Numerical Approach to Creep of Rock Based on the Numerical Manifold Method
Publication: International Journal of Geomechanics
Volume 18, Issue 11
Abstract
The creep behaviors of stressed rock are of great practical significance because time-dependent deformation processes can lead to stable dissipation of energy, thereby reducing violent rockbursts or outbursts in underground mines. The numerical manifold method (NMM) is an effective approach to studying the nonlinear creep deformation of rock since it involves the continuous deformation of intact rock, as well as the discontinuous deformation of cracked rock. In this paper, the incremental viscoelastoplastic constitutive relation based on the extended Nishihara model (ENM) has been incorporated into the NMM to study creep deformation of stressed rock. First, an incremental viscoelastoplastic NMM formulation was derived to perform the treatments in the NMM. Using a time-step–initial strain method, viscous strain and the large timescales of typical creep were divided into a series of incremental time-step values in the improved NMM program to calculate the creep deformation of rocks. Parameter sensitivity analysis, which can reveal the influence of different parameters on the creep of rocks, was performed for the improved NMM program, and then the improved NMM program was validated against experimental data. Finally, the influence of axial stress and confining pressure on the creep of rocks was investigated. The fact that numerical simulations were in good agreement with experimental results shows that improving the NMM by combining it with the ENM is suitable for modeling the creep behavior of rocks.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
Support for this work was provided by the Natural Science Foundation of China (NSFC) (51474051, 41672301, and 51811530312), the National Key Research and Development Program of China (2017YFC1503100), the National Basic Research Program (973) of China (2014CB047100), the Fundamental Research Funds for the Central Universities of China (N150102002), and a Partenariats Hubert Curien (PHC) Cai Yuanpei grant (36605ZB). The authors are grateful for the constructive comments of two anonymous reviewers.
References
An, X. M., Y. J. Ning, G. W. Ma, and L. He. 2014. “Modeling progressive failures in rock slopes with non‐persistent joints using the numerical manifold method.” Int. J. Numer. Anal. Methods Geomech. 38 (7): 679–701. https://doi.org/10.1002/nag.2226.
Ashby, M. F., and C. G. Sammis. 1990. “The damage mechanics of brittle solids in compression.” Pure Appl. Geophys. 133 (3): 489–521. https://doi.org/10.1007/BF00878002.
Atkinson, B. K. 1984. “Subcritical crack growth in geological materials.” J. Geophys. Res. Solid Earth 89 (B6): 4077–4114. https://doi.org/10.1029/JB089iB06p04077.
Atkinson, B. K., and P. G. Meredith. 1987. “The theory of subcritical crack growth with applications to minerals and rocks.” Fract. Mech. Rock 2: 111–166. https://doi.org/10.1016/B978-0-12-066266-1.50009-0.
Baud, P., and P. G. Meredith. 1997. “Damage accumulation during triaxial creep of Darley Dale sandstone from pore volumometry and acoustic emission.” Int. J. Rock Mech. Min. Sci. 34 (3–4): e01–e10. https://doi.org/10.1016/S1365-1609(97)00060-9.
Brantut, N., P. Baud, M. J. Heap, and P. G. Meredith. 2012. “Micromechanics of brittle creep in rocks.” J. Geophys. Res. Solid Earth 117 (B8): B08412. https://doi.org/10.1029/2012JB009299.
Brantut, N., M. J. Heap, P. Baud, and P. G. Meredith. 2014a. “Mechanisms of time-dependent deformation in porous limestone.” J. Geophys. Res. Solid Earth 119 (7): 5444–5463. https://doi.org/10.1002/2014JB011186.
Brantut, N., M. J. Heap, P. Baud, and P. G. Meredith. 2014b. “Rate- and strain-dependent brittle deformation of rocks.” J. Geophys. Res. Solid Earth 119 (3): 1818–1836. https://doi.org/10.1002/2013JB010448.
