Crack Identification and Strain Evaluation from Displacement Fields Measured Using Digital Image Correlation

Abstract

This paper proposes a data processing method for easily and efficiently determining strains near a crack from the displacement field data obtained using digital image correlation. The displacements within a strain window are approximated by polynomials and the strains are obtained as their derivatives. A crack is detected from the difference between the approximated and measured displacement distributions and the correlation coefficients obtained during the displacement measurement. The strains near the crack are obtained by ensuring that the data at the crack does not overlap with the strain window. The proposed method is applied to several displacement fields obtained using digital image correlation and demonstrates its effectiveness. This method makes it possible to easily obtain the strain distributions near a crack and is expected to be applied to various fracture mechanics studies.

1.  Introduction

In researches on the deformation, fracture and strength of solids, various optical displacement, strain and stress measurement methods have been used with considerable success. For example, photoelasticity, caustics, and holography were used to clarify dynamic crack propagation and wave propagation phenomena [1],[2],[3],[4]. Presently, digital image correlation is widely used as a displacement and strain measurement method [5]. However, it is not easy to use this method to measure strains near a crack. In digital image correlation, displacement components at a point are determined by detecting the amount of movement of an area consisting of multiple pixels called a subset from images before and after deformation. Furthermore, the strain components are calculated from the displacements obtained at multiple points, i.e., the distribution of the displacement components in a local region called a strain window. Therefore, it is difficult to accurately determine the displacements when the subset overlaps crack faces that have opposite displacements. Similarly, if a crack is included within the strain window region, the strains are obtained from the displacements that change discontinuously due to the crack opening, resulting in apparent strains that are different from the actual strains. Therefore, a special data processing technique is required to obtain the strains near the crack from the measurement results of the displacements.

Previous studies of fracture problems using digital image correlation have determined strains far from the crack. For example, the displacement data near the crack tip and crack faces were excluded [6],[7],[8]. In this case, the data near the crack cannot be used. Therefore, when calculating the J-integral, the values near the crack surface were extrapolated from the data in the area far from the crack [8]. On the other hand, Poissant and Barthelat [9] proposed a method for dividing the subset along the crack. This method makes it possible to measure displacement near the crack surface, but the problem of determining strains remains unsolved. Yoneyama et al. [10] proposed a method to measure displacements and strains near a crack using global digital image correlation with a finite element mesh containing a crack. This method makes it possible to process data near a crack, but it is necessary to prepare a mesh that includes a crack. The simplest method known to compute strains from displacements obtained by digital image correlation is to approximate the displacements by polynomials in a strain window region and calculate their derivatives. Therefore, if this method can be used to obtain strains around a crack, it would be convenient and highly useful. Chen et al. [11] proposed a method to detect areas of sudden strain change and background areas from the displacement distribution obtained by digital image correlation. They determined the strain computation path and obtained strains utilizing the detected areas. It may be possible to easily and efficiently obtain strains near a crack by detecting areas of sudden strain change and dividing the region in a similar manner to that method.

Based on the above background, this study proposes a data processing procedure for easily and efficiently computing strains from displacement fields near a crack obtained by digital image correlation. As a method for computing strains, a common method is used in which displacements are approximated in a strain window region and strains are determined as their derivatives. A crack is detected from the difference between the approximated displacements and the measured displacements. The correlation coefficients obtained when the displacements are determined are also used for crack detection. The effectiveness of the proposed method is demonstrated by applying it to several displacement fields obtained by digital image correlation. This method makes it possible to easily obtain the strain distribution near a crack and is expected to be applicable to various fracture mechanics studies.

2.  Data Processing Method

The outline of the procedure for evaluating strains around a crack is shown in Fig. 1 whereas the detail of the proposed data processing method is explained with actual data as an example. Figure 2(a) shows the surface images of a single edge notched specimen made of aluminum alloy before and after deformation. The actual dimension of 1 mm corresponds to 26 pixels on the image. The case where the in-plane displacement distributions of the area including the crack are obtained by digital image correlation is considered. Note that when stereo digital image correlation is used, surface displacements can also be obtained, so it can be considered in the same way as two-dimensional digital image correlation. Figure 2(b) shows the in-plane displacement distributions obtained using two-dimensional digital image correlation. The subset size in this case is 21 × 21 pixels, and the data interval is 10 pixels. The details of the two-dimensional digital image correlation method used in this study can be found in Ref. [12]. It can be seen that the displacement in the y direction changes discontinuously at the crack due to the crack opening. At the same time as obtaining these displacement distributions, the distribution of the correlation coefficient is also obtained, as shown in Fig. 2(c). The correlation coefficient distribution shows the low values at the crack, while the values close to 1 are obtained in the areas other than the crack. This correlation coefficient distribution is used later for computing the strains.

