Investigation of Strain Measurements using Digital Image Correlation with a Finite Element Method
 Author: Jian Zhao, Dong Zhao
 Organization: Jian Zhao; Dong Zhao
 Publish: Journal of the Optical Society of Korea Volume 17, Issue5, p399~404, 25 Oct 2013

ABSTRACT
This article proposes a digital image correlation (DIC) strain measurement method based on a finite element (FE) algorithm. A twostep digital image correlation is presented. In the first step, the gradientbased subpixels technique is used to search the displacements of a region of interest of the specimen, and then the strain fields are obtained by utilizing the finite element method in the second step. Both simulation and experiment processing, including tensile strain deformation, show that the proposed method can achieve nearly the same accuracy as the cubic spline interpolation method in most cases and higher accuracy in some cases, such as the simulations of uniaxial tension with and without noise. The results show that it also has a good noiserobustness. Finally, this method is used in the uniaxial tensile testing for Dahurian Larch wood specimens with or without a hole, and the obtained strain values are close to the results which were obtained from the strain gauge and the cubic spline interpolation method.

KEYWORD
Digital image correlation , Finite element , Strain measurement

I. INTRODUCTION
The measurement and analysis of fullfield strain distributions is very important in mechanical engineering, especially in the stress concentration problem in plastic and fracture mechanics. Conventionally, strains are measured by contact strain gauges and extensometers. The disadvantage of these two techniques can only provide the average strain of the local area and are not suitable for fullfield deformation measurements [1]. A traditional strain distribution measurement is as follows: A grid is established on a sample surface before the deformation. We can measure the displacements of the grid after the deformation and analyze the strain distribution. The disadvantage of this method is that it is time consuming at analyzing the data. There are also precise measurement instruments, such as photoelasticity and electrical speckle pattern interferometers, which can be used to measure the surface strain field without contact [23]. But due to the expensive cost and the stability problem those methods cannot be widely used in scientific research and engineering.
A precise measurement of the surface strain distribution can be achieved with the help of the digital image correlation (DIC) technique (also known as the Digital Speckle Correlation Method). As compared to the special requirement of traditional optical measurements, digital image correlation is an easy and cheap method because it takes advantage of the speckle pattern on the specimen surface and only digital images taken by CCD/CMOS camera are processed. The DIC technique is predicated on the maximization of a correlation coefficient that is determined by examining the pixel intensity array subsets of two or more corresponding images and extracting the deformation shape function that relates the images to each other [4]. In a general way, once the displacements at the entire region of interest (ROI) in the deformed image have been calculated, the strains are calculated using this displacement field by using numerical differentiation techniques [5]. Although the relationship between the strain and displacement can be described as a numerical differentiation process in mathematical theory, unfortunately, the numerical differentiation is considered as an unstable and risky operation [6], because it can amplify the noise contained in the computed displacement. Therefore, the resultant strains are untrustworthy if they are calculated by directly differentiating the estimated noisy displacements [7].
A more practical technique for strain estimation is the pointwise local leastsquares fitting technique used and advocated by Wattrisse [8] and Pan [9]. Wattrisse [8] implemented a local leastsquares method to estimate the strains from the discrete and noisy displacement fields computed by the DIC to analyze the strain localization phenomena during the tension of thin flat steel samples. A similar technique was also used by Pan [9] with a simpler and more effective data processing technique for the calculation of strains for the points located at the image boundary, holes, cracks and the other discontinuity areas. Xiang [10] used the moving leastsquares (MLS) method to smooth the displacement field followed by a numerical differentiation of the smoothed displacement field to get the strain fields. Different from above pointwise algorithms, the Q4DIC developed by Besnard [11] and the finite element method proposed by Sun [12] can be classified as a continuum method, in which the surface deformation of the sample is measured throughout the entire image area. These techniques ensure the displacement continuity among pixel points. However, reported results showed that these two methods seem not to provide higher displacement measurement accuracy than gradientbased algorithms. Nevertheless, they can provide new ideas for strain measurement. After obtaining the displacements of points in the region of interest, the finite element method (FEM) is used to estimate continuous strain fields from the continuous displacement fields.
In this study, a twostep DIC strain measurement method is presented. In the first step, the gradientbased subpixel technique is used to search the displacement of a region of interest of the sample, and then the strain can be obtained by utilizing the finite element method in the second step. Both simulation and experiment are carried out to verify the accuracy of this method.
