb the physical process under observation [1]. To achieve this end, several techniques have been developed to apply the grids using scribing, printing, etching, embossing, reflection, cementing, *Corresponding author: Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4L7, Canada. email: [email protected]

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or lithography [1–4]. Recently, Schroeter and McDowell [5] developed a photolithographic technique to make grids of 3–10 mm spacing. Biery et al. [6] developed a technique to evaporate gold on to the sample surface through a fine nickel grid of 59 lines/mm and measured the microstructural-scale strains using scanning electron microscopy (SEM). The major difficulty in utilizing the grid method lies in the complexity of processing the deformed grids to extract the full-field strain distribution. Several deformation mapping systems have been developed in the past. James et al. [7] developed a system to determine the relative displacement fields generated by thermal or mechanical loads by comparing a pair of SEM or optical micrographs, one recorded before the load was applied and the other afterwards. A mechanical linkage system in the positioning device was used for imaging the photographs. The accuracy is J. Strain Analysis Vol. 40 No. 6

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0.5 pixels in the displacement measurement. Davidson and his co-workers developed a strain mapping system, DISMAP, based on the ‘stereoimaging technique’ [8, 9]. This has been intensively applied to various applications [10]. Johnson [11] also proposed a system called ‘SMART’ for fatigue and fracture mechanics applications. Moire´ interferometry using different types of light source can also be used to obtain a full-field displacement distribution over a limited strain with submicrometre resolution. Recent developments in Moire´ interferometry and its broad applications have been reviewed by several authors [6, 12]. A number of novel high-resolution Moire´ methods have been developed recently for microscale strain measurement, such as the electron beam Moire´ method [13], the scanning Moire´ method [14], the logical Moire´ method [15], and the nano-Moire´ method [16]. A brief review is given in reference [16]. The digital image correlation (DIC) method was first developed as an image processing tool in order to analyse speckle diffraction patterns [17]. The technique has been substantially developed [18] with direct correlation of image or speckle patterns prior to and post deformation to measure the full-field strain distribution at subpixel resolution. In recent years, the DIC method has been further enhanced through the development of improved computation algorithms, which have been commercialized for the cases of two- and three-dimensional full-field strain distributions [19–22]. The characteristic features used on the images for DIC are crucial to obtaining correct results. A speckle pattern obtained by either spraying or etching is generally preferred. Chiang et al. [23] developed a so-called ‘micro/nanospeckle technique’ using physical vapour deposition to apply random particles (e.g. gold) to the sample surfaces in the micrometre and submicrometre size range. By combining scanning tunnelling microscopy (STM) [24] and atomic force microscopy (AFM) [25, 26], the DIC method has been applied to strain concentration measurement at the nanometre scale. By growing the grain size to millimetre dimensions, Zhang and Tong [27] were able to use DIC to measure the local plastic strains and rigid-body rotation of the grains. However, it is noted that the AFM or STM observations in these deformation studies are only valid at small strain and limited in their measuring range. Therefore, for strain measurement over a wide range of scales, especially at large strains (of order 1), such as within the necked region in a tensile sample, a more effective and reliable strain measurement technique is still needed. In the present study, a novel microscopic strain mapping technique based on SEM topography image correlation is developed. As validation of the method, the technique is applied to strain measurement J. Strain Analysis Vol. 40 No. 6

within the necked area of an aluminium sheet tensile sample.

2

MICROSCOPIC STRAIN MAPPING USING SEM TOPOGRAPHY IMAGE CORRELATION

Figure 1 shows the principle of DIC as implemented in the commercially available Aramis system [28]. The fundamental principle of DIC is based on relating the distribution of grey-scale values of a rectangular area (known as a ‘facet’ or ‘subset’) in the material before and after an increment of deformation. Thus, a reference point defined in the initial image (see Fig. 1(a)) must also be located as the reference point in the deformed or ‘destination’ image (Fig. 1(b)). The following relation exists between the grey values at these two points [20, 28]

g1 ðx‚ yÞ ¼ g2 ðxt ‚ yt Þ

ð1Þ

where g1 and g2 represent the grey values of initial and destination images respectively. The pixels of the facet in the initial image are then transformed into the destination image as follows [20, 28]

xt ¼ a1 þ a2 x þ a3 y þ a4 xy

ð2Þ

yt ¼ a5 þ a6 x þ a7 y þ a8 xy

ð3Þ

The values of a1 and a5 in equations (2) and (3) above describe the translation of the facet centre, while the others (a2 , a3 , a4 , a6 , a7 , and a8 ) describe the rotation and deformation of the facet. In order to compensate for possible differences of illumination in the images, a linear radiometric transformation is adopted while the images are being matched [28]

g1 ðx‚ yÞ ¼ b1 þ b2 g2 ðxt ‚ yt Þ

ð4Þ

The above parameters (a1 –a8 , b1 , and b2 ) are calculated in such a way that the sum of the quadratic deviation of the matched grey values is minimized. For this purpose, cross-correlation [20] is used as a standard method in DIC for error measurement. A