Brantut, N., M. J. Heap, P. G. Meredith, and P. Baud. 2013. “Time-dependent cracking and brittle creep in crustal rocks: A review.” J. Struct. Geol. 52 (5): 17–43. https://doi.org/10.1016/j.jsg.2013.03.007.
Cruden, D. M., and S. Masoumzadeh. 1987. “Accelerating creep of the slopes of a coal mine.” Rock Mech. Rock Eng. 20 (2): 123–135. https://doi.org/10.1007/BF01410043.
Dawson, P. R., and D. E. Munson. 1983. “Numerical simulation of creep deformations around a room in a deep potash mine.” Int. J. Rock Mech. Min. Sci. Geomechan. Abstr. 20 (1): 33–42. https://doi.org/10.1016/0148-9062(83)91612-1.
Desai, C. S., S. Sane, and J. Jenson. 2011. “Constitutive modeling including creep- and rate-dependent behavior and testing of glacial tills for prediction of motion of glaciers.” Int. J. Geomech. 11 (6): 465–476. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000091.
Fujii, Y., T. Kiyama, Y. Ishijima, and J. Kodama. 1999. “Circumferential strain behavior during creep tests of brittle rocks.” Int. J. Rock Mech. Min. Sci. 36 (3): 323–337. https://doi.org/10.1016/S0148-9062(99)00024-8.
Gao, F. Q., J. Yang, Y. J. Ning, H. Zang, and G. X. Zang. 2009. “Analysis and numerical simulation on the impact failure of rock.” Min. Res. Dev. 29: 76–79.
He, L., X. M. An, X. B. Zhao, and Z. Y. Zhao. 2013. “Investigation on strength and stability of jointed rock mass using three-dimensional numerical manifold method.” Int. J. Numer. Anal. Methods Geomech. 37 (14): 2348–2366. https://doi.org/10.1002/nag.2147.
He, J., Q. S. Liu, and Z. J. Wu. 2017. “Creep crack analysis of viscoelastic material by numerical manifold method.” Eng. Anal. Boundary Elem. 80: 72–86. https://doi.org/10.1016/j.enganabound.2017.04.005.
Heap, M. J., P. Baud, and P. G. Meredith. 2009a. “Influence of temperature on brittle creep in sandstones.” Geophys. Res. Lett. 36 (19): L19305. https://doi.org/10.1029/2009GL039373.
Heap, M. J., P. Baud, P. G. Meredith, A. F. Bell, and I. G. Main. 2009b. “Time‐dependent brittle creep in Darley Dale sandstone.” J. Geophys. Res. Atmos. 114 (B7): B07203. https://doi.org/10.1029/2008JB006212.
Heap, M. J., P. Baud, P. G. Meredith, S. Vinciguerra, A. F. Bell, and I. G. Main. 2011. “Brittle creep in basalt and its application to time-dependent volcano deformation.” Earth Planet. Sci. Lett. 307 (1–2): 71–82. https://doi.org/10.1016/j.epsl.2011.04.035.
Heap, M. J., N. Brantut, P. Baud, and P. G. Meredith. 2015. “Time‐dependent compaction band formation in sandstone.” J. Geophys. Res. Solid Earth 120 (7): 4808–4830. https://doi.org/10.1002/2015JB012022.
King, M. S. 1973. “Creep in model pillars of Saskatchewan potash.” Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 10 (4): 363–371. https://doi.org/10.1016/0148-9062(73)90044-2.
Kranz, R. L., W. J. Harris, and N. L. Carter. 1982. “Static fatigue of granite at 200°C.” Geophys. Res. Lett. 9 (1): 1–4. https://doi.org/10.1029/GL009i001p00001.
Le, T. M., and B. Fatahi. 2016. “Trust-region reflective optimisation to obtain soil visco-plastic properties.” Eng. Comput. 33 (2): 410–442. https://doi.org/10.1108/EC-11-2014-0236.
