Fractal Characteristic of Rock Cutting Load Time Series
A test-bed was developed to perform the rock cutting experiments under different cutting conditions. The fractal theory was adopted to investigate the fractal characteristic of cutting load time series and fragment size distribution in rock cutting. The box-counting dimension for the cutting load time series was consistent with the fractal dimension of the corresponding fragment size distribution, which indicated that there were inherent relations between the rock fragmentation and the cutting load. Furthermore, the box-counting dimension was used to describe the fractal characteristic of cutting load time series under different conditions. The results show that the rock compressive strength, cutting depth, cutting angle, and assisted water-jet types all have no significant effect on the fractal characteristic of cutting load. The box-counting dimension can be an evaluation index to assess the extent of rock crushing or cutting. Rock fracture mechanism would not be changed due to water-jet in front of or behind the cutter, but it would be changed when the water-jet was in cutter.
In rock cutting process, the dynamic characteristic and fragmentation information are recorded in the cutting load time series directly. The irregularity and aperiodicity of the cutting load have close relations to rock crushing [1, 2]. Since the concepts of fractal dimension and fractal geometry were formulated, the fractal theory has been applied in many fields , including the study on rock fragmentation. Xie et al.  proposed a column method to analyze the fractal property of the spatial distribution of acoustic emissions during rock damage and failure process. They found that an increase in fractal dimension corresponds to a decrease in stress and an increase in energy release. Wu et al.  used the fractal theory to study the fractal characteristic of acoustic emission time series when rock failed as a result of uniaxial compression, and they found that the time series had fractal feature at every stage. In addition, the fractal dimension can describe the evolving regularity of microscopic cracks. Wang et al.  investigated the fractal characteristic of electromagnetic radiation time series of coal and rock failure, and they found that the variation of correlation dimension was consistent with coal or rock burst. Huang et al.  researched on the fragments size distribution under DTH hammer reverse circulation drilling and found that the rock fragments size distribution of drillings conform to fractal distribution. As mentioned above, these studies have improved people’s understanding of the fragmentation information on rock crushing or cutting process to a certain extent. However, those research objects were mainly focused on the fractal characteristic of acoustic emission signals, but the research on the relationship between cutting load time series and rock fragmentation by cutter was far from enough. As a consequence, fractal was used to investigate fragments size distribution and the characteristic of cutting load time series, and fractal dimension was calculated and analyzed to provide the basis for studying rock fragmentation and cutting performance under different conditions.
2. Fractal Theory
2.1. Fractal Function of Fragments Distribution
Mandelbrot noticed that several natural phenomena could be well described by a single power law. The exponent of this power law, called fractal dimension, is a nonnegative real number that can assume fractional or nonfractional values . If the size of the fragments is a fractal, then the fractal dimension could be estimated from the size distribution of the fragments. Most applications of fractal concepts to fragments distribution are based on the fragmentation model of Turcotte . Fractal relationship for fragments size distributions have been described relating number of fragments of a defined size to the size as follows: where is the cumulative number of fragments greater than a given comparative sieve size ; is a constant corresponding to the number of fragments of unit length; is the fractal dimension. Equation (1) is also called the power-law relationship of a fractal distribution. In many cases, fragmentation results in a fractal distribution.
The function on fragments number density distribution can be obtained by taking the derivative of (1):
In practical applications, the use of number-based size relationship is inconvenient and the systematic error is introduced into the dimensioning process by assuming the particles are uniform with respect to both shape and density. Tyler et al. suggested that a fractal dimension estimated from the cumulative mass distribution is less sensitive to the assumptions of scale invariant density and size of aggregates compared to number-size distribution [10, 11]. The cumulative mass of fragments size is smaller than a given comparative sieve size : where is the density of fragments, kg/mm3; is the shape coefficient of fragments.
Mandelbrot and Turcotte proposed a logarithmic expression involving the mass and size information for the fragments [8, 9]: where is the cumulative mass of fragments size smaller than a given comparative sieve size ; is the total mass of fragments; is the maximum fragment size as defined by the largest sieve size opening; is the fragmentation fractal dimension.
