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Simulation of the fracture of heterogeneous rock masses based on the enriched numerical manifold method

  • Yuan Wang (College of Water Conservancy and Hydropower Engineering, Hohai University) ;
  • Xinyu Liu (College of Civil and Transportation Engineering, Hohai University) ;
  • Lingfeng Zhou (College of Civil and Transportation Engineering, Hohai University) ;
  • Qi Dong (College of Water Conservancy and Hydropower Engineering, Hohai University)
  • Received : 2023.06.29
  • Accepted : 2023.08.16
  • Published : 2023.09.25

Abstract

The destruction and fracture of rock masses are crucial components in engineering and there is an increasing demand for the study of the influence of rock mass heterogeneity on the safety of engineering projects. The numerical manifold method (NMM) has a unified solution format for continuous and discontinuous problems. In most NMM studies, material homogeneity has been assumed and despite this simplification, fracture mechanics remain complex and simulations are inefficient because of the complicated topology updating operations that are needed after crack propagation. These operations become computationally expensive especially in the cases of heterogeneous materials. In this study, a heterogeneous model algorithm based on stochastic theory was developed and introduced into the NMM. A new fracture algorithm was developed to simulate the rupture zone. The algorithm was validated for the examples of the four-point shear beam and semi-circular bend. Results show that the algorithm can efficiently simulate the rupture zone of heterogeneous rock masses. Heterogeneity has a powerful effect on the macroscopic failure characteristics and uniaxial compressive strength of rock masses. The peak strength of homogeneous material (with heterogeneity or standard deviation of 0) is 2.4 times that of heterogeneous material (with heterogeneity of 11.0). Moreover, the local distribution of parameter values can affect the configuration of rupture zones in rock masses. The local distribution also influences the peak value on the stress-strain curve and the residual strength. The post-peak stress-strain curve envelope from 60 random calculations can be used as an estimate of the strength of engineering rock masses.

Keywords

Acknowledgement

This research was supported by the National Natural Science Foundation of China (U2240210).

