Evaluation of the Joint Roughness Coefficient using the Digital Image Correlation Method

Abstract

The joint roughness coefficient (JRC) is used to estimate the influence of the rock joint roughness on the shear strength. The traditional methods to evaluate the JRC are stylus and laser profilometers. The disadvantage of these traditional methods is that they cost much time or the cost of instrument is too high, and they are not suitable for a in-situ test. Therefore a simple three-dimensional digital image correlation (DIC) method is developed in this research to evaluate the JRC.

In this research, a three-dimensional DIC method is proposed and only one camera is used. The results show that this method can be used to determine the three-dimensional coordinates of the joint surface and the measured JRC values are very close to that measured by laser profilometer.

Keywords: digital image correlation, rock joint, JRC.

1 Introduction

There are always faults, joints, cleavages etc. existed in the natural rock mass. These weak planes have often obvious influence on the mechanical properties of the rock. The shear strength of the rock joint is serious influenced by the rock joint roughness. There are already many related researches. In 1973, Barton presented a new concept to quantify the joint roughness [1]. He defined the joint roughness coefficient (JRC), which can be calculated by using the joint shear strength empirical formula with known shear strength, normal stress, joint wall compression strength and basic friction angle. Barton and Choubey proposed 10 typical roughness profiles in 1977 [2]. They divided the JRC into ten ranges. The JRC can be determined by direct comparing the joint with the standard profiles. Tse and Cruden (1979) enlarged the ten standard profiles 2.5 times. The profiles were digitized and the regression relationship between the calculated JRC and eight parameters were studied. It shows that the root mean square (Z ) and mean square (SF) of the first derivative of the ₂

Paper 229

Evaluation of the Joint Roughness Coefficient using the

Digital Image Correlation Method

S.H. Tung1, M.H. Shih2 and J.C. Kuo3

1 _{Department of Civil and Environmental Engineering}

National University of Kaohsiung, Taiwan

2 _{Department of Construction Engineering}

National Kaohsiung First University of Science and Technology, Taiwan

3 _{Department of Materials Science and Engineering}

National Cheng-Kung University, Tainan, Taiwan ©Civil-Comp Press, 2009

Proceedings of the Twelfth International Conference on Civil, Structural and Environmental Engineering Computing, B.H.V. Topping, L.F. Costa Neves and R.C. Barros, (Editors), Civil-Comp Press, Stirlingshire, Scotland

profile are strongly correlated with JRC. The relationship between these values are [3] 1 2 JRC =32.2+32.47logZ (1) 2 JRC =37.28+16.58logSF (2) Some scholars have also proved that the JRC is influenced by the joint scale. [4,5]

In the past years, the joint profile was digitized with the help of stylus and laser profilometers. The stylus profilometer is a contact type measurement instrument. The scanning speed is slow and the joint surface could be damaged. The laser profilometer is a non-contact type and high precision measurement instrument. But the cost of such instrument is relative high and it’s difficult to be applied in the in-situ measurement.

The development of digital image correlation (DIC) technique is already over two decades. In the recent years, the resolution of digital camera and calculation ability of computer progress very fast. The cost to measure with DIC is getting lower. This makes the DIC technique receive more and more attention. The three-dimensional DIC technique has to determine the three-three-dimensional coordinates of the specimen surface during the analysis. Then the strain field can be calculated by analyzing the displacement of specimen surface. Therefore the three-dimensional DIC technique is applied in this research to determine the three-dimensional coordinates of the joint surface, then equation (1) is used to calculate the JRC value.

2 Digital Image Correlation (DIC) Method

The DIC method can be divided into two categories – two- and three-dimensional. The main hypothesis of two-dimensional DIC method is that the distance between the image acquisition instrument and the specimen keeps constant during the test. Therefore it is suitable for the plane strain test or the test without obvious out of plane deformation. If there exists a large out of plane displacement or the specimen surface is not a plane, then the three-dimensional DIC method is necessary for the measurement. The three-dimensional DIC method utilizes the same image identification principle as the two-dimensional DIC method. The principle of the two-dimensional DIC method and the method to determine the three-dimensional coordinates of the specimen surface will be introduced as below.

