J. Cent. South Univ. (2021) 28: 911-925

DOI: https://doi.org/10.1007/s11771-021-4653-6

Quantitative determination of PFC3D microscopic parameters

LI Zhuo(李卓), RAO Qiu-hua(饶秋华)

School of Civil Engineering, Central South University, Changsha 410075, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2021

Abstract: It is important to calibrate micro-parameters for applying partied flow code (PFC) to study mechanical characteristics and failure mechanism of rock materials. Uniform design method is firstly adopted to determine the microscopic parameters of parallel-bonded particle model for three-dimensional discrete element particle flow code (PFC3D). Variation ranges of microscopic of the microscopic parameters are created by analyzing the effects of microscopic parameters on macroscopic parameters (elastic modulus E, Poisson ratio v, uniaxial compressive strength σ_c, and ratio of crack initial stress to uniaxial compressive strength σ_ci/σ_c) in order to obtain the actual uniform design talbe. The calculation equations of the microscopic and macroscopic parameters of rock materials can be established by the actual uniform design table and the regression analysis and thus the PFC3D microscopic parameters can be quantitatively determined. The PFC3D simulated results of the intact and pre-cracked rock specimens under uniaxial and triaxial compressions (including the macroscopic mechanical parameters, stress-strain curves and failure process) are in good agreement with experimental results, which can prove the validity of the calculation equations of microscopic and macroscopic parameters.

Key words: quantitative relationship of microscopic and macroscopic parameters; uniform design method; three-dimensional particle flow code (PFC3D); rock

Cite this article as: LI Zhuo, RAO Qiu-hua. Quantitative determination of PFC3D microscopic parameters [J]. Journal of Central South University, 2021, 28(3): 911-925. DOI: https://doi.org/10.1007/s11771-021-4653-6.

1 Introduction

In rock mass engineering such as tunnel construction, petroleum and mining exploitation, internal cracks, joints or faults existing in natural rock would propagate under external load and cause potential hazard. Due to inhomogeneity and discontinuity of rock mass, their mechanical behaviors and failure mechanisms are hard to be studied by continuum mechanics theoretically. Furthermore, the crack propagation process is also difficult to be measured in tests, especially in triaxial tests. Therefore, numerical simulation has become an effective method. Since the discrete element method (DEM) based on discontinuous mechanics theory is advantageous over traditional finite element method (FEM) from the viewpoint of microscopic analysis, it has been widely applied to simulate the mechanical characteristics and failure mechanisms of the rock mass. The commonly used discrete element software is particle flow code (PFC) [1-7].

In PFC, the material is modeled as bonded assembly of rigid circular or ball. There are three basic bonded models: contact bond (CB) model, parallel bond (PB) model and flat joint (FJ) model. In the CB model, the particle point-contact bond (without cement) can resist only the normal and shear forces but not the bend moments (free rotation) at the contact point. When the bond breaks, its contact stiffness is still active and the macroscopic stiffness may not be affected significantly, which is not suitable for rocks.

In the PB model, a finite length bond (cement) is used to transmit both force and moment. Since the macroscopic stiffness is decided by both contact stiffness and bond stiffness, the bond breakage will result in the decrease of macroscopic stiffness, indicating that the PB model is more realistic for rock-like materials than the CB model [8]. The FJ model has a similar finite length bond (cement) to the PB model. Differently, when the bond breaks, the cement can still restrict rotation [9, 10]. Although the FJ model has better simulation results of tensile strength and friction angle than the PB model, it is still less applied for rock materials since more microscopic parameters need to be determined compared with the PB model.

The first task of applying PFC is to calibrate its many micro-parameters (≥5) with high precise. The common method is trial-and-error method [11-14], but it is empirical and time-consuming. Therefore, many efforts have been made to seek reasonable mathematical methods. For example, artificial neural network method was utilized to predict the PFC microscopic properties of rock materials, but the calculation equations between microscopic and macroscopic parameters had not been obtained [15, 16]. Mathematical statistical methods, such as the orthogonal design method [17, 18], spherical symmetric design method [19], Plackett-Burman (PB) design combined with central composite design method [20], were used to establish the quantitative equations for microscopic and macroscopic parameters. However, for multi-factor and multi-level experiment design, the orthogonal design and spherical symmetric designs require a lot of trials. The Plackett-Burman design method has only two levels for each factor and cannot meet the accuracy requirement. In addition, the uniform design (UD), proposed by FANG et al [21] based on pseudo Monte Carlo method in number theory, is also a test design method, which only considers the uniform distribution of test points (test numbers) in the test range of parameters (factors), and whose test number depends only on the factor levels. It has much fewer test number and many much more factor levels with higher efficiency and precise of calculation than other design methods (e.g., orthogonal design, PB design).

