J. Cent. South Univ. Technol. (2007)06-0877-06 

DOI: 10.1007/s11771-007-0166-1 

Mathematic simulation and parameters determination of slacking process of weak rocks

LIU Xiao-ming(刘晓明), ZHAO Ming-hua(赵明华), SU Yong-hua(苏永华)

(Institute of Geotechnical Engineering, Hunan University, Changsha 410082, China)

Abstract: Slacking experiments were performed on the red beds weak rock from vicinity of Changsha. Grain size distribution of the slacking rock was tested periodically during the experiments, which can be used to describe the physical transition of red beds weak rock during slacking process. According to the similar characters of many attributions such as environmental moisture, mineralogical composition, grain size and other factors between big rocks before slacking and its slacked product, the self-similar property of big rock and the small one can be induced. Fractal concept was introduced to construct the slacking model of red beds weak rock.  Combining the supposed relationship of time for slacking and grain size of weak rock, the mathematic simulation of slacking process of red beds weak rock was conducted. To simplify the parameters back calculation, the fractal model proposed by Tyler and Wheatcraft was introduced to describe the characters of grain size distribution variation. The results show that the fractal dimension calculated from simulation data meet experiments data closely, which proves that the mathematic simulation method is reasonable and the parameters determination method is effective.

Key words: slacking process; fractal; simulation; weak rock

1 Introduction

    Red beds weak rocks distribute in China widely, characterized by the properties of slacking and their low resistance to short term weathering when exposed to natural environments and/or wetting and drying cycles. The slacking behavior of a rock plays a major role in its failure. During the process of slacking, the mass of red beds weak rock quality decreases continuously, with both reduction of strength and stiffness. Many environmental problems such as slope failure, slope deterioration, ground subsidence, are associated with the slacking process. So slacking property is an important factor to considerate when evaluating the engineering behavior of red beds weak rocks-mass and rock-materials in geotechnical practice.

Without a thorough understanding the slacking mechanism, engineering behavior of weak rock can neither be fully appreciated nor accurately predicted^[1]. The importance of slacking mechanism was investigated deeply^[2]. But the slacking mechanism is not available in deep. The slacking process of weak rock is very complex and is affected by many factors such as environment moisture, mineralogical composition, grain size and other factors^[3-4]. Meanwhile research results also show that the slacking process of weak rock is s kind of physical process for chemical composition of red beds weak rock hardly being changed after slacking^[5]. So the pattern of the physical transition process is very important to understand the slacking mechanism of weak rock. Yet, the study on the slacking process of weak rock is devoid.  In this paper, slacking experiments were carried out on red beds weak rock, based on the analysis of weak rock slacking mechanism, mathematic simulation on weak rock slack process was also conducted.

2 Slacking experiment

Big red beds weak rock will break to small pieces gradually when exposed to the natural environment and/or wetting and drying cycles. The slacking process and the slacking pattern of rock is depicted in Fig.1.

To deeply study the variation rules of weak rock in the continuous slacking process, rock samples were obtained in the vicinity of Changsha, China. With the rapid economic growth and the civil infrastructure continuous digging, excavating and blasting are required. As a result, red beds weak rocks are cut at steep slops and used as filling material of highway embankment. Sampling locations were selected around these areas.

Several pieces of red beds weak rocks with a size range of 200-250 mm were picked up randomly and lay out of door, letting them slack in natural environments. With time going on, the rocks slacked gradually, and then sieved periodically according to the standards of test method of soil for highway engineering (JTJ-051-93)^[6] and the grain size distribution of slacking red beds weak rock was recorded. The results are listed in Table 1.

Fig.1 Sketch of slacking pattern of weak rock

Table 1 Variation of grain size distribution of red beds weak rocks during slacking process

Tests results show that the main variation tendency of all grains with different sizes increases firstly and decreases with time going on, which shows that big rock and small rock share the similar variation tendency rules. Another variation tendency revealed by the tests is that the slacking rate in the early days is far more fast than that in the late days. In the first 30 d the big block rock slacked rapidly, and the percentage of particles less than 2 mm increased from 0 to 90%, but in the late 30 d, not only particles less than 2 mm but also particles less than 0.075 mm increased very little, which show that small weak rock is more difficult to slack than big one.

3 Mathematical model

3.1 Fractal model of slacking pattern of weak rock

As seen from the slacking pattern of weak rock in Fig.1, it is difficult to analyze the slacking pattern of weak rock quantificationally because the slacking process of weak rock is a continuous transform process, in this process the shape and grain size of weak rocks change complexly, which looks like stochastic without any rules and is very difficult to describe precisely. But the experimental results in this paper show that big rocks and small ones share the similar variation tendency rules. Meanwhile, some literatures show that slacking process of weak rock is a kind of physical transition process in essence. So the big rock before slacking must have similar mineralogical compositions, granularities and structures to its smaller descendant slacked materials. Then when exposed to similar environments the smaller ones must have similar slacking pattern to the big ones.

