Modified Hoek-Brown criterion model for laminated rock based on fracture mechanics
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摘要: 本文基于各向同性Hoek-Brown强度准则的断裂力学理论，从层状岩石细观断裂机理出发，分析初始微裂纹沿层理分叉后起裂条件，考虑初始裂纹引发岩石破裂的临界角，建立层理弱面主导裂纹偏转的参数m1各向异性公式，引入考虑岩石沿层理和基质混合破裂修正系数，得到Hoek-Brown强度准则的各向异性修正模型。对比其他各向异性Hoek-Brown模型以及页岩三轴试验结果证明了模型的有效性。修正模型继承了各向同性Hoek-Brown强度准则的断裂力学理论对于岩石破坏特征量的选择，反映了岩石的细观破坏机理，同时考虑层理各向异性的影响，准则中的相关参数物理意义明确。模型中参数m与层理面角度、层理及岩石的抗拉强度、抗压强度、摩擦系数有关，其中层理面摩擦系数的变化会引起参数m极小值点位置及岩石强度特性发生改变。
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关键词:
- 层状岩石 /
- Hoek-Brown准则 /
- 断裂力学 /
- 起裂准则 /
- 各向异性修正
Abstract: Based on the fracture mechanics theory of the isotropic Hoek-Brown strength criterion, this paper established the initiation conditions of the bifurcated microcrack along bedding which begins with the meso-fracture mechanism of laminated rocks. Then the author substituted the critical angle which indicates that the initial crack most likely causes the fracture of the rock into above established conditions and derived the anisotropy expression of the parameter m1 for the weak plane of bedding leading to the crack deflection. Through introducing the correction coefficient for the mixed fracture of the rock along the bedding and the matrix, this paper obtained the anisotropic modified model of Hoek-Brown strength criterion. Compared with the established anisotropic Hoek-Brown model and shale triaxial test results, the validity of the model was proved. The modified model inherits the fracture mechanics analysis of the isotropic Hoek-Brown strength criterion for the selection of rock failure characteristic quantity, and reflects the microscopic failure mechanism of the rock, also considers the effect of bedding anisotropy. The parameters in the modified criterion have a clear physical meaning. The parameter m in the model is related to the angle of the bedding plane, the tensile strength, compressive strength and friction coefficient of bedding and matrix. The change in the friction coefficient of the bedding plane will cause the position of the minimum value of m and the rock strength characteristics changed.-
Key words:
- laminated rock /
- Hoek-Brown criterion /
- fracture mechanics /
- fracture criterion /
- anisotropic correction
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图 1 等围压下单裂纹断裂力学模型[12]
Figure 1. Fracture mechanical model of single crack under constant confining pressure
图 2 等围压下层状岩石含单裂纹断裂力学模型
Figure 2. Fracture mechanics model of bedding rock with single crack under constant confining pressure
图 3 分叉裂纹尖端几何与受力示意图
x′y′—初始裂纹尖端坐标系；r′θ′—初始裂纹尖端极坐标系；x y —分叉裂纹尖端坐标系；r′ θ—分叉裂纹尖端极坐标系；σr′r′—初始裂纹尖端径向应力；σθ′θ′—初始裂纹尖端切向应力；τr′θ′—初始裂纹尖端剪应力
Figure 3. Geometry and stress of the branch crack system
图 6 龙马溪组、牛蹄塘组不同层理角度页岩修正的Hoek-Brown准则曲线及三轴实验结果对比
Figure 6. Comparison of modified Hoek-Brown criterion curve with triaxial experimental results for Longmaxi and Niutitang shales with different bedding angles
图 7 板岩三轴实验结果、修正的Hoek-Brown模型及Pietruszezak模型曲线对比
Figure 7. Comparison of modified Hoek-Brown criterion curve with triaxial experimental results and Pietruszezak model curve for slate
表 1 修正的Hoek-Brown准则相关参数拟合结果
Table 1. The determination of parameters for the modified Hoek-Brown criterion
岩样 m0 m2 θ/(°) 页岩1 5.95 11.14 60.72 页岩2 2.865 6.612 55.79 页岩3 8.179 15.42 61.88 页岩4 1.741 4.913 52.15 板岩 18.39 25.24 61.42 
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表 2 修正的Hoek-Brown准则三轴压缩实验拟合结果
Table 2. Fitting results of modified Hoek-Brown criterion for triaxial compression test
参数 岩样 β=0° β=15° β=30° β=45° β=60° β=75° β=90° 修正系数
α页岩1 1 — 1 — 0 — 0.337 6 页岩2 0.454 5 1 0.830 6 0.762 2 0 0.555 6 0.043 0 页岩3 0.938 7 1 0.904 4 0.609 6 0.000 7 1 0.813 4 页岩4 1 — 0.130 0 0 0.157 5 — 0.919 6 板岩 0.398 7 — — 0 — 0.166 9 1 拟合优度
R2页岩1 0.974 5 — 0.975 3 — 0.976 9 — 0.966 3 页岩2 0.991 0 0.965 7 0.945 9 0.967 6 0.880 6 0.946 0 0.928 4 页岩3 0.934 1 0.889 5 0.955 9 0.950 4 0.968 3 0.946 7 0.902 5 页岩4 0.980 7 — 0.986 0 0.971 9 0.905 8 — 0.883 9 板岩 0.991 4 — — 0.976 4 — 0.988 6 0.988 3
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表 3 Pietruszczak各向异性模型参数拟合结果
Table 3. The determination of parameters for the Pietruszczak modified Hoek-Brown criterion
岩样 a1 a2 Ω0 页岩1 13.43 -5.766 0.322 4 页岩2 3.282 0.921 -0.554 3 页岩3 -701.2 711.4 -0.000 6 页岩4 -227.9 231.3 -0.000 4 板岩 -5 594 5 618 0.000 2
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表 4 Pietruszczak各向异性修正模型对三轴压缩实验拟合结果
Table 4. Fitting results of Pietruszczak model for triaxial compression test
参数 岩样 β=0° β=15° β=30° β=45° β=60° β=75° β=90° 拟合优度
R2页岩1 0.988 3 — 0.943 2 — 0.960 9 — 0.957 6 页岩2 0.936 5 0.899 6 0.933 3 0.941 0 0.835 6 0.900 6 0.924 8 页岩3 0.934 1 0.882 5 0.951 6 0.947 9 0.899 6 0.878 6 0.893 6 页岩4 0.589 4 — 0.922 6 0.881 2 0.857 5 — 0.563 5 板岩 0.986 3 — — 0.972 8 — 0.982 0 0.960 3
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