Computers and Geotechnics 30 (2003) 549–555
www.elsevier.com/locate/compgeo
Modeling of creep in rock materials in terms
of material degradation
J.F. Shaoa,*, Q.Z. Zhua, K. Sub
a
Laboratory of Mechanics of Lille, UMR 8107, EUDIL-USTL, Cité Scientifique, 59655 Villeneuve d’Ascq, France
b
Agence Nationale pour la gestion des déchets radioactifs (ANDRA) 92296 Chatenay Malabry, France
Received 10 March 2003; received in revised form 9 May 2003; accepted 15 May 2003
Abstract
In this paper, we present a constitutive model for creep deformation in rock materials. Starting from an elastoplastic model for
the description of short term behavior, the time-dependent deformation is described in terms of evolution of microstructure, leading
to progressive degradation of elastic modulus and failure strength of material. The proposed model is applied to predict material
responses in creep and relaxation tests. There is a good agreement between numerical simulations and experimental data. The
proposed model is able to describe the main features observed in most cohesive frictional geomaterials (rocks and concrete), such as
plastic deformation, damage, volumetric dilation, pressure sensitivity, rate dependency and creep.
# 2003 Elsevier Ltd. All rights reserved.
Keywords: Creep; Damage; Plasticity; Rock; Microstructure; Constitutive model
1. Introduction
Most geomaterials, like rocks and concrete, exhibit
both instantaneous and time-dependent irreversible
deformations. This has been observed in laboratory
tests and in situ monitoring of existing geotechnical and
civil engineering structures. In the context of rock
mechanics, for example, significant creep and relaxation
effects have been recorded in sedimentary rocks [1,2],
salt rocks [3], as well as hard rocks [4–6]. On a structural
level, a typical manifestation of creep involves a progressive evolution of microcracks, as observed, for
example, around a tunnel in Lac du Bonnet granite [7].
In classic models, the creep deformation is usually
described by using viscoelasticity or viscoplasticity [8–
10], i.e. the time-dependent deformation is entirely
attributed to the viscous effects. However, both these
approaches provide a mathematical description of creep
and do not take into account physical mechanisms
inside. In geomaterials (rocks and concrete), extensive
laboratory investigations suggest that the development
of creep deformation is mainly associated with
* Corresponding author. Tel./fax: +33-3-2043-4626.
E-mail address: jian-fu.shao@eudil.fr (J.F. Shao).
0266-352X/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/S0266-352X(03)00063-6
progressive evolution of microstructure. Typical
mechanisms can involve the sub-critical propagation of
microcracks in hard rocks [11,12], pore collapse in
highly porous rocks [13], dissolution process due to
chemical-mechanical coupling [14].
In this paper, even if the proposed model can be used
for the description of creep in most cohesive frictional
geomaterials, we limit the discussion to one class of
sedimentary rocks, argilites. These rocks are extensively
investigated in the framework of research projects related to the underground laboratory for nuclear waste
storage, operated by the French National Agency for
Radioactive Waste Management (ANDRA). The work
presented here has been developed in the framework of
the European project Modex-Rep, concerning hydromechanical modeling of experiments performed in the
underground laboratory. In these materials, there is a
coupling between plastic flow due to sliding of clay
sheets and damage growth due to propagation of
microcrack around quartz and calcite grains [15].
In the recent works by Pietruszczak et al. [16,17], a
general methodology has been proposed for the
description of creep in anisotropic rocks in terms of
microstructural evolution. In this work, we will apply
such a methodology to propose a unified model for both
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J.F. Shao et al. / Computers and Geotechnics 30 (2003) 549–555
instantaneous and time-dependent elastoplastic behaviors of argilites. The constitutive model for long term
behavior is logically formulated from the extension of
the model for short term behavior. The time-dependent
deformation is considered as a macroscopic consequence of progressive degradation of material structure in microscopic scale. The evolution of
microstructure is quantified by an internal variable,
which is a function of plastic deformation, which in turn
evolves in time. The elastic properties and failure coefficient of material are affected by material degradation.
