State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 90

The areas or volumes associated with each phase can be
written relative to the dimensions of the soil element (i.e.,
dx , dy , and dz ). This is done using a porosity designation for
the volume and area associated with individual phases. The
porosity n of a phase is the ratio of the volume (or surface
area) of each phase relative to the total volume. The volume
porosity is assumed to be equal to the area ratio of each
phase relative to the total cross-sectional area. This concept
is referred to as the “theorem of the equality of volume
and surface porosities in a homogeneous porous medium”
(Faizullaev, 1969). Therefore, the same porosity term can
be applied to the body force as to the surface tractions.

∂ u w
∂ y

n w ( u w + )

dy

dy

n w

r w g

F sy

y dz

3.3.4 Water Phase Equilibrium
The water phase is used to illustrate how the equilibrium
equation can be written for an individual phase (Fig. 3.6).
Let us consider the case of a saturated soil (i.e., a two-phase
system). It is necessary to include an interaction force which
acts between the soil particles and the water when the water
phase is separated from the remainder of the unsaturated
soil element. The interaction force is a body force and can
be given the designation F sy . The water phase equilibrium
in the y -direction also includes the gravity force due to the
volume of water. Summing forces in the y -direction for the
water phase gives
n w

x

z

n w u w

dx

Figure 3.6 Surface tractions and body forces associated with
water phase force equilibrium in y-direction for saturated soil.

Since the sum of the parts is equal to the overall element,
it is possible to subtract the air phase equilibrium equation
from the total or overall equilibrium equation. The resulting
equation is the equilibrium equation for the soil solids when
the pore fluid is air. The resulting equation contains the
surface tractions equal to the difference between total stress
and pore-air pressure.

F sy dx dy dz

∂u w
∂y

+ n w ρ w g + = 0 (3.18)

where:

3.3.6 Contractile Skin Equilibrium
The contractile skin is only a few molecular layers in thick-
ness. However, its presence affects the equilibrium con-
ditions in an unsaturated soil. The contractile skin influ-
ences the soil structure through its ability to exert surface
tension, T s . The contractile skin also allows for equilib-
rium conditions to be maintained between the air and water
phases. That means that constant-degree-of-saturation condi-
tions can be established in an unsaturated soil. It also means
that it is possible for air and water to flow through the soil
in an independent manner. The independent flow of air and
water takes place while the contractile skin maintains a con-
stant degree of saturation in the soil.
The magnitude of the unit surface tension is constant at
a particular temperature. The interdependency of the con-
tractile skin with the other phases causes its equilibrium
equation to be somewhat complex (Fredlund and Rahardjo,
1993a). Details of the analysis are not presented herein
(Fredlund and Morgenstern, 1977).

u w =

pore-water pressure,

ρ w = density of water,

n w = porosity with respect to the water phase, and

F sy

= interaction force (i.e., body force) between

the water phase and the soil particles in the
y -direction.

The interaction force between the water and the soil solids,
F sy , has the form of a “seepage force” associated with water
flow through a saturated soil. Similar equilibrium equations
can be written in the x- and z- directions for the water phase.
Since the sum of the parts (i.e., each of the phases) is equal
to the overall element, it is possible to subtract the water phase
equilibrium equation from the total or overall equilibrium
equation. The resulting equation is the equilibrium equation
for the soil solids. The resulting equation contains the surface
tractions known as the effective stress variable.

3.3.5 Air Phase Equilibrium
Let us consider the case where a soil is completely dry.
Once again the soil is a two-phase system consisting of soil
solids and air. Equilibrium equations can be written for the
air phase in a manner similar to that shown for the water
phase (Fredlund and Rahardjo, 1993a).

3.3.7 Equilibrium for the Soil Structure
(Arrangement of Soil Particles)
The total equilibrium of an unsaturated soil element is equiv-
alent to the resultant of the equilibrium equations for the
individual phases. The equilibrium of the soil structure in