State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 82

moment equilibrium and should also be added to the stress
tensor, as shown in the equation
⎡

In the 1970s (Fredlund, 1973; Fredlund and Morgenstern,
1977), a theoretical equilibrium analysis was formulated for
an unsaturated soil element using concepts consistent with
multiphase continuum mechanics. An unsaturated soil had
generally been viewed as a three-phase system; however,
it was shown that the contractile skin (i.e., the air-water
interface) should be introduced as a fourth and indepen-
dent phase when studying the equilibrium conditions for
each phase. The equilibrium analysis on an unsaturated soil
element provided justification for the use of independent
stress state variables for an unsaturated soil. The soil par-
ticles were assumed to be incompressible and the soil was
treated as being chemically inert. These assumptions have
been historically applied in saturated soil mechanics.
The analysis concluded that any two of three possible
stress state variables can be used to describe the stress state
of an unsaturated soil. The three possible combinations
which can be justified as stress state variables for an
unsaturated soil are (1) σ

⎤

σ x − u w τ yx τ zx

⎣ ⎦

τ xy σ y − u w τ zy (3.2)

τ xz τ yz σ z − u w

The effective stress concept provides a fundamental basis
for studying saturated soil mechanics. The effective stress
concept states that all mechanical behavior in a saturated
soil is governed by effective stresses (and shear stresses) in
each of the three Cartesian coordinate directions. Changes
in volume and shear strength are controlled by changes in
effective stress. An effective stress change (i.e., a change in
pore-water pressure or a change in total stresses) will alter
the equilibrium state of a saturated soil. Consequently, the
effective stress variables qualify as stress state variables.

3.1.3 Background of Stress State for Unsaturated Soils
In 1941, Biot proposed a general theory of consolidation for
an unsaturated soil with occluded air bubbles. The consti-
tutive equations relating stress and strain were formulated
in terms of two independent stress state variables, namely,
effective stress ( σ

− u a and u a − u w ,(2) σ − u w

and u a − u w . Out of the three

possible combinations of stress state variables that can be
justified, it is the σ

u w , and (3) σ − u a and σ −

u w combination that
received the widest acceptance in formulating unsaturated
soil mechanics problems.
The stress state variables for an unsaturated soil take on
the form of two independent stress tensors when consider-
ing a three-dimensional Cartesian coordinate system. The
proposed stress state variables for unsaturated soils were
experimentally tested by Fredlund (1973a) and subsequently
used to formulate constitutive equations to describe shear
strength behavior and volume change behavior.
Stress tensors that contain stress state variables form the
basis for developing a science for both saturated and unsat-
urated soils. It is possible to write first, second, and third
stress invariants for each stress tensor. While it is not imper-
ative that the stress invariants be used in developing consti-
tutive models, the stress invariants should be given consid-
eration because all three Cartesian coordinates are indepen-
dently taken into consideration.
In summary, it is the two independent stress tensors con-
taining stress state variables [e.g., net normal stress ( σ

− u a and u a −

u w ) and pore-water pressure, u w .Itwas
recognized that there needed to be a separation between
the effects of total stress changes and pore-water pressure
changes when attempting to describe unsaturated soil con-
stitutive behavior.
Coleman (1962) suggested the use of three “reduced”
stress variables, namely, σ 1 −

−

u a ,
to represent the axial, confining, and pore-water pressures,
respectively, in the interpretation of triaxial test results. Con-
stitutive relations for volume change in unsaturated soils
were then formulated in terms of the above stress variables.
In 1963, Bishop and Blight reevaluated their previously
proposed effective stress equation for unsaturated soils and

u a , σ 3 − u a , and u w −

noted that a variation in matric suction u a − u w did not

result in the same change in soil behavior as did a change

in the net normal stress σ

u a . Laboratory test results
were presented using three-dimensional graphical plots with
matric suction and net normal stress forming independent
orthogonal axes. In other words, net normal stress and matric
suction were presented as independent stress variables.
Matyas and Radhakrishna (1968) introduced the concept
of “state parameters” in describing the volume change behav-
ior of unsaturated soils. Volume change was presented as a
three-dimensional graphical surface with respect to two state
parameters, namely, σ

−

−

u a ), matric suction ( u a − u w ), and shear stresses] that form

a fundamental basis for the development of a science for
unsaturated soil mechanics. Constitutive relationships con-
necting various state variables can then be used in con-
junction with soil properties (and soil property functions) to
solve practical engineering problems. All proposed consti-
tutive relationships must be tested for uniqueness in the lab-
oratory on a variety of soil types. The laboratory equipment
must be able to independently control each stress component
of the stress state variables.

u w . Changes in degree
of saturation were likewise plotted versus the matric suc-
tion and net normal stress indicating that two independent
constitutive relations were required for the complete char-
acterization of volume-mass behavior. Barden et al. (1969a)
also suggested that volume change in unsaturated soils be ana-
lyzed in terms of separate components of net applied stress

− u a and u a −

3.1.3.1 What about Effective Stress Equations?
There have been numerous equations proposed that relate
one or more of the stress variables to other stress variables

σ

u a and matric suction u a − u w .

−