2.1 Two-Dimensional DIC Method

The principle of two-dimensional DIC method is to determine the local correlation of two images. The local correlation is used to identify the mapping relationship between the images before and after deformation. The structural speckle will be manufactured on the specimen surface. This makes a different grayscale distribution in the image. This grayscale distribution characteristic is utilized to identify the corresponding position of images before and after deformation.

Figure 1: Schematic drawing of relative location of sub-images of deformed and undeformed images on surface

As shown in Figure 1, central point prior to deformation is point P, and then changed to point P* after deformation, the correlation between P and P* can be determined by using the correlation coefficient[6,7]：

2 2 ij ij ij ij g g COF g g Σ = Σ ⋅ Σ   (3)

Where, g and _ij g_ij is grayscale of sub-image A on coordinate

( )

B on coordinate

( )

( )

corresponds to coordinate

( )

coefficient is equal to 1. It means that the sub-image B is exactly the image of

sub-image A after deformation. Therefore we are looking for the position which yields

the maximum value of the correlation coefficient during the analysis.

2.2 Method to Establish the Three-dimensional Coordinate

The principle of image formation by a converging lens is shown as in figure 2. The relationship between object distance ( p ), image distance ( q ) and focal length is

shown as equation (4).

1 1 1

Figure 2: Image formation by a converging lens

If the height of the object is equal to L and the object distance is p , the height _n

of the image L formed by a converging lens is: _n

-n n L f L p f × = (5) If we define the focal point as the origin of a Cartesian coordinate system.

n n

z = p − is the z coordinate of the object. Then the equation (5) can be rewritten f

as n n L f L z × = (6)

Figure 2 shows that the image height can be decided by extending the line connecting this point and focal point. The distance of intersection point of this line and the plane of the optical center to the principal axis is equal to the height of the image. Equation (6) shows that the height of the image is inverse proportional to the z coordinate. Therefore the image position formed by a converging lens of a non-plane object before and after a lateral movement e can be simplified as shown in figure 3.

Suppose that the object is moved a distance e to the left. From the geometrical relationship, it shows that the displacement of the image '

a

f e aa

z

= . Because the unit of a digital image is pixel, this equation is multiplied by γ , which is the number of pixel per actual length. That is to say, the actually measured displacement in the image is _aa' a f e N z γ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠. Let Λ =γ f , then ' aa a N z e Λ = (7)

Figure 3: Image formation of a non-plane object before and after lateral movement by converging lens

Theoretically, Λ can be determined with the measured N , coordinate _aa' z and _a

the lateral displacement e. The z coordinates of the other points can then be determined. But there is no point with known z coordinate, therefore the Λ value can not directly be calculated with the help of equation (7).

The equation (7) can be modified as

' a aa e z N Λ = (8)

The lateral displacement e is known and the displacement in the image can be determined by two-dimensional DIC. Assume that there are N points with different z coordinate. The z coordinate differences of these points relative to point a are known. Then these z coordinate differences can be expressed as

' ' ' ' - , 1,... aa ii i i a aa ii N N z z e i N N N ⎛ ⎞ Δ = − =_{⎜ ⎟}Λ = ⎝ ⎠ (9)

Then the Λ can be express as below.

' ' ' - ' i aa ii aa ii N N e N N ⎛ ⎞ Δ Λ = _{⎜ ⎟} ⎝ ⎠ (10)

According to equation (10), the calibration parameter Λ can be found out with only one known z coordinate difference Δ. Consider the inevitable errors during the test, an average calibration parameter Λ should be calculated by using more than one point.

2.2.2 Revise the x and y coordinate

If an object move parallel to the z axis, the position of a point of this object on the image will also change as shown in figure 4. Therefore the x and y coordinates must be revised, in order to make sure that these coordinates is relative to the same z coordinate.

Figure 4: Schematic Drawing of the influence of different z coordinates on the image formation

Suppose that an object is taken pictures at different z coordinate Z and _a Z . _b

The x coordinates of one point of this object on the image is X and _a X _b

respectively. From the geometrical relation, the x coordinates of a point of the object at different z coordinates can be expressed as follows:

o a a b b

X f = X Z =X Z (11)

Equation (11) can be modified as below:

a a b b Z X X Z = (12)

The x coordinate X of a point on the image as z coordinate equal to _a Z can be _a

revised to the x coordinate according to z coordinate equal to Z with the help of _b

equation (12). The y coordinate can be also revised in a similar way.