Fig. 1 Principle of digital image correlation using Aramis. (After reference [28])

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Fig. 2 Tensile sample for FE-SEM observation

correlation coefficient C for the two facets is defined as P ½g1 ðx‚ yÞ g2 ðxt ‚ yt Þ C ¼1 P ð5Þ P 1=2 2 ½ g1 ðx‚ yÞ g22 ðxt ‚ yt Þ The Aramis software determines the set of grey values that minimize the correlation coefficient C . The Newton–Raphson search method is used for the above calculation [20, 28]. If a facet is nearly white or black, the correlation coefficient C is close to 0. Thus the correlation algorithm cannot match between two successive images and the displacement and corresponding strain data will not be reported [29]. This is regarded as a weak correlation and a ‘hole’ will appear in the corresponding position in the strain map. Usually, a random speckle pattern is applied to the sample surface and is then used to define the facet for the processing steps. This has been used in most DIC studies to date. However, any pattern that contains characteristic features and good grey-level contrast with the background could be used in the correlation calculation in DIC. Typically, abundant characteristic features exist on most SEM images that could be used as input to the correlation algorithm. For example, at an appropriate magnification, polishing scratches, particles, slip lines, and grain structures are all candidates for correlation. It should be noted, however, that some of those features (e.g. scratches) may disappear at large strain, while others may emerge at the same time, such as patterns of local shear or grain boundary widening. Despite these concerns, it is possible to use SEM topography directly for the correlation calculations. As the topographic features may vary with magnification, it is possible to take SEM images at different magnifications at each loading stage and thereby measure strain inhomogeneity simultaneously at a range of scales. For the strain calculation, a series of captured images of the same region are utilized to obtain a map of incremental strain distribution (hereafter referred to as the ‘relative stage method’). The size of macroscopic strain between each successive JSA73 # IMechE 2005

strain increment should be kept at a modest level (say 3–5 per cent). This is justified by practice in order to minimize the effect of emerging new features on the sample surface owing to deformation. An example is given below in the discussion section.

3

EXPERIMENTAL

In order to demonstrate the validity of the technique just described, an in situ tensile test of an aluminium alloy sheet was carried out using a Hitachi S-4800 field-emission scanning electron microscope (FESEM). The material used in the present study was a strip cast AA5754 sheet metal. The average grain size and yield strength of the material were about 17 mm and 108 MPa respectively. The true strain at the onset of necking, as measured by an extensometer with a 25 mm gauge length, was about 0.18. Shear banding was the main deformation mechanism in this Al–Mg alloy [30, 31]. The geometry of the tensile samples for the in situ tensile tests is illustrated in Fig. 2. This was developed following finite element simulations of deformation of various test geometries using ABAQUS [32]. Owing to its symmetry, only one-quarter of the sample was analysed using two-dimensional elastic–plastic formulation. Four-point plane stress shell elements and the von Mises yield criterion were employed for the analysis. For the chosen geometry, the finite element analysis results show that the strain distribution is uniform in a region about 2 mm long in the middle of the gauge sections up to a true strain of about 0.2, i.e. up to about the necking strain (Fig. 3). A simple tensile jig (shown in Fig. 4) was designed to perform in situ uniaxial tensile tests within the SEM chamber. Before testing, all the tensile samples were mechanically polished to a 1 mm finish and then electropolished to achieve a smooth scratch-free surface. In the middle section, diamond-shaped indentations were made with spacings of 0.25 mm along the intended loading direction using a microhardness tester. The local strain was calculated directly by measuring the change in the indent spacings from micrographs recorded in the FE-SEM J. Strain Analysis Vol. 40 No. 6

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Fig. 3 Strain distribution in the proposed sample geometry by finite element analysis. The strain presented here is the von Mises effective strain

Fig. 4 Tensile jig for the tensile test within an FE-SEM chamber

during the test. The accuracy of each spacing measurement is 0.1 mm with the microscope used in this investigation. The true strain is defined as "t [28, 33]
 l ð6Þ "t ¼ ln l0 where l and l0 are the current and initial gauge lengths respectively. Several accelerating voltages and magnifications were tried and the combination of 5 kV and 800 was found to show the best image details. Figure 5 shows a series of FE-SEM images of the same area at different tensile strains. Figures 5(a) and (b) show the in-grain slip lines that develop at small strains prior to the onset of necking. Figures 5(c) to (g) show the development of surface topography within the necked region up to a local true strain of 0.45. The images also show that no void initiation or damage development occurred on the sample surface, even at a large local strain.

