ISRM International Symposium 2008 5th Asian Rock Mechanics Symposium (ARMS5), 24-26 November 2008 Tehran, Iran

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THE EFFECT OF SURFACE IRREGULARITIES ON JOINT CLOSURE BEHAVIOR USING NEW MODELING TECHNIQUES

A.H. GHAZVINIAN1, Z.Y. YANG2, A. TAGHICHIAN1 1Department of Mining Engineering, Faculty of Engineering, Tarbiat Modares University, Tehran, Iran

2Department of Civil Engineering, Tamkang University, Tamsui, Taipei 25137, Taiwan (e-mail of corresponding author: yang@mail.tku.edu.tw)

Abstract It is obvious that surface irregularities of a rock joint play a dominant role in its closure behavior. Therefore, special attention should be paid for the study of surface irregularities consisting of asperities in different scales. The laboratory tests of model joints are mainly conducted on the 2D joint profile because of producing the 3D joint surface is inconvenient. In this paper, a new technology called Laminated Object Manufacturing (LOM) is used to create the 3D surface geometry of joint models. It is proved that using the first five harmonics of Fourier transform will cause the first order asperities to be modeled and summing up higher order components lead the second order asperities to be also modeled. In this way, the effect of asperity order and scale is investigated on the joint closure. Hertz contact theory proposed for metal surfaces is employed to calculate the closure behavior of joint surfaces.

Keywords: rock joint; Fourier transform; asperity order; closure behavior.

1. Introduction

Patton[1] categorized asperity into first-order (waviness) and second-order (unevenness) categories. Hoek and Bray[2] stated that at low normal stress levels the second-order asperity (with highest-angle) controls the shearing process. As the normal stress increases, the second-order asperity is sheared off and the first-order asperity (with a longer base length and lower-angle) takes over as the controlling factor [3].

In 3D joint surface modeling, a depicted mother mold is usually cast from a natural rock surface using silicon material. Then, the multiple replicas cast from the mother mold with identical surface geometry are reproduced [4]. It is difficult to generate another artificial surface from a typical surface. Especially, it is the urgent need in the field of scale effect using similar surfaces in different sizes. The key factor is the difficult of automatically generating a 3D rough joint surface.

This paper proposes a method to generate the 3D joint surface using Laminated Object Manufacturing (LOM) technology [5]. This method can produce a 3D mother surface with a given surface geometry. The ability of

generating two joint surfaces with the same primary roughness but different in micro scale is also possible. This method is helpful to study the effect of the primary and secondary asperity on the joint closure behavior, either.

The present mathematical approaches for calculating the closure behavior or contact area is based on the stochastic concept [6]. Some statistical parameters for averaging the irregular joint surfaces must be obtained. They found that estimating the geometrical features seem to play the essential part for calculating the contact closure. Recently, several researchers [7] had been successfully adopted the Fourier transform concept to characterize the irregular joint surfaces. Superimposing several periodic functions can ideally approximate the irregular joint profile.

In this paper, superimposing twenty components of sine functions that have different wavelengths, amplitudes and phases approximates the joint profile under a normal load [8]. Each component, a regular sine-shaped sub-profile, is subjected to a normal load and then progressively closures. Consequently, the Hertz contact theory is capable of employing for calculating the elastic deformation of each sine-

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shaped asperity. Then, the resulting deformation of the irregular joint profile is summed up proportionally by the twenty regular sub-profiles. In addition, a closure test of 2D artificial joints is used to evaluate the mathematical model.

2. Methodology

In order to investigate the effect of order and scale of asperities on the normal response of rock joints under normal loads, three different steps containing laser scanning, mathematical modeling of joint surface and finally, physical modeling of surface roughness are required.

2.1. Surface topography of the joint

Laser scanning is needed for achieving accurate surface topography of a natural rock joint generating distinct (x,y,z) points corresponding to surface topography of the joint. To prepare the CAD data for automatically manufacturing joint surface geometrical model, a laser profilometer was employed to measure the joint geometry. During the measurement, the joint sample is placed on the machine table which moves along two orthogonal axes (x,y) controlled by a predefined path and measures the elevation in z direction. In this study, the natural joint surface of sandstone in 100 mm diameter (Fig.1) was used as example.

Fig. 1. The morphology of Sandstone joint surface with reconstructed 3D morphology of joint surface

The data points along 100 parallel profiles were scanned for the joint surface at 1 mm sampling interval. The natural joint surface and reconstructed appearances of this type of rock surface by measure data is shown in Fig. 1. The created x, y, and z coordinates in a discrete point format can be used as input into the CAD control system for producing a LOM model of the whole

joint surface.