The fractal dimension can be calculated by the simpler linear equation obtained from the regression line via (4) as shown below:
The linear best fit of (5) produced estimates of and . is found using the slope coefficient, for the linear regression line and the equation:
2.2. Box-Counting Dimension of Time Series
The fractal dimension is a numerical measure of the self-similarity of an object; it can be used to analyze the irregularity of a set of time series. In rock cutting process, the load time series could be regarded as open curve in two dimensions, and its outline has fractal characteristic. The commonly used definition about the fractal dimension in literature is the box-counting dimension given as below [12–14]: where is the smallest number of boxes of side to cover the cutting load time series ; is the fractal dimension called box-counting dimension. Taking logarithm on both sides of the equation, we have According to the feature of cutting load time series, the operative method to determine the fractal dimension in nontrivial cases is counting the number of boxes needed to cover the cutting load curve for different values of . The region of the coordinate plane of the load curve is meshed by those boxes with side as shown in Figure 1, and it is the algorithm basic philosophy of box-counting dimension method.
When , the cutting load time series is discrete sampled on the vertical coordinate. The number of covered boxes adds 1 when , which can be expressed as
Therefore, the number of total boxes, which can cover the cutting load time series totally, is as follows: where is equal to 0, 1, 2, …, ; is equal to 0, 1, 2, …, .
For a series of boxes with different side , the linear best fit of (8) produced estimates of and . is equal to the slope coefficient for the linear regression line by the least square method.
3.1. Acquisition of Cutting Load Time Series
Figure 2 described the rock cutting test-bed, Figure 2(a) was the schematic diagram of the acquisition and processing system of cutting load time series, and Figure 2(b) was the photograph of the test-bed in laboratory. There were mainly four components: (I) rock cutting device, which was used to perform the cutting experiments with a linear feed speed. Meanwhile, many experiments could be realized, such as different cutting depth, cutting angle, rock compressive strength, and assisted water-jet types; (II) fixed beam, which was designed to attach strain gages. Furthermore, the full-bridge circuit of strain gages was selected to collect voltage signal , whose aim was to avoid the change of voltage signal as a result of beam bending and calculate the cutting load time series according to the fixed beam torque; (III) acquisition instrument INV306U, which was developed by Beijing Dongfang Vibration And Noise Technology Research Institute. And analog signal or digital signal all could be collected through it; (IV) signal processing equipment, which was a personal computer with the signal processing software DASP-V10.
(a) Schematic diagram of signal acquisition system
(b) Photograph of rock cutting test-bed
3.2. Experimental Phenomena
The load time series and rock fragments in rock cutting were shown in Figure 3. It was obvious that the sizes of rock fragments were very different from each other. Therefore, the rock cutting could be regarded as a nonlinear and nonperiodic process, and the fragments might be reflected in the cutting load time series. Figure 3(b) showed that there was a great difference among the amplitudes and time intervals of the cutting force peaks (corresponding to big rock fragments). Also it indicated that the cutting load time series was discrete and irregular. Firstly, the crushing zone was caused by the cutter extrusion on rock. Then, the cutting force continued to increase after the crushing zone was formed, and several small peaks showed that local breakings happened in rock cutting process. Lastly, the cutting load decreased suddenly after it achieved the peak value, which shows that a big fragment had formed at this moment. The above-mentioned phenomena were consistent with the previous rock cutting theory and the experimental results [1, 16, 17].
