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Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
independent state variable, identical to ignoring the item of
a
P
in Eq. (10), the conditions and constraints to attain to critical
state can be rewritten as: the volume increment of each phase
should satisfy
and
0
s
v
0
r
S
, the porosity
n
keeps
constant, and the stress variables
p
,
,
,
,
ˆ
p p
q
s
and
w
P
remain constant. 2) When both air phase pressure,
a
P
, and
degree of saturation,
r
, are not considered as independent state
variables, the conditions and constraints to attain to critical state
can be rewritten as: the volume increment should satisfy
, the porosity
n
keeps constant, and the stress variables
S
0
s
v
p
, , ,
q
,
ˆ
p p
s
and
w
P
remain unchangeable. Alonso, Gens
and Josa (1990) developed the Barcelona model, in which they
selected mean net stress
a
ˆ
p p P
, matric suction
s
, deviator
stress
, specific volume
q
, and shear strain
s
s
as state
variables, the conditions to attain to critical state given by them
belong to the second special case described above. In the
models of coupling of hydraulic hysteresis and stress-strain
behaviour developed by Wheeler, Sharma and Buisson (2003),
Sheng, Sloan and Gens(2004), Li (2007), Sun, Sheng, Cui and
Sloan (2007), etc., the conditions to attain to critical state
belong to the first special case described above. Besides when
the degree of saturation,
r
S
, equals to 1, the soil is saturated,
and matrix suction and pore air presser are zero, then the
necessary conditions and constraints for critical state of
unsaturated soils are degraded into those for saturated soils, e.g.
equal to what are given in Eq. (1). Of course, more study is
needed for some special circumstances whether the conditions
and constraints for critical state of unsaturated soils are all
necessary.
Alonso E. E., Gens A. and Josa A. 1990. A constitutive model for
partially saturated soils.
Géotechnique
. 40(3), 405-430.
de Boer R. 2000.
Theory of porous media
. Springer, Berlin.
Gens A. and Alonso E. E. 1992. A framework for the behavior of
unsaturated expansive clays.
Canadian Geotechnical Journal
, 29(6),
1013-1032.
Houlsby G.T. 1997. The work input to an unsaturated granular material.
Géotechnique
, 47(1), 193-196.
Kayadelen C., Sivrikaya O., Taskiran T. and Guneyli H. 2007. Critical-
state parameters of an unsaturated residual clayey soil from Turkey.
Engineering Geology
, 94(1-2), 1-9.
Kuiken G.D.C. 1994.
Thermodynamics of irreversible processes
. John
Wiley & Sons, Chichester.
Li X.S. 2007. Thermodynamics-based constitutive framework for
unsaturated soils. 1: theory.
Géotechnique
. 57(5), 411-422.
Li X.S. and Dafalias Y.F. 2010. Anisotropy at critical state: the role of
fabric.
Proceedings of the 9th HSTAM International Congress on
Mechanics: Vardoulakis mini-symposia
. International Congress on
Mechanics: Vardoulakis Mini-Symposia (9th, HSTAM), Limassol,
Cyprus, 57-60.
Maatouk A., Leroueil S. and La Rochelle P. 1995. Yielding and critical
state of a collapsible unsaturated silty soil.
Géotechnique
, 45(3), 465-
477.
Rampino C., Mancuso C. and Vinale F. 1998. Behavior of a compacted
silty sand during suction controlled tests.
Proceedings of the 2nd
International Conference on Unsaturated Soils
, Beijing, China, 1,
108-113.
It should be noted that when the deformation process of soils
attains to critical state, the soil structure might be anisotropic. Li
and Dafalias (2010) discussed the uniqueness of critical state
line with anisotropic structure and the enhanced critical state
conditions for saturated soils. But more research on critical state
with anisotropic structure and its uniqueness for unsaturated
soils is needed.
Roscoe K.H., Schofield A.N. and Worth C.P. 1958. On the yielding of
soils.
Géotechnique
, 8(1), 22-53.
Schofield A.N. and Wroth C.P. 1968.
Critical state soil mechanics
.
McGraw-Hill, New York.
Sheng D.C., Sloan S.W. and Gens A. 2004. A constitutive model for
unsaturated soils: thermomechanical and algorithmic aspects.
Computational Mechanics
, 33, 453-465.
Sun D.A., Sheng D.C., Cui H.B. and Sloan S.W. 2007. A density-
dependent elastoplastic hydro-mechanical model for unsaturated
compacted soils.
International Journal for Numerical and Analytical
Methods in Geomechanics
. 31(11), 1257-1279.
4 CONCLUSIONS
This paper develops the necessary conditions for the
deformation of unsaturated soils to reach critical state based on
the theory of local equilibrium thermodynamics. The necessary
conditions are developed using state variables and parameters
widely used in soil mechanics and the expression of work
W
for unsaturated soils proposed by Zhao, Liu and Gao (2010).
Comparing with the conditions given by other pioneering
researchers based on the laboratory triaxial tests, the conditions
given in this paper are more complete and generalized with
rigorous theoretical basis, and these conditions are not based on
the laboratory test results on some special samples. In addition,
it is shown that when some variables are not treated as
independent variables, the conditions given by other researchers
are a special case of the conditions presented in this paper.
Toll D.G. 1990. A framework for unsaturated soil behavior.
Géotechnique
, 40(1), 31-44.
Wang Q., Pufahl D.E. and Fredlund D.G. 2002. A study of critical state
on an unsaturated silty soil.
Canadian Geotechnical Journal
, 39(1),
213-218.
Wark Jr., K. and Richards D.E. 1999.
Thermodynamics, 6th Edition
.
McGraw-Hill, New York.
Wheeler S.J. and Sivakumar V. 1995. An elasto-plastic critical state
framework for unsaturated soil.
Géotechnique
, 45(1), 35-53.
Wheeler S.J., Sharma R.S. and Buisson M.S.R. 2003. Coupling of
hydraulic hysteresis and stress-strain behaviour in unsaturated soils.
Géotechnique
, 53(1), 41-54.
Zhao C.G., Liu Y. and Gao F.P. 2010. Work and energy equations and
the principle of generalized effective stress for unsaturated soils.
International Journal for Numerical and Analytical Methods in
Geomechanics
, 34(6), 920-936.
5 ACKNOWLEDGEMENTS
This study was carried out under the support of the National
Natural Science Foundation of China (Grant No. 51278047), the
National Basic Research Program of China (“973” Program)
(Grant No. 2010CB732100) and the Science Foundation of
Beijing (Grant No. 8112024). The support is gratefully
acknowledged.
6 REFERENCES
Adams B.A. and Wulfsohn D. 1997. Variation of the critical-state
boundaries of an agriculture soil.
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