2.2. Mathematical modeling of surface topography of rock joints

A non-periodic function similar to a joint profile can be approximated as well using one or more periodic functions. Thus, the joint profile regarded as a periodic function in the whole joint length can be disintegrated into several simple sine (or cosine) waves, each having a definite wavelength, amplitude and phase. When a function y(x) varies periodically with distance x in period L (Fig. 2(a)), it would be possible to write the Fourier form of function y(x) as:

=

+

+=0

02sin2cos)(

iii xL

iHxL

iGGxy (1)

y

Go x

CLA

y(x)

L (a)Periodic profile

)(sin nnn xba +=

x

y

Ln

a n

n b/

(b) n-th harmonic sub-profile

Fig. 2. A periodic profile and its n-th Fourier component

in which, the centerline averages CLA= G0 and coefficients Gi and Hi can be expressed as:

dxxyL

GL= 00 )(1 (2)

dxL

ixyL

GL

i = 0 2cos)(2 (3) dx

Lixy

LH

Li = 0 2sin)(2 (4) In which, the distance parameters x1 and x2

can be chosen so that x2-x1=L. The above Fourier series can be adopted to describe any discrete function such as the irregular joint profile. In order to define the components of roughness, the irregular joint profile can be separated into sinusoidal components named as harmonics (see Fig. 2(b)) of a fundamental

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frequency within the whole profile length. The amplitude of the i-th harmonic would be given as:

22iii GHA += (5)

2.3. Capability of Fourier series in simulating Bartons standard profiles

The well known JRC standard joint profiles presented by Barton et al. [2] are digitized about 500 data points in 10 cm length before the study. In each profile, the 500 obtained points were then used to back-calculate the coefficients G0, Gi, Hi of Fourier series up to twenty harmonics. The resulted joint profiles summed by the first 5 and 20 harmonics for JRC=14~16 are plotted in Fig. 3 to compare with the standard profile.

0 2 4 6 8 10-1

0

1

0 2 4 6 8 10-1

0

1

N= 5 N= 5

0 2 4 6 8 10-1

0

1

N=20

Fig. 3. The approximated degree of the JRC=14-16 profile by superimposing various harmonic terms (N= number of harmonics)

It is revealed from Fig. 3 that superimposing of the first twenty harmonics is enough to describe the original JRC profile. Thus, the twenty harmonics are summed up to simulate the original Bartons standard profiles in the study. It is worth noting that each rock surface with its certain roughness can be modeled by some separate two dimensional profiles using Fourier series.

2.4. Physical modeling of joint surface using CAD data and LOM technology

In the LOM process, CAD data of the given joint surface geometry (obtained by using laser scanner) goes into the LOM systems process controller and a cross-sectional slice similar to

the iso-altitude contour is created by the LOM software system (Fig. 4). Based on the geometric information provided by the stereo lithography file (STL), an algorithm is developed. A laser beam cuts the first cross-sectional outline in the top layer of multiple sheet papers (which each is 0.1075 mm thick) and then cross hatches the excess paper for later removal. A new layer of the second sheet paper with little heat-melted glue is bonded by a roller heating-and-pressing process to previously cut layer. A new cross section is created cut as before. Once all layer sheet papers have been laminated and cut, excess waste paper is removed to expose the finished LOM model [9].

Fig. 4. Layout of the LOM technology [9]

It is difficult to remove the crosshatch parts in a horizontal bonded sheet paper containing many isolate cutting contours (sills) for remaining the actual joint material in a special elevation. In fact, a 3D joint surface consists of numerous 2D joint profiles. During the cutting of a 3D joint surface, the cutting and bonding direction of the traditional LOM process is changed to the diametrical direction. Each sheet paper was stack-bonded along the diametrical direction for a joint profile. Then, the edge shape in each sheet paper was cut according to one profile altitude of a 3D surface but not the contour in surface altitude. After bonding and cutting the profiles of more or less 930 sheet papers, based on the used order of harmonics, a 3D surface geometry of LOM joint model is obtained as shown in Fig. 5.