In this study, the uniform design method was adopted to quantitatively determine the microscopic parameters of PB model in PFC3D for rock materials. Firstly, a suitable uniform design table was determined by studying the effect of microscopic parameter on macroscopic parameters based on PFC3D. And then, the quantitative equations of microscopic and macroscopic parameters were established by multiple regression analysis of uniform design table. Finally, the new quantitative equations were verified by uniaxial and triaxial compression tests of intact rock and pre-cracked rock specimens.

2 Uniform design process

2.1 Undetermined microscopic parameters

Figure 1 shows a standard cylinder specimen (Φ50 mm×100 mm) of rock under uniaxial compression for determining PFC3D microscopic parameters, where the particle is in yellow. In this paper, the displacement control manner (usually used in laboratory tests) is adopted for conducting the numerical test [22, 23], and the loading rate is 0.05 m/s. In the PB model, there exists two groups of microscopic parameters (11 unknown parameters): grain and cement microscopic parameters (Table 1).

To reduce the unknown parameters, let R_rat=1.66 (R_max=1.66R_min), λ(—)=1, E_c=E(—)_c and k_n/k_s=

k(—)_n/k(—)_s [24]. In addition, the macroscopic mechanical parameters of rock materials tend to constant values when L/d≥25 [25], where L is the smallest length of specimen (L=50 mm) and d is the average particle diameter (d=0.5(R_max+R_min)). Thus, d≤2 mm and R_min≤1.5 mm (R_max=1.66R_min). Let R_min=0.6, 0.8, 1.0, 1.2 and 1.5 mm, respectively. Select other six microscopic parameters (E_c=E(—)_c=20 GPa, k_n/k_s=k(—)_n/k(—)_s=3, σ_n=σ_s=60 MPa and R_σ=0.2, μ=0.5) arbitrarily and calculate the macroscopic mechanical parameters (elastic modulus E, Poisson ratio v, and uniaxial compression strength σ_c) and failure patterns of the rock specimens for different R_min by PFC3D, as listed in Table 2 and Figure 2. It is seen that the macroscopic mechanical parameters are all within the range of most rock materials (Table 3) [26]. When R_min≤1 mm, both these parameters and their failure patterns (concentrated microcrack area) have little change. Considering the calculation precise and efficiency, R_min=0.8mm. The remaining six microscopic parameters (E_c, k_n/k_s, σ_n, σ_s, R_σ and μ) need to be determined in PFC3D.

Figure 1 PFC3D model of rock specimen under uniaxial compression

Table 1 Basic microscopic parameters of PB model for PFC3D

Table 2 PFC3D simulated results of macroscopic mechanical parameters for different R_min

2.2 Standard uniform design table

Uniform design method is an effective experimental design method by distributing the experimental points uniformly and scatteredly in the experimental domain, in order to ensure that the experimental points have statistical properties of uniform distribution, i.e., each level of each factor is tested only once. It is applied by means of the uniform design table in terms of U_n(q^s) or U_n^*(q^s), where s is factor number (i.e., column number of the table) depending on basic microscopic parameters of PFC3D; n is experimental number (usually 3 to 5 times s); and q is level number of each factor (i.e., row number of the table). For each factor (s), there needs only one experiment for one level (i.e., q=n). The superscript _* indicates that U_n^*(q^s) has better uniformity than U_n(q^s), where its uniformity is evaluated by discrepancy (D) [27]. Obviously, the smaller the D is, the better the uniformity is.

Figure 2 PFC3D simulated failure patterns for different R_min:(red: tensile microcrack; blue: shear microcrack; yellow: particle)

Table 3 Test results of main mechanical parameters for most rock materials

Since only six microscopic parameters (E_c, k_n/k_s, σ_n, σ_s, R_σ and μ) need to be determined in Table 1, i.e., s=6 (the test number n>18), the standard uniform design table U₂₀^*(20⁷) is adopted as shown in Table 4, where each value U_ij (i is row number and j is column number) indicates the ordinal number of the j-th factor in the i-th test. For each microscopic parameter, its actual value in U₂₀^*(20⁷) is: U′_ij=S_min+(U_ij-1)(S_max-S_min)/20, where S_max and S_min are the maximum and minimum value of the range for j-th factor, respectively. The equation indicates that the microscopic parameters are divided into the same difference value of two adjacent levels for 20 trials and arranged according to the level ordinal number U_ij in Table 4.

