Calibration method for microscopic parameters in particle flow code models for rock-like materials: semi-analytical lookup table approach
-
摘要: 为了研究类岩石材料颗粒流模型的细观参数标定方法,基于CBM模型,首先进行了理论推导,而后基于大量的数值模拟获得了理论关系式中系数的拟合值,最终形成半解析查表法。研究表明:接触黏结的强度比对数值试件的细观破坏模式有重要影响;代表性单元体与实际颗粒流模型具有相同的宏、细观参数变化规律,且各系数的理论值与拟合值基本相符;单轴抗压强度、黏聚力与接触黏结强度的较小值呈较好的线性相关;细观参数的标定是具有明确的先后顺序的。最后通过标定砂岩的细观参数来证实半解析查表法是能够用于快速的完成参数标定的。Abstract: The calibration method for microscopic parameters in particle flow code (PFC) models of rock materials is investigated based on the contact bond model. Firstly, theoretical deductions of the selected representative unit are made, and the fitting coefficients in the theoretical relationship are obatined through extensive numerical simulations. Finally, the semi-analytic lookup table (SLT) method is established. The researches show that the ratio of contact bond strength has a significant impact on the micro-failure mode of numerical specimens. The representative unit has the same macro-microscopic parameter variation law as the actual PFC models, and the theoretical values of each coefficient are generally consistent with the fitted values. The uniaxial compressive strength and cohesion are linearly correlated with the smaller value between the two contact bond strengths. The calibration of microscopic parameters has a clear sequence. Lastly, the calibration of microscopic parameters for sandstone confirms that the SLT method can achieve a rapid parameter calibration.
-
Keywords:
- particle flow code /
- microscopic parameter /
- calibration method /
- rock-like material /
- CBM
-
-
表 1 均质模型的几何参数
Table 1 Geometric parameters of homogeneous model
Rmax/Rmin θ 颗粒配位数N 小颗粒 大颗粒 平均值 (30°, 45°) 4 6 表 2 颗粒流材料的破坏模式
Table 2 Failure modes of PFC materials
序号 判据 τcs/σcn < 判据 τcs/σcn > 判据 破坏模式 宏观强度与细观参数关系式 破坏模式 宏观强度与细观参数关系式 1 拉致剪裂 拉致拉裂 2 拉致剪裂 拉致拉裂 3 压致剪裂 压致拉裂 4 — 压致剪裂 — — 5 剪致剪裂 剪致拉裂 表 3 细观参数取值表
Table 3 Values of microscopic parameters
细观参数 试验常量取值 几何参数 最小粒径Rmin/mm 0.2 粒径比Rmax/Rmin 1.67 细观强度参数 接触法向黏结强度σcn/MPa 20 接触切向黏结强度τcs/MPa 20/50 颗粒摩擦系数μ 1.7 细观刚度参数 刚度比kn/ks 3.5 颗粒接触模量Ec/GPa 10 表 4 拟合系数KB与KD
Table 4 Fitting coefficients KB and KD
系数 Rmin/
mmRmax/Rmin 1 2 3 4 5 6 7 8 KB 0.1 0.609 0.768 0.997 1.090 1.086 1.368 2.015 2.568 0.15 0.589 0.861 1.137 1.271 1.303 1.317 1.616 2.044 0.2 0.602 0.817 0.987 1.270 1.583 1.949 2.559 1.983 0.25 0.737 0.970 0.953 1.213 1.734 2.099 1.926 1.870 0.3 0.677 0.779 1.120 1.386 1.612 1.692 2.198 2.054 0.35 0.633 0.839 1.183 1.348 1.988 1.478 1.877 2.008 0.4 0.541 1.064 1.155 1.247 1.948 1.795 2.155 2.698 系数 Rmin/ Rmax/Rmin mm 1.2 1.67 2 3 4 5 6 7 KD 0.1 1.320 1.414 1.677 2.175 2.216 2.960 3.443 3.894 0.15 1.234 1.444 1.614 2.009 2.099 2.698 3.260 3.751 0.2 1.308 1.425 1.622 1.971 2.450 2.630 3.204 — 0.3 1.479 1.479 1.605 2.067 2.330 — — — 0.4 1.186 1.425 1.527 1.844 — — — — 0.5 1.267 1.475 1.542 — — — — — 表 5 方案B的拟合结果
