State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 81

an isothermal environment. Temperature is known to be
a state variable, but the assumption is usually made that
isothermal conditions exist. When considering unsaturated
soil behavior, however, it becomes increasingly important
to be aware that temperature is a state variable that plays a
prominent role in some physical processes (e.g., evaporation
from ground surface). Unsaturated soils are usually found
near to the ground surface in an environment subjected to
significant changes in thermal conditions on a daily basis.
The basis for the state variables for a material can be
understood by considering the basic conservative laws of
physics, namely, the conservation of energy and the conser-
vation of mass. If the system under consideration is assumed
to be in an isothermal environment, then the state vari-
ables embedded within the conservation of energy are those
related to the stress state. Thermodynamic considerations
are therefore reduced to the conservation of momentum. A
multiphase system can be assumed to be in static equilib-
rium or subjected to negligible velocities with the exception
of earthquake or seismic loadings. Consequently, the sys-
tem reduces to the consideration of Newton's second law of
statics (i.e., summation of forces and moments).
The stress state variables associated with a multiphase
material can be obtained by considering Newtonian equi-
librium for each phase of a multiphase system. Free-body
diagrams with designated spatial variation can be drawn for
each phase of a multiphase system. The superposition of
coincident equilibrium stress fields for a multiphase system
such as saturated or unsaturated soil systems is presented in
this chapter.
Newtonian equilibrium considerations become a theoret-
ical justification for the choice of acceptable stress state
variables. There are specific requirements that must be sat-
isfied when performing an equilibrium type of analysis. For
example, the summation of forces across a wavy plane can-
not be considered as an equilibrium analysis since a wavy
plane does not constitute a legitimate free-body diagram.
Rather, the starting free-body diagram must be multidimen-
sional and have unbiased, smooth surfaces to be acceptable
within the context of continuum mechanics (Fung, 1965).
The state variables associated with states of deformation
must satisfy the conservation of mass. As such, systems of
mapping the movement of each phase of a multiphase sys-
tem must satisfy the conservation-of-mass requirements for
the system as a whole. There are different types of elements
that can be used when defining the deformation state vari-
ables (e.g., spatial element or referential element). An REV
will be used in this topic when describing both the stress and
deformation state variables for saturated and unsaturated soil
systems.
Continuum mechanics has provided a valuable context in
which to study material behavior for engineering purposes.
Continuum mechanics principles provide an important rela-
tionship between the conservative laws of physics and the
development of a science for the behavior of a multiphase

system. The conservative laws reveal the basic “building
blocks” required for the development of a science. These
building blocks are called state variables.

3.1.2 Background on Stress State for Saturated Soils
The effective stress variable σ

u w commonly used in satu-
rated soil mechanics is a stress state variable towhich saturated
soil behavior can be related. The effective stress variable is
applicable to sands, silts, or clays and it is independent of
the soil properties. The volume change process and the shear
strength characteristics of a saturated soil are both controlled
by effective stress variables. The effective stress state variable
can be independently applied in each of the three Cartesian
coordinate directions. In so doing, effective stress takes on
the form of a stress tensor (i.e., a 3

−

3 matrix).
Soil mechanics as a science has been successfully applied
to many geotechnical problems involving saturated soils.
The success of the stress state variables is largely due to the
ability of engineers to uniquely relate observed soil behavior
to stress conditions in the soil. Terzaghi (1936) described the
stress state variables controlling the behavior of saturated
soils as follows: “The stresses in any point of a section
through a mass of soil can be computed from the total prin-
cipal stresses, σ 1 , σ 2 , σ 3 , which act at this point. If the voids
of the soil are filled with water under a stress, u w , the total
principal stresses consist of two parts. One part, u w , acts in
the water and in the solid in every direction with equal inten-
sity. It is called the neutral (or the pore-water) pressure. The
balance, σ 1 =

×

u w , σ 2 = u w , and σ 3 = u w

represents an excess over the neutral stress, u w , and it has its
seat exclusively in the solid phase of the soil. All the mea-
surable effects of a change in stress, such as compression,
distortion, and a change in shearing resistance, are exclu-
sively due to changes in the effective stress σ 1 , σ 2 , and σ 3 .”
Effective stress is a stress state variable for a saturated
soil. Effective stress has also been expressed in the form of
an equation:

σ 1 − σ 2 − σ 3 −

σ =

σ − u w (3.1)

where:

σ =

effective normal stress,

σ =

total normal stress, and

u w = pore-water pressure.

Effective stress is the term used to describe the general
stress state that can subsequently be used to describe the
physical behavior of saturated soils. Experimental evidence
has shown that the effective stress variable (i.e., σ

u w )is
sufficient to describe the mechanical behavior of a saturated
soil. A more complete understanding of the stress state for a
saturated soil is achieved by writing the effective stress vari-
ables for each of the three orthogonal directions to form a
tensor. Shear stress components arise from consideration of

−