Table 4 Uniform design table U₂₀^*(20⁷)

According to the applied table of U₂₀^*(20⁷) (Table 5), when s=6, selection of columns 1, 2, 4, 5, 6, 7 has better uniformity with the minimum discrepancy (D_min) than the other six columns. Therefore, the column 3 in U₂₀^*(20⁷) (Table 4) must be deleted for test arrangement.

Table 5 Applied table of U₂₀^*(20⁷)

2.3 Microscopic parameter ranges and actual uniform design table

To obtain the actual uniform design table of microscopic parameters (E_c, k_n/k_s, σ_n, σ_s, R_σ and μ), the microscopic parameter range needs to be firstly determined, based on analyzing the effect of microscopic parameters (normalization) on macroscopic mechanical properties, as shown in Figures 3-6. For example, when E_c is changed within a certain range (E_c=10-80 MPa) and the other five microscopic parameters are unchanged (k_n/k_s=3, σ_n=σ_s=60 MPa, R_σ=0.2, and μ=0.5), the stress-strain curves and macroscopic mechanical properties (E, v, σ_c and σ_ci/σ_c) varying with E_c are calculated by PFC3D, where the crack initiation stress (σ_ci) defined as the axial stress at which there is a specified ratio (R) of the crack number to the total crack number (corresponding to σ_c) [24]. R is governed by a subroutine variable “pk_ci_fac” in PFC3D and R=1% in this study. It needs to be explained here that, for simplicity, let σ_n=σ_s first (since σ_s is usually larger than or equal to σ_n [20]) and then determine the ratio of σ_s and σ_n in the following Figure 7. Normalization of microscopic parameters in Figures 3-6 means the variation values of microscopic parameters divided by their initial values (E₀=10 MPa, k_no/k_so=1, σ_so=σ_no=40 MPa, R_σ0=0.1). For example, when E_c is changed within a certain range (E_c=10-80 MPa), the normalization of E_c is equal to 1-8, i.e., E_c=10-80 MPa divided by the initial value E₀=10 MPa.

Figure 3 Effect of microscopic parameters on E

Figure 4 Effect of microscopic parameters on v

Figure 5 Effect of microscopic parameters on σ_c

Figure 6 Effect of microscopic parameters on σ_ci/σ_c

Figure 7 Relationship between σ_c and σ_s/σ_n

It is seen from Figure 3 that E_c has much greater effect on E than k_n/k_s, but both σ_n and R_σ have almost no effect on E. Therefore, the range of E_c could be determined by the range of E (E=10-70 GPa), i.e., E_c=8-75 GPa. Similarly, Poisson ratio v is affected significantly by k_n/k_s (Figure 4) and thus the range of k_n/k_s could be obtained by the range of v (v=0.15-0.3), i.e., k_n/k_s=1.1-4.0.

Figure 5 indicates that the uniaxial compressive strength σ_c is mainly related to σ_n (σ_s=σ_n). Thus the range of σ_n could be obtained by the range of σ_c(50-200 MPa), i.e., σ_n=40-175 MPa. It is also found from Figure 7 that σ_c varies with σ_s/σ_n only when 1≤σ_s/σ_n≤1.5. For obtaining the value ranges of both σ_n and σ_s influencing on σ_c, let σ_s/σ_n=1.0-1.5 and thus σ_s=40-262.5 MPa.

Figure 6 shows that the ratio of the crack initiation stress to uniaxial compressive strength (σ_ci/σ_c) is greatly affected by R_σ rather than the other microscopic parameters. The range of R_σ can be achieved by the range of σ_ci/σ_c (σ_ci/σ_c=0.2-0.6) [28, 29], i.e., R_σ=0.1-0.29.

Particle friction coefficient (μ) has only influence on the post-peak response since once a contact bond between two contacted particles is broken, the slip model is activated and slips of the two particles is governed only by μ [24]. Since the post-peak response is difficultly measured and described quantitatively for calibration of μ, the non-zero value of μ is usually regarded as a reasonable value. Let μ=0.1-0.9 in this study.