Table 5 Fitting results of Plan B
Rmin/
mm拟合结果 Rmax/Rmin 拟合结果 Rmax/Rmin 1.67 2 3 4 1.67 2 3 4 0.1 aD 1.3 1.47 1.83 — cD 0.78 0.73 0.62 — bD 0.65 0.85 1.07 — dD 1.54 1.01 0.62 — R2 0.961 0.86 0.85 — R2 0.92 0.93 0.72 — 0.15 aD 1.32 1.39 1.78 1.83 cD — — 0.61 — bD 0.48 0.62 0.97 1.26 dD — — 0.57 — R2 0.89 0.74 0.98 0.99 R2 — — 0.94 — 0.2 aD 1.27 1.41 — 2.16 cD 0.76 0.70 — — bD 0.67 0.76 — 1.19 dD 1.14 0.67 — — R2 0.99 0.92 — 1 R2 0.99 1.00 — — 0.3 aD — 1.41 1.77 2.1 cD — 0.71 0.61 0.55 bD — 0.89 1.03 1.17 dD — 0.77 0.55 0.26 R2 — 0.92 0.9 0.99 R2 — 0.97 0.81 1.00 0.4 aD 1.26 1.36 1.61 — cD 0.76 0.68 — — bD 0.71 0.65 0.99 — dD 1.05 1.06 — — R2 0.99 0.92 0.99 — R2 0.99 0.88 — — 0.5 aD 1.34 1.32 — — cD 0.77 0.70 0.57 — bD 0.66 0.92 — — dD 0.92 0.89 0.42 — R2 0.94 0.96 — — R2 0.87 0.96 0.99 — 表 6 拟合系数的理论值与模拟值
Table 6 Theoretical and simulated ranges of fitting coefficients
试验类型 系数 理论值范围 拟合值范围 直接拉伸试验 aD 2~3.46 1.26~2.16 bD 2.31~4 0.48~1.26 cD 1.15~1.41 0.55~0.78 dD 0.33~1 0.26~1.54 单轴压缩试验 aU 1.41~3.46 1.11~1.93 bU 5.67~9.24 1.01~2.53 cU 1.33~4 1.74~7.17 dU 1~3 0.08~0.77 eU 0.33~1.04 0.25~0.86 fU 0.77~2 2.02~4.5 gU 0.33~1 1.95~3.45 hU 1~3 1.698~2.21 表 7 单轴抗压强度的拟合结果(方案C)
Table 7 Fitting results of uniaxial compressive strength (Plan C)
τcs/σcn 拟合结果 a b R2 0.25~0.428 0.0398 1.7570 0.999 0.5~0.714 0.1392 1.4726 0.999 1 a+b=1.388 0.997 1.4~2 1.591 -0.0740 0.998 2.33~5 1.549 -0.0318 0.999 6~14 1.419 -0.0010 1 表 8 方案D单轴抗压强度拟合结果
Table 8 Fitting results of uniaxial compressive strength (Plan D)
τcs/σcn Rmax/Rmin Rmin/mm 式(13) cU dU R2 0.5 1.67 0.2 2.949 0.331 0.95 2 0.15 3.924 0.165 0.88 2 0.3 3.377 0.251 0.98 3 0.1 4.333 0.331 0.99 4 0.15 4.696 0.589 0.91 5 0.15 7.170 0.267 0.87 1 1.2 0.1 2.435 0.081 0.99 1.2 0.4 1.736 0.342 0.99 1.67 0.3 2.358 0.270 0.98 1.67 0.5 2.735 0.098 0.99 2 0.1 3.907 0.283 0.99 2 0.3 2.302 0.395 0.86 2 0.4 2.689 0.244 0.98 3 0.1 3.464 0.200 0.98 3 0.4 2.941 0.358 0.98 4 0.1 4.628 0.350 0.93 4 0.2 5.166 0.147 0.98 5 0.1 5.526 0.140 1.00 5 0.15 4.638 0.617 0.91 6 0.15 6.239 0.330 1.00 6 0.2 4.615 0.767 0.99 7 0.15 7.654 0.485 0.99 τcs/σcn Rmax/Rmin Rmin/mm 式(12) aU bU R2 2 1.67 0.2 1.932 1.007 0.99 2 0.15 1.723 1.077 0.71 2 0.3 1.461 2.190 0.72 3 0.1 1.210 2.186 0.83 3 0.15 1.482 1.047 0.99 3 0.2 1.116 2.301 1.00 3 0.3 1.133 2.529 0.99 表 9 拟合参数KE及其拟合优度R2
Table 9 Fitting coefficient KE and its goodness of fit R2
颗粒几何参数 拟合参数及R2 Rmax/Rmin Rmin/mm KE R2 1.67 0.1 1.5596 0.9945 0.2 1.5052 0.9938 2 0.15 1.6553 0.9947 0.3 1.632 0.9939 3 0.1 2.272 0.99 0.3 2.0795 0.9921 4 0.15 2.5885 0.9938 0.2 2.4767 0.9889 表 10 方案D中弹性模量的拟合系数及其拟合优度R2