Fig. 1

Outline of procedure for evaluating strains near crack

Fig. 2

(a) Before and after deformation images of single edge notched specimen, (b) displacement distributions obtained using digital image correlation, and (c) correlation coefficient distribution

Generally, the inclusion of cracks in the subset reduces the measurement accuracy in the cracked areas, so strains are computed from data from areas other than the cracked areas, where accurate displacements are obtained. The in-plane displacement distributions, u_x, and u_y, in the strain window area obtained by a measurement are approximated by the following linear functions.

  

（1）

Here, x and y are the position in a strain window area, and a₁₀, a₀₁, a₀₀, b₁₀, b₀₁ and b₀₀ are the coefficients of the linear functions. The coefficients a₁₀, a₀₁, a₀₀, b₁₀, b₀₁ and b₀₀ can be determined by a least-squares method. Therefore, the in-plane strain components, ε_x, ε_y, γ_xy, in the strain window area can be obtained from the coefficients of the functions as

  

（2）

Figure 3(a) shows the displacement distributions and Fig. 3(b) shows the strain distributions obtained when the strain window size is 8 × 8 data points (71 × 71 pixels). As mentioned above, the displacement in the y direction is discontinuous at the crack, but it is treated as continuous in Fig. 3. Therefore, it can be seen that the large strain, i.e., an apparent strain, appears in the normal strain in the y direction due to the suddenly changing displacement in the y direction. Also, from Fig. 3(a), it can be seen that the discontinuous displacement distribution in the y direction changes somewhat smoothly due to the approximation of the displacement in the strain window.

Fig. 3

(a) Displacement distributions and (b) strain distributions obtained using local least-squares with linear functions

Displacement discontinuity regions such as a crack surface and regions near a crack tip where strain changes suddenly are detected in this study. In these regions, the displacement distributions within the strain window cannot be adequately approximated by Eq. (1). Therefore, the displacements obtained by the measurement and those approximated by Eq. (1) are compared. The residual RES between them can be obtained by the following equation.

  

（3）

Here, u_i^exp is the displacement obtained by the measurement, u_i^approx is the displacement approximated by Eq. (1), and M represents the number of data. Since there are two displacement components in the x and y directions, M is twice the number of data points in the strain window. If the residual is larger than a certain threshold, it can be considered that the displacement distributions in the strain window cannot be adequately approximated by Eq. (1). At the same time, regions that do not contain enough data in the strain window region are distinguished as the boundary of the data region. Figure 4 shows the result of the domain partitioning. The threshold for the residual in this case is set to 0.02 pixels (approximately 0.77 μm). The basis for this value is that the accuracy of the displacement obtained by digital image correlation is approximately 0.02 pixels [12]. In Fig. 4, the domain A1 in the domain partitioning A is near the boundary, the domain A2 is a region where the residual is small, so it is possible to approximate the displacements obtained by the measurement and compute the strains, and the domain A3 is a region where the residual is larger than the threshold and the strains are not obtained. In this example, if the distance between the center of the strain window and the centroid of the data contained in the strain window area is more than 1/5 of the strain window size (15 pixels or more), it is determined to be near the boundary, i.e., the domain A1. At the same time, the other region divisions are also performed as shown in the domain partitioning B. When the strains are obtained in the domain A2, the region included in the strain windows is the domain B1, and the rest is the domain B2. In other words, this indicates that the data in the domain B1 can be used to compute strains. The displacement and normal strain distributions in the y direction obtained in the domain A2 are shown in Fig. 5. In this way, the strains in the domain A2 can be obtained.

Fig. 4

(a) Domain partitioning A and (b) domain portioning B

Fig. 5

Displacement and normal strain distributions in y direction in domain A2 obtained using linear function

In regions where the strain changes rapidly, such as near the crack tip, the displacement distributions may be approximated by using quadratic equations. The displacement distributions are approximated by the following quadratic equations for the data in the strain window centered in the domain A3.

  

（4）

Here, a₂₀~a₀₀ and b₂₀~b₀₀ are the coefficients of the functions as in the case of Eq. (1), and are determined by a least-squares method. The residual between the approximated displacements and the measured displacements within the strain window is again evaluated, and if the residual is lower than the threshold, the displacements in that region are considered to have been approximated by Eq. (4). In that case, the strain components at the center of the strain window region can be obtained as in the above case. If the displacement distributions cannot be approximated using Eq. (4), that is, if the residual is greater than the threshold, it is considered that the strain window contains the crack. Based on this judgement, the domain partitioning in Fig. 4 and the displacement and strain distributions in Fig. 5 are updated. Figure 6 shows the updated displacement and strain distributions. There is no change in the region division near the crack surface. In the vicinity of the crack tip, on the other hand, the region in which strain can be obtained is expanded because the displacement distribution near the crack tip can be approximated by the quadratic equation.