To sum up, the ranges of six microscopic parameters (E_c, k_n/k_s, σ_n, σ_s, R_σ and μ) are determined completely: E_c=8-75 GPa, k_n/k_s=3,σ_n=40-175 MPa, σ_s=40-262.5 MPa (σ_s/σ_n=1.0-1.5), R_σ=0.1-0.29, μ=0.1-0.9 (μ>0), in which only five microscopic parameters are independent.

For obtaining the actual uniform design table of microscopic parameters (E_c, k_n/k_s, σ_n, σ_s, R_σ and μ), the range of each microscopic parameter (S_min≤S≤S_max) is firstly divided into 20 equal parts, (S_max-S_min)/20. Then each U′_ij is calculated, U′_ij=S_min+(U_ij-1)(S_max-S_min)/20. Finally, U′_ij is placed into U₂₀^*(20⁷) with deletion of the column 3 (Table 4) by the ordinal number, as listed in Table 6.

Table 6 Actual uniform design table of microscopic parameters

3 Calculation formula of microscopic and macroscopic parameters

In the actual uniform design table of microscopic parameters (Table 6), each row represents one test. By substituting the six microscopic parameters of each row into the PFC3D model of rock specimen (Figure 1), corresponding strain-strain curves and macroscopic mechanical parameters (E, v, σ_c, σ_ci) are obtained, as listed in Table 7. The PFC3D simulated value ranges are E=10.38-79.67 GPa, v=0.154-0.348, σ_c=46.85-238.68 MPa and σ_ci/σ_c=0.20-0.61, which are in good agreement with test results in Table 3 (without σ_ci/σ_c).

Table 7 PFC3D calculation results of macroscopic parameters

According to Table 7, each macroscopic parameter (E, v, σ_c, σ_ci) is related to six microscopic parameters (E_c, k_n/k_s, σ_n, σ_s, R_σ, μ) and their relation equations can be established by multiple regression analysis method, Eq. (1), where the interactive term of E_c and k_n/k_s is considered for improving calculation precise since both E_c and k_n/k_s affect E in different degree (Figure 3). The relation equations have very high fitting accuracy (R²>98.7%).

 (1)

Actually, there are only five independent microscopic parameters (σ_s/σ_n=1-1.5). Considering that μ can be determined as any non-zero constant (μ=0.1-0.9 in this study), the remaining four unknown microscopic parameters (E_c, k_n/k_s, R_σ, σ_n) can be completely determined by the four known macroscopic parameters (E, v, σ_c, σ_ci), based on Eq. (1).

4 Experimental verification

Uniaxial and tri-axial compression tests of intact and pre-cracked rock specimens were adopted to verify the validity of the above equations between the microscopic and macroscopic parameters as follows.

4.1 Uniaxial compression tests of intact rock specimens

4.1.1 Measurement of macroscopic mechanical parameters

Uniaxial compression tests of seven types of rock materials with standard cylinder specimens (Φ50 mm×100 mm) were adopted for PFC3D simulation: red sandstone (tested on MTS815 equipment by our group), Lac du Bonnet granite [24], BS granite [1], Carrara marble [30], Wonju granite [31], Hwangdeung granite [32] and rock-like material (cement: sand: water=1: 0.8: 0.35) [2, 33]. Table 8 lists their main macroscopic mechanical parameters.

Table 8 Main macroscopic mechanical parameters of different rocks

4.1.2 Calculation of microscopic parameters

According to these macroscopic mechanical parameters of rock (E, v, σ_c and σ_ci/σ_c), the microscopic parameters of PFC3D (E_c, k_n/k_s, σ_n, σ_s=kσ_n, R_σ, where k is the ratio of parallel bond shear to normal strength) can be calculated by Eq. (1) for two cases: 1) k=1, i.e., σ_s=σ_n, both the microscopic tensile and shear failures are possible; 2) k=1.2, i.e., σ_s=1.2σ_n, the microscopic tensile failure is possible), where μ=0.5 (μ=0.1-0.9), as listed in Table 9. For simplication, consider this two cases (k=1 and 1.2) only for two rock materials (Red sandstone and Lac du Bonnet granite).

