The Measurement of Strain Based on Finite Element Theory and Image Method

Fu Jie, Xiong Shishu School of Civil Engineering and Mechanics

Huazhong University of science and technology Wuhan China

fujiegege@gmail.com

Abstract—A large strain measurement program has been

developed in Labview, which is based on digital image and finite

element theory. It is a new method for strain measurement. This

new method includes several steps: marking some circles on

surface of the specimen, taking some pictures of the specimen

before and after deformation, calculating the exact position of

the mark center by the program, comparing the center position,

and getting the strain according to the finite element theory.

Finally, this paper gives experimental results, which proves the

correctness of this method.

Key words: Digital image; Strain measure; Finite element;

Image method; Labview

I. INTRODUCTION

Using digital image to measure strain is a non-contact optical measurement method. The advantages of this method is noticeable: it has wide range of application and wide range of sensitivity, and can realize the measurement from macroscopical to microcosmic deformation by choosing different lens. In the past, many studies have been carried out by using Digital Image Correlation (DIC), which was independently proposed by W. H. Peter, W. F. Ranson[1] and I. Yamaguchi[2]. Its basic principle is that it records the light intensity distribution graph of the specimen before and after deformation by digital camera, and compares the correlation of tow digital images, then obtains the deformation of the specimen[3].

In this study, a method to measure deformation is presented which is different from DIC, and the plane strain can be achieved by combing the image processing and finite element theory. The software which is used in this study is developed by the author under Labview.

II. BASIC METHOD

The finite element method is a numerical method which advances with the development of computer technology. In this method, continuous elastic body is discretized into a lot of small units. Assuming its displacement mode, the displacement of any point can be calculated by interpolation and infinite unknown displacements in the whole area can be expressed by displacements of limited nodes. This method avoids the difficulty of solving the displacement function for entire region in elasticity mechanics. For example, for a triangular element, the relationship between unit strain and node displacement is:

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2012 International Conference on Control Engineering and Communication Technology

978-0-7695-4881-4/12 $26.00 © 2012 IEEE

DOI 10.1109/ICCECT.2012.25

37

2012 International Conference on Control Engineering and Communication Technology

978-0-7695-4881-4/12 $26.00 © 2012 IEEE

DOI 10.1109/ICCECT.2012.25

37

strain in triangular unit is constant, the strain values are equal at any point in the same unit. Obviously, according to the finite element theory if the specimen is discredited using quadrangle unit, the measured strain value would be more precise.

Considering the characteristics of finite element method above, in this study several displacements of the points on specimen are obtained by digital image processing techniques, in order to solve the unit strain, it need to discrete the specimen base on these points and then use the finite element theory which reflects the relationship between the strain and displacement. Fig. 1 shows the test system.

Figure 1. The test system

III. PROGRAM DESIGN

A. Digital image processing

Fig. 2(a) shows the image of specimen. Before the test, some white round spots are marked on the surface of the specimen, which are the unit nodes. In order to calculate the position of these nodes accurately, the digital image must be processed. Fig. 3 shows the program flow char.

Figure 2. The test image

Figure 3. Image processing flow chat

1) Image grayscale. The images which are obtained by camera are true color bitmap of 24 bits, and it represents a pixel color with three color components which are red, green and blue(RGB). Processing color information is relatively complex, long processing times and large memory space, so it is better to remove the color information from the digital image at first, using grayscale image. This study uses the method of weighted mean, the equation is:

BGRBGR 11.059.03.0 ����� (2) It should be noted that the coefficients of RGB are determined by human’s visual sensitivity.

2) Image denoising. It is inevitable to generate some noise in the course of image acquisition, so it is necessary to reduce the image noise before the next processes. There are several filtering algorithms based on different theories, such as wavelet analysis or Fourier Transforms Method. In this study adaptive smoothing filter is adopted, which is a basic denoising method with the purpose of maintaining a sharp image edge and this character is helpful to improve the accuracy of the measurement.

3) Image Binarization. The program inspects each pixel in the image, when the bright value is greater than threshold value, it turns the pixel to pure white (RGB=0xFF), otherwise it turns the pixel to pure black (RGB=0x00). The algorithm function is:

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(a) Original image (b) Processing results

Gray scale Image Denoising

Image Binarization

Connecting Area label

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the threshold value which must ensure the integrity of the marker and no noise interference around. It can keep a good lighting condition usually in experiment, so it is easy to select an appropriate threshold value. In this study the value is selected by person. At last the program reverses the color of the image. Fig 2(b) is the processing result.