The proposed model is applied to simulate short term
triaxial compression tests, creep tests and relaxation
tests. Comparisons between simulations and experimental data are provided. In this work, we only present
the modeling of rock behaviors in dry condition (or
saturated condition without pore pressure or with constant pore pressure). The extension of the model to
coupled poromechanical behaviors in saturated and
unsaturated conditions will be presented in other
papers.
2. Constitutive model for short term behavior
In this section, we start with a brief presentation of
the constitutive model for short term behavior. Based
on experimental data on sedimentary rocks, we propose
to use an elastoplastic model for the description of fundamental behaviors in short term analysis. As mentioned above, in this work, we only present the
description of mechanical behavior of dry materials.
The extension of this model to poromechanical behavior
of saturated and unsaturated materials is discussed in
another paper [18]. Further, the main topic of this paper
is the description of creep for long term analysis, the
instantaneous damage due to applied stresses is
neglected. However, we have proposed an extended
version of the model by including an isotropic damage
for the description of short term behavior in [19]. Thus,
with the assumption of small strains, the total strain
tensor is decomposed into an elastic part and plastic
part:
"ij ¼ "eij þ "pij ; d"ij ¼ d"eij þ d"pij
ð1Þ
The general form of stress–strain relation is written as
follows:
0
"kl "pkl
ij ¼ Cijkl
ð2Þ
0
being the components of initial elastic stiffness
with Cijkl
tensor.
2.1. Description of plastic behavior
Most frictional geomaterials exhibit high pressure
sensitivity. The material strength, plastic flow and
damage evolution depend on the mean stress. The failure criterion can not be accurately described by a
Mohr–Coulomb type linear function. In this work, a
quadratic function is used to describe plastic yielding
and failure surface:
f ¼ q2 þ A hðÞp ðp C0 Þp0 ¼ 0
p¼
kk
3
;
Sij ¼ ij q¼
pffiffiffiffiffiffiffi
3J2 ;
1
J2 ¼ Sij Sij
2
kk
ij
3
ð3Þ
;
ð4Þ
where p is the mean stress, q deviatoric stress and Lode’s angle. The function h() defines the dependency
of yield function on the Lode angle. The specific form of
h() can be determined from experimental values of
yield stress on the deviatoric plane. However, the
emphasis of this work is the description of time dependent behavior. For the reason of simplicity, we have
taken h()=1. The parameter C0 represents the material
cohesion and A the frictional angle of failure surface.
The fixed reference pressure p0=1 MPa is used to keep
the parameter A without dimension. The plastic hardening is described by the increasing function p of the
generalized plastic shear strain p. Based on experimental data obtained from hard argilites [15], we have
proposed the following specific form:
ð rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 p p
p
0
0
de de ;
p ¼ p 1 p
; p ¼
3 ij ij
B þ p
ð5Þ
1
epij ¼ "pij trð"p Þij
3
The initial value of yield function, 0p , defines the
initial plastic yield threshold and p=1 corresponding
to the failure surface. The parameter B determines
plastic hardening rate.
Plastic flow rule is determined by the plastic potential.
In most geomaterials like rocks and concrete, the plastic
flow does not verify the normal dissipation rule. Plastic
volumetric strain generally exhibits a transition from
compressibility to dilation according to loading path.
Therefore, we need to define a non-associated plastic
flow rule. Based on experimental data from the argilites
studied [15] and on the elastoplastic model proposed for
concrete [20], the following function is used as plastic
potential:
g ¼ q þ AhðÞðp C0 Þln
p C0
¼0
I0
ð6Þ
J.F. Shao et al. / Computers and Geotechnics 30 (2003) 549–555
The variable I0 < 0 is the stress state at the intersection
between the plastic potential surface and the mean stress
axis p. By using this potential, the stress space is divided
into two zones, respectively corresponding to plastic
compressibility and dilation. The boundary between
these two zones is defined by the condition @g =@p ¼ 0.
In this model, we assume that this boundary can be
approximated by a linear function:
fb ¼ q AhðÞðp C0 Þ ¼ 0
ð7Þ
The parameter is the slope of the boundary line
between compressibility-dilation zones. In Fig. 1, we
show the initial yield surface, failure lotus and compressibility-dilation transition line, compared with
experimental data from hard argilites [19].