4.1.3 Comparison of simulation and test results

Figure 8 shows stress-strain curves of red sandstone, BS granite and rock-like material obtained by numerical simulation and experiment, respectively. Stress-strain curves of different k of red sandstone are selected for comparing with test results. They have similar characteristics: linear (elastic) stage, non-linear (elastic-plastic) stage with positive slope and non-linear (failure) stage with negative slope. Compared with the tested stress-strain curve, no compaction stage (i.e., slope is increasing) is found since the rock material is assumed to be homogeneous and continuous in PFC3D calculation model. In the true test, pre-existing micro-cracks in natural rock material would be closing in the first stage of compression, i.e., formation of compaction stage. Noticeably, the straight line of simulated and test curves are almost parallel and their peak stresses are very close to each other, and thus the simulated values of E and σ_c are in good agreement with the test values. Meanwhile, the simulated stress-strain curves of red sandstone for different k has very small and simulated curves are in good agreement with the test results.

Table 9 Calculation results of microscopic parameters for four kinds of rocks

Figure 8 PFC3D simulated σ-ε curves compared with test results:

Table 10 lists simulated and experimental results of macroscopic mechanical parameters as well as error for seven rock materials under uniaxial compression. Most of errors are less than 5%, i.e., the simulated results have good precision. It is proved that the calculation equation (Eq. (1)) deduced by the uniform design method is effective and feasible for determining the microscopic parameters of PFC3D.

4.2 Uniaxial compression tests of pre-cracked rock specimens

4.2.1 PFC3D model

Uniaxial compression tests of standard cylinder specimens (Φ50 mm×100 mm) were adopted for PFC3D simulation of two types of brittle materials: red sandstone with single crack of 2a=30 mm and α=45° (tested on MTS815 equipment by our group, Figure 9), rock-like material with double cracks of 2a=12 mm, 2b=16 mm and α=30° [2] (Figure 10). Their macroscopic and microscopic parameters are listed in Tables 8 and 9 above.

4.2.2 Stress-strain curve

Figure 11 shows stress-strain curves of the single-cracked red sandstone specimen with different k (k=σ_s/σ_n) and double-cracked rock-like specimen obtained by numerical simulation and experiment, respectively. It is seen that the simulated stress-strain curves consist of three stages: elastic stage (oa), damage stage (ab for crack stable propagation) and failure stage (bd for crack unstable propagation, bcd for crack coalescence and crack unstable propagation) without compaction stage of test curve (red sandstone). The elastic moduli (slope of the linear stage) of simulated and test curves are almost the same, and the simulated values of peak strength for red sandstone (σ_c=24.90 MPa for k=1 and σ_c=23.28 MPa for k=1.2) and rock-like material (σ_c=37.00 MPa for k=1) are also very close to the test values (σ_c=24.18 MPa for red sandstone and σ_c=33.60 MPa for rock-like material) with small error of 2.9%-3.7% (k=1-1.2, red sandstone) and 10% (rock-like material). k has only effect on peak strength and no effect on elastic modulus.

Table 10 Simulated and experimental results of macroscopic mechanical parameters for different rocks under uniaxial compression

Figure 9 Single-cracked red sandstone specimen:

Figure 10 Double-cracked rock-like material specimen:

Figure 11 PFC3D simulated σ-ε curve compared with its test result:

4.2.3 Failure process

Figures 12 and 13 show the FPC3D simulated failure processes of single-cracked red sandstone specimen (k=1, 1.2) at different stages. They have the same characteristics: crack initiation (point a), stable propagation (ab), peak stress (point b), and final failure (point d). For different k, micro-cracks are both initiated at the crack tips (Figures 12(a) and 13(a)) and propagate at the angle of about 80° deviating from the original crack plane (Figures 12(b) and 13(b)), i.e., formation of wing crack. At peak stress (Figures 12(c) and 13(c)), two secondary cracks appear: one is in the coplanar or quasi-coplanar direction of the original crack plane (secondary crack 1), and the other one is in the opposite direction of the wing crack (secondary crack 2). Finally, the secondary crack 1 propagates to the failure of the specimens (Figures 12(d) and 13(d)).

