State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 93

as pore-water pressure becomes equal to pore-air pressure.
This is true regardless of the pore-air pressure. When matric
suction goes to zero, the pore-air pressure term in the first
stress tensor becomes equal to the pore-water pressure u w
and soil behavior can be described using the stress state
variables for a saturated soil (i.e., σ

Unit area

Pore-water
pressure, u w

u w ).
There is a hierarchy of limiting stress conditions related
to the magnitude of the individual stress components for an
unsaturated soil. The total stress must always be the largest
value and the pore-water pressure must be the smallest value.
The pore-air pressure must be intermediate or equal to the
pore-water pressure or the total stress:

−

Wavy plane through a
soil mass

Area of contact, a Contacts

σ ≥ u a ≥ u w (3.26)

Figure 3.10 Consideration of force equilibrium across wavy
plane.

The above hierarchy must be maintained in order to ensure
stable equilibrium conditions. Limiting stress state condi-
tions occur when one of the stress state variables becomes
zero. For example, if the pore-air pressure u a is momentar-
ily increased in excess of the total stress σ , an “explosion”
of the soil may occur. In other words, once the σ

as a proof or statical equilibrium. Rather, it is necessary to
introduce spatial variation into the stress fields in order to
write legitimate force equilibrium equations.
Continuum mechanics and the mechanics of statics require
that equilibrium considerations must commence with a legit-
imate free-body diagram. The starting point must make use
of an REV that has unbiased, planar sides in three orthog-
onal directions. In a multiphase system, it is also necessary
for the volume phase percentages of the element to be equal
to the area phase percentages. Only in this way can the
fundamental definition of a continuum be maintained.
The definition of a continuum dictates that the density
function for the mixture under consideration must remain
constant as the size of the REV is reduced. Then the results
of an equilibrium analysis cannot be biased by the analyst.

u a
variable goes to zero, a limiting stress state condition is
reached. This limiting stress condition occurs when testing
unsaturated soils in a pressure plate apparatus.
Another limiting stress state condition occurs when matric
suction u a −

−

u w vanishes. If the pore-water pressure is
increased and approaches the pore-air pressure, the degree of
saturation of the soil approaches 100% and matric suction
goes toward zero. The behavior of the soil can then be
described in terms of one stress state variable (i.e., σ

u w ).
A smooth transition from the unsaturated case to the saturated
case takes place under the limiting stress state condition
where the pore-air pressure becomes equal to the pore-water
pressure.
A limiting condition also occurs in saturated soils when
the σ

−

u w stress state variable (i.e., effective stress) reaches
zero. Any attempt to force effective stress to become nega-
tive results in the soil structure of the saturated soil becom-
ing unstable. The soil is said to “quick.” A further increase
in the pore-water pressure results in a “boil” being formed.

−

Horizontal
surface
Wavy surface through
contacts

(a)

3.3.12 Consideration of Equilibrium
across Wavy Plane
There have been numerous attempts to prove the form of
an appropriate effective stress equation for both saturated
and unsaturated soil by passing a wavy plane through a soil
mass (Fig. 3.10). The wavy plane verification analysis is
unacceptable for a number of reasons even though it is found
in many soil mechanics textbooks. Consideration of a biased
plane or a wavy plane is unacceptable within the context of
a Newtonian equilibrium-type analysis (Fig. 3.11).
Force equilibrium across a plane is comparable to New-
ton's third law, which states that “action must be equal to
reaction.” A wavy plane allows one set of stresses to be
set equal to another variable; however, this does not qualify

Vertical view of
mineral cut by
horizontal surface

(b)

Vertical view of minerals
cut by wavy surface

(c)

Figure 3.11 Shortcomings associated with passing plane through
multiphase medium.