Le, T. M., B. Fatahi, H. Khabbaz, and W. Sun. 2017. “Numerical optimisation applying trust-region reflective least squares algorithm with constraints to optimise the non-linear creep parameters of soft soil.” Appl.Math. Modell. 16: 236–256. https://doi.org/10.1016/j.apm.2016.08.034.
Li, H. F., G. X. Zhang, G. H. Shi, and X. C. Peng. 2010. “Manifold cut and generation of three-dimensional manifold element under Fe mesh cover.” [In Chinese.] Chin. J. Rock Mech. Eng. 29 (4): 731–742.
Liu, J., and Q. Chen. 2012a. “A numerical manifold method for simulating creep of rocks.” [In Chinese.] Rock Soil Mech. 33 (4): 1203–1209.
Liu, J., and Q. Chen. 2012b. “Study of high-order numerical manifold method in viscoelastic creep of rock mass.” [In Chinese.] Rock Soil Mech. 33 (7): 2174–2180.
Lockner, D. A. 1998. “A generalized law for brittle deformation of Westerly granite.” J. Geophys. Res. B Solid Earth 103 (B3): 5107–5123. https://doi.org/10.1029/97JB03211.
Luo, S. M., X. W. Zhang, W. G. Lü, and D. R. Jiang. 2005. “Theoretical study of three-dimensional numerical manifold method.” Appl. Math. Mech. 26 (9): 1027–1032. https://doi.org/10.1007/BF02507721.
Ma, G. W., X. M. An, and L. He. 2010. “The numerical manifold method: A review.” Int. J. Comput. Methods 7 (1): 1–32. https://doi.org/10.1142/S0219876210002040.
Ngwenya, B. T., I. G. Main, S. C. Elphick, B. R. Crawford, and B. G. D. Smart. 2001. “A constitutive law for low‐temperature creep of water‐saturated sandstones.” J. Geophys. Res. Solid Earth 106 (B10): 21811–21826. https://doi.org/10.1029/2001JB000403.
Nicolas, A., J. Fortin, J. B. Regnet, B. A. Verberne, O. Plümper, A. Dimanov, C. J. Spiers, and Y. Guéguen. 2017. “Brittle and semi‐brittle creep of Tavel limestone deformed at room temperature.” J. Geophys. Res. Solid Earth 122 (6): 4436–4459. https://doi.org/10.1002/2016JB013557.
Ning, Y. J., X. M. An, and G. W. Ma. 2011. “Footwall slope stability analysis with the numerical manifold method.” Int. J. Rock Mech. Min. Sci. 48 (6): 964–975. https://doi.org/10.1016/j.ijrmms.2011.06.011.
Nishihara, M. 1955. “A method for predicting creep characteristics.” Doshisha Kogaku Kaishi 6 (2): 89–95.
Phienwej, N., P. K. Thakur, and E. J. Cording. 2007. “Time-dependent response of tunnels considering creep effect.” Int. J. Geomech. 7 (4): 296–306. https://doi.org/10.1061/(ASCE)1532-3641(2007)7:4(296).
Shi, G. H. 1992. “Discontinuous deformation analysis: A new numerical model for the statics and dynamics of deformable block structures.” Eng. Comput. 9 (2): 157–168. https://doi.org/10.1108/eb023855.
Terada, K., and K. Mao. 2004. “Performance assessment of generalized elements in the finite cover method.” Finite Elem. Anal. Des. 41 (2): 111–132. https://doi.org/10.1016/j.finel.2004.05.001.
Voight, B. 1989. “A relation to describe rate-dependent material failure.” Science 243 (4888): 200–203. https://doi.org/10.1126/science.243.4888.200.
Wang, Q. Y., W. C. Zhu, T. Xu, L. L. Niu, and J. Wei. 2016. “Numerical simulation of rock creep behavior with a damage-based constitutive law.” Int. J. Geomech. 17 (1): 04016044. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000707.
