Stress-Deformation Analysis for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 683

in Fig. 13.7. A positive or negative sign is associated with
the deformation or stress state variable change in order to
indicate an increase or decrease. The following sign conven-
tion can be formulated for multiphase materials. A positive
change in the state variable (i.e., deformation or stress)
refers to an increase in the state variable, whereas a negative
change indicates a decrease in the state variable. The signs
of the deformation state variables and the stress state
variables determine the sign for the volumetric deformation
coefficients. For example, the stable-structured soil behavior
shown in Fig. 13.7 has a negative sign for the modulus of
elasticity E . This means that an increase in the net normal

stress, ( + ) d σ − u a , causes a decrease in volumetric

strain, ( − ) dε v [i.e., ( − ) E = ( − ) dε v / ( + ) d σ − u a ].
The elastic modulus H also has a negative sign [e.g.,

( − ) H = ( − ) dε v / ( + ) d u a − u w ].
The relationship between the water volume change and
the stress state variable change for a stable-structured soil
is similar to that shown in Fig. 13.7. The water volumetric
moduli E w and H w would also have negative signs. There-
fore, it can be concluded that a stable-structured soil has
negative volumetric deformation moduli associated with the
soil structure and water phase. The negative sign applies
similarly to all coefficients of volume change, m 1 , m 2 , m 1 ,
and m 2 , which are used in the compressibility form of the
constitutive equations. The a t , a m , b t , and b m coefficients
associated with the soil mechanics form of the constitutive
equations are also negative for a stable-structured soil.
A collapsing soil is referred to as a metastable-structured
soil. A decrease in matric suction results in swelling for a
stable-structured soil whereas it may cause a volume decrease
in a metastable-structured soil (Barden et al., 1969a). The m 2
and m 2 coefficients of volume change have a positive sign
for collapsing soils (Eqs. 13.53 and 13.54). A metastable-
structured soil can be referred to as a soil that has one or
more positive volumetric deformation moduli.

Figure 13.13 Testing for uniqueness of soil structure constitutive
surface: constitutive surface verification using (a) three identical
soil specimens and (b) a single soil specimen.

increase in net normal stress, d σ − u a , by increasing
the total normal stress while the matric suction u a − u w is
maintained constant (i.e., 0-1 loading path in Fig. 13.13a).
The measured volumetric strain change dε v can be used to
compute the coefficient of volume change m 1 :

13.4.5 Verification of Uniqueness of Constitutive
Surfaces Using Small Stress Changes
Let us suppose that several so-called identical unsaturated
soil specimens are subjected to the same total normal stress
and the same pore-air and pore-water pressures. All spec-
imens will be at the same state point in space and have
the same initial volume-mass properties. Let each specimen
then be subjected to different stress path changes while the
volume changes are monitored. If the constitutive surface is
essentially planar near a state point, the volumetric deforma-
tion moduli associated with any two orthogonal directions
can be used to describe the deformation produced by other
stress state variable changes.
A somewhat simpler test for uniqueness can be described
as follows. Suppose that three so-called identical specimens
are prepared at the same initial volume-mass properties
and subjected to the same initial stress state variables
(Fig. 13.13a). Let one specimen be subjected to a small

dε v

m 1 =

d σ − u a

(13.60)

A second specimen could be subjected to a small increase

in matric suction, d u a − u w , by decreasing the pore-water
pressure u w while maintaining a constant net normal stress
(i.e., 0-2 loading path in Fig. 13.13a). The measured change
in volumetric strain, dε v , can be used to calculate the m 2
coefficient of volume change:

dε v

m 2 =

d u a − u w

(13.61)

A third specimen could then be subjected to small increases

in both the net normal stress d σ − u a and the matric suc-

tion d u a − u w . This can be achieved by increasing the
total normal stress σ and decreasing the pore-water pres-
sure u w . The anticipated volume change can be predicted