Study on the correlation of macro and meso parameters of parallel bond model sandstone
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摘要: 为研究砂岩细观层面的力学响应及破坏特征，本文首先开展室内岩石压缩试验和基于平行黏结模型的PFC2D单轴宏细观参数关联性数值试验；然后采用试错法确定各细观参数对宏观力学指标的影响程度排序，并进行显著因子交互作用分析；最终利用回归分析建立宏细观参数间函数关系；运用控制变量法研究岩样裂纹演化及破坏形式的主要影响因素。研究结果表明：①单轴抗压强度σc与平行黏聚力c、黏结法向强度σc、黏结内摩擦角φ呈非线性关系；②弹性模量E与黏结模量E*、黏结刚度比k*呈多项式关系；③泊松比v与k*、摩擦系数μ呈线性关系；④抗拉强度σt与c、σc呈较好的线性关系；⑤岩样裂纹演化与破坏形式主要受黏结强度比σc/c影响，其与拉伸裂纹数目负相关，与剪切裂纹数目正相关；随σc/c增大，岩样的破坏形式呈拉伸破坏—共轭破坏—剪切破坏的变化特征。室内试验和数值模拟的单轴压缩σ-ε演化规律及破坏形式基本一致。Abstract: In order to study the mechanical response and failure characteristics of sandstone at the meso level, this paper first carried out indoor rock compression test and PFC2D uniaxial macro-meso parameter correlation numerical test based on parallel bond model.Then, the trial-and-error method is used to determine the order of the influence degree of each mesoscopic parameter on the macroscopic mechanical index, and the interaction analysis of significant factors is carried out.Finally, the regression analysis is used to establish the functional relationship between macroscopic and mesoscopic parameters.The main influencing factors of crack evolution and failure mode of rock samples are studied by using the control variable method.The results show that: ①The uniaxial compressive strength σc has a complex nonlinear relationship with parallel cohesion c, bond normal strength σc and bond internal friction angle φ.; ② Elastic modulus E has a polynomial relationship with bond modulus E * and bond stiffness ratio k *; ③ Poisson's ratio v is linear with k * and friction coefficient μ; ④The tensile strength σt has a good linear relationship with c and σc.; ⑤The crack evolution and failure mode of rock samples are mainly affected by the bond strength ratio σc/c, which is negatively correlated with the number of tensile cracks and positively correlated with the number of shear cracks. With σc/c increases, the failure mode of rock samples is characterized by tensile failure-conjugate failure-shear failure.The evolution law and failure form of uniaxial compression σ-ε in laboratory test and numerical simulation are basically the same.
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图 2 PFC2D模型平行黏结的失效包络线
Figure 2. Failure envelope of parallel bonding in the PFC2D model
图 5 单轴抗压强度UCS与各显著因子间的关系
Figure 5. Relationship between UCS of uniaxial compressive strength and significant factors
图 6 弹性模量E与各显著因子间的关系
Figure 6. Relationship between elastic modulus E and significant factors
图 7 泊松比v与各显著因子间的关系
Figure 7. Relationship between Poisson's ratio v and significant factors
图 9 不同σc/c比下数值岩样的破坏形式
Figure 9. Failure modes of numerical rock samples at different σc/c ratios
图 11 数值单轴压力下不同刚度比k*的岩石破坏形式
Figure 11. Failure modes of rocks with different stiffness ratios k* under numerical uniaxial compression
图 13 数值单轴压力下不同内摩擦角φ的岩石破坏形式
Figure 13. Rock failure mode with different internal friction angle φ under numerical uniaxial pressure
图 14 数值单轴压力下不同μ与E*的裂纹数目
Figure 14. Numerical number of cracks in different μ and E* under uniaxial pressure
图 15 数值单轴压力下不同摩擦系数μ的岩石破坏形式
