State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 83

through the inclusion of a soil property. It is important to
differentiate between the role of these equations and the
description of the stress state at a point in a continuum.
There are theoretical and formulation limitations associated
with the use of effective stress equations. Little attention
is given to these equations in this topic; however, a brief
summary is given of effective stress equations that have
been proposed.
The oldest and most-often-referred-to single-valued effec-
tive stress relationship is that proposed by Bishop (1959).
The equation is commonly referred to as Bishop's effective
stress equation for unsaturated soils and has the form
σ = σ

where:

σ ij =

total stress tensor,

δ ij =

Kroneker delta or substitution tensor,

σ ij = Bishop's average soil skeleton stress, and

S = degree of saturation.

The degree of saturation has been substituted for the χ
soil parameter in Eq. 3.4. The equation is once again empir-
ical and constitutive in character. Consequently, the equation
must face the rigor of tests for “uniqueness” in order to
determine whether it proves satisfactory for geotechnical
engineering practice. It is the philosophical and theoretical
justifications that present the greatest hurdles with regard to
acceptance of Eqs. 3.3 and 3.4.
It can be reasoned that the need for an effective stress
equation for unsaturated soils has been replaced through
the use of independent stress tensors containing stress
state variables. Reexamination of proposed effective stress
equations has led many researchers to conclude that inde-
pendent stress state variables (e.g., σ

u a + χ u a − u w

− (3.3)

where:

σ =

effective stress and

χ = a soil parameter related to degree of saturation and

ranging from 0 to 1.

u w )
provide the increased flexibility needed for describing
unsaturated soil behavior. However, there continues to
be ongoing attempts to form a simple, so-called effective
stress relationship between total stresses and soil suction.
The simplified stress component relationship is then used
to develop shear strength and volume change constitutive
relations. This approach places serious constraints on
subsequent formulations and violates the basic assumption
inherent to classical continuum mechanics.

− u a and u a −

Bishop's equation relates net normal stress to matric suc-
tion through the incorporation of a single-valued soil prop-
erty, χ . Bishop's equation should not be referred to as a
fundamental description of stress state for an unsaturated
soil. The equation contains a soil property and should be
referred to as a constitutive equation. Within the context of
continuum mechanics it is not proper to elevate the Bishop
equation to the status of a stress state variable for an unsat-
urated soil.
Morgenstern (1979) explained the limitations of Bishop's
effective stress equation as follows: Bishop's effective
stress equation “proved to have little impact on practice.
The parameter, χ , when determined for volume change
behavior was found to differ when determined for shear
strength. While originally thought to be a function of degree
of saturation and hence bounded by 0 and 1, experiments
were conducted in which χ was found to go beyond these
bounds. The effective stress is a stress variable and hence
related to equilibrium considerations alone.”
Morgenstern (1979) went on to explain: Bishop's effec-
tive stress equation “contains the parameter, χ , that bears
on constitutive behavior. This parameter is found by assum-
ing that the behavior of a soil can be expressed uniquely
in terms of a single effective stress variable and by match-
ing unsaturated soil behavior with saturated soil behavior in
order to calculate χ . Normally, we link equilibrium consid-
erations to deformations through constitutive behavior and
do not introduce constitutive behavior into the stress state.”
Other equation forms similar to the Bishop (1959)
equation have been proposed by several researchers for
the development of unsaturated soil behavior models (e.g.,
Jommi, 2000).

3.1.4 Designation of Deformation State Variables
Deformation state variables are necessary for describing rel-
ative volume changes and distortions of the various phases
comprising an unsaturated soil. Deformation state variables
may take on a variety of forms but must always satisfy conti-
nuity, compatibility, and conservation of mass requirements
for a multiphase system (i.e., Fredlund, 1973a).
The mapping of deformation state changes has histori-
cally started with the definition of selected volume-mass soil
properties such as void ratio e , gravimetric water content
w , and degree of saturation S . These variables are related
to one another through the following basic volume-mass
relationship:

Se = wG s (3.5)

where:

G s = specific gravity of the soil solids.

The basic volume-mass relationship shows that it is nec-
essary to have at least two independent constitutive relations
in order to predict phase deformation state changes for an
unsaturated soil. Changes in void ratio (i.e., distortion and

σ ij = σ ij − ( Su w + ( 1 − S)u a )δ ij (3.4)