422
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
ph
ph
ph
b dT a
d
~
~
~
(Liu & Yu, 2011). Frost heave can be
inhibited by overburden stress (Konrad, 2005; Michalowski &
Zhu, 2006). This sensitivity of volumetric expansion to
overburden stress can be expressed as
,
conserving increment of volumetric strain from Eq. (11).
a
i
ph
i
d
d
Incremental relations of the effective stress and strain can be
expressed as follows:
(12)
Plastic flow rule determines increment of mechanical plastic
deformation in a direction perpendicular to plastic potential
function with a magnitude of non-negative scalar multiplier
.
~
~
g d
d
mp
(13)
3.5. Hydraulic characteristics of frozen soils
Darcy's law using the slope of total head has been adopted to
describe fluid flow in porous material. Many experimental
results on frozen soils showed liquid flow in the direction to
lower temperature even at the same total head (Hoekstra, 1966;
Mageau & Morgenstern, 1980). Water flow due to thermal
gradient could be estimated by introducing cryogenic suction
from water-fluid interfacial tension (Thomas et al., 2009;
Hansson et al., 2004; Liu & Yu, 2011) or segregation potential
(Tan et al., 2011).
This study used segregation potential method which can
directly calculate thermal water flow through the laboratory tests.
And Darcy's law considering abundance of liquid water in frozen
soils can be summarized as follows.
T SPS g P
k
q
l
l
l
l
l
l
~
0
~
~
~
(14)
where is segregation potential, the ratio of fluid velocity
of unfrozen water to thermal gradient (Konrad & Morgenstern,
1981). The effect of applied stress is well considered in the
variation of porosity as a function of mean stress (Konrad &
Morgenstern, 1984).
0
SP
The relationship between temperature and volumetric water
content in frozen soil is called freezing characteristic function. It
can be estimated from empirical equation through laboratory
experiments (12a, Andersland & Ladanyi, 2004) and using
similarity with SWCC in unsaturated soils (22b, van Genuchten,
1980; Nithimura et al., 2009; Tan et al., 2011). Empirical formula
(22a) was partially modified due to inordinate prediction in
temperature range
(Thomas, et al., 2009).
C TC
0.0
0.1
)]
(1[
0
TT
S
l
(15a)
1
1
0
1
P
PP
S
l
i
l
(15b)
where is freezing temperature of water, and
0
T
is
determined from pore size and chemical composition of the pore
fluid (Thomas et al., 2009). and
0
P
are material parameters in
van Genuchten model (van Genuchten, 1980), and .
Ice pressure
in frozen soil can be calculated from
Clausius-Claperyron (Eq. 23), assuming that thermodynamic
equilibrium is satisfied at the contact surface between ice and
liquid water in the pore (Henry, 2000).
i
P
)15. 273 /
ln(
T L P P
f
w
i
l w
l
w
i
i
(16)
where
is specific latent heat of fusion of water,
.
f
L
kg
/
J
10 34.3
5
4. CONCLUSIONS
Recent increasing interest in the frozen soil has raised the
demand to advance theoretical establishment and numerical tools
to interpret fully coupled thermal-hydro-mechanical phenomena
y or artificially freezing ground. However, previous
numerical analysis of freezing soil usually disregarded
mechanical characteristics or assumed freezing porous material
as a linear elastic material.
in naturall
]
[:
~
~
~ ~
~ ~
ph
T
mp
e
d
d
d d D d
In this paper, THM elastoplastic constitutive model for porous
materials is derived for the frozen-unfrozen soil consisted of soil
particles, unfrozen water, and ice. The new stress variable, the
sum of skeletal stress and ice pressure, has continuity in the
frozen-unfrozen transition condition, and it can easily be
applicable to unsaturated freezing soil. Stress increment due to
strain and temperature change was derived as the form of
incremental formulation. Proposed mechanical model was
implemented into THM FEM code, and it was applied to
numerical examples to confirm stability of solution and
applicability of the model. Numerical results effectively
described complex THM phenomena related with frozen soil,
where governing equations had high nonlinear and constitutive
models were inter-coupled.
5. REFERENCES
1. Alonso, E. E., Gens, A., and Josa, A. (1990), “A constitutive model
for partly saturated soils”, Géotechnique, Vol.40, No.3, pp.405-430.
2. Andersland, O. B., Ladanyi, B. (2004), “Frozen Ground
Engineering”, John wiley & Sons.
3. Bear, J., Bachmat, Y. (1991), “Introduction to modeling of transport
phenomena in porous media”, Kluwer Academic Publisher, p. 553.
4. Konrad, J. M., Morgenstern, N. R., (1981), “The segregation
potential of a freezing soil”, Can. Geotech. J., Vol.18, pp.482-491.
5. Nishimura, S., Gens, A., Olivella, S. and Jardine, R. J. (2009),
“THM-coupled finite element analysis of frozen soil: formulation
and application”, Géotechnique, Vol.59, No.3, pp.159-171.
6. Shin, H. S. (2011), “Formulation of Fully Coupled THM Behavior in
Unsaturated soil”, Journal of Korean Geotechnical Society, Vol.27,
No.3, pp.75-83.
7. Terzaghi, K. (1936), “The shear resistance of saturated soils”,
Proceedings for the 1st. International Conference on Soil Mechanics
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