2.2. Simulation of short term responses
In this section, we briefly outline the general methodology proposed for the determination of model’s
parameters for short term behaviors. The proposed
model involves eight parameters, which can be entirely
determined step by step from direct interpretation of
triaxial compression tests. The initial elastic constants
(E0 and 0) can be obtained from the linear part of
stress–strain curves in a triaxial compression test. The
values of elastic constants may depend on the confining
pressure. This feature is not considered in this work and
we use the average values obtained from various tests.
The parameters included in the failure criterion (A and
C0) can be deduced by drawing the locus of peak stresses
in p–q plane, while the parameter for the initial yield
surface 0p is captured from the locus of stresses at onset
of inelastic strains. We can obtain the value of parameter by identifying stress points corresponding to
:
"pv ¼ 0 on stress–strain curves. The plastic hardening
parameter B can be obtained by drawing the function p
versus plastic deviatoric strain p. For the argilite studied, we have used the following values of parameters:
Fig. 1. Illustration of plastic yielding surface, failure surface and
compressibility/dilation boundary in p–q plane for argilite during
triaxial compression tests (after [19]).
551
E=5800 MPa, =0.14, 0p ¼ 0:017, A0=50, C=7 MPa,
B=0.0021, =0.0242.
We present now numerical simulations of two triaxial
tests by using the developed model. These tests have
been performed on hard argilites from eastern region of
France [15]. In Fig. 2, we show comparisons between
the numerical simulations and experimental data. There
is a good agreement. However, as this kind of tests have
been used in the determination of model’s parameters.
These comparisons represent a simple verification of
model formulation and identification procedure. Further, the emphasis of this work is the description of
creep which will be presented in the next section. For
the reason of clarity, we have neglected the material
damage due to applied stresses in the simulations presented here. Validation of the model against other
experimental data can be found in Bourgeois [19].
Nevertheless, the proposed model has the capability to
describe basic behaviors of cohesive–frictional geomaterials (rocks and concrete), such as plastic deformation,
material damage, pressure dependency, transition from
plastic compressibility to dilation.
3. Constitutive model for long term behavior
According to the general framework defined in [17],
we use an unified approach for the description of both
short and long term behavior of material. Thus, the
constitutive model for long term behavior is logically
formulated by extension of the short term model. For
this purpose, we consider that the time dependent
deformation is a macroscopic consequence of evolution
of microstructure (microcracking, clay sheet sliding,
dissolution, etc.). The evolution of microstructure is a
time dependent progressive damage process. However,
when the emphasis is put on the description of long
term behavior, we can assume that the instantaneous
damage can be neglected. The material damage is
entirely a differed process.
Fig. 2. Simulation of two triaxial compression tests under different
confining pressures.
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J.F. Shao et al. / Computers and Geotechnics 30 (2003) 549–555
We are developing a phenomenological model in this
work. For the purpose of quantitative description of
evolution of microstructure, introduce an internal variable , which is a scalar valued function of plastic
deformation which in turn, depends on time. Thus,
ðtÞ ¼ "pij ðtÞ; t
ð8Þ
Taking into account the degradation of failure surface
parameter, we can write the plastic consistency condition,
df ¼
@f
@f @p
@f
dA ¼ 0
dij þ
dp þ
@ij
@p @p
@A
and the plastic flow rule,
d"pij ¼ dl
Further, assume that for t!1 there is ! . Thus
for a prescribed load history, represents a stationary
state associated with microstructure equilibrium. After
Pietruszczak et al. [17], it is postulated that the kinetics
of microstructure evolution can be described in terms of
the
from the equilibrium state, measured by
deviation
. The evolution law may be expressed in a simple
linear form:
:
¼ ð9Þ
where is a material constant, 2 ½0;1 , while 2 0; .
The function can be formally defined by using the
Laplace transforms and convolution theorems [17].