(a) Rock fragments
(b) Cutting load time series
4. Results and Discussion
4.1. Fractal Characteristic
The fragments in Figure 3(a) were sieved into six different sizes, ranging from 0 to 5 mm, from 5.1 to 10 mm, from 10.1 to 15 mm, from 15.1 to 20 mm, from 20.1 to 25 mm, and from 25.1 to 30 mm. The cumulative mass rate of the fragments was shown in Figure 4, and its variation was consent with Power law showing that the cumulative mass rate followed fractal distribution. According to (5), the fractal dimension of fragments size distribution could be obtained by first developing an versus on ln-ln plots and determining the slope of best-fit line through the data points by linear regression method as shown in Figure 5, the correlation coefficient of 0.994 showing the cumulative mass rate and fragments size had the fractal characteristic again. According to (6), it was easy to get the fractal dimension of fragments size distribution which was equal to 1.42. According to (8), the double logarithmic curves for calculating the box-counting dimension of the time series corresponding to the fragments in Figure 3(a) was shown in Figure 6, and the correlation coefficient of 0.998 shows that the cutting load time series also followed the fractal distribution in space. The difference between the fractal dimension of fragments size distribution and the box-counting dimension of cutting load time series was smaller than 3%, which indicated that there were inherent relations between rock fragmentation and the load time series in rock cutting process. Therefore, the box-counting dimension could be as the fractal dimension of fragments size distribution, and it provided a new idea to investigate the rock crushing. As a result of the tedious process of the fragments classification, the box-counting dimension was used to investigate the fractal characteristic of the load time series and fragments size distribution under different cutting conditions below.
4.2. Effect of Rock Compressive Strength on Fractal Characteristic
The rock compressive strength has significant effect on cutting load time series and fragments. The box-counting dimensions for the compressive strengths 2.5 MPa, 4 MPa, and 5 MPa were obtained by first developing an versus on log-log plots and determining the slope of best-fit line through the data points using linear regression as shown in Figure 7(a). The correlation coefficient of these fitting lines were all greater than 0.95; therefore, it was concluded that the cutting load time series under different rock compressive strength all had fractal characteristic. Moreover, the difference between the box-counting dimensions in these three conditions was little, and it indicated that the rock fracture mechanism would not change when the rock compressive strength was different. For the size distribution of fragments, the rock of low compressive strength (LCS) was easier to be crushed than that of high compressive strength (HCS). In addition, the box-counting dimension of LCS rock was greater than that of HCS; it shows that the cutting load time series of LCS had better space filling ability. Box-counting dimension of cutting load time series approximately linearly decreased with compressive strength as shown in Figure 7(b). Through the above analysis, the box-counting dimension of the load time series could be used to estimate the extent of rock crushing in rock cutting process.
(a) log-log plots
(b) Box-counting dimension
4.3. Effect of Cutting Depth on Fractal Characteristic
According to the calculational method of box-counting dimension as (8), the log-log plots for estimating the box-counting dimension under cutting depths of 10 mm, 15 mm, and 20 mm were shown in Figure 8(a). The correlation coefficients of fitting straight lines using the least square method were 0.996, 0.998, and 0.998, respectively, which indicated that the time series had fractal distribution in space. And the difference of cutting depth had no effect on the fractal feature of rock cutting load. However, the box-counting dimension of cutting load was different with different cutting depths. The smaller cutting depth increased the box-counting dimension, thus it indicated that the cutting load time series had more self-similarity in space when cutting depth was smaller. For the rock fragments, the smaller box-counting dimension was corresponding to larger number and smaller size of rock fragments, which meant that rock was seriously broken. Oppositely, the big box-counting dimension produced smaller number and larger size of rock fragments. Box-counting dimension of cutting load time series approximately linearly decreased with cutting depth as shown in Figure 8(b). Therefore, the box-counting dimension of the time series could be regarded as an evaluation index for the extent of rock cutting or crushing.
(a) log-log plots
(b) Box-counting dimension
4.4. Effect of Cutting Angle on Fractal Characteristic
The for the cutting angles 40°, 45°, and 50° were obtained by first developing an versus on log-log plots and determining the slope of best-fit line through the data points using linear regression as shown in Figure 9(a). The values from different cutting angles of 40°, 45°, and 50°were 1.41, 1.427, and 1.47, respectively; the correlation coefficient in these cases were 0.997, 0.999, and 0.997. The box-counting dimension and correlation coefficient all indicated that the cutting time series of different cutting angles had fractal characteristic with similar rock-cutting mechanism. According to rock fragments in these experiments, the extent of rock crushing increased as the cutting angle. The reason was that the rock failure was mainly in tensile when the cutting angle was small, but it was mainly in shear or compression failure when the cutting angle was big. For the box-counting dimension, it was greater of small cutting angle than that of big cutting angle, and the box-counting dimension of the load time series also could be used to estimate the fragmentation extent of rock in rock cutting. Moreover, box-counting dimension of cutting load time series approximately linearly decreased with cutting angle as shown in Figure 9(b).