4) Connecting area label. The program identifies which black pixels belong to a same marker, and calculate how many markers in the image. In this study a pixel labeled algorithm based on recursive method [4] is used. This algorithm mainly includes two steps: firstly, scanning the binary image to get the whole connecting labeled area and the equal marks list; then analyze the equal marks list by recursive method to ascertain the right distribution and count of connecting areas.

5) Calculate the coordinates of the nodes. Some related studies often use grayscale center of gravity to obtain the coordinates [5], which is a general sub-pixel technology, however, the accuracy of it is affected by noise, specimen smoothness and other factors. This study utilizes least squares method to curve fit then the marker central position can be achieved. When the marker is a circle, it will turn to ellipse after being forced, so the program uses ellipse function to fit. As Fig. 4, node position is the center of ellipse.

Figure 4. The ellipse fitting result

B. Specimen discretization

Trough the processes above, the program can achieve the coordinates of the nodes, compare the coordinates before and after being forced and get displacements of nodes. The next process is to discretize the specimen into several units based

on nodes above. This study utilizes Delaunay triangle network which possesses the characteristic that the sum of minimal angles is the maximal. It can approach equilateral triangle automatically and avoid the appearance of deformed unit, thus it can guarantee the stability and reliability of finite element method. This part of functions is programmed under Delphi and compiled to Dynamic Link Library (DLL) and called under LabVIEW. The basic steps of classical Delaunay triangulation algorithm proposed by Lawson [6] are:

1) Create a large triangle to include all of the points and put them into the triangle list.

2) Insert one points into the triangle list, then find the triangles which’s circumcircles includes the points, and delete common edges of these triangles. Connect all the vertexes of these triangles and the inserted points, then accomplish the inserting of this point in Delaunay triangle list. This is the most vital step, as Fig. 5.

Figure 5. Delaunay triangulations Algorithm diagram

3) Circularly perform the Step 2 until all the points are inserted.

Figure 6. The processing result and the program interface

Finally, it can achieve strain value of every unit by substituting the coordinates and displacements of unit nodes into (1). Fig. 6 shows the operating result

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and program interface.

IV. PRECISION ANALYSIS

Utilizing the formula (4), it generates a standard circle graphic. As the formula (4), h and k are the center coordinates, R1 and R2 are inner and outer radiuses which is equal to 61 pixels and 71 pixels, and let the center not completely overlap to a certain pixel. Table 1 shows the position results of the program.

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As the Table I, when the center coordinates changes by 0.1 pixel, the maximum of the deviation is 0.028 pixel. Thus, if the test range is 2000�2000 pixels, the accuracy could reach 50 micro-strains. Obviously, improving the resolution of the camera is the essential method to ameliorate test accuracy.

TABLE I. THT POSITIONING RESULTS OF FITTING

Actual coordinates

x, y (Pixel

Testing coordinates

x, y (Pixel)

Deviation

x, y (Pixel)

100.1,150.0 100.108,150.000 0.008,0.000

100.2,150.0 100.228,150.000 0.028,0.000

100.3,150.0 100.309,150.000 0.009,0.000

100.4,150.0 100.420,150.000 0.020,0.000

100.5,150.0 100.500,150.000 0.000,0.000

100.1,150.2 100.114,150.226 0.014,0.026

100.2,150.2 100.223,150.223 0.023,0.023

100.3,150.2 100.317,150.212 0.017,0.012

100.4,150.2 100.408,150.209 0.008,0.009

100.5,150.2 100.500,150.195 0.000,-0.005

V. EXPERIMENTAL INSTANCE

Rubber specimens of 10cm, 15cm and 20cm are axially stretched by universal testing machine with accuracy of 0.001mm and the testing images are collected by Canon PwoerShot S5 digital camera with resolution of 3264�2488 pixels. The loading procedure is divided into five levels and in each of them there is a 5000�� deformation. Fig. 1 (a) shows the photo of program analysis and Fig. 5 shows the grid division. For lack of space, it just lists the particular result of specimen NO.1, as Table II shows. In this test, it is axially stretching, so the shear strain should be zero and the test

result is corresponsive to the theory. The test results of the other two specimens are similar to the specimen NO.1, and Table III shows all the average value of unit strains on each specimen.

Analyzing the dates of Table 2, it is knowable that the maximal deviation of unit strain is 10.9% and the deviation of average values of unit strains are all less than 5%, and the deviations of units on specimen edges are obvious larger. So it can be considered that non-desired constraint is the main reason of forming deviation, and the results are accuracy and effective. In applications, it can improve testing accuracy and stability by augmenting units.

