1209
Technical Committee 106 /
Comité technique 106
paper, the variables and parameters widely used in soil
mechanics are selected as the state variables and parameters, the
same as those used in Zhao, Liu and Gao (2010), e.g., the total
stress
and the dual variables in Eq. (6), effective stress
and strain
s
of soil skeleton that determines soil deformation,
suction
s
and degree of saturation , and air pressure and
air volume strain
r
S
a
P
a
v
.
Based on porous media theory, it is assumed that the solids
and the pore water are incompressible, neither heat nor mass is
transferred among the three phases, and the velocities of
seepage and airflow are sufficiently small such that the
diffusion effects on internal energy, stress, heat and entropy are
all negligible, then, as in Zhao, Liu and Gao (2010), the work
for unsaturated soils can be expressed as:
[tr(
)
]
s
a a
r
a
v
W
snS P n
]
v
TS
)
(6)
Houlsby (1997) gave a similar form as Eq. (6). Substituting Eq.
(6) into Eq. (4) results in the following:
[tr(
s
a a
r
a
i
snS P n
U TS
(7)
where
s
is soil skeleton strain,
a
w
s P P
is suction,
a
P
is
the pore air pressure,
w
P
is the pore water pressure,
n
is the
porosity of soil,
r
S
is degree of saturation,
n n
is
volume fraction of air phase,
a
v
(1
)
a
r
S
is volume strain of air phase
and
is the effective stress of unsaturated soils, as given by
Zhao, Liu and Gao (2010) as:
[
(
)
r w
r a
S P
P
1
S
]
(8)
where
is unit tensor.
In soil mechanics, some fundamental concepts and
constitutive models are usually developed based on the results
of triaxial tests, and the conditions and constraints for critical
state of saturated soils given by Eq. (1) and Eq. (2) are in three-
dimensional axisymmetric space. Following this convention, Eq.
(8) is rewritten in the three-dimensional axisymmetric stress and
strain space as:
[
(1 ) ]
s
s
a
v
s
r
a
v
i
nS P n
TS U TS
r
S
p q
s
(9)
where
s
v
and
s
s
are volume and deviator strains, respectively,
of solid phase in triaxial strain space,
11
22
33
1/ 3(
p
)
[
(
r w
1
r
)
a
]
p S P S
P
is the mean effective pressure dual to
s
v
, and
11
22
33
1/ 3(
)
p
n S
is the mean total pressure.
The objective of this paper is to discuss the stress and strain
conditions and constraints of each phase to attain to the ultimate
state or steady state of deformation process for unsaturated soils
from thermodynamic perspective. According to the theory of
thermodynamics as mentioned in the above section, when the
unsaturated soil continually being sheared to reach the ultimate
point of the deformation process, a local steady equilibrium
state should be attained, and then the local state variables should
not change with time, and Helmholtz free energy,
, reaches
the lowest value, i.e.
. In other words, all state variables
except shear strain should be constant. When soil deformation
attains to steady state, according to Eq. (5), Eq. (9) equals to
zero. Following the principle proposed by Roscoe, Schofield
and Wroth (1958), and Schofield and Wroth (1968), the shear
strain energy is entirely dissipated at steady state, and then Eq.
(9) can be decomposed into:
0
[ ]
[
(1 ) ]
0;
0;
s
s s
s
s
a
v
r
a
r v s
s
s s
s
s
s
s
q
T
P
U T S
U S
0;
0
i
S
p snS
(10)
The first equation in Eq. (10) demonstrates that when the
deformation process of unsaturated soils attains to steady state,
its state variables should not include shear strain. The shear
strain energy is supposed to be the only dissipated energy, and it
should be dissipated (assume that the elastic shear strain is
sufficiently small and its corresponding energy is ignored) to
make the system entropy increase. The system entropy reaches
the maximum value, or the Helmholtz free energy reaches the
lowest value at steady state. From the second equation in Eq.
(10), it can be learnt that because the pressure of each phase is
not null when the deformation process attains to steady state,
the volume increment of each phase in the left side of the
equation must be zero. This result is very important since
currently there is not a clear definition about whether the
volume or volume fraction of each phase keeps constant at the
steady state or critical state. Some researchers might believe
intuitively the above point, but there has not been any rigorous
theoretical proof. Based on the theory of thermodynamic, it has
been proven theoretically that the volume or volume fraction of
each phase must keep constant at steady state. So according to
the second equation in Eq. (10), the necessary conditions and
constraints for steady state of unsaturated soils can be expressed
as:
0;
0;
0;
0
s
a
v
r
v
s
S
s
(11)
where is the porosity of the soil. It should be constant in
order to keep the volume of each phase constant, otherwise the
change of
n
would make
n
s
v
,
and
a
r
S
v
change, which is
contradictory to Eq. (11).
On the other hand, when the deformation process of
unsaturated soil attains to steady state, in addition to
s
v
,
r
a
a
v
S
nd
k ping constant, the stress variables
ee
p
,
q
,
s
a d
a
n
P
in
second equation of Eq. (10) should also keep constant,
otherwise the deformation process would not attain to steady
state in light of thermodynamics. If
the
s
and
p
are cons t at
steady state, it is obvious that
w
tan
P
,
p
an n pressure
ˆ
d et
a
p p P
constant.
must be
From above discussion, a thermodynamic process evolves
continuously and finally attains to steady state. For the
deformation process of unsaturated soils, steady state or steady
balance means that deformation process of unsaturated soils
attains to the ultimate state, namely the critical state in soil
mechanics or the steady state in thermodynamics, and at this
state all the state variables do not change. The necessary
conditions and constraints for critical state of unsaturated soils
based on thermodynamics are being stated as: 1) the volume
change of each phase should satisfy the requirement of Eq. (11)
and porosity
n
should keep constant; 2) the stress variables:
p
,
,
q
s
and
a
P
(also including
w
P
,
and
p
ˆ
p
) should also
be constant. Comparing with the conditions and constraints for
critical state of saturated soils, i.e. Eq. (1), more conditions and
constraints are needed for critical state of unsaturated soils.
Toll (1990), Wheeler and Sivakumar(1995), Maatouk et al.
(1995), Adams and Wulfsohn (1997), Rampino et al. (1998),
Wang et al. (2002), and Kayadelen et al. (2007) have conducted
some pioneering work on critical state of unsaturated soils by
laboratory triaxial tests, and suggested some conditions required
for critical state of unsaturated soils. Comparing the conditions
from these researchers, the conditions proposed in this paper are
more complete and generalized with rigorous theoretical basis.
In some laboratory tests on samples of unsaturated soils, at the
end of the deformation processes, the conditions and constraints
given in this paper are not all satisfied. This does not mean that
the conditions and constraints provided in this paper are
incorrect, since they are established based on the universally
applicable laws of thermodynamics. The explanation may be
that these tests might be limited by laboratory equipment and
experimental conditions, and the deformation of unsaturated
soil samples might not be able to attain to critical state, just as
those with saturated soils.
Based on above the necessary conditions and constraints for
critical state of unsaturated soils, two special cases need further
discuss: 1) When air pressure,
a
P
, is not considered as an
