Study on correlation of macro and microstructural parameters of hollowcylindrical grey sandstone based on PFC3D
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摘要:
为从细观层面揭示主应力量值变化并为应力主轴旋转组合条件下围岩的破坏机理研究提供一种有效的数值模拟方法，基于空心圆柱灰砂岩单轴压缩和常规三轴压缩试验特性，分别采用单因素分析法、响应曲面分析法和回归分析法，研究了离散元软件PFC3D中平行黏结模型宏细观参数的敏感性特征。结果表明：依靠单因素分析法确定出的细观参数可靠性极低，基于响应曲面分析法和回归分析法获取的细观参数在低围压下与实验室结果表现出良好的拟合性；空心圆柱灰砂岩数值模型的抗压强度$ {\mathrm{\sigma }}_{\mathrm{c}} $与黏聚力$ \bar{c} $呈正相关；弹性模量$ E $分别与有效黏结模量$ {\bar{E}}_{{\mathrm{c}}} $、平行黏结刚度比的二次方（$ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $）2呈正相关，与$ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $呈负相关；泊松比v与平行黏结刚度比$ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $呈正相关；剪切裂纹比$ {\alpha } $与（$ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $）2呈线性相关。最后，采用迭代法对响应曲面分析法和回归分析法获取的细观参数进行了优化，优化后的数值模拟结果与实验室结果在不同围压下的试样峰值强度、变形参数和破坏形态方面吻合度较高。迭代法为PFC3D中平行黏结模型的细观参数标定提供了一种有效的方法，通过这一优化方法可以简化标定步骤，提升标定结果的适用性，提高模型的准确性和可靠性。
Abstract:In order to provide an effective numerical simulation method to reveal the damage mechanism study of surrounding rock under the combination of the change of principal stress value and the rotation of the stress principal axis from the fine-scale level, based on the uniaxial compression and conventional triaxial compression test characteristics of hollow cylindrical gray sandstone, the sensitivity of the macro fine-scale parameters of the parallel bonding model in the discrete element software PFC3D was studied by using the one-factor analysis method, the response surface analysis method, and the regression analysis method, respectively. characteristics of the parallel bonding model in the discrete element software PFC3D. The results show that: the reliability of the macroscopic parameters determined by the one-factor analysis method is extremely low, and the macroscopic parameters obtained based on the response surface analysis and regression analysis method show a good fit with the laboratory results at low perimeter pressure; the numerical model compressive strength $ {\sigma }_{\mathrm{c}} $ is positively correlated with cohesion $ \bar{c} $, and the elastic modulus E is positively correlated with the quadratic of the effective modulus of adhesion $ {\bar{E}}_{{\mathrm{c}}} $ and stiffness ratio $ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $, and positively correlated with stiffness ratio $ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $ positively correlated with the stiffness ratio $ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $ and negatively correlated with the quadratic of the stiffness ratio $ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $, Poisson's ratio v is positively correlated with the stiffness ratio $ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $, and the shear cracking ratio α is linearly correlated with the quadratic of the stiffness ratio $ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $. Finally, the fine-scale parameters obtained by response surface analysis and regression analysis were optimized using the iterative method, and the optimized numerical simulation results were in good agreement with the laboratory results in terms of the peak strength, deformation parameters and damage morphology of the specimens under different circumferential pressures. The iterative method provides an effective method for the calibration of the fine apparent parameters of the parallel bonding model in PFC3D, and this optimization can simplify the calibration steps, as well as enhance the applicability of the calibration results, and improve the accuracy and reliability of the model.
