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Application of Fractal Theory in Analyzing Character of Joints and Cracks Inside a Rock-mass by Fu Helin, Li Liang, Liu Baochen, Civil Engineering College, Changsha Railway University, China Hou Zhengmeng, and K-H Lux Claustahal Technological University, Germany |
ABSTRACT
Fractal Geometry is a theory to analysis nonlinear phenomena, which has been founded in recent 10 years. It regards irregular geometry as research objects, it was founded by B.B Mandelbrot in 1976. He thought, fractal dimension could applied in disposing of crack and irregular problems, it is beyond concept of traditional integral dimension. According to research results, rock-mass structure appears irregular, factually it has formal regularity, i.e. self similar statistics, it is a typical structure. So this paper apply fractal geometry in analyzing characters of rock-mass, and obtain good results.
KEYWORDS: Fractal Geometry, Irregular Problems, analysis characteristics of rock-mass
INTRODUCTION
Among the slope slide affairs in rock engineering, most of the slide affairs are influenced by Joints and cracks in rock-mass. The joints and cracks deeply concern with sliding plane. The study of joints and cracks becomes main content of rock mechanics. The joints and cracks formed in long geological history, and became somewhat trench, somewhat scale, somewhat shape, which influencing the rock-mass character. On the basis of analyzing slope slide in rock-mass, the reasons why slope slides take place are normally divided into two types, i.e. geologic elements and no geologic elements.The geologic elements include characters of Joints and cracks, character of rock block, ground water, and ground stress. The no geologic element includes rain, earthquake, explosion, and vibration of train. According to former study result of rock-mass stability in slope, the crack rock-mass stability coefficient is in this formula, the coefficient is influenced by
|
(1) |
are the parameters of joints and cracks.
According to the results of Barton’s study, ,
f: basic friction angle, i: joint roughness angle.
| (2) |
JRC: Joint roughness coefficient, JCS: axial pressure strength of joint,
normal stress in joints
is mainly influenced by character of joints and cracks and its existing environment.
The ground water is not only influenced by rain, but also influenced by penetration
of joints and cracks, i.e the ground water is influence by character of joints and
its existing environment. and in sometime, it is mainly controlled by characters of
joints and cracks. In order to analysis the slope stability, it is necessary to
analysis character of joints and cracks first.
BACKGROUND
Research on character of rock-mass
The characters of joints and cracks of rock-mass include: trench, shape, scale, distance, width, roughness coefficient, joint strength, and fillings. Since 70, many experts and scholars have been systemically studying about these characters, and have leaded rock mechanics and engineering geology to develop quickly. But through detail analysis, because of irregularity of joints and cracks, and because of complication of jointed net, the forehead research is restricted.
Factually, in natural there are many objects and affairs appear irregular in crack time state or space state. For example, the side of the cloud, the appearance of earth, the fault of rock-mass, and space effect of explosion, these phenomena are difficult to describe in Euro space, so it is necessary to find other way to analysis these phenomena.
Fractal Geometry is a theory to analysis nolinear phenomena, which has been founded in the last ten years. It regards irregular geometry as research objects, it was founded by Mandelbrot in 1976. He thought, fractal dimension could applied in disposing of crack and irregular problems, it is beyond concept of traditional integral dimension. According to research results, rock-mass structure appears irregular, factually it has formal regularity, i.e self similar statistics, it is a typical structure. So application of fractal geometry in analyzing characters of rockmss will obtain good results.
Basic concepts of fractal geometry
Fractal geometry is one of the research branches that study cracks and their forming.
The research content of fractal geometry includes:
At present, the most typical type used is the linear fractal.
Self-similarity is part of a object represents the whole character of the whole object. The parameter reflects the self-similarity is fractal dimension.
Background on fractal rock mechanics
Rock mechanics combined with fractal geometry is called fractal rock mechanics. Its research content includes many rock mechanics problems existing in fractal area. Although the fractal rock mechanics is still in premier period, great development is obtained in many research field, shown in table 1.