3 Experiment

3.1 Preparation of specimen

A natural joint surface is duplicated using the silicon rubber at first. Then the replica is used to produce the gypsum specimen. This method can good simulate the natural joint surface. Before the test, the specimen surface will be marked with spray paint in order to compare the images by using DIC method.

Figure 5: Gypsum specimen

3.2 Experimental Instruments

A Canon EOS 400D DSLR camera equipped with CANON EF-S 60mm f2.8 Macro USM lens is applied to capture the digital images of the gypsum specimen. The camera and a high precision X-Y-Z table are fixed on a copy stand. The specimen is putted on the X-Y-Z table. The X-Y-Z table is used to control the displacement of the specimen in order to get two images of the specimen from two different positions. A digital dial gauge and a micrometer head are used to control the movement of the X-Y-Z table in horizontal and in vertical direction. Both the digital dial gauge and the micrometer head have the precision of 0.001 mm. Two light sources are also equipped. The full installation of the instruments is shown in figure 6.

(a) Local installation (b) Full installation Figure 6: Installation of the instruments

3.3 Experimental Process

The three-dimensional coordinates of the joint surface and the JRC value can be determined with the following process.

1. Calibrate Λ : Two or more images with different z coordinates are necessary to calibrate Λ . These z coordinates must be known. Therefore the specimen for calibration is putted on the X-Y-Z table. The focus of the camera is set at this plane. Then the specimen will be moved to 5mm± in the z direction. And the specimen will be moved 10 mm horizontally at these two elevations. Four images are taken and the Λ value can be calculated with the method stated in the preceding section.

2. Take the image of the gypsum specimen: Put the gypsum specimen on the X-Y-Z table at a specified elevation. Move the specimen 10 mm horizontally and take two pictures before and after movement.

3. Determine the three-dimensional coordinates of the joint surface: A grid is defined over the image surface of the analyzed region. The positions of the nodes before and after displacement will be identified using two-dimensional DIC method. The z coordinates of every node can be calculated using the method of preceding section and then the x and y coordinates will be revised. 4. Calculate JRC: The JRC value can be calculated with the help of equation (1)

after determining the joint surface coordinates.

4 Results and Discussions

(1) Calibrate Λ

Three points are selected on one of the four images. Then the two-dimensional DIC method is applied to determine the positions of these three points on the other three images. The displacements of these three points at two different elevations can be calculated. These displacements can be substitute into equation (10) in order to determine the Λ . The average value of Λ is 17006.77.

(2) Determine the three-dimensional coordinates of the joint surface

A about 50 mm × 50 mm region (as shown in figure 7) on the specimen surface was selected to be analyzed. 100 × 100 points in this region was analyzed. The z coordinates and the revised x and y coordinates of these points are determined. The joint surface obtained by using DIC method is shown in figure 8(a). In order to prove the accuracy of this result, about the same region is also scanned with a three-dimensional laser scanner (as shown in figure 8(b)). Comparing figure 8(a) and 8(b), we can find that these two results are very similar but not identity. The reason is that the region and the measured points are not the same. The analysis using DIC method uses a subimage, whose center is the point to be analyzed, to determine the position of a point before and after displacement. The positioning results will be influenced by the size of the subimage. The rapid elevation variation within a subimage can not be exactly simulated with a bilinear interpolation function. If we want to determine the more detailed elevation variation, we have to select a smaller

subimage relative to the actual specimen. The image of the same specimen is also taken with a shorter object distance. Compared with the preceding pictures, the specimen in the image is enlarged. Therefore the subimage includes a smaller region relative to the preceding analysis. The result is shown in figure 9. Compared with figure 8(a), we can find that the detailed elevation variation can be better simulated.