Wang, S. 1981. “On the mechanism and process of slope deformation in an open pit mine.” Rock Mech. 13 (3): 145–156. https://doi.org/10.1007/BF01239035.
Wang, Z. C., and R. C. K. Wong. 2016. “Strain-dependent and stress-dependent creep model for a till subject to triaxial compression.” Int. J. Geomech. 16 (3): 04015084. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000583.
Xu, T., C. A. Tang, and J. Zhao. 2012a. “Modeling of rheological deformation of inhomogeneous rock and associated time-dependent response of tunnels.” Int. J. Geomech. 12 (2): 147–159. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000130.
Xu, T., C. A. Tang, J. Zhao, L. Li, and M. J. Heap. 2012b. “Modelling the time‐dependent rheological behaviour of heterogeneous brittle rocks.” Geophys. J. Int. 189 (3): 1781–1796. https://doi.org/10.1111/j.1365-246X.2012.05460.x.
Xu, T., Q. Xu, C. A. Tang, and P. G. Ranjith. 2013. “The evolution of rock failure with discontinuities due to shear creep.” Acta Geotech. 8 (6): 567–581. https://doi.org/10.1007/s11440-013-0244-5.
Xu, T., G. L. Zhou, M. J. Heap, W. C. Zhu, C. F. Chen, and P. Baud. 2017. “The influence of temperature on time-dependent deformation and failure in granite: A mesoscale modeling approach.” Rock Mech. Rock Eng. 50 (9): 2345–2364. https://doi.org/10.1007/s00603-017-1228-9.
Yang, S. Q., and Y. Z. Jiang. 2010. “Triaxial mechanical creep behavior of sandstone.” Min. Sci. Technol. (China) 20 (3): 339–349. https://doi.org/10.1016/S1674-5264(09)60206-4.
Yang, S. Q., P. Xu, and T. Xu. 2015. “Nonlinear visco-elastic and accelerating creep model for coal under conventional triaxial compression.” Geomech. Geophys. Geo-Energy Geo-Resour. 1 (3–4): 109–120. https://doi.org/10.1007/s40948-015-0014-y.
Yang, Y. T., X. H. Tang, H. Zheng, Q. S. Liu, and L. He. 2016. “Three-dimensional fracture propagation with numerical manifold method.” Eng. Anal. Boundary Elem. 72: 65–77. https://doi.org/10.1016/j.enganabound.2016.08.008.
Ye, G. L., T. Nishimura, and F. Zhang. 2015. “Experimental study on shear and creep behaviour of green tuff at high temperatures.” Int. J. Rock Mech. Min. Sci. 79: 19–28. https://doi.org/10.1016/j.ijrmms.2015.08.005.
Zhao, B. Y., D. Y. Liu, Y. R. Zheng, and H. Liu. 2013. “Uniaxial compressive creep test of red sandstone and its constitutive model.” [In Chinese.] J. Min. Saf. Eng. 30 (5): 744–747.
Zhao, L., L. Y. Xu, and K. Nikbin. 2017a. “Predicting failure modes in creep and creep-fatigue crack growth using a random grain/grain boundary idealised microstructure meshing system.” Mater. Sci. Eng. A 704: 274–286. https://doi.org/10.1016/j.msea.2017.08.035.
Zhao, Y. L., Y. X. Wang, W. J. Wang, W. Wan, and J. Z. Tang. 2017b. “Modeling of non-linear rheological behavior of hard rock using triaxial rheological experiment.” Int. J. Rock Mech. Min. Sci. 93: 66–75. https://doi.org/10.1016/j.ijrmms.2017.01.004.
Information & Authors
Information
Published In
International Journal of Geomechanics
Volume 18 • Issue 11 • November 2018
Copyright
© 2018 American Society of Civil Engineers.
History
Received: Dec 28, 2017
Accepted: May 18, 2018
Published online: Sep 11, 2018
Published in print: Nov 1, 2018
Discussion open until: Feb 11, 2019
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.