1

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, , and the petrographic constant, mi. 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 , , without a prime. The empirically determined parameters mb, s and a are given by

(2)

(3)

(4)

T

D

or

T

H

F

S

In

gi

st

st

ap

The rock m

Diederichs,20

r, if the intac

0.Typical value

Hoek-Brown

Figure 1. (a)

Stress upda

n order to ob

iven using a

tandard meth

tress is then

pply, as can

Return to th

Return to th

Return to th

Return to th

mass modulu

006) .ct rock modu

02 es of Poisso

criterion in

The Hoek-B

ate for Ho

btain unknow

an increment

hods. In prin

n back trans

be seen on F

he yield surfa

he curve l1

he curve l2

he apex

us of elastic

10 ulus, Ei, is kn ons ratio, full three-di

(a)

Brown criter

(b

oek-Brown

wn stress in

tal elastic str

ncipal stress

formed into

Figure 1b,

face

city, Erm, c

nown

, for rock mmensional p

ion in princi

b) the four di

n plasticity

ncrement, the

ress-strain la

space the st

xyz-space.

2

can be estim

masses are g

principal stre

ipal stress sp

ifferent stres

y

e predictor s

aw. The princ

tress is then

For Hoek-B

mated usin

given in (E.

ess space can

pace. The hy

ss returns.

stress state i

cipal predict

returned to

Brown plasti

g the follo

Hoek, E. T

n be seen on

(b)

ydrostatic str

in the gener

tor stresses,

the yield su

icity four di

owing (E. H

T. Brown, 19

Figure 1a.

ress axis is d

al stress spa

, are thenurface and th

ifferent stre

Hoek, M.

(5)

(6)

997). The

denoted p.

ace, , is n found by

he updated

ss returns

3

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 (or shear) Poisson's ratio (or shear modulus)

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 a DLL file of the ModHB model compiled with Microsoft Visual C++ 2005 at 32bit for FLAC v6.0.

modelModHB2D32.dll a DLL file of the ModHB model compiled with Microsoft Visual C++ 2005 at 32bit for FLAC v7.0.

modelModHB3D32.dll a DLL file of the ModHB model compiled with Microsoft Visual C++ 2005 at 32bit for FLAC-3D v.4.00.32 32bit.

modelModHB3D64.dll a DLL file of the ModHB model compiled with Microsoft Visual C++ 2005 at 64bit for FLAC-3D v.4.00.32 64bit.

modelModHB3D32.dll a DLL file of the ModHB model compiled with Microsoft Visual C++ 2010 at 32bit for FLAC-3D v.5.00.86 32bit.

modelModHB3D64.dll a DLL file of the ModHB model compiled with Microsoft Visual C++ 2010 at 32bit for FLAC-3D v.5.00.86 64bit.

Example2D.dat example input file test for FLAC2D.

Example3D32.dat example input file test for FLAC3D-32bit.

Example3D64.dat example input file test for FLAC3D-64bit.

4

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: roozbehg@berkeley.edu / sitar@ce.berkeley.edu

AALBORG University Department of Civil Engineering, Division of Structural Mechanics Ass. Prof. Johan Clausen Sohngrdsholmsvej 57, 9000, Aalborg Denmark Phone: 9940 7234 Fax: 9940 8552 e-mail: jc@civil.aau.dk

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,

Berkeley and Itasca Consulting Group Inc. Programs FLAC2D and FLAC3D were generously made

available by Itasca Consulting Group Inc. under collaborative research agreements.

5

References

J. Clausen, L. Damkilde, An exact implementation of the HoekBrown 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), 11651186.

E. Hoek, M. S. Diederichs, Empirical estimation of rock mass modulus, International Journal of Rock Mechanics & Mining Sciences 43 (2006), 203215.

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), 5565.