Figure 12 PFC3D simulated failure process (k=1) of single-cracked red sandstone specimen:

Figure 13 PFC3D simulated failure process (k=1.2) of single-cracked red sandstone specimen:

The simulated fracture trajectories agree well with theoretic analysis results [34] and test results (Figures 12(e) and 13(e)), which can verify the validity of calculation equation (1). Since the observed fracture trajectory is only on the specimen surface while the simulated one is presentation of the whole specimen, there is a little difference in the secondary crack 2, which is not obvious in the test results. Comparison of Figures 12 and 13 indicates that the failure process of single-cracked red sandstone specimen is less influenced by k (k=σ_s/σ_n).

Figure 14 shows the FPC3D simulated failure processes of double-cracked rock-like material specimen at different stages (horizontal crack 1, inclined crack 2): crack initiation (point a), peak stress (point b), crack coalescence (point c) and final failure (point d). It is seen that micro-cracks are initiated at the four crack tips simultaneously (Figure 14(a)) and then propagated stably to peak load at two tips of the crack 1 (almost in direction of perpendicular to the crack 1) and at the lower tip of the crack 2 (almost in direction of perpendicular to crack 2), as shown in Figure 14(b). The upper tip of the crack 2 is slowly propagated due to interaction of two cracks. After peak load (Figure 14(c)), it is gradually propagated upwards at about 100° deviating from the original crack plane and tends to link up with the left tip of the crack 1, due to stress redistribution. New micro-crack is also initiated at middle point of the crack 1 parallel to the loading direction and other cracks continue to extend. Finally, the specimen fails when the two cracks coalesce (Figure 14(d)). Obviously, the simulated fracture trajectories coincide well with test results, which can prove again the effectiveness of calculation equations.

4.3 Triaxial compression tests of intact rock specimens

The intact cylinder specimen (Φ50 mm×100 mm) of rock-like material used in triaxial compression test (confining pressure σ₃=5 MPa) is the same as that in uniaxial compression test. Figure 15(a) shows the stress-strain curves of simulation and experiment [2]. Both of them have the same stages: linear elastic stage (with the equal slope of straight line, i.e., elastic modulus E), damage stage and failure stage (with about 10% error of compressive strengths σ_c). In addition, the simulated and tested values of σ_c are increased with σ₃ (Figure 15(b)) and very close to each other, (except for σ₃=5 MPa), due to heterogeneity of the rock-like material. Comparatively, the intact specimen in the triaxial compression test has higher values of E and σ_c than that in uni-axial compression test.

4.4 Triaxial compression test of pre-cracked rock specimen

4.4.1 Stress-strain curve

The double-cracked cylinder specimen (Φ50 mm×100 mm) of rock-like material used in triaxial compression test (confining pressure σ₃=5 MPa) is the same as that under uniaxial compression test (Figure 10(b)). Figure 16 shows the PFC3D simulated and tested stress-strain curves [2]. They are also divided into the three stags: linear elastic stage (oa), damage stage (ab, crack stable propagation) and failure stage (bcd, crack coalescence and crack unstable propagation), where the elastic modulus E (E=19.24 GPa) and compressive strength σ_c (σ_c=42.01 MPa) are almost the same. Comparatively, the double-cracked specimen in the triaxial compression test has higher values of E and σ_c than that under uni-axial compression test (E=13.41 GPa and σ_c=35.98 MPa).

Figure 14 PFC3D simulated failure process of double-cracked rock-like material specimen:

Figure 15 PFC3D simulated results as well as test results [2]:

Figure 16 PFC3D simulated and test σ-ε curves of double-cracked rock-like material specimen under triaxial compression (confining pressure σ₃=5 MPa)

4.2.2 Failure process

Figure 17 shows PFC3D simulated failure process of double-cracked rock-like material specimen (σ₃=5 MPa) at different stages (horizontal crack 1, inclined crack 2): crack initiation (point a), peak stress (point b), crack coalescence (point c) and final failure (point d). Similarly to the fracture process in uniaxial compression test, the micro-cracks initiate at the four crack tips simultaneously (Figure 17(a)) and then propagate stably to peak load at two tips of the crack 1 (almost in direction of perpendicular to crack 1) and at the lower tip of the crack 2 (almost in direction of perpendicular to crack 2), as shown in Figure 17(b). The upper tip of the crack 2 propagates slowly. After the peak load, it gradually propagates upwards at about 100° deviating from the original crack plane and tends to connect with the left tip of the crack 1. Crack coalescence occurs at the left and right tips of the cracks 1 and 2 firstly, respectively (Figure 17(c)), and then crack coalescence occurs at the left tips of the crack 1 and the right tips of the crack 2 (Figure 17(d)). Finally, the specimen fails when the cracks coalesce. Compared to rock-like material specimen under uniaxil compression, there exsits few wing crack due to confining compression. Obviously, the simulated fracture trajectories coincide well with test results, which can prove again the effectiveness of Eq. (1).