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图 3 宏观力学参数与细观参数对应关系
Figure 3. Correspondence between macroscopic mechanical properties and fine parameters
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图 4 基于单因素试验确定的细观参数数值模拟应力应变曲线与实验室结果对比
Figure 4. Comparison of numerically simulated stress-strain curves based on single-factor tests with laboratory results for fine-scale parameters
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图 5 细观参数对宏观力学特性的响应曲面
Figure 5. Response surface of fine parameters to macroscopic mechanical properties
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图 6 常规三轴压缩试验应力−应变曲线
Figure 6. Conventional triaxial compression test stress-strain curve
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图 7 平行黏结摩擦角与宏观黏聚力和内摩擦角对应关系
Figure 7. Correspondence of parallel cohesive friction angle with macroscopic cohesion and internal friction angle
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图 8 常规三轴压缩试验应力−应变曲线
Figure 8. Conventional triaxial compression test stress-strain curve
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图 9 常规三轴压缩试验应力−应变曲线
Figure 9. Conventional triaxial compression test stress-strain curve
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表 1 单因素试验分析法计算方案
Table 1 Calculation scheme of single-factor experimental analysis method
方案 $ {\bar{E}}_{\mathrm{c}} $/GPa $ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $ $ \bar{c} $/MPa k $ \bar{\varphi } $ 围压/MPa 1 8 2.9 10 0.7 12.5 0 2 10 2.9 10 0.7 12.5 0 3 12 2.9 10 0.7 12.5 0 4 14 2.9 10 0.7 12.5 0 5 10 1.0 20 0.7 12.5 0 6 10 2.5 20 0.7 12.5 0 7 10 2.0 20 0.7 12.5 0 8 10 3.0 20 0.7 12.5 0 9 10 2.9 20 0.7 12.5 0 10 10 2.9 30 0.7 12.5 0 11 10 2.9 40 0.7 12.5 0 12 10 2.9 50 0.7 12.5 0 13 10 2.9 20 0.7 10.0 0，5，10 14 10 2.9 20 0.7 20.0 0，5，10 15 10 2.9 20 0.7 30.0 0，5，10 16 10 2.9 20 0.7 40.0 0，5，10 
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围压/MPa 抗压强度/MPa 弹性模量/GPa 泊松比 0 65.820 8.982 0.476 5 101.477 12.231 0.330 10 130.400 14.165 0.280 20 162.219 14.679 0.199
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表 3 响应曲面试验设计
Table 3 Response surface experimental design
水平 −1 0 1 $ {\bar{E}}_{\mathrm{c}} $/GPa 9.00 10.50 12.00 $ \bar{{k}_{\mathrm{n}}}/\bar{{k}_{\mathrm{s}}} $ 2.00 2.50 4.00 $ \bar{c} $/MPa 35.00 40.00 45.00 $ \bar{{\varphi }} $/（°） 15.00 12.00 25.00
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表 4 宏观参数及其主要影响因素
Table 4 Macro parameters and main influencing factors
宏观参数 主要影响因素 抗压强度$ {\mathrm{\sigma }}_{\mathrm{c}} $ $ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $、$ \bar{c} $、$ {\bar{E}}_{c} $2、$ ({\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}}) $2、$ \bar{c} $2、$ \bar{\mathrm{\varphi }} $2 弹性模量E $ {\bar{E}}_{c} $、$ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $、$ {\bar{E}}_{c} $×$ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $、$( {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} )$2 泊松比$v $ $ \bar{{k}_{n}}/\bar{{k}_{s}} $ 剪切裂纹比$ \mathrm{\alpha } $ $ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $、$ \bar{c} $、$ \bar{\mathrm{\varphi }} $、$ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $×$ \bar{\mathrm{\varphi }} $、$ {\bar{E}}_{c} $2、$( {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}}) $2、$ \bar{c} $2
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表 5 灰砂岩宏细观参数之间回归关系式及拟合系数
Table 5 Table of regression equations and fitting coefficients between fine view parameters and macroscopic parameters of sandstone
编号 拟合公式 拟合系数 1 $ {\bar{E}}_{\mathrm{c}}=-3.443-0.013{\mathrm{\sigma }}_{\mathrm{c}}+0.876E+16.471v $ 0.987 2 $ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}}=0.788-0.008{\mathrm{\sigma }}_{\mathrm{c}}-0.004E+10.485v $ 0.994 3 $ \bar{c}=-6.267+0.408{\mathrm{\sigma }}_{\mathrm{c}}+28.719v $ 0.782 4 $ \bar{\varphi }=- $65.076$ + $3.312$ \varphi $ 0.999
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表 6 试验1组细观参数更名及取值
Table 6 Name change and value of fine parameters of test group 1
细观参数 细观参数更名 细观参数取值 $ {\bar{E}}_{\mathrm{c}} $/GPa emodLast 11.579 $ {\bar{k}}_{\mathrm{n}}/{\bar{k}}_{\mathrm{s}} $ kratio 3.359 $ \bar{c} $/MPa cohLast 44.826 $ \bar{{\varphi }} $ faLast 23.460
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表 7 实验室试验及试验1组试验结果
Table 7 Results of laboratory tests and test group 1 tests
围压$ {P}_{i} $/MPa 抗压强度/MPa 轴向应变 室内试验结果$ {\sigma }_{i} $ 试验1组试验结果$ {\sigma }_{i}^{\prime} $ 室内试验结果$ {\varepsilon }_{i} $ 试验1组试验结果$ {\varepsilon }_{{i}}^{\prime}$ 5($ {P}_{1} $) 101.477($ {\sigma }_{1} $) 101.000($ {\sigma }_{1}^{\prime}$) 0.0078 ($ {\varepsilon }_{1} $)0.0082 ($ {\varepsilon }_{1}^{\prime} $)10($ {P}_{2} $) 130.400($ {\sigma }_{2} $) 119.140($ {\sigma }_{2}^{\prime} $) 0.0089 ($ {\varepsilon }_{2} $)0.0085 ($ {\varepsilon }_{2}^{\prime} $)20($ {P}_{3} $) 162.219($ {\sigma }_{3} $) 150.918($ {\sigma }_{3}^{\prime} $) 0.0103 ($ {\varepsilon }_{3} $)0.0097 ($ {\varepsilon }_{3}^{\prime} $)
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