Fig. 6

Displacement and normal strain distributions in y direction updated using quadratic functions

Next, the strains in the domain A3 where strains cannot be obtained in previous step are computed by excluding one side of the crack surface from the strain window. Here, it is necessary to identify the crack position. It is judged that the crack exists at the position where the apparent strain shown in Fig. 3(b) is maximum or the position where the correlation coefficient shown in Fig. 2(c) is below the threshold. If the crack exists within the strain window region, the strains are computed without using the data farther from the center of the strain window than the crack. Since the displacements near the crack surface do not change suddenly, unlike near the crack tip, the linear functions of Eq. (1) are used. This makes it possible to obtain the strains at some points in the domain A3. Figure 7 shows the updated displacement and strain distributions. It can be seen that the area where the strains are not obtained near the crack surface has narrowed. The domain partitioning is also updated. It is noted that the threshold for the correlation coefficient is set to 0.99 here. This value is thought to depend on the quality of the random pattern on the surface in the image. In this study, the threshold value is determined empirically. At present, the standard values for the residual threshold and the correlation coefficient threshold are 0.02 pixels and 0.99, respectively. However, these thresholds are data-dependent and need to be adjusted while checking the results.

Fig. 7

Displacement and strain distributions updated so as not to overlap one of crack face in strain window

Finally, the strains in the domains A1 and A3, where the strains have not yet been obtained are determined. The data in the part of the domains A1 and A3 that overlaps with the domain B1 are included in the strain window area when the strains in the domain A2 are obtained. Therefore, the strains in these domains A1 and A3 are determined from the approximated functions in the strain window centered at the domain A2. It is noted that each point in the domain B1 is included in multiple strain window areas. Therefore, the approximation functions in the strain window when the distance to the center of the strain window area is the closest are applied. The displacement and strain distributions obtained by the above procedure is shown in Fig. 8. In this figure, the data for the part where the correlation coefficient is below the threshold is excluded. Therefore, the crack tip is a data missing part, but the strain values near the crack are obtained. As shown in the figure, the two normal strain components are concentrated in the local area in front of the crack tip. The area in front of the crack tip is a plastic deformation region [13]. Therefore, it is known that the strain in this area is extremely large compared to other locations, and it can be considered that the results obtained are appropriate. In addition, the shear strain is distributed symmetrically above and below the crack, which is also the appropriate distribution.

Fig. 8

(a) Displacement and (b) strain distributions obtained using the proposed method

The procedure for determining strains is summarized below.

Step 1: The displacement distributions in the strain window areas are approximated by linear functions using the local least-squares method (Fig. 3). The area is divided (Fig. 4) and the strain distributions in the domain A2 are obtained (Fig. 5).

Step 2: The displacements in the strain window areas centered in the domain A3 are approximated by quadratic functions using the local least-squares method, and the strains in the part of the domain A3 are obtained (Fig. 6) and the domain partitioning is updated.

Step 3: The position of the crack is identified from the apparent strain distribution and the correlation coefficient. The strains in the part of the domain A3 are determined so that only one of the crack faces overlaps the strain window region (Fig. 7).

Step 4: The strains near the boundary A1 and near the crack A3 are obtained using the functions used to obtain the strains in the domain A2 (Fig. 8).

3.  Application Examples

3.1  Delamination at interface of dissimilar materials

This section shows an example of applying a tensile load to an epoxy resin specimen containing steel disk inclusions, and measuring the strain distributions around the area where the steel inclusions and epoxy resin peeled off. The specimen with the inclusions is made of carbon steels with the diameter of 8 mm and the longitudinal elastic modulus of 200 GPa and the epoxy resin with the longitudinal elastic modulus of 2 GPa. A specimen mold is made of silicon resin. After placing the carbon steel disks on the mold, epoxy resin is poured in and cured at the room temperature for 24 hours. The specimen is a strip of 25 mm wide, 200 mm long, and 5 mm thick.