References

  1. Ayatollahi, M.R., Aliha, M.R.M. and Hassani, M.M. (2006), "Mixed mode brittle fracture in PMMA-an experimental study using SCB specimens", Mater. Sci. Eng. : A, 417(1-2), 348-356. https://doi.org/10.1016/j.msea.2005.11.002. 
  2. Azarafza, M., Ghazifard, A., Akgun, H. and Asghari-Kaljahi, E. (2019), "Development of a 2D and 3D computational algorithm for discontinuity structural geometry identification by artificial intelligence based on image processing techniques", Bull. Eng. Geol. Environ., 78(5), 3371-3383. https://doi.org/10.1007/s10064-018-1298-2. 
  3. Azarafza, M., Nanehkaran, Y.A., Akgun, H. and Mao, Y. (2021), "Application of an image processing-based algorithm for riverside granular sediment gradation distribution analysis", Adv. Mater. Res., 10(3), 229-244. https://doi.org/10.12989/amr.2021.10.3.229. 
  4. Baud, P., Wong, T.F. and Zhu, W. (2014), "Effects of porosity and crack density on the compressive strength of rocks", Int. J. Rock Mech. Min. Sci., 67, 202-211. https://doi.org/10.1016/j.ijrmms.2013.08.031. 
  5. Bocca, P., Carpinteri, A. and Valente, S. (1990), "Size effects in the mixed mode crack propagation: softening and snap-back analysis", Eng. Fract. Mech., 35(1-3), 159-170. https://doi.org/10.1016/0013-7944(90)90193-K. 
  6. Chen, S., Yue, Z.Q. and Tham, L.G. (2004), "Digital image-based numerical modeling method for prediction of inhomogeneous rock failure", Int. J. Rock Mech. Min. Sci., 41(6), 939-957. https://doi.org/10.1016/j.ijrmms.2004.03.002. 
  7. Chiou, Y.J., Lee, Y.M. and Tsay, R.J. (2002), "Mixed mode fracture propagation by manifold method", Int. J. Fract., 114(4), 327-347. https://doi.org/10.1023/A:1015713428989. 
  8. Cook, N.G. (1965), "The failure of rock", Int. J. Rock Mech. Min. Sci. Geomech., Pergamon, December.
  9. Goodman, R.E., Taylor, R.L. and Brekke, T.L. (1968), "A model for the mechanics of jointed rock", J. Soil Mech. Found. Division, 94(3), 637-659. https://doi.org/10.1061/JSFEAQ.0001133. 
  10. Huang, D., Liu, Y., Yang, Y., Li, Z. and Meng, Q. (2021), "Experimental study on three-point-bending characteristics of hard and soft rock-like materials under different loading rates", Arabian J. Geosci., 14, 1-12. https://doi.org/10.1007/s12517-021-08284-9. 
  11. Jiao, Y., Huang, G., Zhao, Z., Zheng, F. and Wang, L. (2015), "An improved three-dimensional spherical DDA model for simulating rock failure", Science China Technol. Sci., 58(9), 1533-1541. https://doi.org/10.1007/s11431-015-5898-9. 
  12. Katona, M.G. (1983), "A simple contact-friction interface element with applications to buried culverts", Int. J. Numer. Anal. Method. Geomech., 7(3), 371-384. https://doi.org/10.1002/nag.1610070308. 
  13. Li, J., Khodaei, Z.S. and Aliabadi, M.H. (2017), "Spectral bem for the analysis of wave propagation and fracture mechanics", J. Multiscale Modell., 8(3-4), 1740007. https://doi.org/10.1142/S1756973717400078. 
  14. Lim, I.L., Johnston, I.W. and Choi, S.K. (1993), "Stress intensity factors for semi-circular specimens under three-point bending", Eng. Fract. Mech., 44(3), 363-382. https://doi.org/10.1016/0013-7944(93)90030-V. 
  15. Liao, Z.Y., Zhu, J.B. and Tang, C.A. (2019), "Numerical investigation of rock tensile strength determined by direct tension, Brazilian and three-point bending tests", Int. J. Rock Mech. Min. Sci., 115, 21-32. https://doi.org/10.1016/j.ijrmms.2019.01.007. 
  16. Liu, X., Hu, C., Liu, Q. and He, J. (2021), "Grout penetration process simulation and grouting parameters analysis in fractured rock mass using numerical manifold method", Eng. Anal. Bound. Elem., 123, 93-106. https://doi.org/10.1016/j.enganabound.2020.11.008. 
  17. Ma, G.W. and An, X.M. (2008), "Numerical simulation of blasting-induced rock fractures", Int. J. Rock Mech. Min. Sci., 45(6), 966-975. https://doi.org/10.1016/j.ijrmms.2007.12.002. 
  18. Ma, G.W., An, X.M., Zhang, H.H. and Li, L. (2009), "Modeling complex crack problems using the numerical manifold method", Int. J. Fracture, 156(1), 21-35. https://doi.org/10.1007/s10704-009-9342-7. 
  19. Ma, G., An, X. and He, L.E.I. (2010), "The numerical manifold method: a review", Int. J. Comput. Method., 7(1), 1-32. https://doi.org/10.1142/S0219876210002040. 
  20. Ma, G.W., Wang, X.J. and Ren, F. (2011), "Numerical simulation of compressive failure of heterogeneous rock-like materials using SPH method", Int. J. Rock Mech. Min. Sci., 48(3), 353-363. https://doi.org/10.1016/j.ijrmms.2011.02.001. 
