Removal of Singularities in Hoek-Brown Criterion and Its Numerical Implementation and Applications
Publication: International Journal of Geomechanics
Volume 18, Issue 10
Abstract
The representation of the Hoek-Brown criterion is an irregular curved hexagonal pyramid in the principal stress space, which leads to the occurrence of numerical singularities on the edges of the pyramid. With the aim of achieving a physical approximation to the pyramid, this study used both the C1 and C2 smoothing artifices on its sharp edges, and it was found that they were able to successfully eliminate the singularity and ensure the convexity of the yield surface. Meanwhile, to reflect the characteristic of the poor tensile strength of a rock mass, a tension cutoff surface was employed to form the whole modified combined yield surface. To facilitate comprehension and programming, the initial Hoek-Brown criterion and the smoothing and tension cutoff yield functions were all expressed in terms of stress invariants. The fully implicit backward Euler integral regression algorithm was employed to form the consistent stiffness matrix to ensure the high accuracy and fast convergence of numerical computations. In accordance with the failure zone in which a trial stress may fall, it may be pulled back to the initial Hoek–Brown yield surface, the transitional rounding yield surface, the tension cutoff yield surface, or the vertices that are the intersections of the former two with the latter. Furthermore, to facilitate the application of this modified Hoek-Brown criterion, a three-dimensional (3D) user-defined material behavior subroutine was developed in a finite-element program, and its reliability and applicability were verified through the numerical simulations of the triaxial compression and uniaxial tension tests of a rock and the excavation of a rock tunnel.
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Acknowledgments
The authors are grateful for the financial support of the Dept. of Transport of Fujian Province, China, for the research project (Grant 20130002).
References
Abbo, A. J., A. V. Lyamin, S. W. Sloan, and J. P. Hambleton. 2011. “A C2 continuous approximation to the Mohr-Coulomb yield surface.” Int. J. Solids Struct. 48 (21): 3001–3010. https://doi.org/10.1016/j.ijsolstr.2011.06.021.
Alejano, L. R., and A. Bobet. 2012. “Drucker-Prager criterion.” Rock Mech. Rock Eng. 45 (6): 995–999. https://doi.org/10.1007/s00603-012-0278-2.
Anyaegbunam, A. J. 2015. “Nonlinear power-type failure laws for geomaterials: Synthesis from triaxial data, properties, and applications.” Int. J. Geomech. 15 (1): 0401436. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000348.
Baker, R. 2004. “Nonlinear Mohr envelopes based on triaxial data.” J. Geotech. Geoenviron. Eng. 130 (5): 498–506. https://doi.org/10.1061/(ASCE)1090-0241(2004)130:5(498).
Belytschko, T., W. K. Liu, and B. Moran. 2000. Nonlinear finite elements for continua and structures. Chichester, UK: Wiley.
Benz, T., R. Schwab, R. A. Kauther, and P. A. Vermeer. 2008. “A Hoek-Brown criterion with intrinsic material strength factorization.” Int. J. Rock Mech. Min. Sci. 45 (2): 210–222. https://doi.org/10.1016/j.ijrmms.2007.05.003.
Borst, R. D., M. A. Crisfield, J. J. C. Remmers, and C. V. Verhoosel. 2012. Non-linear finite element analysis of solid and structures. 2nd ed. Chichester, UK: Wiley.
Carranza-Torres, C. 2004. “Elasto-plastic solution of tunnel problems using the generalized form of the Hoek-Brown failure criterion.” Int. J. Rock Mech. Min. Sci. 41 (1): 629–639. https://doi.org/10.1016/j.ijrmms.2004.03.111.
Carranza-Torres, C., and C. Fairhurst. 1999. “The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek-Brown failure criterion.” Int. J. Rock Mech. Min. Sci. 36 (6), 777–809. https://doi.org/10.1016/S0148-9062(99)00047-9.
Charles, J. A., and M. M. Soares. 1984. “The stability of slopes with nonlinear failure envelopes.” Can. Geotech. J. 21 (3): 397–406. https://doi.org/10.1139/t84-044.
Clausen, J., and L. Damkilde. 2008. “An exact implementation of the Hoek–Brown criterion for elasto-plastic finite element calculations.” Int. J. Rock Mech. Min. Sci. 45 (6): 831–847. https://doi.org/10.1016/j.ijrmms.2007.10.004.
Clausen, J., L. Damkilde, and L. Andersen. 2006. “Efficient return algorithms for associated plasticity with multiple yield planes.” Int. J. Numer. Methods Eng. 66 (6): 1036–1059. https://doi.org/10.1002/nme.1595.
Dawson, E., K. H. You, and Y. J. Park. 2000. “Strength-reduction stability analysis of rock slopes using the Hoek-Brown failure criterion.” In Proc., Geo-Denver 2000, 65–77. Reston, VA: ASCE.
Drucker, D. C., and W. Prager. 1952. “Soil mechanics and plastic analysis or limit design.” Q. Appl. Math. 10 (2): 157–165. https://doi.org/10.1090/qam/48291.
Fu, W., and Y. Liao. 2010. “Non-linear shear strength reduction technique in slope stability calculation.” Comput. Geotech. 37 (3): 288–298. https://doi.org/10.1016/j.compgeo.2009.11.002.