In conclusion, the simulated stress-strain curves and fracture processes of the intact and pre-cracked specimens under uniaxial and triaxial compression are in good agreement with the tested results, which verifies the validity of Eq. (1) for determining the microscopic parameters of PFC3D (PB model) based on the uniform design method. The new method can be also applied for other bonded models of PFC, e.g., CB, FJ model.

5 Conclusions

1) Uniform design method is firstly adopted to establish the quantitative equations of microscopic and macroscopic parameters for PB model of PFC3D. It has the advantages of fewer trials, more levels and higher precision over the orthogonal design and Plackett-Burman design, and can be also applied for other CB and FJ models of PFC.

Figure 17 PFC3D simulated failure process of double-cracked rock-like material specimen in triaxial specimen at:

2) Effects of the microscopic parameter (E_c, k_n/k_s, σ_n(σ_s) and R_σ) on the macroscopic parameters (E, v, σ_c and σ_ci/σ_c) indicate that E is greatly influenced by E_c and slightly by k_n/k_s; v is mainly affected by k_n/k_s; σ_c is mostly related to σ_n(σ_s), and σ_ci/σ_c is primarily dependent of R_σ.

3) The relationship expression between the microscopic and macroscopic parameters is established by the actual uniform design table and the regression analysis, where the interactive term of E_c and k_n/k_s is taken into account for improving fitting precise. It has very simple form and high accuracy for calculating the PFC3D microscopic parameters conveniently and efficiently.

4) The PFC3D simulated stress-strain curves of the intact and pre-cracked rock specimens under uniaxial and triaxial compression have almostly the same deformation characteristics of linear, non-linear and failure. The simulated failure process of double-cracked specimen shows that the micro-cracks initiate and propagate at four crack tips simultaneously in the direction perpendicular to the original crack plane. The three tips of two cracks (the upper tip of inclined crack and two tips of horizontal crack) coalesce to failure under uniaxial compression, while four tips of two cracks (the left tip of horizontal crack and two tips of inclined crack, the right tip of horizontal crack and the upper tip of inclined crack) coalesce to failure under triaxial compression. The PFC3D simulated results agree well with the experimental results, which can verify the validity of the new established relationship expression of microscopic and macroscopic parameters by the uniform design method.

Contributors

RAO Qiu-hua provided the concept of manuscript and simulation calculation, and edited the draft of manuscript. LI Zhuo conducted the literature review, wrote the first draft of the manuscript and conducted the laboratory tests. All authors replied to reviewers’comments and revised the final version.

Conflict of interest

LI Zhuo and RAO Qiu-hua declare that they have no conflict of interest.

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(Edited by ZHENG Yu-tong)

中文导读

定量确定PFC3D细观参数

摘要：细观参数的标定对于应用PFC研究岩石材料的力学特性和破坏机理具有重要意义。本文首次采用均匀设计的方法来确定三维离散元颗粒流程序中平行粘结模型的细观参数。通过分析细观参数对宏观参数(弹性模量E、泊松比v、单轴抗压强度σ_c、起裂应力与单轴抗压强度比σ_ci/σ_c)的影响规律，得到细观参数的变化范围，从而建立了细观参数均匀设计表。通过均匀设计表和回归分析，建立了岩石材料细观参数与宏观参数的计算公式，从而可以定量确定PFC3D细观参数。完整岩石试件和预制裂纹岩石试件在单轴和三轴压缩下的PFC3D模拟结果(包括宏观力学参数、应力应变曲线和破坏过程)与实验结果吻合较好，验证了该细观参数与宏观参数计算公式的有效性。

关键词：宏-细观参数定量关系；均匀设计方法；三维颗粒流(PFC3D)；岩石

Foundation item: Projects(51474251, 51874351) supported by the National Natural Science Foundation, China

Received date: 2020-09-21; Accepted date: 2021-01-25

Corresponding author: RAO Qiu-hua, PhD, Professor; Tel: +86-13787265488; E-mail: raoqh@csu.edu.cn; ORCID: https://orcid.org/ 0000-0003-1171-647X