Research of application fractal geometry in analyzing character of joints and cracks
It is briefly concluded, there are 6 fractal measure methods to analysis character of joints and cracks. They are yard measure method, modified yard measure method, box-dimension method, self-affinely method, power method, periphery- area method.
Researchers so far have always concentrated on the study of joint roughness coefficient. Joint roughness coefficient is an important factor influencing rock-mass character, but there are many other factors that influence rock-mass character, for example, trench, shape, scale, distance, width of joint, etc.
Study of fractal character of joint scale and density
The scale of joints can be signed with length of joints, for example, total length, haft length. Joints length is divided into three types, linear density, plane density, body density. It can be obtained by the joints numbers in a determined area. Joint density can be obtained from distance between joints. Width between joints can be directly measured with ruler. The bigger the ruler, the more inexact the measure result will be obtained. The smaller the ruler, the more exact the measure result. This statistic information is similar as a Cantor collection. So man can apply Cantor collection in calculating fractal dimension of joints.
The method of calculating fractal dimension is: If N (e) is a number can represent the dot collection in a small ball, which diameter is e, the fractal dimension of dot collection is:
|
(3) |
where e is the yard measure.
Table 1. Main development of fractal rock mechanics
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fractal damage mechanics | fractal flexible | fractal pore | fractal of rock structure | fractal hydraulic rock mechanics | fractal mechanical character |
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The process of rock damage can be regarded as fracture. Fractal dimension reflects damage content. Through founding damage model, the connection between damage density and fractal dimension can be obtained | The process of rock flexible can be regarded as fractal, fractal dimension reflects energy during rock flexible taking place. Through founding flexible model, the connection between flexible and fractal dimension can be obtained | There are many pores inside the rock-mass, different grade pores can be regarded as fractal structure, application of fractal geometry in analyzing pore inside rock-mass can reflect character of rock-mass | Rock mass shear strength concerns deeply with fractal dimension of rock-mass structure, application of fractal geometry can infer the rock deformation | There is fractal character of penetration in irregular rock-mass, application fractal dimension can infer rock-mass penetration character and change appearance. | Rock-mass under pressure, when rock-mass take place deformation, it sends out Acoustic Emission. This phenomena has fractal character, application of fractal geometry can predict rock-mass collapse. |
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e: yard measure
N(e): measure number
There are many methods to calculate fractal dimension, the basic method is box fractal method. Its calculation steps are:
(a) For the area select a square with a side length of L, where L is 1m, 2m, 5m, 10m, 20m, 40m etc.
(b) Inside the LL square calculate the number of the joints whose length is longer than L.
(c) Connect the middle point of each side, dividing the square into 4 parts, i.e, 4 L/2 net. Calculate the number of the joints their length longer than L/2. Add the 4 parts; the number of joints their length longer L/2 is N(L/2).
(d) Do the same transformation n times, then obtain the number of joints with lengths longer than L/2n-1 is N(L/2n-1)
(e) Regard each time N(L/2n-1) as X-coordinate, Ln(L) as y-coordinate, connect all points obtain a line, the slope of the line is fractal dimension of Joint length or distance.
SIMULATION OF FRACTAL DIMENSION OF JOINT ROUGHNESS COEFFICIENT
The shape of joints and cracks can be described by angle of climb and roughness coefficient. Former is signed i, the behind can be signed JRC.
Summary of study of angle of climb
Barton(1966) put up forward study of shear strength of non-filling joints with JRC. Through a serious of experiment, Barton obtained the connection between shear stress () and normal stress (
) :
|
(4) |
: basic friction angle
: angle of climb
Experimenting on more than 100 artificial joints, Barton(1973) obtained:
|
(5) |
: axial strength of inner plane in joint
Equation (5) is called Barton Formula.