  21. Melin, S. (1989), "Why are crack paths in concrete and mortar different from those in PMMA?", Mater. Struct., 22(127), 23-27. https://doi.org/10.1007/BF02472691. 
  22. Ouchi, H., Katiyar, A., Foster, J.T. and Sharma, M.M. (2015), "A peridynamics model for the propagation of hydraulic fractures in heterogeneous, naturally fractured reservoirs", Proceedings of the SPE Hydraulic Fracturing Technology Conference and Exhibition, SPE, February. 
  23. Patil, R.U., Mishra, B.K. and Singh, I.V. (2019), "A multiscale framework based on phase field method and XFEM to simulate fracture in highly heterogeneous materials", Theor. Appl. Fract. Mech., 100, 390-415. https://doi.org/10.1016/j.tafmec.2019.02.002. 
  24. Shi, G.H. (1991), "Manifold method of material analysis", Proceedings of the Transactions of the 9th army conference on applied mathematics and computing, Minneapolis, June. 
  25. Schlangen, E. and Van Mier, J. (1992), "Experimental and numerical analysis of micromechanisms of fracture of cement-based composites", Cement Concrete Compos., 14(2), 105-118. https://doi.org/10.1016/0958-9465(92)90004-F. 
  26. Shemirani, A.B., Haeri, H., Sarfarazi, V. and Hedayat, A. (2017), "A review paper about experimental investigations on failure behaviour of non-persistent joint", Geomech. Eng., 13(4), 535-570. https://doi.org/10.12989/gae.2017.13.4.535. 
  27. Shi, J.W., Fu, Z.Z. and Guo, W.L. (2019), "Investigation of geometric effects on three-dimensional tunnel deformation mechanisms due to basement excavation", Comput. Geotech., 106, 108-116. https://doi-org/ 10.1016/j.compgeo.2018.10.019. 
  28. Shi, J.W., Chen Y.H., Lu, H., Ma, S.K. and Ng, C.W.W. (2022), "Centrifuge modeling of the influence of joint stiffness on pipeline response to underneath tunnel excavation", Can. Geotech. J., 59(9), 1568-1586. https://doi.org/10.1139/cgj-2020-0360. 
  29. Shi, J.W., Wang, J.P., Chen Y.H., Shi, C., Lu, H., Ma, S.K. and Fan, Y.B. (2023), "Physical modeling of the influence of tunnel active face instability on existing pipelines", Tunn. Undergr. Sp. Tech., 140, 105281. https://doi.org/10.1016/j.tust.2023.105281. 
  30. Strouboulis, T., Copps, K. and Babuska, I. (2000), "The generalized finite element method: an example of its implementation and illustration of its performance", Int. J. Numer. Meth. Eng., 47(8), 1401-1417. https://doi.org/10.1002/(SICI)10970207(20000320)47:8<1401::AID-NME835>3.0.CO;2-8. 
  31. Strouboulis, T., Copps, K. and Babuska, I. (2001), "The generalized finite element method", Computer Method. Appl. M., 190(32-33), 4081-4193. https://doi.org/10.1016/S0045-7825(01)00188-8. 
  32. Tang, C. (1997), "Numerical simulation of progressive rock failure and associated seismicity", Int. J. Rock Mech. Min. Sci., 34(2), 249-261. https://doi.org/10.1016/S0148-9062(96)00039-3. 
  33. Tang, C.A., Yang, W.T., Fu, Y.F. and Xu, X.H. (1998), "A new approach to numerical method of modelling geological processes and rock engineering problems-continuum to discontinuum and linearity to nonlinearity", Eng. Geol., 49(3-4), 207-214. https://doi.org/10.1016/S0013-7952(97)00051-3. 
  34. Tang, C.A., Liu, H., Lee, P.K.K., Tsui, Y. and Tham, L. (2000), "Numerical studies of the influence of microstructure on rock failure in uniaxial compression-part I: effect of heterogeneity", Int. J. Rock Mech. Min. Sci., 37(4), 555-569. https://doi.org/10.1016/S1365-1609(99)00121-5. 
  35. Tang, C.A., Tham, L.G., Lee, P.K.K., Tsui, Y. and Liu, H. (2000), "Numerical studies of the influence of microstructure on rock failure in uniaxial compression-part II: constraint, slenderness and size effect", Int. J. Rock Mech. Min. Sci., 37(4), 571-583. https://doi.org/10.1016/S1365-1609(99)00122-7. 
  36. Tang, X. and Zhang, C. (2009), "Meso-scale modeling of concrete: Effects of heterogeneity", J. Hydroelectric Eng., 28(4), 56-62. 
  37. Tang, S. (2011), "Applications of rock failure process analysis (RFPA) method", J. Rock Mech. Geotech. Eng., 3(4), 352-372. https://doi.org/10.3724/SP.J.1235.2011.00352. 
  38. Tsang, Y.W. and Witherspoon, P.A. (1981), "Hydromechanical behavior of a deformable rock fracture subject to normal stress", J. Geophys. Res. Solid Earth, 86(10), 9287-9298. https://doi.org/10.1029/JB086iB10p09287. 
  39. Wang, S.L., Feng, X.T. and Ge, X.R. (2003), "Study on crack propagation modeling by high order manifold method", Rock Soil Mech.-Wuhan-, 24(4), 622-625. https://doi.org/10.16285/j.rsm.2003.04.033. 