4

RESULTS AND DISCUSSION

SEM images, such as those shown in Fig. 5, were loaded into the Aramis software and correlated using the DIC method. The facet size used for the calculation was 50  50 pixels, which corresponded to an actual size of 5 mm. It was found that the relative stage method is the best way to deal with this batch of images and the results for each strain increment are shown in Fig. 6. Here, the label (a) ! (b) means that the reference image in Fig. 5(a) is correlated with the destination image in Fig. 5(b), and so on. The J. Strain Analysis Vol. 40 No. 6

fields covered by the strain maps in Figs 6(b) to (g) correspond to the image fields in Figs 5(b) to (g). Although all three in-plane strain components were calculated, strain y (transverse strain) and the inplane shear strain are much smaller than strain x (strain along the loading direction) (Fig. 7). Thus only the axial strain is reported for the rest of the results of this paper. As mentioned earlier, the strains were calculated from the indentation spacings which were originally 0.25 mm apart (referred to as "ind ). On this scale the strains were relatively uniform along the gauge length. The corresponding strains calculated from the DIC measurement were based on a spacing of 0.1 mm (referred to as "DIC ). This value is then the average strain in the region shown in Fig. 5. The choice of the spacing for DIC analysis was based on the size of SEM images that could be acquired. The correlation between the two measurements of strain is shown in Fig. 8, suggesting good agreement between these two methods except for the last point, which corresponds to a stage close to final fracture. The results also suggest that strain measured over a gauge length of 0.1 mm is independent of the grain size. This is understandable as the spacing size is about 6 times the grain size in the material, although this may well depend on the area selected for study. The inhomogeneity in the strain field on a scale finer than this is clearly evident in the strain maps shown in Fig. 6. For example, consider the specific location labelled as Area 1 in Figs 5(g) and 6(g). The local strain is compared with the global strain "ind in Fig. 9. It develops rapidly (and more or less linearly) at about 2.3 times the rate of the macroscopic strain, reaching a maximum true strain of 1.06. This is the place of highest local strain found in this region of the specimen. This result is consistent with the critical true strain involved in the formation of shear bands within this material, which might range from 0.50 to 1.5 as measured by the offsets of linear features (e.g. scratches or grain boundaries) at shear bands [30]. It is important to assess the accuracy of this measurement. Before the calculation, the allowable accuracy was set to 0.015 pixels for all cases and JSA73 # IMechE 2005

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Fig. 5 FE-SEM observations of an in situ tensile test of an aluminium alloy sheet metal

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Fig. 6 Full-field strain distribution using DIC via the Aramis system. The field corresponds directly to the images in Fig. 4

validated for all of the calculations. In order to warrant the accuracy [29], the correlation algorithm of Aramis tries to find the same four corner points of a facet in two successive images and analyses their corresponding grey-level distribution. Then the algorithm does the iterations to reposition the corner points until no better positions can be reached within the set accuracy. In other words, the resulting accuracy of 0.015 pixels means that the corner points can move about 0.015 pixels without getting a different accuracy value. The accuracy reported here is consistent with the analysis by Smith et al. [34] on an early DIC system. The spacing from centre to centre of the neighbouring facets is 50 pixels. Thus, the maximum J. Strain Analysis Vol. 40 No. 6

error
in the strain calculation based on the facet size of 50  50 pixels is


¼

0:015 ¼ 0:0003 ¼ 0:03% 50

ð7Þ

Measures were taken to minimize the error of the strain calculation during the testing and data processing. For example, by using an accurate fiveaxis motorized eucentric sample stage, the same observation area within the SEM chamber was automatically resumed. Then at least three characteristic points were selected and their spacings were monitored to ensure the same location and magnification continued during the testing. Thus, the error induced by magnification changes can be minimized. JSA73 # IMechE 2005

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Fig. 6 (continued over)

