1208
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
with saturated soils, the stress states and the phase volume
changes of unsaturated soils are more complicated. The
effective stress for unsaturated soils is not only related to the
pore water pressure, but also related to the pore air pressure and
even the degree of saturation (Zhao, Liu and Gao 2010).
Therefore the critical state of unsaturated soils could not be
accurately defined by Eq. (1). For this reason it is important to
determine which conditions and constraints should be satisfied
when the unsaturated soils attain to critical state, and which
additional conditions and constraints should be added to Eq. (1)
to completely and accurately describe critical state for
unsaturated soils. It is impossible for the theory of critical state
unsaturated soil mechanics and the corresponding constitutive
models to be further developed if these problems are not fully
solved.
Until now most researches on critical state of unsaturated
soils have been based on laboratory triaxial tests, such as Toll
(1990), Wheeler and Sivakumar (1995), Maatouk et al.(1995),
Adams and Wulfsohn (1997), Rampino et al. (1998), Wang et al.
(2002), Kayadelen et al. (2007), etc. By means of laboratory
triaxial tests, these researches have been focused on whether the
critical state for unsaturated soils exists, and if the critical state
for unsaturated soils exists, then what necessary conditions and
constraints should be satisfied when unsaturated soils attain to
critical state. Many researchers, such as Wheeler and Sivakumar
(1995), Maatouk et al. (1995), Adams and Wulfsohn (1997),
etc., selected the mean net stress, the deviator stress, the suction
and the specific volume as the state variables to describe critical
state of unsaturated soils. In addition to the above-mentioned
state variables, other researchers such as Wang et al. (2002),
Kayadelen et al. (2007), etc., suggested an additional variable,
specific water volume
w
or degree of saturation
r
S
, to define
critical state of unsaturated soils. Some laboratory test results,
such as from Wheeler and Sivakumar (1995), showed that
w
did not attain to the steady value ultimately, possibly due to the
limitations of equipment and experimental conditions. Whereas
Rampino et al. (1998) reported that
w
could keep constant at
the end of deformation test for unsaturated soils. Wang et al.
(2002) pointed out that it may not be reliable to use
w
as an
indicator of critical state for unsaturated soils, and more data
from tests and more researches are required before a conclusion
can be made on this matter.
Thermodynamics is a universally applicable theory. The
deformation process for unsaturated soils must follow
thermodynamic laws. By using thermodynamic theory, the
deformation process and behavior of unsaturated soils can be
investigated with more common situations and on more general
perspectives in order to reveal the complex soil behaviors and
properties. The objective of this paper is to establish the
necessary conditions and constraints for unsaturated soils to
reach critical state based on thermodynamic theory.
With more general perspectives based on the theory of
thermodynamics, critical state for unsaturated soil is studied in
this paper. Steady state in thermodynamic process is more
common and general than critical state in soils. Steady state
describes the final state of a thermodynamic process, and
critical state of soils is a special case of it. This paper
demonstrates that steady state in a deformation process of
unsaturated soils is the final state of the process just as the
critical state in soil mechanics, and it includes the more state
variables and restrictions than those in critical state of saturated
soils.
2 THERMODYNAMIC CONDITIONS FOR STEADY
STATE
It is assumed that unsaturated soils satisfy the assumption of
local equilibrium thermodynamics, and the theory of local
equilibrium thermodynamics can be used to approximately
describe the irreversible deformation process of unsaturated
soils. The theory of local equilibrium thermodynamics (Kuiken
1994, Wark and Richards 1999) demonstrates that under the
environmental force disturbance, a closed system that keeps
constant temperature but cannot exchange both mass and heat
energy with its surroundings can evolve from a non-equilibrium
state to a steady equilibrium state with system entropy reaching
the maximum. In accordance with the assumptions widely used
in unsaturated soil mechanics, it is assumed that a representative
volume element (RVE) of unsaturated soils is a closed system
that cannot exchange mass with its surroundings (actually the
real system can exchange mass with its surroundings. If the
system were not supposed to be a closed system, Gibbs’s
thermodynamic theory would not be applied to it. As usual
applied thermodynamics, here a representative volume element
(RVE) of unsaturated soils is supposed to be a closed system).
Therefore, a RVE evolves from a non-equilibrium state to a
steady equilibrium state with the maximum entropy. Under the
isothermal and isometric conditions, the Helmholtz free energy
of the RVE reaches the minimum value and remains
constant at steady state according to the theory of local
equilibrium thermodynamics (Kuiken 1994, Wark and Richards
1999).
The first law and the second law of thermodynamics (Wark
and Richards 1999) are given as following:
First law
/
0 Second law
i
W Q U
S S Q T
(3)
where
,
,
,
,
i
and are work, heat supplied,
internal energy, temperature, internal entropy and total entropy,
respectively. From Eq. (3), the following equation can be
developed:
W
Q
U
T
S
S
i
W TS U TS
(4)
In light of Gibbs’s theory of local equilibrium
thermodynamics (Kuiken 1994, Wark and Richards 1999) when
the RVE attains to steady equilibrium, Helmholtz free energy,
, reaches the minimum value, i.e.
, and all state
variables must keep constant, which can be expressed as:
0
0
s i
s
s s
s
W T S U T S
(5)
where the subscript
s
stands for steady state.
3 CONDITIONS AND CONSTRAINTS TO ATTAIN TO
CRITICAL STATE FOR UNSATURATED SOILS
To analyze the problems of soil mechanics, it is important that
some independent state variables should be selected at first. In
continuum mechanics, total stress
and corresponding strain
are used to describe material mechanical properties. As there
are solid, liquid and gas phases in soils, and each phase has its
dual variables, stress and the volume fraction, the properties of
soils cannot be analyzed strictly by continuum mechanics.
However, theory of porous media can be used to describe
behaviors and properties of soils. According to theory of porous
media, such as de Boer (2000), the volume fraction and the
stress of each phase are selected as the dual state variables. In
theory of porous media, the velocity is defined as a mass
weighted average quantity, but in soil mechanics or hydrology,
the velocity of soil skeleton
s
v
is generally used as a reference
configuration to construct seepage or other equations. The
control equations based on the theory of porous media may
have different forms from the ones developed in some
engineering fields, such as soil mechanics and hydrology, due to
the selection of different basic kinematical variables. In this