If taking the big rock as large magnifications of objective, and the small one as small magnifications of objective, obviously the slacking pattern fitly suits to the fractal concept with self-similar of different magnifications on complex and irregular objective. The concept of fractal dimension was first introduced by MANDELBROT^[7] and XIE^[9]. From then on, fractal geometry has been developed as a new theory to study complex and irregular phenomena. MANDELBROT^[7] thought that patterns or distributions at different scales could be related to each other by a power law function with an exponent termed the fractal dimension D.

According to the fractal concept, combining the similar properties of slacking pattern of weak rock, a fractal model of slacking pattern for the red beds weak rocks can be constructed as follows. When long term exposed to natural environments, a rock fragment with grain size R always slacks to N_i particles with granularity size of 1/k_i of its own size, that is to say, the slacked rock

materials have the grain size of , with the count numbers of particle of {N₁, N₂, …, N_i,…, N_l}, respectively.

Furthermore, according to the self-similar principle, every piece of rock slacks following this pattern. So at the second time, one of the slacked rock fragment stripes from the rock with size R. For example, the fragment with size R/k_i will slack to the grain with size of

 with count numbers of  respectively. This slacking pattern of rock is conceptually shown in Fig.2.

Fig.2 Sketch map of fractal model for slacking pattern of weak rock

If the rock fragment with grain size R can slack for n times, then the slacked product follows the slacking sequence , its granula- rity should be , and its count number should be with the total mass . Besides, because the volume or mass of big rock material before slacking equals to summation of all smaller slacked rock materials, so the grain size ratio k_i and count numbers N_i should meet the following equation:

       (1)

3.2 Relationship between needed time for slacking and grain size of weak rock

No matter the rock is big or small, they would slack inevitably when long term exposed to the natural environment and/or wet and dry cycles. The distinctive difference is that the bigger rock needs shorter time to slack into small pieces than the smaller ones. The durable test results for rock slacking show that the smaller the weak rock sample size, the more the needed time to slack, which should be due to the smaller piece containing less crannies such as crack and joint. At the same time, experiment results show that the smaller the rock size, the higher the unconfined compressive and shear strength of the rock. So the needed time for rock to slack should correspond to its grain size. Therefore, the relationship between needed time for slacking of weak rock and its grain size can be reasoned out: the smaller the particle size, the more the needed time for slacking.

To construct the relationship between needed time and grain size, ENGIN^[9] conducted a great deal of slacking and unconfined compressive strength test on Kentucky shale, and the following linear equation of slacking durable index and unconfined compressive strength is obtained.

p_CS= 658I_D2+9 081, R=0.63       (2)

where  p_CS is the unconfined compressive strength；I_D2 is the second cycle slacking durability index.

MCGOWN et al^[10] conducted a great deal of shear tests on soil samples with different sizes, and used several kinds of regression curves to fit the relationship between shear strength and size of sample, the following equation is the most satisfying one:

            (3)

where  f is the shear strength of soil sample；r is the radius of soil sample; a₁, a₂ and a₃ are regression coefficients.

From Refs.[9-10], the hypothetic relationship can be induced as follows. If a rock with grain size of R has already slacked for n times, and one of its slacked production is a fragment with granularity of the needed time for the fragment  with size to slack continuously is T_n+1. The hypothetic relationship between T_n+1 and  meeting the following equation is reasonable.

         (4)

where  b₁, b₂ and b₃ are regression coefficients. So the total needed time for the grain with size of R to slack for n times and product the slacked material with granularity of should be

    (5)

If the rock with granularity of R could slack for n times, the total time that it exposed to natural environments should be greater than T, and if slacked material with grain size ofwont keep on slacking continuously, no less than t_n+1 is needed additionally.

4 Simulation method

4.1 Mathematic simulation method

Based on the fractal model of slacking pattern and the relationship between the needed time and grain size, simulation on slacking process of weak rock can be conducted with mathematic method. Using t(time) as independent variable, construct a set of data stack named S(t) by using data structure {r，N , t_n } to record rock fragments information, where r represents grain size of a rock fragment, N represents its count number and t_n represents needed time of the original rock with size R to slack to the rock with size r. When t=0, the only rock fragment is the original grain with size R and count number of 1, so {R, 1, 0} is the unique record in the data stack S(0). Starting from t=1 d, conduct the simulation according to the following step:

Step 1 Choose a record by sequence from the data stack S(t-1), its grain size is r, check whether (t-t_n) is greater than T_n+1 or not, and then calculate T_n+1 by Eqn.(4)by =r.

