An exact implementation of the Hoek-Brown criterion in FLAC (2D or 3D) Abstract This report presents a plastic stress update algorithm for the exact generalized Hoek-Brown criterion (modhb) including the apex and corner singularities. The model builds on the constitutive efforts of Johan Clausen and Lars Damkilde (2008). The plastic flow rule is taken to be non-associated with a plastic potential which are similar to the yield criterion. Perfect plasticity and isotropic linear elasticity are assumed. The stress update algorithm belongs to the class of algorithms termed return mapping, backward Euler or implicit integration.
The Hoek-Brown criterion The material parameters for the rock mass are derived from two parameters relating to the intact rock material, coupled with two parameters which characterize the quality of the in-situ rock mass. The intact rock parameters are the uniaxial compressive strength of the intact rock material, constant,
mi.
, and the petrographic
The first in-situ parameter is the Geological Strength Index, GSI, which is a qualitative
classification number for rock masses, see e.g. reference (Marinos
et al.,
2005). The second in-situ
parameter is the disturbance factor, D, which ranges from 0 to 1 (Hoek et al., 2002). For undisturbed rock masses D = 0. Based on these parameters the failure criterion is written as (J. Clausen and L. Damkilde, 2008):
́ ́ ́ ́ ́ (1) where ́ ́ ́ are the effective principal stresses. In Eq. (1) compression is taken as positive, which is often the case in rock mechanics and geotechnical engineering. Later on in this paper tension will be taken as positive and this is denoted by mb, s
, , without a prime. The empirically determined parameters
and a are given by
(2) (3) (4)
1
The rock mass modul s of elasti ity, E rm,
an be estimated using the following (E.
oek, M.
iederichs,2 06)
⁄ ⁄ . 10
(5)
o , if the inta t rock mod lus, E i, is k own
⁄ 0.02 ⁄ Typical valu s of Poisson’s ratio,
(6) , for rock asses are iven in (E. Hoek, E.
ν
. Brown, 1997). The
oek-Brown criterion in full three-dimensional principal stre ss space ca be seen on Figure 1a.
(a)
(b)
igure 1. (a) The Hoek- rown criter ion in princi pal stress s ace. The h drostatic str ess axis is d enoted p. ( ) the four d fferent stre s returns.
Stress upd te for H ek-Brow plasticit I order to o tain unknown stress increment, th predictor tress state n the gener al stress sp ce, g ven using an incremental elastic str ess-strain law. The prin ipal predic or stresses,
, are the
, is
found by
s andard met ods. In pri cipal stress space the s ress is then returned to the yield surface and the updated s ress is the back transformed into xyz-space. For Hoek- rown plasticity four d fferent stress returns a ply, as can be seen on igure 1b, – Return to t e yield surf ce – Return to t e curve l1 – Return to t e curve l2 – Return to t e apex
2
The first step is to determine whether the stress should be returned to the apex. If this is the case the updated stress is simply the apex stress. If the stress is not to be returned to the apex, a yield surface or the edges return is initiated. A detailed description of the constitutive model, its simulative potential is given in (Clausen, 2007).
Model input parameters Model Parameters (modhb) Name
Description
young (or bulk)
Young's modulus (or bulk modulus) Poisson's ratio (or shear modulus)
poisson (or shear) comp
Uniaxial compressive strength of the intact rock
m
"friction" parameter of the rock mass
s
Hoek-Brown parameter
a
Curvature parameter in the Hoek-Brown criterion
mg
"dilation" parameter of the rock mass
sg
Hoek-Brown plastic potential parameter
ag
Curvature parameter in the Hoek-Brown plastic potential
Included documents / files modelModHB2D32.dll modelModHB2D32.dll modelModHB3D32.dll modelModHB3D64.dll
a DLL file of the ModHB model compiled with Microsoft Visual C++ 2005 at 32bit for FLAC v6.0. a DLL file of the ModHB model compiled with Microsoft Visual C++ 2005 at 32bit for FLAC v7.0. a DLL file of the ModHB model compiled with Microsoft Visual C++ 2005 at 32bit for FLAC-3D v.4.00.32 32bit. a DLL file of the ModHB model compiled with Microsoft Visual C++ 2005 at 64bit for FLAC-3D v.4.00.32 64bit.
Example2D.dat
a DLL file of the ModHB model compiled with Microsoft Visual C++ 2010 at 32bit for FLAC-3D v.5.00.86 32bit. a DLL file of the ModHB model compiled with Microsoft Visual C++ 2010 at 32bit for FLAC-3D v.5.00.86 64bit. 2D example input file test for FLAC .
Example3D32.dat
example input file test for FLAC -32bit.
Example3D64.dat
example input file test for FLAC -64bit.
modelModHB3D32.dll modelModHB3D64.dll
3D 3D
3
Contact address
University of California, Berkeley Civil and Environmental Engineering, Geoengineering Department PhD. Candidate Roozbeh Geraili Mikola / Prof. Nicholas Sitar Davis Hall UC Berkeley Berkeley, California 94720-1710 Phone: (510) 643-8623 Fax: (510) 642-7476 e-mail:
[email protected] /
[email protected]
AALBORG University Department of Civil Engineering, Division of Structural Mechanics Ass. Prof. Johan Clausen Sohngårdsholmsvej 57, 9000, Aalborg Denmark Phone: 9940 7234 Fax: 9940 8552 e-mail:
[email protected]
Acknowledgments This work was performed with funding from NSF-NEES-CR Grant No. CMMI-0936376: Seismic Earth Pressures on Retaining Structures through collaborative project Between University of California, 2D
Berkeley and Itasca Consulting Group Inc. Programs FLAC
3D
and FLAC
were generously made
available by Itasca Consulting Group Inc. under collaborative research agreements.
4
References
J. Clausen, L. Damkilde, An exact implementation of the Hoek–Brown criterion for elasto-plastic finite element calculations in International Journal of Rock Mechanics and Mining Sciences 45 (2008) , 831-847.
J. Clausen, Efficient non-linear finite element implementation of elasto-plasticity for geotechnical problems. Ph.D. thesis, 2007, (http://vbn.aau.dk/files/14058639/JCthesis.pdf ).
E. Hoek, E. T. Brown, Practical estimates of rock mass strength, International Journal of Rock Mechanics & Mining Sciences 34 (8) (1997), 1165–1186.
E. Hoek, M. S. Diederichs, Empirical estimation of rock mass modulus, International Journal of Rock Mechanics & Mining Sciences 43 (2006), 203–215.
E. Hoek, C. Carranza-Torres, B. Corkum, Hoek-Brown failure criterion - 2002 edition, in: Proceedings of the North American Rock Mechanics Society Meeting in Toronto in July 2002, 2002.
V. Marinos, P. Marinos, E. Hoek, The geological strength index: applications and limitations, Bulletin of Engineering Geology and the Environment 64 (2005), 55–65.
5
