Inuence of Particle Distribution on Macroscopic Properties of Particle Flow Model

The particle flow discrete element models for uniaxial compression and tensile tests of rocks are established to study the influence of the particle distribution randomness on the macroscopic mechanical properties of such model. The results of macroscopic mechanical properties show strong discreteness due to the variance of particle distributions, in which compressive strength, tensile strength and Poisson's ratio follow the normal distribution and the Young's modulus follows the negative skewness distribution. The average values of the macroscopic strength obtained based on multiple calculations with different particle distributions should be used for the calibration of microscopic parameters. According to the relationship between sample size and deviation of macro strength averages, the minimum calculation number required to obtain high-precision macro strength with different confidence levels is given.

The basic principle of the particle flow discrete element method is to use the collection of 2 particles to simulate the rock and soil, and to duplicate the macroscopic mechanical behavior 3 of such materials through the interaction between particles. It is one of the most popular particle distribution may have a significant influence on the macroscopic mechanical properties 12 of the model because the change of particle locations will affect the crack initiation and 13 propagation in the rock model. Therefore, a one-time simulation under a given particle 14 distribution may not be able to obtain the accurate relationship between microscopic 15 parameters and macroscopic strength in calibrations. This paper studies the influence of the 16 randomness of particle distribution on the macroscopic mechanical parameters of the discrete 17 element model, to improve the calibration accuracy of the particle flow discrete element model.  compressive strength is linear. The stiffness ratio has the greatest influence on Poisson's ratio, 28 but only has minor influence on other macro parameters. The particle friction coefficient 29 merely affects the post-peak response of the particle flow model. Wang  that the internal friction angle increased with the increase of particle stiffness and friction 37 coefficient, while the friction coefficient had little influence on the system stiffness.  However, there are still some unsolved problems. The heterogeneity and randomness of natural 41 8 changing particle distribution yet, to obtained the benchmark for the subsequent study. The 85 macroscopic mechanical properties of the model after calibration are shown in Table 1 and the  86 corresponding microscopic parameters are shown in Table 2.

97
To illustrate the influence of particle distributions on the macroscopic mechanical 98 parameters of the model, ten simulations of uniaxial compression and tensile tests with different MPa, accounting for 30% of the experimental measured value. The tensile strength is also 103 strongly affected by the randomness of particle distributions; the maximum difference of 104 simulated tensile strength in Fig. 2(b) is more than 4.5 MPa, accounting for about 70% of the 105 experimental measured value. The above-mentioned results illustrate the necessity of a 106 systematically investigation on the statistical rule of the particle distribution effect on the 107 macroscopic mechanical properties of the particle flow discrete element model.

Normality test 122
The normality test is commonly used to determine whether a sample follows the normal 123 distribution. The popular normality test methods including skewness and kurtosis coefficient 124 The parameters required for the normality tests are shown below. 137

1) Skewness and kurtosis coefficient test 138
Skewness is a parameter that reflects the degree and direction of data distribution skewed, 139 which is defined as (Keskin, 2006; (Trust, 2016): distribution curve at the mean, which is defined as (Keskin, 2006 ); and the significance level P (P is the significance 187 level corresponding to the statistics when the variable are tested by LL and S-W) of both the 188 LL test and the S-W test is 0.015 and 0, respectively, which is less than the corresponding 189 confidence standard α=0.05. It can be seen from Fig. 3 (a) and Fig. 4 (a)

Determination of the minimum sample size for high-precision averages 243
It has been demonstrated that the randomness of particle distribution has a significant 244 effect on the results of macroscopic mechanical properties of the discrete element model. So in 245 model calibrations, the averages obtained from multiple calculations with different particle 246 distributions should be used, instead of the results from one calculation without considering 247 particle distributions. The approach to determine the number of computations (sample size) is 248 of significant. Apparently, the larger the sample size, the smaller the deviations of averages 249 (Naing, 2003); but in practice, due to the limitation of computing resource, it is impossible to 250 perform infinite calculations to obtain the true averages. Therefore, the correlation between the 251 sample size and the precision of the corresponding average values needs to be studied, then a 252 reasonable calculation number that balances the computing accuracy and efficiency can be 253

determine. 254 255
According to the fundamental theory of random sampling, the minimum sample size 256 required to obtain an average for a given precision can be determined according to the total 257 sample variance, allowable error (accuracy) and confidence (Naing, 2003): The standard deviations of the population samples of each item can be found in Table 3. 262 The selection confidence is 95%. The minimum sample size of each macroscopic properties 263 under different accuracy are calculated According to Eq. 12, and the results are shown in Fig.  264 7 (a). The results indicate that the minimum sample size becomes larger when the preset error 265 decreases. The fitted curves are also shown in Fig. 7 (a); the minimum sample size for all 266 macroscopic parameters are inversely proportional to the square of error. By comparing the 267 minimum sample size of each item with the same precision, it can be found that the minimum 268 sample size of tensile strength is always the largest. The confidence level also affects the 269 minimum sample size. As Fig. 7(b) shows, at the same precision, the minimum sample size 270 increases with the confidence level.

Verification of the "three in five tests for the mean" method 278
The "three in five tests for the mean" method (which is shortened as "three in five" method 279 in the following) is popular in the investigations involving stochastic problems. In this method, 280 each test will be repeated five times, and the middle three values are adopted to calculate the 281 average. To verify the applicability of the "three in five" method in calculating the averages 282 Poisson's ratio show a certain degree of dispersion, but the relative errors of the averages 289 obtained by the "three in five" method are largely smaller ±5%. The maximum error in Fig. 8  290 is about 7.5%. As shown by Fig. 7 (b), when sample size is five, the relative error of the average 291 is about 8.7% which is larger than that of the average obtained by the "three in five" method 292 where the effects of maximum and minimum are eliminated. In summary, the "three in five" 293 method works well for the problem of calculating the averages that used for calibrations. In this paper, the statistical rule of the influence of particle distributions on the 301 macroscopic mechanical property discreteness of the particle flow model is studied. The major 302 conclusions are as follows: 303 304 a. The particle distributions have a significant effect on the macroscopic mechanical 305 properties of particle flow model, and the simulation results of compressive strength, tensile 306 strength and Poisson's ratio with different particle distributions follows normal distributions, 307 while that of Young's modulus follow negative skewness distributions. Therefore, the averages 308 obtained from multiple calculations with different particle distributions should be used for 309 model calibrations, instead of the results from one calculation without considering particle 310 distributions. c. The "three in five" method (calculating five times with different particle distributions 317 and calculate the average value after removing the maximum and minimum) is adopted to 318 obtain the averages of macroscopic mechanical properties that used for model calibrations. The 319 deviations of the averages provided by this method is largely within the range of ±5% while 320 the maximum of deviations is not over ±10%, suggesting that the "three in five" method is    Histogram of macroscopic property result after Young's modulus data conversion Determination of the minimum sample size :(a) the minimum sample size of each item varies with the accuracy level when the con dence is 95%; (b) the minimum sample size of tensile strength with given accuracy level under different con dence levels