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
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 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 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 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 -
[1] 唐杰, 吴国忱. 低孔隙度泥页岩应力依赖的各向异性裂纹演化特性研究[J]. 地球物理学报, 2015, 58(8): 2986-2995. https://www.cnki.com.cn/Article/CJFDTOTAL-DQWX201508031.htmTang Jie, Wu Guochen. Stress-dependent anisotropy of mudstone and shale with low porosity[J]. Chinese Journal of Geophysics, 2015, 58(8): 2986-2995. https://www.cnki.com.cn/Article/CJFDTOTAL-DQWX201508031.htm [2] Kanfar M F, Chen Z, Rahman S S. Effect of material anisotropy on time-dependent wellbore stability[J]. International Journal of Rock Mechanics and Mining Sciences, 2015, 78: 36-45. doi: 10.1016/j.ijrmms.2015.04.024 [3] Kuila U, Dewhurst D N, Siggins A F, et al. Stress anisotropy and velocity anisotropy in low porosity shale[J]. Tectonophysics, 2011, 503(1/2): 34-44. [4] Heng S, Guo Y T, Yang C H, et al. Experimental and theoretical study of the anisotropic properties of shale[J]. International Journal of Rock Mechanics and Mining Sciences, 2015, 74: 58-68. doi: 10.1016/j.ijrmms.2015.01.003 [5] 杨国梁, 毕京九, 张志飞, 等. 被动围压下层理角度对页岩动态强度及耗能的影响[J]. 矿业科学学报, 2021, 6(2): 188-195. doi: 10.19606/j.cnki.jmst.2021.02.006Yang Guoliang, Bi Jingjiu, Zhang Zhifei, et al. The influence of the bedding angle under passive confining pressure on the dynamic strength and energy consumption of shale[J]. Journal of Mining Science and Technology, 2021, 6(2): 188-195. doi: 10.19606/j.cnki.jmst.2021.02.006 [6] Jaeger J C. Shearfailure of anistropicrocks[J]. Geological Magazine, 1960, 97(1): 65-72. doi: 10.1017/S0016756800061100 [7] 刘亚群, 李海波, 李俊如, 等. 基于Hoek-Brown准则的板岩强度特征研究[J]. 岩石力学与工程学报, 2009, 28(S2): 3452-3457. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX2009S2028.htmLiu Yaqun, Li Haibo, Li Junru, et al. Study on strength characteristics of slates based on hoek-brown criterion[J]. Chinese Journal of Rock Mechanics and Engineering, 2009, 28(S2): 3452-3457. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX2009S2028.htm [8] Hoek E, Brown E T. Underground excavations in Rock[M]. London: The Institution of Mining and Metallurgy, 1980. [9] Hoek E., Kaiser P K, Bawden W F. Support of underground excavations in hard rock[M]. Rotterdam, A.A. Balkena, 1995. [10] 李全生, 徐祝贺, 张勇, 等. 基于Hoek-Brown准则的薄基岩厚松散层覆岩变形破坏特征研究[J]. 矿业科学学报, 2019, 4(5): 417-424. http://kykxxb.cumtb.edu.cn/article/id/241Li Quansheng, Xu Zhuhe, Zhang Yong, et al. Study on deformation and failure characteristics of overlying strata with thick loose layers and thin bedrock based on Hoek-Brown criterion[J]. Journal of Mining Science and Technology, 2019, 4(5): 417-424. http://kykxxb.cumtb.edu.cn/article/id/241 [11] Zuo J P, Li H T, Xie H P, et al. A nonlinear strength criterion for rock-like materials based on fracture mechanics[J]. International Journal of Rock Mechanics and Mining Sciences, 2008, 45(4): 594-599. doi: 10.1016/j.ijrmms.2007.05.010 [12] 李洪涛, 左建平, 李辉. Hoek-Brown强度准则的断裂力学理论研究[J]. 