TABLE II. THE TEST RESULTS FOR SPECIMEN NO.1

Unit Level 1 Level 2 Level 3 Level 4 Level 5

Strain Deviation Strain Deviation Strain Deviation Strain Deviation Strain Deviation

4040

1 4550 -9.0% 5382 7.6% 4754 -4.9% 5528 10.6% 4630 -7.4%

2 4539 -9.2% 4751 -5.0% 4550 -9.0% 4794 -4.1% 4738 -5.2%

3 4621 -7.6% 4657 -6.9% 4651 -7.0% 4762 -4.8% 4653 -6.9%

4 4770 -4.6% 5338 6.8% 4580 -8.4% 5463 9.3% 4699 -6.0%

5 4720 -5.6% 5247 4.9% 4747 -5.1% 4755 -4.9% 4829 -3.4%

6 5197 3.9% 5201 4.0% 5198 4.0% 5126 2.5% 5183 3.7%

7 5015 0.3% 5255 5.1% 5195 3.9% 5236 4.7% 5133 2.7%

8 5045 0.9% 4995 -0.1% 4999 0.0% 4964 -0.7% 4965 -0.7%

9 4830 -3.4% 5219 4.4% 4849 -3.0% 4935 -1.3% 5146 2.9%

10 5342 6.8% 5118 2.4% 4991 -0.2% 5284 5.7% 4983 -0.3%

11 5274 5.5% 5131 2.6% 5058 1.2% 5299 6.0% 5334 6.7%

12 4961 -0.8% 4884 -2.3% 4938 -1.2% 4569 -8.6% 5103 2.1%

13 4943 -1.1% 5428 8.6% 5094 1.9% 5256 5.1% 5375 7.5%

14 4901 -2.0% 4795 -4.1% 4480 -10.4% 5544 10.9% 4554 -8.9%

15 4627 -7.5% 5092 1.8% 4862 -2.8% 5522 10.4% 4693 -6.1%

16 4752 -5.0% 4735 -5.3% 4510 -9.8% 5449 9.0% 4578 -8.4%

Mean 4880 -2.4% 5077 1.5% 4841 -3.2% 5155 3.1% 4912 -1.8%

TABLE III. THE AVERAGE STRAIN OF UNITS FOR THRESS SPECIMENS

Specime

n

Number

Level 1 Level 2 Level 3 Level 4 Level 5

Strain Deviation

Strain Deviation

Strain Deviation

Strain Deviation

Strain Deviation

1 4880 -2.4% 5077 1.5% 4841 -3.2% 5155 3.1% 4912 -1.8%

2 4801 -4.0% 5133 2.7% 5195 3.9% 5109 2.2% 5235 4.7%

3 4836 -3.3% 4972 -0.6% 5081 1.6% 5179 3.6% 5083 1.7%

VI. CONCLUSION

Image strain measurement proposed by this study is suitable to large strain measurement. Compared to the traditional strain gauge, it has particular advantages, such as simple equipments and low consume. Analysis software which is developed under Labview can automatically process the images and obtain the strain values, which have extensive application values in mechanical tests. In addition, in aspects of improving recognition accuracy of sub-pixel and optimization of algorithm, it is valuable to have deeper research.

REFERENCES

[1] W. H. Peters, W. F. Ranson, “Digital image

techniques in experimental stress analysis,” Optical Engineering, vol. 21, pp. 427-431, May-Jun 1982. [2] Yamaguchi I. “Advances in the laser speckle strain gauge,” Optical Engineering, vol. 27, pp. 214-218, March 1988. [3] Zhang Dongsheng. “Digital image correlation: theory and applications,” Journal of Medical Biomechanics, vol. 24, pp. 85-88, 2009. [4] Xu Zhengguang, Ba oDonglai, Zhang Lixin. “Pixel Labeled Algorithm Based on Recursive Method of Connecting Area in Binary Images,” Computer Engineering, vol. 32, pp. 186-188, Dec. 2006. [5] Li Yuanhai, Zhu Hehua, UENO Katsutoshi, MOCHIZUKI Akitoshi. “Application of Image-based Measuring Deformation Method Using Target

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Markers in Physical Model Test,” Journal of TongJi University, vol. 10, pp. 1141-1145, 2003. [6] Lawson C L. “Software for C1 surface

interpolation,” Mathematical Software III, vol. 3, pp. 161-194, 1977.

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