Figure 7: Analyzed region

(a) DIC method

(b) Three-dimensional laser scanner

Figure 8: Surface measured by two different methods Analyzed region

x

y

Figure 9: Surface measured by DIC method with smaller subimage relative to the actual specimen

(3) Calculation JRC

Equation (1) is applied to calculate the JRC value. The JRC values are calculated with different spacing and along different directions. The results are shown in figure 10. It shows that the JRC value measured by laser scanner is larger than that measured by DIC with spacing 0.5 mm. But the tendency is very similar. As the spacing increases to 1 mm, we found that the JRC values measured by two methods are almost identical. Figure 11 shows the JRC values measured by DIC method with different spacing. We can find that as the spacing increases, the JRC values reduce slightly. The reason is that the high frequency elevation variation can no more be registered. The surface is smoother and this makes the JRC smaller.

The above results show that the three-dimensional DIC method can be used to measure the JRC value. The precision is a little lower than the three-dimensional laser scanner. But the cost of this method is much lower that the three-dimensional laser scanner and it can be applied to measure the in-situ JRC with only a small modification. Therefore the three-dimensional DIC method has more utility value than the three-dimensional laser scanner

0 10 20 30 40 50 distance(mm) 0 5 10 15 20 25 JR C DIC Laser 0 10 20 30 40 50 distance(mm) 0 4 8 12 16 JRC DIC Laser

0 10 20 30 40 50 distance(mm) 0 4 8 12 16 20 JRC DIC Laser 0 10 20 30 40 50 distance(mm) 0 4 8 12 16 JR C DIC Laser

(c) Spacing 0.5 mm, along y direction (d) Spacing 1 mm, along y direction Figure 10: JRC with different spacings and along two different directions

0 10 20 30 40 50 distance (mm) 0 4 8 12 16 JRC DIC_X (0.5 mm) DIC_X (1 mm) DIC_X (2 mm) 0 10 20 30 40 50 distance (mm) 0 4 8 12 16 JRC DIC_Y (0.5 mm) DIC_Y (1 mm) DIC_Y (2 mm)

(a) Along x direction (b) Along y direction

Figure 11: The JRC value measured by DIC method with different spacing

5 Conclusions

The following conclusions can be drawn according to the analysis results:

1. The DIC method is a non-contact measurement technique. It will not disturb the specimen. Therefore it can avoid damaging the specimen. Then the measurement error due to specimen damage can also be ignored.

2. The analysis results show that the JRC value measured by three-dimensional DIC method very close to the JRC value measured by three-dimensional laser scanner. It shows the correctness of three-dimensional DIC method.

3. The three-dimensional DIC method can be used to measure the JRC value of different scale. It can also be applied to measure the in-situ joint roughness. Therefore the three-dimensional DIC method has more utility value than the three-dimensional laser scanner.

References

[1] Barton, N., “Review of a New Shear strength Criterion For Rock Joints”, Engineering Geology, Vol. 7, pp. 287-332, 1973.

[2] Barton, N. and Choubey, V., “The Shear Strength of Rock Joints in Theory and Practice”, Rock Mechanics, Vol.10, pp.1-54, 1977.

[3] Tse R. and Cruden D. M., “Estimating Joint Roughness Coefficients, ” International Journal of Rock Mechanics Mining Scicnce and Geomechanics Abstracts , Vol. 16, pp.303-307, 1979.

[4] Bandis S., Lumsden A.C. and Barton N., “Experimental Studies of Scale Effects on The Shear Behavior of Rock Joints”, International Journal of Rock Mechanics Mining Science and Geomechanics Abstracts, Vol.18, pp.1-21, 1981.

[5] Fardin, N., Stephansson, O. and Lanru Jing, “The Scale Dependence of Rock Joint Surface Roughness” International Journal of Rock Mechanics and Mining Science, Vol.38, No.5, pp.659-669, 2001.

[6] Chu, T.C., Ranson, W.F., Sutton, M.A. and Peters, W.H., “Application of Digital-Image- Correlation Techniques to Experimental Mechanics”, Experimental Mechanics, 25(3), p.232, 1985.

[7] Russell, S.S. and Sutton, M.A., “Strain field analysis aquired through correlation of x-ray radiography of a fiber reinforced composite laminate”, Experimental Mechanics, 29, p.237-240, 1989.