In Barton formula, the most difficult is to obtain JRC. After many experiment, Barton put up forward curve of JRC. Compared with curve of JRC, the JRC of factual joint can be inferred. Because of great variance geologic survey, numerical simulation to determine JRC of joints put up forward. From (5), 10 Barton curves of joints can be inferred, shown in Figure 1.
Summary of study of numerical simulation to determine JRC of joints
It is well known that any function x(t) can be decomposed into a series of harmonic functions where each tem has a different frequency, different amplitude, and phase angle.
|
(6) |
where w denotes the angular frequency and
through Fourier transformation.
|
(7) |
In fact the curve can be expressed as follows.
|
(8) |
Fig 1. Barton curve of joints
Fig 2. A roughness curve
Figure 2 is a Barton curve. In same distance, amplitude of vibration
, so can obtain
from (8), thus,
|
(9) |
can be obtained by least squares method:
|
(10) |
All 10 Barton curves are shown in Table 2.
Because JRC is a trace of joint, only a JRC value can not represent real appearance of joints and cracks, so it is necessary to apply fractal geometry in describing joints.
Fractal analysis of character of joints and cracks of rock-mass
According to fractal character of joints, the connection between length L of joint and yard measure e is
|
(11) |
N is the measure steps when yard measure is e, and Dis the fractal dimension
when L=1, make logarithm calculation to formula(11), then obtain:
|
(12) |
When measuring a Barton curve with yard measures ˘20mm, ˘60mm, ˘200mm and ˘600mm, the fractal dimension is D1when the same Barton curve is measured with ˘2mm, ˘6mm, ˘20mm and ˘60mm, the fractal dimension is D2.
D1 = D2. This is called self similar in fractal geometry.
Now measure 10 Barton curves with yard measure ˘2mm, ˘6mm, ˘20mm and ˘60mm, obtain their fractal dimension, shown in table 3.
Description of joint width
Joint width is a important factor that influences greatly on rock-mass penetration. Barton and Bakhtar derived the formula to simulate joint width.
|
(13) |
, JCS -- are axial pressure strength of rock-mass
influenced by natural element(JRC), but also influenced by normal stress and shear stress
.
|
(14) |
Table 2. Standard expression of JRC
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Standard expression of JRC | Amplitude of vibration |
JRC value |
original JRC value |
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x(t)=1.61-0.00513cos720t+0.0089sin720t x(t) = 1.158 - 0.013 cos12t x(t) = 1.127 + 0.0224 cos240t - 0.000157 sin240t x(t) = 1.051 + 0.0318 cos240t + 0.0393 sin240t + 0.045 cos120t x(t) = 1.178 - 0.1097cos120t - 0.0257 sin120t x(t) = 1.072 - 0.1274cos120t - 0.1359 sin120t x(t) = 0.9544 - 0.0685cos120t - 0.1762 sin120t x(t) = 0.8517 + 0.0445cos120t + 0.1453 sin120t + 0.1940 cos240t + 0.0247 sin240t x(t) = 0.872 + 0.2117cos120t - 0.1543 sin120t x(t) = 1.056 - 0.363cos120t - 0.0694 sin120t - 0.0588 cos240t - 0.0366 sin240t |
0.010 0.01 0.02 0.04 0.11 0.18 0.19 0.20 0.26 0.36 |
0-2 2-4 4-6 6-8 8-10 10-12 12-14 14-16 16-18 18-20 |
2-4 0-2 4-6 6-8 8-10 16-18 12-14 14-16 10-12 19-20 |
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Table 3. Fractal dimension of inside appearance of Barton joints
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JRC |
fractal dimension |
|
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0-2 2-4 4-6 6-8 8-10 10-12 12-14 14-16 16-18 18-20 |
1.0 1.0019 1.0027 1.00499 1.0054 1.0045 1.0077 1.0070 1.0104 1.0170 |
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|
(15) |
u: normal displacement, d: shear displacement, through (14)-(15), we obtain,
|
(16) |
When u = 0, E = Km bm |
(17) |
So under normal stress, the average width is joint is:
|
(18) |
-- premier normal rigidity of joint
6.5 The connection between grad flow inside joint and width
Under normal stress, grad flow inside joint is very sensitive to width of joints. The very small change of width will lead great change of flow inside joints. Snow’ formula is
|
(19) |
----horizontal penetration ratio of joint
----penetration ratio under premier stress
----pressure stress
---normal rigidity
-----width of joint
s---area of joint
Combine formula (18)(19), the connection between grad flow inside joint and width can be obtained.
