1105
Numerical study of damage in unsaturated bentonite with
θ
-stock finite element
code
Étude numérique d'endommagement pour les milieux poreux non saturés
avec le code des éléments finis θ-stock
Fathalikhani M.
Department of Civil Engineering, University of Tehran, Tehran, Iran.
Gatmiri B.
Center of Excellence for Engineering and Management of Infrastructures (CE-EMI), Department of civil engineering,
University of Tehran, Tehran, Iran. ANDRA, Direction Scientifique; Mécanique des Fluides et des Solides, Paris, France.
ABSTRACT: A coupled thermohydromechanical damage model of Arson and Gatmiri (THHMD model) in an unsaturated quasi-
brittel rock mass is briefly indicated in this paper. The model is based on the use of independent state variables (net stress, suction,
and thermal stress). The approach has been mixed thermodynamic and micromechanical theories. The stress-strain thermodynamic
relations have been derived from the free energy, which has been written as the sum of damaged elastic deformation energies and of
residual strain potentials; moreover, the damage rigidities have been computed by applying the Principal of Equivalent Elastic Energy
for each stress state variable. Damage has been assumed to have an isotropic influence on air and heat flows, through the inelastic
component of volumetric strains. The influence of damage on liquid water and vapor transfers has been accounted for by introducing
internal length parameters, related to specific damage-induced intrinsic conductivities. The THHMD model has been implemented in
θ-Stock Finite Element code. A numerical study is conducted on the impact of the thermal loading on the response of the unsaturated
bentonite.
RÉSUMÉ : un modèle d’endommagement introduit par Arson et Gatmiri (THHMD model) pour un milieu non saturé fragile est
brièvement présenté dans cet article. Ce modèle est développé en utilisant des variables d’état indépendants et en combinant les
approches thermodynamique et micromécanique. La relation de déformation-contrainte a été obtenue en dérivant l’énergie libre qui a
été considéré comme la somme de l’énergie de déformation élastique endommagée et le potentiel de déformation résiduelle. La
rigidité du milieu endommagé est calculée par le principe de l’énergie équivalant élastique (PEEE). L’influence de l’endommagement
sur les flux de l’air et de la chaleur est considéré isotrope. L’impact de l’endommagement sur le transfert de l’eau et de la vapeur a été
introduit en utilisant les paramètres de la longueur interne qui affecte la conductivité intrinsèque endommagée. Ce modèle a été
implanté dans le code des éléments finis θ-Stock. Des études numérique et paramétrique sont menées afin de clarifier les effets du
chargement sur la réponse de la bentonite non saturé.
KEYWORDS: Damage, Finite element method, Poromechanics, Thermohydromechanical coupling.
1
INTRODUCTION
Damage modeling has become a crux point in the study of the
Excavation Damaged Zone (EDZ) (Martino and Chandler,
2004, Mertens and Bastiaens 2004). In the context of nuclear
waste storage, cracking effects have to be accounted for in the
constitutive laws of non-isothermal unsaturated porous media
(Gens et al. 1998). The geological barriers, often made of quasi-
brittle material like granite or clay-rock, undergo damage during
the excavation phase. Hydro-mechanical interaction may occur
in the neighborhood of the engineered barrier, which is
generally constituted of unsaturated compacted clay.
Most of the existing damage models proposed to unsaturated
porous media are formulated by means of an effective stress,
defined in same way as Bishop's stress (Arson and Gatmiri
2008.a). These framework are not satisfactory to represent some
aspects of the behavior of unsaturated soils, like wetting
collapse (Houlsby 1997, Fredlund and Morgenstern 1977).
Alternatively, the THHMD model involves independent state
variables (net stress, suction and thermal stress), in order to
emphasize the role of suction rigidity. Waste is a heat source
which can generate traction, and thus cracks. After a brief
presentation of the theoretical frame in the first part of this
paper, the numerical parametric study which is inspired from a
laboratory test is performed in order to determine the main
parameters controlling the generation of damage in THHMD
model.
2
OUTLINE OF THE MODEL
2.1
Damage representation
It should be mentioned that in this paper, presentation of the
main concepts of damage is just for help us to deduce and adapt
the deductions to the theoretical concepts of damage.
In the following, the damage variable Ω will be defined as crack
density tensor expressed in a principal base:
k
jn
k
k
inkd
ij
1
3
(1)
Stress and damage will be assumed to have the same
principal directions. Physically, the damaged behavior of the
RVE is modeled by three meso-cracks representing three main
families of fissures. Each meso-crack is characterized by a
direction
n
k
(normal to the crack plane) and a volumetric
faraction
d
k
.
2.2
Phenomenological approach
Following the modeling approach of Gatmiri (Gatmiri and
Arson 2008.b), it has been assumed that thermal and capillary
phenomena are isotropic. Assuming each strain contribution
may encompass an elastic (e) and an inelastic (d) part leads to:
ij
d
Tv d e
Tv d
ij
d
Sv d e
Sv d
d
ijMd e
ijMd ij d
)
(
3
1 )
(
3
1 )
(
(2)