Step 2 If (t-t_n) is greater than T_n+1, the rock with size r will slack, and transform to be smaller fragments with grain size of  and count number of  respectively. Add the n records information of the smaller rock fragments which are  … … into data stack S(t). If (t-t_n) is not greater than T_n+1, the rock with will not slack, and then add the record {r，N, t_n} into data stack S(t).

Step 3 Repeat step 1 and step 2 till all records in data stack S(t-1) are checked.

Step 4 According the records in the data stack S(t), calculate the grain size distribution of the slacked materials at t by the following equation:

 if    (6)

where is the mass of particles with size between R_i and R_i-1 at t; m_T is the total mass.

Step 5 Set t=t+1, repeat step1 to step 5 till t reaches the demand simulation time.

4.2 Determination of parameters of model

There are (2l+3) parameters to be determined before mathematic simulation, they are grain size ratio coefficients  count numbers of slacked product and  of Eqn.(4). Too many parameters make it impossible to determinate them by experiments method, so it is inevitably to determinate them by back method. According to the results of slacking experiments, the back calculation goal function may be:

             (7)

where  f_min is the goal function value;  is the mass of particles with size between R_i and R_i-1 at t. But the calculation results show that using Eqn.(7) as the goal function can hardly converge because the slacking pattern of weak rock may be not determinate, but statistical to some extent and/or for some other reasons. It is difficult to back calculate the parameters using the raw grain size distribution data directly. The raw data obtained from experiments need to be pre-processed before being used in back calculation. In this paper, values of the fractal dimension proposed by TYLER   et al^[11] was used to do this job.

TURCOTTE^[12] suggested a relationship for grain size distribution based on fractal concepts in the form:

               (8)

where  N_(r>R) is the total number of particles with size bigger than R; and D is the fractal dimension. Because counting the number of particles with a particular size is impractical, the number－size distribution is altern- atively obtained from mass－size distribution, as mass is easily measurable. By this approach, the mass of particle grains with size range between an upper and lower bound defined by the sieve diameters was measured. TYLER  et al^[11] derived a mass-based relation that contains the conventional expression describing particle size distribution data, that is, percentage of mass less than or equal to  is

          (9)

where  R_T is the upper size limit of the particle from sieve analysis; m(ris the mass of particles with size less than R; and mT is the total mass. In practice, D is estimated from the above expression by logarithm transforming both sides of the equation and employing a linear regression fit. Fig. 3 shows an example plot of cumulative mass m(r＜R) versus particle with size R on a double logarithmic scale at t=4 d.

Fig.3 Fractal dimension of grain size of rock material after slacking for 11 d

Fig.4 shows the plot of fractal dimension of slacking rock materials versus the slacked time. The curve shows that fractal dimension of slacking rock materials increases fast in the early days, and then the increasing rate decreases and the fractal dimension trends to be stable value.

Fig.4 Variation of fractal dimension of weak rock vs slacking time

Using the fractal model can describe the characters of grain or particle size distribution rightly, which can simplify the parameters back process. Therefore, the back calculation goal function may be expressed as:

 (10)

where  Dt is fractal dimension calculated from experiment results; Dt^* represents fractal dimension calculated from simulation results. In this paper genetic algorithms (GA’s) were chosen to search the minimal value of goal function, because GA’s take advantage of an entire set of solutions spread throughout the solution space, all of which are experimenting upon many potential optima.

5 Simulation results

According to the experiment results and the mathematic simulation method established in this paper, performing back calculation with goal function of Eqn. (10) and GA, the parameters of the models are obtained: n=3; {k1, k2, k3}={1.800, 2.989, 7.000}; {N1, N2, N3}={4, 8, 5};{b1, b2, b3}={2, 50, -0.06}. Using these parameters, mathematic simulation was conducted. Based on this result, the fractal dimensions versus the slacking time are shown in Fig.5.

Fig.5 shows the comparison of fractal dimension calculated based on experiment results and mathematic simulation ones. Fractal dimension calculated from simulation results meet that from experiments results closely, which proves the that mathematic simulation method is reasonable.

Fig.5 Comporison of fractal dimension of weak rock vs slacked time

6 Conclusions

1) According to the similar characters between big rocks before slacking and its slacked product, such as environmental moisture, mineralogical composition, grain size and other factors such as self-similar property of big rocks and the small ones can be induced. Fractal concepts are be introduced to construct the slacking model of red beds weak rock.

2) Using the fractal model proposed can describe the characters of grain or particle-size distribution rightly, which can simplify the parameters back process.

3) The relationship between needed time to slack and size of weak rock is proposed, mathematic simulation of slack process of red beds weak rock is also conducted. Fractal dimension calculated from simulation results meet experiments ones closely, which proves that the mathematic simulation method is reasonable.

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(Edited by YANG Hua)

Received date: 2007-04-02; Accepted date: 2007-06-18

Corresponding author: LIU Xiao-ming, PhD; Tel: +86-731-8821659; E-mail: Liu_705@tom.com