On the basis of fractal dimension of JRC, under normal stress, the average width of joint can be inferred. The connection between grad flow inside joint and width can also inferred. This play important role to analysis rock-mass penetration.
APPLICATION OF FRACTAL GEOMETRY IN SITE ENGINEERING
Xiangqian Railway is a main railway in China. Its geologic condition is very complicate. Especially in Dizhuang area. When it rains, slope slide normally takes place. it often hinds traffic, leads to great loss to our economic. In order to solve the slope stability problem, it is necessary to study the characters of joints and crack in slope.
On the basis of geologic survey, select K303+600, k303+800, k303+850 as analysis project. Fractal dimension if applied in analyzing character of joint density, length, width and JRC.
Fractal study of joint density, length and width
On the basis of geologic survey result, sign ln(x)
(x represents joint density, length, width and JRC) as x coordinate,
sign (ln(xi)(N(xi)) is joint numbers which density, length and width greater than ) as y-coordinate, so fractal dimension of joint density, length and width is obtained, shown in Figure 3, fig, 4, Figure 5 and table 4.
Figure 3. Fractal dimension of joint density
Figure 4. Fractal dimension of joint length
Figure 5. Fractal dimension of joint width
Table 4. Connection between fractal dimension and geometric parameters of joint
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site fractal relation regression parameters |
||
dimension |
coefficient |
equation |
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density (distance) |
K303 2.048 .9528 |
y=-1.59176+2.048x |
length |
+600 -.8772 .9267 |
y=11.2389-0.8772x |
width |
+850 .8180 .9634 |
y=5.5392-.8280x |
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Fractal study of JRC
Step 1. Connection between JRC and D
On the base of’ study result, statistic analysis is made, the following equation id obtained.
|
(20) |
Application of equation (20)in statistical analysis of 10 Barton artificial Joints. Equation (21) (22) are obtained, show in Figure 6.
Figure 6. Connection between JRC and D
|
(21) |
|
(22) |
Step 2. Calculate fractal dimension of JRC and angle of climb i
The fractal dimension of (Joints and fault) JRC is slope k303+600—k303+850 is calculated, as shown Figure 7 and Figure 8.
Figure 7. Calculation of fractal dimension of fault
Figure 8. Calculation of fractal dimension of joints
So the angle of climb of the fault is
and the angle of joints is
On the basis of mechanical experimenal work, cohesion of fault and joints are fb = 30°, s0 = 45 MPa. According to other analysis result in the slope in k303+600—k303+850, existing block that maybe slide. At the site of blocks, the stress is , so angle of climb of complex slides plane is
. The stability of the slope can be analyzed using Equation (1).
CONCLUSION
It is a new approach to apply fractal theory in analyzing characters of joints and cracks inside rock-mass. Following conclusions can be derived from this work:
References
Helin, Fu, Theoretical Analysis Model for Block Crack Rockmass Slope Stability and its Application, Dr. Degree Disseration, South Central University, China, 2000.9
Arr J., Fractal Character of joint surface roughness in welded ruff at Yucca Mountain, Nevada, 30th US Symposium on Rock Mechanics, publ. Rotterdam, pp. 193-200, 1989
ACKNOWLEDGMENT
This paper is sponsored by China Scholarship Council and Hunan Natural Science Fund Council, here we thank them.
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