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Fig. 5. The LOM object of 3D joint surfaces (only with the first 5-order harmonics)

2.5. Duplicating the Mother Mold Using Silicon Rubber

LOM paper is frequently used for silicon rubber molding to make urethane or epoxy cast plastic parts. As the LOM objects do not undergo phase changes and are resistant to shrinkage, the accuracy of the master is predictable regardless of geometry. Also, LOM masters do not react with silicon rubber. Using the LOM as a mold for silicon casting, a mother mold mixed by the silicon rubber (RTV-533, Product of Wacker, Japan) and silicon catalyst in the ratio of 25:1 would be able of duplicating the joint surface geometry (Fig. 6).

(a) Smoother surface (only 1st order asperities) (b) Rough surface (1st and 2nd order asperities)

Fig. 6. The surplices of artificial joint specimens of plaster and water

It is important noting that modeling of first order or second order asperities is totally dependent on the used number of harmonics in Fourier series (input CAD data to LOM system). The 5-order (Fig. 6-a) and 20-order harmonics (Fig. 6-b) are used in modeling of first and second order asperities, respectively.

2.6. Mathematical modeling of closure response in 2D joint profiles

2.6.1. Closure between a flat plane and rough profile

The Greenwood-Williamson model (GW model) is applicable to the elastic contacts of two metal surfaces, one of which is rough and the other is smooth. It is presumed that the surface roughness is a set of spherical summits having the same radius but random heights. When we force two surfaces close together (Fig. 7), then the material near the present summit contacts would deform and the new contacts would appear.

dReference line

Contact portion

Flat plan

Fig. 7. The contact portion of asperity between a flat plane and a rough surface approaching

If the deformation of summits is elastic, the Hertz solution can give the basic solution between the nominal load P and the relative approach d (with respect to a reference line).

( ) dzzdzENwEPd

)(34

34 2

3

21

23

21

== (7) In which, N is the number of asperity summits

on the nominal flat plane, Ei and vi are elastic modulus and Poisson ratio for the two surfaces and the equivalent modulus E is as below:

12

221

21 ]/)1(/)1[(

+= EEE (8) The function of (z) is the probability density

function (PDF) of asperity height distribution, the integration of which is not easy to be obtained. Therefore, Swan[10] first applied this theory to rock joints and proposed a discrete numerical technique to facilitate the integral of Eq. (7) during joint closure. He concluded that this technique could give reasonable quantitative predictions with the experimental results. After studying the case of two misaligned surfaces in contact, they observed that the force between a pair of peaks with given heights is not dependent

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on the individual heights but instead is associated with their sum. Therefore, Brown et al[11] extended this theory for two rough surfaces by adopting a composite topography concept. Among these, describing the geometrical features seems to play the essential part of calculating the contact closure. In this paper, the original irregular joint surface y(x) is first decomposed into twenty harmonic regular sub-profiles by Fourier transformation in which each harmonic sub-profile contains several sine-shaped asperity with the same height and radius (Fig. 8).

0 20 40 60 80 100(mm)

-1.0

0.0

1.0

(mm

) i = 1

0 20 40 60 80 100(mm)

-0.4

0.0

0.4

(mm

) i = 4

0 20 40 60 80 100(mm)

-0.2

0.0

0.2

(mm

) i = 20

Fig. 8 The n-th harmonic sine-shaped sub-profiles with different wavelength and amplitude

Then every summit of the sine-shaped asperity on each two-dimensional harmonic sub-profile is modeled as a circle with a curvature of (radius R) given as

Ry

y 1

)1( 232

=+= (9)

Therefore, the elastic contact theory presented by Hertz between two spherical bodies with different radius can be applied to calculate the contact deformation. If two circular summits (Fig. 9) are forced to close by a nominal load P, they would satisfy the following relation:

3/1

21

212

2122 )()(

169

++=

RRRRkkP (10)

Fig. 9. Definition of the relative closure between two approaches circles.

The contact area between two bodies would be a circular area with a radius a,

3/1

21

2121 ))((4

3

+

+=RR

RRkkPa (11) where Ri is the radius of the two circular

bodies and ki=(1-vi2)/Ei is related to elastic constants. Let the elastic constants of E, v for the joint profiles identical (i.e. E1=E2, v1=v2), the normal compliance during closure deformation C=/P is also derived as,

EavC

21= (12) For the special contacting case of a flat plane

and circular body, one can assume the radius 1R and expressing the relative closure as,

3/1

2

221

22 )(16

9

+=

RkkP (12)

Through the above equations (9) and (12), the normal load P and the relative closure of each contact between a flat plane and a sine-shaped harmonic sub-profile can be obtained. However, a real joint profile is composed of twenty harmonic sub-profiles. Therefore, superimposing the closure of each sub-profile will cause the total closure of the joint. As the contact surface of an unmated joint is composed of two rough profiles, the Browns concept of composite profile will be adopted to deal with two rough profiles. As shown in Fig. 10, the surface heights are measured from parallel reference plane fixed in each surface. The so-called composite topography of a real unmated joint is defined by summing the heights of both surfaces at each point along the joint. Consequently, a flat plane and the composite rough profile (see Fig.11) can

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simulate the closure behavior of the two rough profiles as previously mentioned.

