Stress-Deformation Analysis for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 670

In 1977, Fredlund and Morgenstern proposed semiempir-
ical constitutive relations for an unsaturated soil by using
any two of the three independent stress state variables. The
proposed equations were similar in form to those proposed
by Biot (1941) and Coleman (1962). The deformation state
variables used to describe volume changes needed to satisfy
continuity among the phases of an unsaturated soil (Fred-
lund, 1973a). The proposed stress and deformation state
variables were used to form constitutive relations for the
soil structure, air phase, and water phase. However, only two
of the three constitutive relations were required to describe
changes in the relative volume of each phase. The constitu-
tive relations for the soil structure (i.e., arrangement of soil
particles) and the water phase appeared to be most appro-
priate for use in a volume change analysis. Volume changes
associated with the soil structure and the water phase are
often written in terms of void ratio change and water con-
tent change in geotechnical engineering practice. Volume
changes associated with the air phase can be computed as
the difference between the overall volume change (i.e., soil
structure constitutive surface) and the water volume change.
The proposed constitutive relations can be graphically pre-
sented in the form of a deformation state variable versus two
independent stress state variables. The proposed constitutive
surfaces were experimentally tested for uniqueness near a
point by Fredlund and Morgenstern (1976). Four series of
experiments were performed involving undisturbed Regina
clay and compacted kaolin. The specimens were tested under
K 0 and isotropic loading conditions using a modified oedome-
ter and a triaxial apparatus, respectively. The total, pore-air,
and pore-water pressures were independently controlled dur-
ing the tests. The results indicated uniqueness for the consti-
tutive relations as long as the volume change of each phase
was monotonic. Uniqueness of the constitutive surfaces under
larger stress increments was experimentally tested by Matyas
and Radhakrishna (1968) and Barden et al., (1969a).
A number of volume-mass constitutive models based on
the critical state concepts of elastoplasticity have been pro-
posed in the literature (Alonso et al., 1987, 1990; Wheeler
and Sivakumar, 1995; Delage and Graham, 1995; Chiu and
Ng, 2003; Tamagnini, 2004; Thu et al., 2007; Sheng et al.,
2008a). These models are considered to be outside the scope
of this topic.
Pham and Fredlund (2011a) proposed a volume-mass con-
stitutive model for unsaturated soils (Pham, 2005, Fredlund
and Pham, 2006; Pham and Fredlund, 2011a). The soil speci-
mens started as a slurry material and considered the indepen-
dent behavior of the void ratio and water content constitutive
relations over a wide range of loading and unloading stress
paths. Hysteresis associated with loading and unloading as
well as drying and wetting was taken into consideration.
The model for hydraulic hysteresis was consistent with those
developed in soil physics. The hysteresis model was incor-
porated into a stress path-dependent constitutive model. The

concepts of the Pham and Fredlund (2011a) model are pre-
sented later in this chapter.
It is advantageous to commence a fundamental study of
the constitutive behavior of an unsaturated soil by starting
with a saturated soil. The saturated soil can then be sub-
jected to ever-increasing matric suction values to study the
change in overall volume as well as changes in the relative
volumes of air and water. It is important that a smooth tran-
sition exists between saturated and unsaturated soil states
as matric suction is increased and the soil desaturates. The
reverse is also true; namely, there should be a smooth tran-
sition as a soil moves toward the saturated state through a
decrease in matric suction. The Pham and Fredlund (2011a)
model is capable of (i) predicting water content change, (ii)
predicting void ratio change taking into account both elastic
and plastic strains, and (iii) taking into account hysteretic
behavior of the SWCC when following a wide range of
possible stress paths. Isotropic loading conditions were mod-
eled. A differentiation is made between elastic and plastic
strains; however, the model does not take into consideration
the independent influence of σ 1 ,σ 2 , and σ 3 . In this sense,
the model is not a critical state or elastoplastic model.

13.2 CONCEPTS OF VOLUME CHANGE
AND DEFORMATION

Volume changes in an unsaturated soil can be expressed in
terms of deformations or relative movement of the phases
of the soil (i.e., relative volumes of the various phases). The
deformation state variables need to be consistent with multi-
phase continuum mechanics principles. A change in the rel-
ative position of points or particles in a body forms the basis
for establishing deformation state variables. These variables
should produce the displacements of the body under consid-
eration when integrated over the body. This concept applies
to a single or multiphase system and is independent of the
physical properties.
Two sets of deformation state variables are required to
adequately describe the volume changes associated with an
unsaturated soil. The deformation state variables associated
with the soil structure and the water phase are commonly
used in a volume change analysis (Biot, 1941; Coleman,
1962; Matyas and Radhakrishna, 1968). Void ratio changes
or porosity changes can also be used as deformation state
variables representing soil structure deformation. Changes
in the volume of water in the soil can be considered as the
deformation state variable for the water phase.
There are several ways to describe the relative move-
ment or deformation in a phase from the standpoint of con-
tinuum mechanics kinematics. Two of these descriptions
are mentioned for consideration relative to unsaturated soil
behavior. The position of each particle is described as a
function of its initial position and time for a referential
description. In other words, the original position and time
are the independent variables. A fixed element of mass is