Table 10 Fitting coefficient of elastic modulus and its goodness of fit R2(Plan D)
颗粒几何参数 拟合参数及R2 Rmax/Rmin Rmin/mm eU fU gU R2 1.2 0.15 0.339 2.201 2.616 0.999 0.2 0.294 2.565 3.086 0.998 0.3 0.295 2.579 3.174 0.997 0.4 0.295 2.629 3.446 0.999 0.5 0.314 2.026 2.507 0.981 1 0.254 2.634 3.360 0.996 1.67 0.1 0.436 2.271 2.414 0.998 0.15 0.352 2.815 3.063 0.996 0.2 0.385 2.411 2.573 0.992 0.3 0.329 2.931 3.214 0.992 2 0.1 0.460 2.567 2.374 0.998 0.15 0.423 2.617 2.508 0.997 0.2 0.422 2.453 2.274 0.998 0.4 0.365 3.090 3.207 0.991 3 0.1 0.541 3.387 2.518 0.996 0.15 0.581 2.770 2.184 0.973 0.2 0.466 3.776 2.827 0.996 0.3 0.557 2.865 2.176 0.981 4 0.1 0.672 3.854 2.186 0.996 0.15 0.641 4.311 2.758 0.993 0.2 0.701 3.387 1.955 0.993 0.3 0.662 3.610 2.270 0.988 5 0.1 0.864 4.030 2.428 0.953 0.15 0.788 4.547 2.394 0.989 表 11 方案D中泊松比的拟合系数hU及其R2(方案D)
Table 11 Fitting coefficient hU and its goodness of fit R2(Plan D)
Rmin/mm 强度比 Rmax/Rmin 1.2 1.67 2 3 4 5 6 0.1 0.5 — — — 1.960 — — — 1 2.127 2.210 2.017 1.916 1.906 1.980 2.050 2 — 2.161 2.125 1.987 2.015 2.125 — 0.15 0.5 — — 2.039 — 2.054 2.027 — 1 1.982 1.914 1.960 1.94 1.962 1.981 1.933 2 — 2.026 1.987 1.983 2.096 2.052 — 0.2 0.5 — 1.987 — — — — — 1 1.933 1.896 1.871 2.019 1.766 1.804 2.104 2 — 1.948 1.983 2.093 1.964 1.910 — 0.3 0.5 — — 2.055 — — — — 1 1.901 1.851 1.916 1.858 1.883 — — 2 — 2.001 1.938 1.953 1.935 — — 0.4 0.5 — — — — — — — 1 1.822 1.941 1.904 1.698 — — — 2 — 2.016 2.072 — — — — 0.5 0.5 — — — — — — — 1 1.788 1.851 1.874 — — — — 2 — 1.909 1.991 — — — — 表 12 系数a,b
Table 12 Fitting coefficients a and b
切法向强度比 拟合公式:c =σcn×(a+b×τcs/σcn) a b R2 0.25~0.428 0.013 0.551 0.997 0.5~0.714 0.005 0.516 0.991 1 a+b=0.414 0.998 1.4~2 0.474 -0.035 0.999 2.33~5 0.412 0.004 0.991 6~14 0.426 -0.002 0.996 表 13 系数bS,cS,dS的拟合结果(方案H)
Table 13 Fitting results of coefficients bS, cS and dS (Plan H)
颗粒几何参数 拟合系数及R2 Rmin /mm Rmax/Rmin bS cS dS R2 0.15 1.67 0.6482 0.8027 2.1002 0.639 0.2 2 0.6644 0.1074 0.4247 0.723 0.3 3 0.8122 0.9226 2.3140 0.968 表 14 砂岩的力学特性及细观参数的选取
Table 14 Mechanical properties of sandstone and selection of microscopic parameters
细观参数取值 巴西试验 单轴压缩试验 三轴试验强度/MPa Rmin/
mmRmax/
Rminσcn/
MPaks/kn Ec/
GPaτcs/
MPaµ σt/MPa fc/MPa E/GPa 围压1 MPa 围压6 MPa 值 误差 值 误差 值 误差 值 误差 值 误差 值 误差 砂岩 — — — — — — — 5.99 — 64 — 9.195 — 0.226 — 71.89 — 95.45 — 1号 0.2 2 34.896 2.172 14.165 69.792 1.7 4.77 -20.4 50.69 -20.8 9.005 -2.1 0.239 +5.8 — — 2号 0.2 2 51.235 2.172 14.165 51.235 1.7 6.96 +16.2 60.35 -5.7 9.005 -2.1 0.239 +5.8 — — 3号 0. 2 2 45 2.172 14.165 90 1.7 6.58 +9.8 67.31 +5.2 8.972 -2.4 0.234 +3.5 72.55 +0.9 91.18 -4.5 -
[1] HUANG H. Discrete Element Modeling of Tool-Rock Interaction[D]. Minnesota: University of Minnesota, 1999.
