State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 87

Pressure after a 68.9-kPa increase

.3 Null test ( N - 37)

= 615.4 kPa
= 541.2 kPa
= 411.3 kPa

s

.2

u a

Water volume change

.1

u w

0

.1

Total volume change

.2

.2

Total volume change

.1

0

Pressure after a 68.9-kPa decrease

.1

= 549.4 kPa
= 477.1 kPa
= 347.6 kPa

s

Water volume change

.2

u a

.3

u w Null test ( N - 38)

0.1 1.0 10

Elapsed time, t (min)

100 1000 10000

Figure 3.4 Results of null tests N-37 and N-38 on compacted kaolin (from Fredlund, 1973a).

the combination of all the component stress fields. In 1977,
Fredlund and Morgenstern used the concept of multiphase
continuum mechanics to write the equilibrium equations that
act on the soil particle phase (or soil structure) of an unsat-
urated soil. The principle of superposition of coincident
equilibrium stress fields was used as described in continuum
mechanics (Truesdell and Toupin, 1960; Green and Naghdi,
1965; Truesdell, 1966).
The assumption was made that an independent continuous
stress field was associated with each phase of the multi-
phase system. The number of independent force equilibrium
equations that can be written for a multiphase system is
equal to the number of Cartesian coordinate directions mul-
tiplied by the number of phases constituting the continuum.
Equilibrium equations can also be written for various combi-
nations of phases, but the number of independent equations
is limited by the number of phases.
The stress variables appearing in the equilibrium equations
for the soil structure can be taken as the stress state vari-
ables. The stress state variables must be expressed in terms
of the measurable stresses, such as total stress σ , pore-
water pressure u w , and pore-air pressure u a in the case of
an unsaturated soil. It was observed that the surface trac-
tions appearing in the force equilibrium equations for the
soil structure were the same as those for the equilibrium
equations for the contractile skin (i.e., air-water interface). It
was also observed that three possible sets of stress state vari-
ables could be combined to form two stress tensors where
each set of tensors depended on the reference phase used
when deriving the equilibrium equations.

The force equilibrium equations for the multiphase sys-
tem support the concept of independent stress state vari-
ables. However, the derivation cannot be viewed as exclu-
sive proof of a particular set of stress state variables. It
would appear that the selection of the most suitable com-
bination of stress state variables must be based on practical
preference considerations rather than theoretical considera-
tions. There has been a general consensus that σ

− u a and

u a − u w form the most easy-to-use combination of stress

state variables for geotechnical engineering practice. This
combination of stress state variables allows for the indepen-
dent consideration of the effects of external total stresses and
the effect of internal pore-water pressure u w . Each of these
stress variables are referenced to the pore-air pressure u a ,
which in most real-world situations is equal to atmospheric
pressure.

3.3 STRESS STATE VARIABLES
FOR UNSATURATED SOILS

The mechanical behavior of soils is controlled by the same
stress variables that control the equilibrium of the soil struc-
ture. Therefore, the stress variables required to describe the
equilibrium of the soil structure can be taken as the stress
state variables for the soil. The following sections illustrate
the role of equilibrium considerations in providing a clear
understanding of stress state variables in the development
of an engineering science for multiphase systems.

3.3.1 Equilibrium Analysis for One-Phase Solid
There are two types of forces that can act on an element of
soil. These are body forces and surface forces (i.e., surface