One of the most important issues in this calculation is the selection and processing of the SEM images. The characteristic features appearing on the SEM images depend significantly on the magnification of the images. Some features may appear in the initial stages and disappear in the later stages. Some features may come out at low magnification but disappear at high magnification, or vice versa. Two approaches were used to address this in the present study. Firstly, SEM images were collected at different stages using several magnifications and attention was paid to the features that remained over several stages at least. Secondly, to eliminate the loss of data at a certain stage owing to the disappearance of characteristic features, the SEM images were JSA73 # IMechE 2005

processed by the relative stage method in DIC. Thus, the incremental strain distribution could be determined and the total strain calculated by summing the results. To illustrate this idea, an analysis was performed using the image in Fig. 5(d) as a reference. The accumulated strain from Figs 5(d) to (f) and from Figs 5(d) to (g) are shown in Fig. 10 [the accumulated strains from Figs 5(d) to (e) were shown earlier in Fig. 6(e)]. It can be seen that, as the strain increments become larger, parts of the image can no longer be analysed owing to the emergence of new features that lead to a weak correlation of the corresponding facets. The facet size is another critical parameter that has a significant influence on DIC calculations [18, 19, 24, J. Strain Analysis Vol. 40 No. 6

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Fig. 7

In-plane strain components calculated from step a ! b in Fig. 6

28]. On the one hand, facet size should be small enough to ensure that the deformation within the facet is uniform. On the other hand, the facet size should be large enough to include the characteristic features of the SEM image. Therefore, an optimum facet size needs to be carefully chosen to balance these two requirements. As a series of images can be

taken under different magnifications in the SEM, this offers a more flexible choice for the selection of a proper facet size for SEM topography correlation. Through a simple calibration, the facet size can be converted into an actual size. Thus, the most important issue is that the facet size should be smaller than the average grain size in order to reveal the

Fig. 8 Local strain measured by indentation spacing versus DIC measurement. Data shown here are calculated starting from Fig. 5(a), which corresponds to a uniform true strain of 0.135

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Fig. 9

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Local strain evolution in Area 1, as labelled in Figs 5(g) and 6(g)

strain distribution at the grain level. Recently, Schreier and Sutton reported that higher-order shape functions can be used to mitigate the requirement for small subsets [35]. Figure 11 shows an example for the calculation from Figs 5(b) to (c) using larger facet sizes of 60  60 pixels and 80  80 pixels. It was found that the strain distribution in Area 1 cannot be determined. The results are therefore misleading as the new position of maximum strain appears to be at Area 2 in the bottom of the image (see Fig. 11(a)).

Figure 12 shows the results of a strain scan for different facet sizes taken along the centre-line as marked in Figs 6(c) and 11. The results are quite similar for the three cases. It is apparent, however, that more details are found in the strain distribution as the facet size decreases, while the strain distribution tends to be flatter with increasing facet size. This is understandable since the facet size behaves in a manner similar to the grid size in the grid method. Therefore, the facet size needs to be studied in a wide range in order to obtain reliable results.

Fig. 10 Strain mapping using the image in Fig. 5(d) as the only reference image. The other conditions are kept the same as in Fig. 6

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Fig. 11 Strain mapping using different facet sizes. Results shown here are obtained using the same conditions as in Fig. 6 except for a larger facet size of (a) 60  60 pixels and (b) 80  80 pixels

Indeed, when the facet size is increased to 100  100 pixels, no correlation was found between these two images and thus no strain results were obtained. On the contrary, it is not clear what effect further reductions in facet size below 50 pixels would reveal. Figure 12 suggests that the differences would be small except in regions with high local strain gradients (e.g. at

shear band edges). However, further reductions in facet size introduce problems of accuracy. Moreover, once the facet dimension is comparable with or less than the average spacing of the features used for DIC (slip steps in this case) there is a loss of correlation. Thus, there is a characteristic size of facet dimension for which good results are obtained. If,

Fig. 12 Linear strain distribution for different facet sizes. Results are from the centre-line as marked in Figs 6(c) and 11

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however, there is a need to probe the strain distribution on a scale much below 50 times the pixel size, it will be necessary to move to higher magnification images, provided that appropriate surface features useful for DIC still exist. The kind of features, namely surface slip steps, used in this study represent one of the more challenging types of all those that might be used for DIC. This is because, as deformation proceeds, new slip steps emerge and the existing ones grow. Therefore, the successful use of such features suggests that this technique is both flexible and robust. It should be possible to utilize both naturally occurring features such as particles or machining scratches and artificially imposed features such as surface deposits with equal or greater success.

5

CONCLUSIONS

A novel technique of microscopic strain mapping using SEM topography image correlation has been developed. The technique has been applied to SEM images taken from an in situ tensile test on aluminium alloy sheet. These images were used to map local strains over a broad area. The results correlate well with conventional strain measurements made using a row of indents. The results also reveal that extensive local strain of the order developed in the specimen studied is consistent with the critical strain for the formation of shear bands in this material. The SEM topography image correlation is capable of measurements of such strains provided that an optimum facet size is used. The method could be readily applied to deformation on any scale provided that the surface has persistent features on the same scale as can be imaged by optical or scanning electron microscopy.

