1106
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
M, S and T subscripts refer to a thermodynamic conjugation
to net stress, suction and thermal stress, respectively. The free
energy breakdown used by Dragon and his coworkers (Dragon
et al. 2000), for damaged dry materials, is generalized and
extended to multiphase media:
Tv ji
ij Tg
Sv ji
ij Sg
ijMij Mg Tv ij
TTv
Sv ij
s Sv
M ij
ijkl eDM ij
Tv Sv Ms
ij
ij
3
3
) (
2
1
) (
2
1
:) (
:
2
1 ) ,
,
,
(
(3)
The first three terms are the mechanical, capillary and
thermal degraded elastic energies respectively. They depend
respectively on damage mechanical, capillary and thermal
rigidities (
D
e
,
and
respectively). The
second three terms are residual strain potentials, which quantify
the remaining openings due to cracks after unloading. The
derivation of the free energy
S
, provides the
whole stress/strain relations. The damage stress
, conjugated
to damage, writes:
) (
ij
) (
ij
s
) (
ij
T
,
,
Tv Sv Mij
) ,
(
ij
dY
ij Tv Tg
ij Sv sg
MMg
Tv
ij
kl
T Tv
Sv
ij
kl
s Sv
M
qp
kl
eD
M dY
ij
pq
nmij
mn
ij
3
3
)
(
2
1
)
(
2
1
)
(
2
1
(4)
The damage evolution function is assumed to depend on the
tensile strains that develop the skeleton. As in many models
(Dragon et al. 2000, Homand-Etienne et al. 1998, Shao et al.
2005) (among others), a very simple damage evolution function
is used:
ji
ij CC
ijdY
ijdY
ij
dijYdf
1 0 1 1 2
1 ) ,
(
(5)
C
0
is the initial damage-stress rate that is necessary to trigger
damage. C
1
controls the damage increase rate. The damage
evolution law is computed by an associative flow rule (Arson
and Gatmiri, 2010).
2.3
Micro-mechanical approach
The elastic components of the strain tensor are determined by
computing the damaged rigidities
,
and
.
Damaged Stress state variables are defined (damaged net stress,
damaged suction and damaged thermal stress), by using the
forth-order operator of cordebois and Sidoroff (1982) (noted
):
) (
eijkl
D
) (
s
) (
T
) (
ijkl M
2/1)
(
2/1)
(
) (
lj
kl
ik
lk
ijkl M
(6)
The Principle of Equivalent Elastic Energy is applied on the
three elastic potentials
Mlk
eijkl
DMji
) (
2
1
,
Sv
s Sv
) (
2
1
and
Tv
TTv
) (
2
1
. The final expressions of the damaged rigidities
are:
2] 1)
[(
09
) (
2] 1)
[(
09
) (
) (
:
0 :1) (
) (
ji
ij
T
T
ji
ij
s
s
T
qpkl
M emnpq D
ijnmM q
eijkl D
(7)
0
eijkl
D
,
and
are the mechanical, capillary and thermal
rigidities in the intact state, respectively.
0
s
0
T
2.4
Moisture Transfer laws
The details of the modeling of isothermal transfers in porous
media may be found in (Gatmiri & Arson 2008.b).
Liquid water and vapour transfers are assumed to be diffusive.
Hydraulic conductivity is modeled by a second-order
permeability tensor Kw
ij
:
)
,(
int
) ( )(
pq n ij
KwSrkTTk ij Kw
(8)
Only the intrinsic water permeability
,
depending on porosity n, and thus on the behavior of the solid
skeleton, may be influenced by damage. A specific crack related
component k2ij is introduced in order to model the influence of
damage on liquid water transfer:
)
,( int
pq n ij
K
)
,
(2
100 )
,( int
ij
frac n ij k ij
ew
wk rs n ij
K
rev
(9)
The intrinsic permeability related to fracturing is thus a
function of the crack densities, d
k
:
3
1
3/5
23/43/2
12
,
2
k
jnin ij
kd
b
w
w
frac n ij k
(10)
w
and
w
are the volumetric weight and the dynamic viscosity
of liquid water respectively. b is the characteristic dimension of
the REV and plays the role of an internal length parameter
(Arson and Gatmiri, 2010).
3
NUMERICAL RESULTS OF UNSATURATED
BENTONITE
3.1
The Simulation
The numerical simulation has been inspired by the laboratory
test of Pintado (Pintado et al. 2002) , and the simulation has
been performed by damage model integrated in θ-Stock Finite
Element code (Gatmiri and Arson 2008b). In the Pintado
laboratory test, a thermal source is installed between two
cylinder-shaped bentonite samples with diameters of 38mm and
heights of 76mm which are both wrapped in isolate foam. The
bottom being maintained at a constant temperature (Gatmiri et
al. 2010). The calculations are performed through axial
symmetry. The initial saturation degree Sw0 is equal to 0.63
like in the experiment conditions. After a heating period of one
week, a relaxation period of seven weeks is observed. All of the
imposed boundary conditions are given in Figure 1.
3.1.1
Discussion and results
Here we are interested in the effects of different Van Genuchten
water retention curves introducing by A to E in Figure 3, on the
variation of different parameters of damage model. Table 1
represents the parameters chosen in the five water retention
curves.
Figure 4 shows the damage force for specimens A to E in
different times. The strains resulting from heating and drying
are thus isotropic, which leads to isotropic permeability, damage
and thus isotropic damage force. That is, the reason why only
the radial components of permeability, damage and damage
force are represented in this paper. It can be seen that there are
some values showing the height of the region with damage force
getting less in comparison with the previous time; further, the