II. FUNDAMENTAL THEORY
2.1. The Principle of Digital Image Correlation
Digital Image Correlation (DIC) is based on a comparison between two digital images of a sample surface captured at two different levels of loading. The discrete pixels of these digital images and their grey values for intensity were recorded and transferred as a twodimensional array. Under the assumption that there is a onetoone correspondence on the intensity pattern of two images taken before and after deformation, one could deduce the deformation information from comparison.
In this study, in order to reduce the noise from the CMOS camera and the backlight, a 2D adaptive noiseremoval smoothing technique was applied to the obtained digital image. This function could smooth the grey level of a pixel according to the average value and standard variation of its neighbor pixels. The average value could be obtained as follow.
The considered region is denoted as
η that consists ofM ×N pixels, andn _{1},n _{2} represents a point of this region. The original grey level is represented asI _{a}. The standard variance is calculated asThen the smoothed grey level could be obtained as
where
ν ^{2} is the average standard variance of all regions.After the smoothing treatment, these two smoothed images, which characterize the original and deformed surfaces of a material subjected to a known loading, were used to evaluate the deformation. A square reference subset of (2
M + 1) >× (2M + 1) pixels centered at the point (x_{0}, y_{0}) is arbitrarily chosen from the reference image and a reference point (x ,y ) is selected from this subset, the position (x′ ,y′ ) of the reference point after the deformation increment could be described aswhere u_{0} and v_{0} are the integral pixel displacements in the x and y directions, which can be accurately determined with ease and are assumed as known values; ∆u and ∆v are the corresponding subpixel displacements in
the x and y directions, respectively. The whole subset is assumed to have the same displacements. Under this assumption, both the subset and the deformation increment should be small.A sumsquared difference (SSD) correlation function is used for deriving a theoretical model of the displacement measurement accuracy of DIC. The SSD correlation function is defined as [11]
where f(x, y) is the grayscale intensity value at point (x, y) of the reference image, and g(x′, y′) is the grayscale intensity value at point of the deformed image.
After neglecting the highorder terms, the firstorder Taylor expansions of g(x′, y′) yields
where g_{x} and g_{y} are the firstorder derivatives of grayscale intensities, and in this study they are calculated by central difference of neighboring points in
the x andy directions, respectively.Minimization of the SSD correlation function would provide the best estimate of the desired subpixel displacements ∆u and ∆v. Thus we have
The solution of Eq. (7) can be expressed in the following closed form
Once the location of the target subset in the deformed image is found, the displacement components of the reference and target subset centers can be determined. More details of the error estimation of the DIC technique can be found in Ref. [13].
2.2. Calculation of the Strain Field
After obtaining the displacements of points in a prescribed region, the concept of fournode quadrilateral (Q4) finite element as shown in Fig. 1(a) is utilized to calculate the strains in the region defined by four node points. Consider an arbitrary point with the coordinates
x andy in this region, its displacements along thex andy direction arewhere
a_{i} andb_{i} are constants that could be obtained from the displacements of the four node points. Then the strains in this region could be calculated by the following equationswhere
ε_{xx} andε_{yy} are normal strains andγ_{xy} is the shear strain. In addition to the fournode quadrilateral (Q4) finite element, the concept of ninenode quadrilateral (Q9) finite element as shown in Fig. 1(b) is also used to calculate the strain from the displacement information. Then, the displacements of an arbitrary point in the element could be described aswhere
a_{i} andb_{i} are constants that could be obtained from the displacements of the eight node points. Then the strains in this region could be calculated by the following equationsIII. RESULTS AND ANALYSIS
To verify the efficiency of the proposed method, a typical tension deformation with and without noise are designed to explore the precision and noiserobustness of the method. Then, a real tension deformation with DIC is processed to validate the ability of the real data.
3.1. Simulation of Strain Measurement with Simulated DIC Images
3.1.1. Simulation without Noise
In the precision comparison experiments, the speckle images before and after deformation are generated using the algorithm developed by Zhou [14]. The simulated image is 512×512 pixels with 1200 speckles, each 4 pixels in size, as shown in Fig. 2. The subset is 41×41 pixels, displacement fields are calculated using the gradientbased method.
To test the precision of the DICFE strain measurement method, the experimental results are compared with the results obtained by the cubic spline interpolation method [15], as shown in Table 1. All of the results are the average of 20 independent experiments on the center point.