Taking (0)=0, we obtain:
ðt
ðtÞ ¼ ð Þe ðt Þ d
ð10Þ
0
so that, upon integration by parts,
ðt
@ ðt Þ
e
d
ðtÞ ¼ ðtÞ 0 @
ð11Þ
Thus, the degradation parameter depends on the
history of the time derivative of , whereas the exponential term represents the memory effect. In the present
model, the parameter for stationary state of microstructure evolution is identified as the plastic hardening
function ¼ p .
Similarly to Pietruszczak et al. [17] and based on the
concept of damage mechanics [21], we assume that the
time-dependent degradation affects the elastic modulus
and failure surface parameter A:
E ¼ ð1 ÞE0 ;
A ¼ ð1 1 ÞA0
ð12Þ
where , 1 are two model parameters. Introducing the
degradation of elastic modulus into the constitutive Eq.
(2), we obtain:
0
ij ¼ ð1 ÞCijkl
"kl "pkl
@g
@ij
ð16Þ
Using Eqs. (12) and (14) into the consistency condition (15), we obtain,
ð1 Þ
dl ¼
@f 0
@f 0 e
@f
A0 1 d
Cijkl d"kl Cijkl "kl þ
@ij
@ij
@A
H"d
ð17Þ
The plastic hardening modulus H"d is given by:
@f 0 @g
@f @p
C
@pq pqrs @rs @p @p
"
#1=2
2
@g
@g
dev
dev
3
@ ij
@ ij
H"d ¼ ð1 Þ
ð18Þ
We notice that the plastic flow can be decomposed
into two parts, which respectively describes the instantaneous plastic deformation due to applied load (d"kl)
and the plastic creep deformation due to material
degradation (d). It is clear that in the absence of material degradation d=0, we recover the short term plastic
behavior described by the basic plastic model.
Introducing the expression of plastic multiplier and
plastic flow rule into the constitutive equation, we
obtain the increment of stresses for a prescribed increment of strains:
:
epd
dij ¼ Cijkl
d"kl þ "ij dt
ð19Þ
epd
is the tangent elastoplastic stiffness tensor of
where Cijkl
damaged material and C"ij a second order tensor defining the variation of stresses during relaxation process,
which are given by:
epd
Cijkl
¼ ð1 Þ
1
@f @g 0
0
0
Cijkl ð1 ÞCijpq
C
H"d
@pq @mn mnkl
ð20Þ
ð13Þ
1
H"d
@g @f
@f 0 @g
0
C
Cijpq
C 0 "e þ
A0 1
@pq @mn mnkl kl @A ijkl @kl
"ij ¼ ð1 Þ
The incremental form of constitutive equation is,
0
d"kl d"pkl
dij ¼ ð1 ÞCijkl
0
dCijkl
"kl "pkl
ð15Þ
ð14Þ
0 e
Cijkl
"kl
ð21Þ
J.F. Shao et al. / Computers and Geotechnics 30 (2003) 549–555
We notice that this incremental form of constitutive
Eq. (19) can be easily implemented into a numerical
integration algorithm using the finite element method,
with nodal displacements as principal unknowns. From
this equation, we can see that the first term corresponds
to the instantaneous variation of stresses due to applied
load, whereas the second term gives the time dependent
variation of stresses during relaxation process induced
by material degradation.
3.1. Application to creep test
In the case of creep tests, the history of (effective)
stresses is prescribed. We have to determine the variation of strains as functions of time. For this purpose, we
give here the formulation of the proposed model for the
creep path. The stress–strain relation is first inverted:
1
1
"ij ¼
D 0 kl þ "pij ; D 0 ¼ C 0
ð22Þ
1 ijkl
where D0 is the initial elastic compliance tensor of intact
material in drained condition. The differentiation of this
equation leads to the incremental form of strain–stress
relation:
1
0
D 0 dkl þ
Dijkl
kl d
1 ijkl
ð1 Þ2
1
@f
@g
1 @f @g
dmn
A 0 1
d
þ
Hd @mn
@ij
Hd @A @ij
553
4. Numerical simulations
The proposed model is now applied to simulate creep
tests and relaxation tests, performed on the argilites
(data base provided by ANDRA in the framework of
the European project Modex-Rep). These rocks have
being studied in the framework of research projects
related with the underground laboratory for nuclear
waste storage operated by ANDRA. For the simulation
of time dependent responses, we have used the following
values of parameters: =1, 1=0.65 and =105/s.