Fig. 10. The concept of composite topography for the contact between two rough profiles [Brown et al., 1985].

Fig. 11. The new contact type of two rough joint profiles after simplifying as that of a composite profile and a plate.

2.6.2. The effect of asperities on normal closure of joints

A rock-like model material of plaster and water in proportions of 1: 0.9 by weight was selected in this experimental study. All the model specimens were cured at a temperature of 25oC for about one day. The mechanical properties of the model material are: uniaxial compressive strength, c= 4 MPa; elastic modulus E=650 MPa and Poisson ratio v = 0.3.

2.6.3. Closure behavior between a rigid plate and cores

In order to evaluate the validation of GW model applying to this model material, two types of closure tests between a rigid steel plate and plaster core were performed (Figs. 12) before the joint closure. For this case, it is presumed that the joint surface contains only a single asperity with a circular summit. A core diameter of 54 mm or 38 mm is selected acting as the summit of asperity on the plane. Moreover, to clarify the superimpose principle between different asperity sizes, test on another joint sample consisting of

two circular summits in 38mm and 54mm with the same peak height is also performed (Fig. 14). It is revealed from the preliminary test results in Figs. 14 that the closure behavior of the elastic material in this research is reproducible and agrees to the predicted results (Fig.13). The maximum closure of small asperity (38mm) is larger than that of large asperity at the same stress levels. It implies that the smaller asperity (second order) under loading will be completely closured ahead of the larger (first order) one. The test results of composite profile containing the 54mm and 8mm asperity summits (Fig.14), shows that the mathematical model can well predict the closure behavior of composite profile by superimposing method. As each harmonic sub-profile will contain many identical asperity, the rough joint consisting of one, two and three circular summits with the same size is also tested (Fig.15). For example, the first harmonic sub-profile has a single asperity summit and the second has two summits. Figs. 15 both show that the joint plane containing more asperity will deform less than the single asperity case at the same loading. Because every summit in a harmonic sub-profile will take over the whole external loading, it will slightly deformed under the partial loading of P/n. The new approach of can be obtained by substituting the load P/n for each asperity in Eq. (8).

32

)1(n

= (14) where n is the number of asperity summits in

a sub-profile. It can also be derived that the amount of joint closure for a plane with different n circular summits is proportional to (1/n)2/3. This relationship derived from the elastic contact theory is also observed in experimental results as compared in Fig. 15.

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0.00 0.02 0.04 0.06 0.08

0

50

100

150

200

P (k

g)

Exp. No1

Exp. No2

Exp. No3

Exp. No4

(cm)

(E=6500 KSC)

Dia=54mm

0.00 0.02 0.04 0.06 0.08

0

50

100

150

200

P (k

g)Exp. No1

Exp. No2

(cm)

Dia=38mm

(E=6500 KSC)

Fig. 12. Experimental closure behavior for the joint with a single circular asperity:(a) 54mm; (b)38mm core.

0.00 0.02 0.04 0.06 0.08

0

50

100

150

200

P (k

g)

Model (Dia=38)

Exp.(Dia=38 )

Model (Dia=54)

Exp.(Dia=54)

(cm) Fig. 13. Compare the closure behavior of experimental and predicted results.

0.00 0.02 0.04 0.06

0

50

100

150

200

P (k

g)

(cm)

Dia=54+38 mm

Model

Exp. No1

Exp. No2

Exp. No3

0.00 0.02 0.04 0.06 0.08

0

50

100

150

200

P (k

g)

(cm)

Dia=54+38 mm

Model

Exp. No1

Exp. No2

Exp. No3

Fig. 14. Comparison of experimental and predicted closure behavior for a composite joint plane by superimposing each closure according to the radius ratio

0.00 0.02 0.04 0.06 0.08

0

50

100

150

200

P (k

g)

Dia = 54 mm

Model

N=1 Exp.

N=2 Exp.

N=3 Exp.