In the simulated tension deformation experiment, the displacement gradient (strain) ε_{y} is 0.001~0.02. Table 1 shows the absolute error and standard deviation of the central points obtained by the proposed and cubic spline interpolation methods, respectively. The comparisons show the strain precision of the proposed method being better than 2×10^{4}ε, higher than that of the cubic spline interpolation method. Both Q4 and Q9 types of the DICFE method have very close results. The comparisons demonstrate that the proposed method can achieve a high accuracy strain measurement.
3.1.2. Simulation with Noise
To explore noiserobustness of the proposed method, two Gaussian noises (with means of 0 and variance of 0.2 and 0.5) are added to the same simulated image in “Simulation without noise”. Fig. 3 shows the noiseadded simulated speckle images (with variance of 0.2 and 0.5) before deformation. The subset is also 41×41 pixels.
Tables 2 and 3 show the absolute error and standard deviation of the central points of 20 simulated images obtained by the proposed and interpolation methods. The comparisons show that both can get the relative proper results and the precisions of the proposed and interpolation methods decline with the increase of noise. It can been seen from Table 3 that the precision of the proposed method is better than 0.015, a little higher than that of the cubic spline interpolation method. For the lower noise level, the precision showed in Table 2 is higher Table 3.
3.2. Experiments of Strain Measurement with Real DIC Images
3.2.1. Tensile Testing of Specimen
To evaluate the effectiveness of the proposed DICFE strain measurement method in real data processing, dogbone specimens of Dahurian Larch wood were used in a uniaxial tensile test (Fig. 4). The surface of the specimen was sprayed uniformly with white paint on the region of interest, and then black paint was sprayed randomly (Fig. 5(a)). To have reference strain values, a strain gauge was attached at the back surface of the specimen (Fig. 5(b)). Preparation of the testing equipment occurs after the speckle pattern application to produce a testing configuration as shown in Figure 6. The uniaxial tensile testing was conducted on a tensile testing machine and the crosshead loading rate was 3 mm/min. During the testing process, two LED lights were used to shine on the specimen, and the image was taken every 10 seconds. The digital camera used was a CMOS industrial camera with the resolution of 1024×1280 pixels, focal length 8 mm, and
f /1.8. The image magnification was about 0.04549 mm/pixel.Strains at rectangle speckle cells with the coordinate
x from 590 to 690 pixels andy from 490 to 540 pixels with an interval of 5 pixels are obtained by the DICFE method. The strain gauge was applied at the same zone of the back surface of the specimen to obtain the strain information for comparison with those obtained by the present DICFE method. When both the fournode element and the ninenode element are used, the comparison is shown in Fig. 7. It is evident that the present results have good agreement with those from the strain gauge, and the two types of elements have very close results. In considering the calculation time, the present DICFE method with fournode element should be a good choice.3.2.2. Tensile Testing of Specimen with a Hole
In this testing, strain gauge was also applied to obtain the reference strain values of a specimen with a hole (Fig. 8). In this case, the strains around a hole have a high gradient. Hence, this specimen could represent the application of the present method to a high strain gradient region.
As shown in Fig. 9, the present method with fournode element could obtain reasonable strains in the high strain gradient region as compared to the strain gauge and spline method.
From the above analysis, it is concluded that the proposed DICFE4Q strain measurement method can be used in experimental cases including strain deformation, and it is able to obtain nearly the same results and calculating time when compared with the cubic spline interpolation method.
IV. CONCLUSIONS
This article introduces a finite element based strain measurement algorithm to DIC computation, a twostep DIC is presented. In the first step, the gradientbased subpixels technique is used to search the displacement of a region of interest of the specimen, and then the strain can be obtained by utilizing the finite element method in the second step. Simulated results show that its precision is a little higher than that of the cubic spline interpolation method. Finally, this method is used in the uniaxial tensile testing for Dahurian Larch wood specimens with artificial painting on the surface, and the obtained strain values are close to the results obtained from the strain gauge. When this method is applied to measure the strain of a specimen with a hole, it shows that this DIC method with fournode element could obtain correct strains even in the high strain gradient region.

[FIG. 1.] The fournode (a) and ninenode (b) quadrilateral element geometry.

[FIG. 2.] Simulated speckled image.

[TABLE 1.] Errors in a preassigned tension (εy=0.001~0.02ε)

[FIG. 3.] Noiseadded simulated speckle images (a) mean is 0 and variance is 0.2, (b) mean is 0 and variance is 0.5.

[TABLE 2.] Errors in a preassigned tension with noise (Variance is 0.2)

[TABLE 3.] Errors in a preassigned tension with noise (Variance is 0.5)

[FIG. 4.] Specimen configuration

[FIG. 5.] (a) surface and (b) back surface of the Dahurian Larch wood specimen.

[FIG. 6.] Digital Image Correlation Testing Setup.

[FIG. 7.] Comparison of strains obtained by DICFE method and strain gauge.

[FIG. 8.] (a) surface and (b) back surface of the Dahurian Larch wood specimen with a hole.

[FIG. 9.] Strains for tensile specimen with a hole.