Figure 9(a) shows the images of the specimen surface before and after the displacement load of 4 mm is applied. The actual dimension of 1 mm corresponds to approximately 47 pixels on the image. A random pattern for digital image correlation can be observed. In addition, the delamination between the inclusion and the resin can be vaguely observed in the after deformation image. However, the boundary between the delaminated and non-delaminated parts cannot be recognized from the image. Figure 9(b) shows the displacement distributions obtained by applying digital image correlation to these images. The subset size is 21 × 21 pixels, and the data interval is 10 pixels. It can be seen that the part of the displacements are distributed discontinuously between the steel inclusions and the resin. In other words, this result means that there are regions where the inclusions and the resin are peeled off. The interface between the peeled resin and the inclusions is in a similar state to a crack, and the strains are obtained using the proposed method.

Fig. 9

(a) Before and after deformation images of epoxy strip specimen with streel disk inclusions and (b) displacement distributions

Figure 10 shows the strain distributions obtained by the proposed method when the strain window size is 8 × 8 data points (71 × 71 pixels). The threshold for the residual is set to 0.02 pixels, and the threshold for the correlation coefficient is set to 0.99. It is observed that the strains are concentrated at the boundary between the delaminated and non-delaminated regions, that is, the tip of the interface crack. It is also seen that this strain concentration occurs on the resin side. On the other hand, the strains in the steel part, which is the inclusion, are uniformly distributed and the deformation is small. This is due to the difference in the material properties, and it is considered that the appropriate results are obtained.

Fig. 10

Strain distributions of epoxy strip specimen with streel disk inclusions under tension

3.2  Drying shrinkage of paste material

A paste made by mixing magnesium carbonate with an average particle size of 6 mm and water is used as the test material [14]. The powder mass ratio of the paste used in this study is 30%, and the mass is 100 g. This paste is placed in a petri dish with a diameter of 120 mm and dried on a heater at 343 K to generate cracks. Iron powder is scattered on the paste surface to create a random pattern for digital image correlation. The displacement field due to drying shrinkage and the generated crack are photographed by a camera from above the petri dish. The image dimension is 1024 × 768 pixels, the color depth is 8 bits, and the actual dimension of 1 mm is approximately 38 mm on the image.

An example set of the images obtained before and after the crack generation is shown in Fig. 11(a). The random pattern of the iron powder can be observed. It can also be seen that the crack generated by drying shrinkage exists. The displacement fields are measured by applying digital image correlation to these images before and after the crack generation. The subset dimension is 21 × 21 pixels, and the data interval is 10 pixels. The displacement distributions are shown in Fig. 11(b). The discontinuous displacement distribution in the y direction is obtained due to the crack extension. In addition, the minute opening displacement of about 0.3 mm is observed.

Fig. 11

(a) Before and after deformation images of desiccating paste and (b) displacement distributions

The proposed method is applied to the displacement fields in Fig. 11(b). The residual threshold is set to 0.02, and the correlation coefficient threshold is set to 0.99, and the strain window size is 8 × 8 data points (71 × 71 pixels). Figure 12 shows the distribution of the strain components. As shown in this figure, no apparent strain appears on the crack surface, and the appropriate strain distributions concentrated in the vicinity of the crack are obtained. As mentioned above, it is very difficult to obtain such a strain distribution without considering the presence of a crack, and the proposed data processing procedure for strain computation can be considered to be effective.

Fig. 12

Strain distributions around crack in desiccating paste

4.  Conclusions

In this study, we propose a data processing method for easily and efficiently obtaining strains from displacement field data near a crack obtained by digital image correlation. The displacement components are approximated by functions in the strain window region, and the strains are obtained by differentiating the functions. When strains are computed from displacement fields obtained by digital image correlation, a large apparent strain appears at the crack. Therefore, the region where the strains cannot be obtained is identified by comparing the displacement fields obtained by the measurement with the displacement fields approximated by linear functions in the strain window. Next, quadratic functions are used to obtain the strains near the crack tip where the linear functions are not appropriate to approximate the displacement fields. Then, the strains are determined so that only one of the crack faces overlaps the strain window. This allows the strains near the crack surface to be obtained. Finally, the data in the strain window regions used when calculating the strain are reused to obtain the strains in the area where the strains have not yet been obtained. This allows the strains to be obtained up to the vicinity of the crack. It is difficult to obtain the strain distribution from the displacement distribution without considering the presence of cracks, and the proposed data processing procedure for computing strains is effective. This method makes it possible to easily obtain the strain distributions near a crack and is expected to be applicable to various fracture mechanics studies.

Acknowledgements

The authors would like to thank Mr. Tsujii and Ms. Iketa, former students of Aoyama Gakuin University, for their help in collecting experimental data. The authors are grateful to Ditect Co. Ltd., for providing the motivation for this study.

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