(a) log-log plots
(b) Box-counting dimension
4.5. Effect of Assisted Water-Jet Types on Fractal Characteristic
High-pressure water-jet technology has been widely used in rock breaking or cutting mechanism, such as roadheader and shearer drum. However, the fracture mechanism by mechanical cutter assisted with water-jet is still not clear as a result of the opaqueness and instantaneity of rock fragmentation. Therefore, the box-counting dimensions of cutting load time series with three different water-jet assisted types were investigated. And their position relations between mechanical cutter and water-jet were shown in Figure 2(a), including water-jet in front of cutter, water-jet behind cutter, and water-jet in cutter. The log-log plots for calculating box-counting dimension under different water-jet assisted types were shown in Figure 10. The correlation coefficients of fitting lines were greater than 0.95 which shows that the assisted water-jet types had little effect on the fractal characteristic of rock cutting load time series. The difference of box-counting dimension between the water-jet in front of cutter and water-jet behind cutter was very small, and the two values were nearly with the box-counting dimension of cutting load time series without water-jet. However, as was known to us, the fractal dimension of cutting load would obviously change once the rock fracture mechanism changed . Therefore, it showed that rock cutting mechanism would not change with the water-jet in front of or behind the cutter. The decreased cutting load was due to the weakened rock strength caused by the impact and damage of the water-jet in front of the cutter . And the water-jet was mainly used for clearing away rock fragments and lubricating cutter when it was set behind the cutter. On the contrary, the box-counting dimension of cutting load time series was 0.87 when the water-jet was in cutter. Therefore, it indicated that the fracture mechanism of rock cutting by mechanical cutter assisted with water-jet in cutter was changed.
Based on the rock cutting load time series and fragments, the fractal characteristics of the time series and fragments size distribution were investigated by fractal theory and then some main conclusions were obtained as follows.
The sizes of rock fragments are very different from each other, which show that the rock cutting process is a nonlinear and nonperiodic process. There is a great difference among the amplitudes and time intervals of the cutting force peaks, and it indicates that the cutting load time series has discreteness and irregularity. The crushing zone, local breaking, and the avalanche of rock fragments occur alternately.
The difference between the fractal dimension of fragments size distribution and the box-counting dimension of cutting load time series was smaller than 3%, which shows that there are inherent relations between rock fragmentation and the corresponding cutting load time series. The box-counting dimension could be regarded as the fractal dimension of fragments size distribution, and it provides a new idea to investigate the rock fragments size distribution.
The correlation coefficients of fitting straight lines are all greater than 0.95 under different conditions showing that the difference of compressive strength, cutting depth, and cutting angle all cannot change the fracture mechanism of rock cutting. Meanwhile, rock cutting mechanism would not change with the water-jet in front of or behind cutter. However, fracture mechanism of rock cutting by mechanical cutter assisted with water-jet in cutter was changed. The box-counting dimension of cutting load time series can be an evaluation index for the extent of rock cutting or crushing, and it has a negative relationship with rock fragments size.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to acknowledge the Foundation of National 863 Plan of China (2012AA062104), the National Natural Science Foundation of China (51375478), the project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (SZBF2011-6-B35), and the Graduate Education Innovation Project of Jiangsu Province (CXLX12_0948).
