State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 86

null test should show zero overall volume change and zero
water volume change. The null tests were interpreted in view
of the following criteria justifying independent stress state
variables.
A suitable set of independent stress state variables are
those variables that produce no distortion or volume change
of an element when the individual components of the stress
state variables are modified but the stress state variables
themselves are maintained constant (Fredlund, 1973a). Thus
the proposed stress state variable should maintain equilib-
rium conditions. The equilibrium conditions should apply
to the overall soil structure and the contractile skin: the two
solid phases of an unsaturated soil. Equilibrium relative to
the contractile skin is maintained if there is no change in the
degree of saturation when the null-type test is performed. In
other words, it is the contractile skin which constitutes an
equilibrium boundary between the air and water phases.
A total of 19 null tests were performed on compacted
kaolin (Fredlund, 1973a). The soil specimens were com-
pacted according to the standard AASHTO procedure. Two
types of equipment were used in performing the null tests.
The first apparatus was a modified one-dimensional enclosed
oedometer. The second apparatus used isotropic loading in
a modified triaxial cell. The axis translation technique was
used in both cases.
The measured volume changes of the overall soil specimen
and the water inflow or outflow were measured during each
test. The results from one test are presented in Fig. 3.4. The
results show essentially no volume change in the overall spec-
imen and little water flow during the null tests. The stress state
variables are therefore “tested” in the sense that equilibrium
conditions were not changed for the unsaturated soil specimen
while the stress components were changed by equal amounts.
Consequently, the suggested stress state variables qualify for
describing the mechanical behavior of unsaturated soils.
The measured overall volume changes and water volume
changes were approximately 2% of the volume changes that
would have occurred if only one stress component had been
changed. These tests demonstrate that it is the differences
between the stress components that form stress state variables
that can be used to describe the behavior of an unsaturated
soil. Stated another way, it is the difference between the stress
components that forms stress state variables that maintain
equilibrium conditions in the unsaturated soil specimen.
It should also be noted that it appears to be impossible to
experimentally verify that there is an exclusive combination
of stress variables for the description of the stress state. Other
shear strength and volume change laboratory studies on unsat-
urated soils have also supported the use of independent stress
state variables (Escario, 1980; Gulhati and Satija, 1981).

200

150

Loess

Weathered
slate

100

50

Sandy clay

0

− 50

− 100

0 50 100 150 200 250 300

Air pressure, kPa

Figure 3.2 Determination of matric suction using axis translation
technique (after Hilf, 1956).

- Bring specimen to equilibrium
- Then change all stresses
by the same amount
- Observe whether a
process occurs

s 1

u a

s 3

s 3

u w

s 1

Figure 3.3 Null-type triaxial test for verification of stress state
variables.

of test can be successfully used to measure the stress state
of an unsaturated soil.
In 1977, Fredlund and Morgenstern reported a series
of null-type tests on compacted and natural clays. The
null tests subjected an unsaturated soil specimen to equal-
magnitude changes in total pressure, pore-air pressure, and
pore-water pressure while the overall volume and water
volume of the specimens were monitored (Fig. 3.3). The
stress components were varied by equal magnitudes but the

stress state variables [i.e., σ u a , u a − u w σ u w ]

− −

were kept constant:

σ x = σ y = σ z = u w = u a (3.9)

The stress components were either increased or decreased
by an equal amount while overall volume changes and water
volume changes were monitored. If the proposed stress state
variables are valid, there should not be any change in the
overall volume of the soil specimen, and the degree of satu-
ration of the soil specimen should remain constant through-
out the null test. In other words, positive results from the

3.2.2 Theoretical Evidence of Stress State Variables
A multiphase system can be subdivided into a series of com-
ponent phases where the sum of the parts must constitute the
combined or overall system. A total stress field applies to