Figure 15. Rock failure modes with different friction coefficients μ under numerical uniaxial pressure
图 16 数值单轴压力下不同黏结模量E*的岩石破坏形式
Figure 16. Failure modes of rocks with different bond model E* under numerical uniaxial compression
图 18 室内试验和数值试验的单轴压缩σ-ε曲线对比
Figure 18. Uniaxial compression for laboratory and numerical tests σ-ε curve comparison
图 19 数值试验与室内试验岩样的破坏形态
Figure 19. Failure modes of rock samples from numerical and laboratory test
表 1 PFC2D平行黏结模型主要宏细观参数表
Table 1. Main macro and micro parameters of PFC2D parallel bonding model
宏观参数 细观参数 平行黏结细观参数 颗粒细观参数 单轴抗压强度 σc/MPa 平行黏结法向强度 σc/MPa 颗粒摩擦系数 μ 抗拉强度 σt/MPa 平行黏结切向强度 τc/MPa 颗粒最小半径 Rmin/mm 弹性模量 E/GPa 平行黏结模量 E*/GPa 颗粒密度 ρ/(kg·m-3) 泊松比 v 平行黏结内摩擦角 φ/(°) 颗粒刚度比 k*=kn/ks 内摩擦角 φ/(°) 平行黏结力 c/MPa 颗粒半径比 Rmax/Rmin 平行黏结半径乘子 λ 颗粒接触模量 EC/GPa 平行黏结刚度比 k*=kn/ks 
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表 2 PFC2D平行黏结模型细观参数水平
Table 2. Level table of meso parameters of PFC2D parallel bonding model
细观参数 因子水平 1 2 3 4 5 σc/MPa 10 30 50 70 90 E*/GPa 10 20 30 40 50 φ/(°) 15 25 35 45 55 k*=kn/ks 1 2 3 4 5 c/MPa 15 30 45 60 75 μ 0.15 0.25 0.35 0.45 0.55
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表 3 基于PFC2D平行黏结模型宏细观参数关系数值模拟结果
Table 3. Numerical simulation results of fine macro parameter relationship based on PFC2D parallel bonding model
试验编号 细观参数影响因素及因子水平取值 宏观参数模拟结果 σc/MPa E*/GPa φ/(°) k*=kn/ks c/MPa μ σc/MPa E/GPa v σt(σt/σc)/MPa 1 10 10 15 5 15 0.15 20.12 13.03 0.38 2.11 (0.104) 2 30 20 35 3 75 0.15 61.4 24.5 0.32 7.11 (0.115) 3 70 50 55 4 75 0.45 110.12 58.30 0.31 19.65 (0.178) 4 50 10 35 4 30 0.45 65.0 45.0 0.29 6.45 (0.100) $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ $ \vdots $ 46 50 30 15 4 30 0.35 70.70 34.80 0.33 6.29 (0.088) 47 10 20 35 1 30 0.55 20.00 31.50 0.06 3.85 (0.192) 48 90 10 35 4 60 0.45 125.87 12.18 0.31 25.21 (0.200)
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表 4 PFC2D不同黏结强度比下数值单轴压缩岩石破坏形式
Table 4. Numerical uniaxial compression rock failure modes under different bond strength ratios of PFC2D
黏结强度比σc/c 0.25 0.5 0.75 1.0 1.33 1.66 2.0 2.3 2.7 3.0 3.5 4.0 破坏形式 张拉 张拉 张拉 张拉 张拉 共轭 共轭 共轭 共轭 共轭 剪切 剪切
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表 5 平行黏结模型细观参数
Table 5. Microscopic parameter of parallel bond model
试验编号 σc/MPa E*/GPa φ/(°) k* c/MPa μ A1 35.5 9.47 15 3.2 76.85 0.55 B1 25.8 15.1 35 2.1 57.33 0.55 B2 18.8 4.40 35 2.5 30.55 0.50 C1 14.0 3.90 15 3.0 26.30 0.55 C2 36.1 5.20 25 3.0 68.30 0.52 C3 27.5 7.60 20 2.0 49.50 0.30
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表 6 岩石的宏观力学参数试验值与模拟值对比
Table 6. Comparison between experimental and simulated values of macro mechanical parameters of rock
试验编号 数值模拟 室内试验 误差率/% σc/MPa E/GPa v σt(σt/σc) σc/MPa E/GPa v σt/MPa σc E v σt A1 98.98 17.43 0.19 7.11(0.071) 96.11 16.58 0.18 6.52 2.89 4.87 5.15 9.04 B1 68.53 11.55 0.18 6.12(0.089) 70.02 12.02 0.17 5.83 2.12 3.91 6.06 4.73 B2 33.43 8.84 0.19 4.95(0.148) 35.85 8.51 0.21 4.51 6.95 3.87 9.52 8.88 C1 23.41 4.61 0.20 2.27(0.096) 25.15 4.98 0.21 2.41 6.77 7.42 4.76 6.16 C2 74.72 9.91 0.24 6.83(0.091) 72.61 9.81 0.23 6.31 2.82 1.01 4.16 8.24 C3 60.23 15.9 0.18 4.89(0.081) 61.70 15.33 0.19 4.46 2.38 3.71 3.15 9.64
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