C. Labra, J. Rojek, E. Oñate, and F. Zarate, “Advances in discrete element modelling of underground excavations,” Acta Geotechnica, vol. 3, no. 4, pp. 317–322, 2008.View at: Publisher Site | Google Scholar
J. Rojek, E. Oñate, C. Labra, and H. Kargl, “Discrete element simulation of rock cutting,” International Journal of Rock Mechanics and Mining Sciences, vol. 48, no. 6, pp. 996–1010, 2011.View at: Publisher Site | Google Scholar
H. Xie, J.-A. Wang, and W.-H. Xie, “Fractal effects of surface roughness on the mechanical behavior of rock joints,” Chaos, Solitons and Fractals, vol. 8, no. 2, pp. 221–252, 1997.View at: Google Scholar
H. P. Xie, J. F. Liu, Y. Ju, J. Li, and L. Z. Xie, “Fractal property of spatial distribution of acoustic emissions during the failure process of bedded rock salt,” International Journal of Rock Mechanics and Mining Sciences, vol. 48, no. 8, pp. 1344–1351, 2011.View at: Publisher Site | Google Scholar
Z. X. Wu, X. X. Liu, Z. Z. Liang et al., “Experimental study of fractal dimension of AE serials of different rocks under uniaxial compression,” Rock and Soil Mechanics, vol. 33, no. 12, pp. 3561–3569, 2012.View at: Google Scholar
C. Wang, J. K. Xu, X. X. Zhao et al., “Fractal characteristic s and its application in electromagnetic radiation signals during fracturing of coal or rock,” International Journal of Mining Science and Technology, vol. 22, no. 2, pp. 255–258, 2012.View at: Google Scholar
Y. Huang, L. H. Zhu, K. Yin et al., “Grain size distribution law of cuttings in DTH hammer reverse circulation drilling,” Journal of Jilin University, vol. 42, no. 4, pp. 1119–1124, 2012.View at: Google Scholar
B. B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, Calif, USA, 1982.View at: MathSciNet
D. L. Turcotte, “Fractals and fragmentation,” Journal of Geophysical Research, vol. 91, no. 2, pp. 1921–1926, 1986.View at: Google Scholar
S. W. Tyler and S. W. Wheatcraft, “Fractal scaling of soil particle-size distributions: analysis and limitations,” Soil Science Society of America Journal, vol. 56, no. 2, pp. 362–369, 1992.View at: Google Scholar
P. L. Yang, Y. P. Luo, and Y. C. Shi, “Fractal feature of soil on expression by mass distribution of particle size,” China Science Bulletin, vol. 38, pp. 1896–1899, 1993.View at: Google Scholar
T. Vicsek, Fractal Growth Phenomena, World Scientific, Teaneck, NJ, USA, 1989.View at: MathSciNet
P. Soille and J.-F. Rivest, “On the validity of fractal dimension measurements in image analysis,” Journal of Visual Communication and Image Representation, vol. 7, no. 3, pp. 217–229, 1996.View at: Publisher Site | Google Scholar
D. L. Turcotte, Fractals and Chaos in Geology and Geophysics, Cambridge University Press, Cambridge, UK, 2nd edition, 1997.View at: MathSciNet
D. M. Stefanescu, Handbook of Force Transducers-Principles and Components, Springer, Berlin, Germany, 2011.
Y. Nishimatsu, “The mechanics of rock cutting,” International Journal of Rock Mechanics and Mining Sciences and, vol. 9, no. 2, pp. 261–270, 1972.View at: Google Scholar
K. E. Rånman, “A model describing rock cutting with conical picks,” Rock Mechanics and Rock Engineering, vol. 18, no. 2, pp. 131–140, 1985.View at: Publisher Site | Google Scholar
X. Duan, L. Yu, and D. Z. Cheng, “Chaotic dynamical characteristic s of rock breaking mechanism with self-controlled hydro-pick,” Chinese Journal of Rock Mechanics and Engineering, vol. 14, pp. 484–491, 1995.View at: Google Scholar
H. X. Jiang, C. L. Du, S. Y. Liu et al., “Experimental research of influence factors on combined breaking rock with water jet and mechanical tool,” China Mechanical Engineering, vol. 24, no. 8, pp. 1013–1017, 2013.View at: Google Scholar