岩土工程学报, 2004, 26(2): 212-215. doi: 10.3321/j.issn:1000-4548.2004.02.011Li Hongtao, ZuoJianping, Li Hui. Theoretical research of Hoek-Brown empirical strength criterion[J]. Chinese Journal of Geotechnical Engineering, 2004, 26(2): 212-215. doi: 10.3321/j.issn:1000-4548.2004.02.011 [13] Zuo J P, Liu H H, Li H T. A theoretical derivation of the Hoek-Brown failure criterion for rock materials[J]. Journal of Rock Mechanics and Geotechnical Engineering, 2015, 7(4): 361-366. doi: 10.1016/j.jrmge.2015.03.008 [14] Saroglou H, Tsiambaos G. A modified Hoek-Brown failure criterion for anisotropic intact rock[J]. International Journal of Rock Mechanics and Mining Sciences, 2008, 45(2): 223-234. doi: 10.1016/j.ijrmms.2007.05.004 [15] Lee Y K, Pietruszczak S. Application of critical plane approach to the prediction of strength anisotropy in transversely isotropic rock masses[J]. International Journal of Rock Mechanics and Mining Sciences, 2008, 45(4): 513-523. doi: 10.1016/j.ijrmms.2007.07.017 [16] Ismael M A, Imam H F, El-Shayeb Y. A simplified approach to directly consider intact rock anisotropy in Hoek-Brown failure criterion[J]. Journal of Rock Mechanics and Geotechnical Engineering, 2014, 6(5): 486-492. doi: 10.1016/j.jrmge.2014.06.003 [17] Shi X C, Yang X, Meng Y F, et al. Modified Hoek-Brown failure criterion for anisotropic rocks[J]. Environmental Earth Sciences, 2016, 75(11): 1-11. [18] 高岳. 各向异性多孔材料的强度与断裂问题研究[D]. 北京: 清华大学, 2018. [19] Hou C, Jin X C, Fan X L, et al. A generalized maximum energy release rate criterion for mixed mode fracture analysis of brittle and quasi-brittle materials[J]. Theoretical and Applied Fracture Mechanics, 2019, 100: 78-85. doi: 10.1016/j.tafmec.2018.12.015 [20] Williams M L. On the stress distribution at the base of a stationary crack[J]. Journal of Applied Mechanics, 1957, 24(1): 109-114. doi: 10.1115/1.4011454 [21] Wang Y, Li C H, Hu Y Z. Experimentalinvestigation on the fracture behaviour of black shale by acoustic emission monitoring and CT image analysis during uniaxial compression[J]. Geophysical Journal International, 2018, 213(1): 660-675. doi: 10.1093/gji/ggy011 [22] 贾长贵, 陈军海, 郭印同, 等. 层状页岩力学特性及其破坏模式研究[J]. 岩土力学, 2013, 34(S2): 57-61. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX2013S2010.htmJia Changgui, Chen Junhai, Guo Yintong, et al. Research on mechanical behaviors and failure modes of layer shale[J]. Rock and Soil Mechanics, 2013, 34(S2): 57-61. https://www.cnki.com.cn/Article/CJFDTOTAL-YTLX2013S2010.htm [23] 何柏, 谢凌志, 李凤霞, 等. 龙马溪页岩各向异性变形破坏特征及其机理研究[J]. 中国科学: 物理学力学天文学, 2017, 47(11): 107-118. https://www.cnki.com.cn/Article/CJFDTOTAL-JGXK201711012.htmHe Bo, Xie Lingzhi, Li Fengxia, et al. Anisotropic mechanism and characteristics of deformation and failure of Longmaxi shale[J]. Scientia Sinica: Physica, Mechanica & Astronomica, 2017, 47(11): 107-118. https://www.cnki.com.cn/Article/CJFDTOTAL-JGXK201711012.htm [24] 桑宇, 杨胜来, 赵飞, 等. 南方海相页岩各向异性及压裂破坏特征研究[J]. 钻采工艺, 2015, 38(2): 71-74, 10. doi: 10.3969/J.ISSN.1006-768X.2015.02.20Sang Yu, Yang Shenglai, Zhao Fei, et al. Research on anisotropy and failure characteristics of southern marine shale rock[J]. Drilling & Production Technology, 2015, 38(2): 71-74, 10. doi: 10.3969/J.ISSN.1006-768X.2015.02.20 [25] 陈天宇, 冯夏庭, 张希巍, 等. 黑色页岩力学特性及各向异性特性实验研究[J]. 岩石力学与工程学报, 2014, 33(9): 1772-1779. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201409007.htmChen Tianyu, Feng Xiating, Zhang Xiwei, et al. Experimental study on mechanical and anisotropic properties of black shale[J]. Chinese Journal of Rock Mechanics and Engineering, 2014, 33(9): 1772-1779. https://www.cnki.com.cn/Article/CJFDTOTAL-YSLX201409007.htm