Hoek, E. 1983. “Strength of jointed rock masses.” Géotechnique 33 (3): 187–223. https://doi.org/10.1680/geot.1983.33.3.187.
Hoek, E. 1994. “Strength of rock and rock masses.” ISRM News J. 2 (2): 4–16.
Hoek, E., and E. T. Brown. 1980. “Empirical strength criterion for rock masses.” J. Geotech. Eng. Div. 106 (9): 1013–1035.
Hoek, E., and E. T. Brown. 1997. “Practical estimates of rock mass strength.” Int. J. Rock Mech. Min. Sci. 34 (8): 1165–1186. https://doi.org/10.1016/S1365-1609(97)80069-X.
Hoek, E., C. Carranza-Torres, and B. Corkum, 2002. “Hoek-Brown failure criterion—2002 edition.” In Proc., NARMS-TAC Conf., 267–273. Toronto: North American Rock Mechanics.
Koiter, W. T. 1953. “Stress-strain relations uniqueness and variational theorems for elastic-plastic materials with a singular yield surface.” Q. Appl. Math. 11 (3): 350–354. https://doi.org/10.1090/qam/59769.
Lade, P. V., and J. M. Duncan. 1975. “Elastoplastic stress-strain theory for cohesionless soil.” J. Geotech. Eng. Div. 101 (10): 1037–1053.
Lefebvre, G. 1981. “Fourth Canadian geotechnical colloquium: Strength and slope stability in Canadian soft clay deposits.” Can. Geotech. J. 18 (3): 420–442. https://doi.org/10.1139/t81-047.
Matsuoka, H., and T. Nakai. 1974. “Stress-deformation and strength characteristics of soil under three different principal stresses.” Proc. Jpn. Soc. Civ. Eng. 1974 (232), 59–70. https://doi.org/10.2208/jscej1969.1974.232_59.
Menetrey, P. H., and K. J. Willam. 1995. “Triaxial failure criterion for concrete and its generalization.” Struct. J. 92 (3): 311–318. https://doi.org/10.14359/1132.
Mogi, K. 2007. Experimental rock mechanics. Abingdon, UK: Taylor & Francis.
Nayak, G. C., and O. C. Zienkiewicz. 1972. “Convenient form of stress invariants for plasticity.” J. Struct. Div. 98 (4): 949–953.
Owen, D. R. J., and E. Hinton. 1980. Finite elements in plasticity: Theory and practice. Swansea, Wales: Pineridge.
Ramamurthy, T., G. V. Rao, and K. S. Rao. 1988. “A non-linear strength criterion for rocks.” In Proc., 5th Australia-New Zealand Conf. on Geomechanics: Prediction versus Performance, 247–252. Kolkata, India: Institution of Engineers.
Ristinmaa, M., and J. Tryding. 1993. “Exact integration of constitutive equations in elasto-plasticity.” Int. J. Numer. Methods Eng. 36 (15): 2525–2544. https://doi.org/10.1002/nme.1620361503.
Simo, J. C., and R. L. Taylor. 1985. “Consistent tangent operators for rate-independent elastoplasticity.” Comput. Methods Appl. Mech. Eng. 48 (1): 101–118. https://doi.org/10.1016/0045-7825(85)90070-2.
Singh, M., A. Raj, and B. Singh. 2011. “Modified Mohr-Coulomb criterion for non-linear triaxial and polyaxial strength of intact rocks.” Int. J. Rock Mech. Min. Sci. 48 (4): 546–555. https://doi.org/10.1016/j.ijrmms.2011.02.004.
Thorsen, K. 2013. “Analytical failure prediction of inclined boreholes.” Int. J. Geomech. 13 (3): 318–325. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000208.
Trivedi, A. 2013. “Estimating in situ deformation of rock masses using a hardening parameter and RQD.” Int. J. Geomech. 13 (4): 348–364. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000215.
Van Eekelen, H. A. M. 1980. “Isotropic yield surfaces in three dimensions for use in soil mechanics.” Int. J. Numer. Anal. Methods Geomech. 4 (1): 89–101. https://doi.org/10.1002/nag.1610040107.
Wan, R. G. 1992. “Implicit integration algorithm for Hoek-Brown elastic-plastic model.” Comput. Geotech. 14 (3): 149–177. https://doi.org/10.1016/0266-352X(92)90031-N.
Yi, X., P. P. Valkó, and J. E. Russell. 2005. “Effect of rock strength criterion on the predicted onset of sand production.” Int. J. Geomech. 5 (1): 66–73. https://doi.org/10.1061/(ASCE)1532-3641(2005)5:1(66).
Zou, J. F., and S. Yu. 2016. “Theoretical solutions of a circular tunnel with the influence of the out-of-plane stress based on the generalized Hoek-Brown failure criterion.” Int. J. Geomech. 16 (3): 06015006. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000547.
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© 2018 American Society of Civil Engineers.
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Received: Mar 2, 2017
Accepted: Feb 1, 2018
Published online: Jul 25, 2018
Published in print: Oct 1, 2018
Discussion open until: Dec 25, 2018
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