ACKNOWLEDGEMENTS The authors are grateful for the help of Mr David Hoyle at Hitachi High Technologies Canada, Inc. in carrying out the FE-SEM observations and Mr M Klein at GOM, Germany, Mr J Tyson and Mr T Schmitdt at Trilion, USA for the technical support on the Aramis system. The financial support from General Motors of Canada and NSERC is also acknowledged.

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2 Dally, J. and Riley, W. Experimental stress analysis, 1978 (McGraw-Hill, New York). 3 Li, C. S. and Orlecky, L. J. Fiducial grid for measuring microdeformation ahead of fatigue crack tip near aluminum bicrystal interface. Expl Mechanics, 1983, 33(4), 286–292. 4 Takeda, N. Evaluation of microscopic deformation in CFRP laminates with delamination by micro-grid methods. J. Composite Mater., 32(1), 83–100. 5 Schroeter, B. M. and McDowell, D. L. Measurement of deformation fields in polycrystalline OFHC copper. Int. J. Plasticity, 2003, 19, 1355–1376. 6 Biery, N., De Graef, M., and Pollock, T. M. A method for measuring microstructural-scale strains using a scanning electron microscope: applications to  -titanium aluminides. Metallurg. Mater. Trans. A, 2003, 34(A), 2301–2313. 7 James, M. R., Morris, W. L., and Cox, B. N. A high accuracy automated strain-field mapper. Expl Mechanics, 1990, 40(1), 60–67. 8 Williams, D. R., Davidson, D. L., and Lankford, J. Fatigue-crack-tip plastic strains by the stereoimaging techniques. Expl Mechanics, 1980, 30(1), 134–139. 9 Franke, E. A., Wenzel, D. J., and Davidson, D. L. Measurement of microdisplacements by machine vision photogrammetry (DISMAP). Rev. Scient. Instrums, 1991, 62(5), 1270–1279. 10 Page, R. A., Chan, K. S., Davidson, D. L., and Lankford, J. Micromechanics of creep-crack growth in a glassceramic. J. Am. Ceram. Soc., 1990, 73(10), 2977–2986. 11 Johnson, D. A. Automated deformation mapping in fatigue and fracture. In Applications of automation technology in fatigue and fracture testing and analysis: fourth volume (Eds A. A. Braun, P. C. McKeighan, A. M. Nicholson, and R. D. Lohr), 2002, ASTM STP 1411, pp. 220–232 (American Society for Testing and Materials, West Conshohocken, Pennsylvania). 12 Post, D., Han, B., and Ifju, P. High sensitivity Moire´: experimental analysis for mechanics and materials, 1994 (Springer-Verlag, New York). 13 Dally, J. W. and Read, D. T. Electron-beam Moire´. Expl Mechanics, 1993, 33, 270–277. 14 Read, D. T., Dally, J. W., and Szano, M. Scanning Moire´ at high magnification using optical methods. Expl Mechanics, 1993, 33, 110–116. 15 Asund, A. and Yung, K. H. Phase shifts and logical Moire´. J. Opt. Soc. Am., 1991, A18, 1591–1600. 16 Xie, H. M., Liu, Z. W., Fang, D. N., Dai, F. L., Gao, H. J., and Zhao, Y. P. A study on the digital nano-Moire´ method and its phase shifting technique. Measmt Sci. Technol., 2004, 15, 1716–1721. 17 Peters, W. H. and Ranson, W. F. Digital imaging techniques in experimental stress analysis. Opt. Engng, 1982, 21(3), 427–431. 18 Sutton, M. A., Wolters, W. J., Peters, W. H., Ranson, W. F., and McNeil, S. R. Determination of displacements using an improved digital correlation method. Image and Vision Computing, 1983, 1(3), 133–139. 19 Sutton, M. A., Cheng, M., Peters, W. H., Chao, Y. J., and McNeill, S. R. Application of an optimized digital correlation method to planar deformation analysis. Image and Vision Computing, 1986, 4(3), 143–150.

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APPENDIX Notation

a1 ‚ a2 ‚ . . . ‚ a8 constants in equations (2) and (3) b1 ‚ b2 constants in radiometric transformation for illumination compensation C correlation coefficient g1 ‚ g2 grey values in initial and destination images l original spacing between two points l0 current spacing between two points
l relative displacement ¼ l  l0 x‚ y position of pixels in the initial image xt ‚ y t position of pixels in the destination image
"DIC "ind "t

strain measurement error true strain over a gauge length of 0.1 mm in DIC analysis true strain over a gauge length of 0.25 mm measured by indentations true strain

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