  40. Wang, X., Yuan, W., Yan, Y.T. and Zhang, X. (2020), "Scale effect of mechanical properties of jointed rock mass: a numerical study based on particle flow code", Geomech. Eng., 21(3), 259-268. https://doi.org/10.12989/gae.2020.21.3.259. 
  41. Wu, Z., Fan, L., Liu, Q. and Ma, G. (2017), "Micro-mechanical modeling of the macro-mechanical response and fracture behavior of rock using the numerical manifold method", Eng. Geol., 225, 49-60. https://doi.org/10.1016/j.enggeo.2016.08.018. 
  42. Wu, Z., Yu, F., Zhang, P. and Liu, X. (2019), "Micro-mechanism study on rock breaking behavior under water jet impact using coupled SPH-FEM/DEM method with Voronoi grains", Eng. Anal. Bound. Elem., 108, 472-483. https://doi.org/10.1016/j.enganabound.2019.08.026. 
  43. Wu, W., Zheng, H. and Yang, Y. (2019), "Enriched three-field numerical manifold formulation for dynamics of fractured saturated porous media", Comput. Method. Appl. M., 353, 217-252. https://doi.org/10.1016/j.cma.2019.05.008. 
  44. Wu, W., Yang, Y. and Zheng, H. (2020), "Enriched mixed numerical manifold formulation with continuous nodal gradients for dynamics of fractured poroelasticity", Appl. Math. Modell., 86, 225-258. https://doi.org/10.1016/j.apm.2020.03.044. 
  45. Wu, W., Yang, Y. and Zheng, H. (2020), "Hydro-mechanical simulation of the saturated and semi-saturated porous soil-rock mixtures using the numerical manifold method", Comput. Method. Appl. Mech. Eng., 370, 113238. https://doi.org/10.1016/j.cma.2020.113238. 
  46. Xiong, X., Li, B., Jiang, Y., Koyama, T. and Zhang, C. (2011), "Experimental and numerical study of the geometrical and hydraulic characteristics of a single rock fracture during shear", Int. J. Rock Mech. Min. Sci., 48(8), 1292-1302. https://doi.org/10.1016/j.ijrmms.2011.09.009. 
  47. Xue, Y.C., Sun, W.B. and Wu, Q.S. (2020), "The influence of magmatic rock thickness on fracture and instability law of mining surrounding rock", Geomech. Eng., 20(6), 547-556. https://doi.org/10.12989/gae.2020.20.6.547. 
  48. Yang, Y., Tang, X., Zheng, H., Liu, Q. and He, L. (2016), "Three-dimensional fracture propagation with numerical manifold method", Eng. Anal. Bound. Elem., 72, 65-77. http://dx.doi.org/10.1016/j.enganabound.2016.08.008. 
  49. Yang, Y., Tang, X., Zheng, H., Liu, Q. and Liu, Z. (2018), "Hydraulic fracturing modeling using the enriched numerical manifold method", Appl. Math. Modell., 53, 462-486. https://doi.org/10.1016/j.apm.2017.09.024. 
  50. Yang, Y., Xu, D., Liu, F. and Zheng, H. (2020), "Modeling the entire progressive failure process of rock slopes using a strength-based criterion", Comput. Geotech., 126, 103726. https://doi.org/10.1016/j.compgeo.2020.103726. 
  51. Yang, B., Cao, X., Han, T., Li, P. and Shi, J. (2022), "Effect of heterogeneity on the extension of ubiquitiformal cracks in rock materials", Fractal Fractional, 6(6), 317. https://doi.org/10.3390/FRACTALFRACT6060317. 
  52. Yu, C.Y., Zheng, F., Guo, B.C. and Liu, Q.Y. (2020), "A generalized cover renewal strategy for multiple crack propagation in two-dimensional numerical manifold method", J. Central South Univ., 27(8), 2367-2381. https://doi.org/10.1007/s11771-020-4455-2. 
  53. Zhang, G.X., Sugiura, Y., Hasegawa, H. and Wang, G. (2002), "The second order manifold method with six node triangle mesh", Struct. Eng. Earthq. Eng., 19(1), 1-9. https://doi.org/10.2208/jsceseee.19.1s. 
  54. Zhang, H.H., Li, L.X., An, X.M. and Ma, G. (2010), "Numerical analysis of 2-D crack propagation problems using the numerical manifold method", Eng. Anal. Bound. Elem., 34(1), 41-50. https://doi.org/10.1016/j.enganabound.2009.07.006. 
  55. Zhou, X.P., Cheng, H. and Feng, Y.F. (2014), "An experimental study of crack coalescence behaviour in rock-like materials containing multiple flaws under uniaxial compression", Rock Mech. Rock Eng., 47(6), 1961-1986. https://doi.org/10.1007/s00603-013-0511-7. 
  56. Zhu, W.C. and Tang, C.A. (2004), "Micromechanical model for simulating the fracture process of rock", Rock Mech. Rock Eng., 37, 25-56. https://doi.org/10.1007/s00603-003-0014-z. 
  57. Zhu, W.C. and Tang, C.A. (2006), "Numerical simulation of Brazilian disk rock failure under static and dynamic loading", Int. J. Rock Mech. Min. Sci., 43(2), 236-252. https://doi.org/10.1016/j.ijrmms.2005.06.008. 
  58. Zi, G. and Belytschko, T. (2003), "New crack-tip elements for XFEM and applications to cohesive cracks", Int. J. Numer. Meth. Eng., 57(15), 2221-2240. https://doi.org/10.1002/nme.849.