(cm)

12N = 3

0.00 0.02 0.04 0.06 0.08

0

50

100

150

200

P (k

g)

Dia = 38 mm

Model

N=1 Exp.

N=2 Exp.

N=3 Exp.

(cm)

2 1N= 3

(a) 54 mm (b) 38 mm

Fig. 15. Comparison of closure behavior for the joint plane with multiple identical asperity.

2.6.4. Closure behavior of unmated joint surfaces

A pair of irregular 2D joint profiles generated based on the fractal theory was used as the model joint surface (Fig.16-a) in which the asperity heights are Gaussian distributed. A pair of joint surfaces of steel mould was prepared from the mother mould cut by a wire-cut machine. A series of model joint (Fig.16-b) are then cast from the mother mould for the closure testing. The asperity heights of the top and bottom profiles are then summed up to get the composite profile theoretically. Then the composite profile is also separated into twenty sine-shaped sub-profiles and its closure behavior was predicted based on the above mathematical model. Some typical experimental results are shown in Fig.17.

(a)

(b)

0 2 4 6 8 100

1

2

Fig. 16. Two cases of unmated joints cast from model joints.

0.00 0.02 0.04 0.06 0.08 (cm)

0

50

100

150

200

P (k

g)

Exp. Mismatch (1)

Exp. Mismatch (2)

intact rock

0.00 0.02 0.04 0.06 0.08 (cm)

0

50

100

150

200

P (k

g)

Mismatch case (1)

Theory

Exp. no1

0.00 0.02 0.04 0.06 0.08 (cm)

0

50

100

150

200

P (k

g)

Mismatch case (2)

Theory

Exp. No1

Fig. 17. Comparison of experimental and predicted

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closure behavior for the unmated joints.

It is evident from Fig. 17 that the tendency of prediction is agreed to experimental results. It should be noted that the closure of a 3D joint surface shall be the summation of several 2D joint profiles. However, more experimental tests are needed to prove the 3D approach.

3. Conclusion

LOM (Laminated Object Manufacturing) process of rapid prototyping is a new technology using sheet papers to build 3D physical prototypes. The LOM system produces the 3D joint surface model is based on the input CAD data of surface geometry. Employing the mathematical theory to transform the point data of an original surface geometry, the LOM system can easy manufactures another joint surface model according to the new input geometry data. Therefore, this method can duplicate the three-dimensional geometry of natural rock joints and similar brother surface only with the primary surface structures. Also, by applying the Fourier transform method to regularize the irregular surface, the elastic contact theory of Hertz can thus be adopted to predict the closure behavior of irregular joint surfaces. None statistical parameter about the joint topography is required. From comparing the experimental results, it is found that the second order asperity will closure completely ahead of the larger asperity under the same loading. The closure of composite profile can be used to represent the two unmated joint profiles. The amount of closure for a plane containing n identical asperity is proportional to (1/n)2/3. More details of experimental work of 3D joint closure are however required.

References

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2. Hoek, E. and J. W. Bray, 1981. Rock slope engineering. London: IMM.

3. Yang, Z.Y., C.C. Di and K.C. Yen, 2001. The

effect of asperity order on the roughness of rock joints, Int. J. Rock Mech. Min. Sci. 38(5), 745752.

4. Hutson, R.W. and C.H. Dowding, 1987. Computer controlled cutting of multiple identical joints in real rock, Rock Mech. Rock Engng 20, 3955.

5. Helisys Company, 1991. Technology focus: laminated object manufacture, Rapid Prototyping 1(1), 69.

6. Greenwood, J. A. and J.B.P. Williamson, 1966. Contact of nominally flat surfaces, Proc. Royal Soc. London, Math. and Phys. Sci., 300319.

7. Indraratna, B., A. Herath and N. Aziz, 1995. Characterization of surface roughness and its implications on the shear behavior of joints, Proc. 2nd conf. on the Mech. of Jointed and Faulted Rock, Vienna, pp.515520.

8. Raja, J. and V. Radhakrishnan, 1977. Analysis and synthesis of surface profile using Fourier series, Int. J. Mach. Tools, 245251.

9. Helisys Company, 1991. Making parts from papers and Helisys First Product Review, Rapid Prototyping 1(2), 46.

10. Swan, G., 1985. Methods of roughness analysis for predicting rock joint behavior, Proc. Int. Symp. on Fundamentals of Rock Joints, Bjorkliden, pp.1520.

11. Brown, S. R. and C.H. Scholz, 1985. Closure of random elastic surfaces in contact, J. Geol. Res., 5531554.