[2] YANG B D, JIAO Y, LEI S T. A study on the effects of microparameters on macroproperties for specimens created by bonded particles[J]. Engineering Computations, 2006, 23(6): 607-631. doi: 10.1108/02644400610680333
[3] 丛宇, 王在泉, 郑颖人, 等. 基于颗粒流原理的岩石类材料细观参数的试验研究[J]. 岩土工程学报, 2015(6): 1031-1040. doi: 10.11779/CJGE201506009 CONG Yu, WANG Zaiquan, ZHENG Yingren, et al. Experimental study on microscopic parameters of brittle materials based on particle flow theory[J]. Chinese Journal of Geotechnical Engineering, 2015(6): 1031-1040. (in Chinese) doi: 10.11779/CJGE201506009
[4] 刘新荣, 傅晏, 郑颖人, 等. 颗粒流细观强度参数与岩石断裂韧度之间的关系[J]. 岩石力学与工程学报, 2011, 30(10): 2084-2089. LIU Xinrong, FU Yan, ZHENG Yingren, et al. Relation between meso-parameters of particle flow code and fracture toughness of rock[J]. Chinese Journal of Rock Mechanics and Engineering, 2011, 30(10): 2084-2089. (in Chinese)
[5] 陈亚东, 于艳, 佘跃心. PFC^(3D)模型中砂土细观参数的确定方法[J]. 岩土工程学报, 2013, 35(增刊2): 88-93. https://cge.nhri.cn/article/id/15363 CHEN Yadong, YU Yan, SHE Yuexin. Method for determining mesoscopic parameters of sand in three-dimensional particle flow code numerical modeling[J]. Chinese Journal of Geotechnical Engineering, 2013, 35(S2): 88-93. (in Chinese) https://cge.nhri.cn/article/id/15363
[6] YOON J. Application of experimental design and optimization to PFC model calibration in uniaxial compression simulation[J]. International Journal of Rock Mechanics and Mining Sciences, 2007, 44(6): 871-889. doi: 10.1016/j.ijrmms.2007.01.004
[7] WANG M, LU Z X, WAN W, et al. A calibration framework for the microparameters of the DEM model using the improved PSO algorithm[J]. Advanced Powder Technology, 2021, 32(2): 358-369. doi: 10.1016/j.apt.2020.12.015
[8] WANG M, LU Z X, ZHAO Y L, et al. Calibrating microparameters of DEM models by using CEM, DE, EFO, MFO, SSO algorithms and the optimal hyperparameters[J]. Computational Particle Mechanics, 2024, 11(2): 839-852. doi: 10.1007/s40571-023-00656-0
[9] 石崇, 徐卫亚. 颗粒流数值模拟技巧与实践[M]. 北京: 中国建筑工业出版社, 2015. SHI Chong, XU Weiya. Numerical Simulation Skills and Practice of Particle Flow[M]. Beijing: China Architecture & Building Press, 2015. (in Chinese)
[10] MIAO L L, LIU X R, FU Y. Approach for determining micro-strength parameters of rock based on particle flow code[J]. Scientific Reports, 2024, 14(1): 11230. doi: 10.1038/s41598-024-61410-x
[11] 傅晏. 干湿循环水岩相互作用下岩石劣化机理研究[D]. 重庆: 重庆大学, 2010. FU Yan. Study on the Mechanism of Rock Deterioration under the Interaction of Dry-Wet Circulating Water and Rock[D]. Chongqing: Chongqing University, 2010. (in Chinese)
-
其他相关附件