In Figs. 3–5, we show the simulations of two triaxial
creep tests with different loading conditions. The confining pressure is 12 MPa. In the first case (Fig. 3), the
axial stress (then deviatoric stress) is increased with a
large step of 9.85 MPa followed by a small perturbation
of 0.2 MPa. We can see that the proposed model correctly describes the instantaneous response, transient
deformation as well as stationary state. The numerical
simulation is also sensitive to the small stress perturbation. In the second case (see Fig. 4), the axial stress is
applied with several loading phases and one unloading
phase. Again, we have a good agreement between
experimental data and numerical simulations. The
unloading response is correctly reproduced. In the test
d"ij ¼
ð23Þ
with the plastic hardening modulus for stress-prescribed
loads:
"
#1=2
@f @p 2
@g
@g
dev
dev
ð24Þ
Hd ¼ @p @p 3
@ ij
@ ij
We can rewrite the strain–stress relation in the following form:
:
epd
dkl þ ij dt
ð25Þ
d"ij ¼ Dijkl
Fig. 3. Simulation of a creep test (E5697-5) with two monotonic phases of deviatoric loading.
where Depd is the tangent elastic compliance tensor of
damaged material and a second order tensor defining creep deformation:
epd
Dijkl
¼
ij ¼
1
1 @g @f
0
Dijkl
þ
1 Hd @ij @kl
1
@f @g
D0 A0 1
2 ijkl kl
Hd
@A @ij
ð1 Þ
ð26Þ
ð27Þ
We can see that the total increment of strains is composed of an instantaneous part due to applied stresses
and a time dependent part during creep process induced
by material degradation.
Fig. 4. Simulation of a creep test (E5697-5) with phases of deviatoric
loading and unloading.
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J.F. Shao et al. / Computers and Geotechnics 30 (2003) 549–555
Fig. 5. Simulation of a creep test (E104-2394-6) with various phases of
deviatoric loading until failure.
such as plastic deformation, induced damage, plastic
compressibility–dilation transition. The time dependent
deformation is seen as a macroscopic consequence of
the evolution of microstructure. Thus, we have used a
unified formulation for both short and long term behaviors. The evolution of microstructure is quantified
through an internal variable, which can be related to
time dependent process of degradation of different nature (mechanical, poromechanical, chemical–mechanical
etc.). We have applied the proposed model to simulate
triaxial compression tests, creep tests and relaxation
tests. There is a general good agreement between
experimental data and numerical predictions. The proposed model is able to describe the main features
observed in experiments. Finally, even if only the saturated condition is concerned in this work, the proposed
model can be extended and used for the description of
argilites behavior in unsaturated condition.
Acknowledgements
Fig. 6. Simulation of a relaxation test.
shown in Fig. 5, we have applied the axial stress by
several small steps until macroscopic failure of sample.
The proposed model provides a good simulation of
creep strain during different step, and correctly predicts
the material failure by accelerated creep deformation
due to unstable degradation process. Further, with the
proposed model, we can predict the diminution of elastic modulus with time, as a macroscopic consequence of
microstructural evolution.
In Fig. 6, we present the simulation of a triaxial
relaxation test with a confining pressure of 10 MPa.
Once again, there is a good agreement between numerical simulations and experimental data. The proposed
model predicts well the progressive decrease of the axial
stress due to creep process.
5. Conclusion
A new constitutive model is proposed for mechanical
behavior of cohesive frictional geomaterials (rocks and
concrete). It is applied to the argilites studied in the
framework of the underground research laboratory
operated by ANDRA. In the proposed model, we have
taken into account the main features of these materials,
This present work was performed as part of the European Project FIKW-CT2000-00029. The support from
the European Commission and the French Agence
Nationale pour la gestion des Déchets Radioactifs
(ANDRA) are gratefully acknowledged. J. F. Shao
would like to thank Professor S. Pietruszczak from
McMaster University and Professor D. Lydzba from
Wroclaw University of Technology, for fruitful discussions and helpful suggestions.
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