Stress-Deformation Analysis for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 685

for the soil structure. This was attributed, in part, to the
difficulty of measuring small water volume changes over
long periods of time (Fredlund and Morgenstern, 1976). It
was also discovered that some moisture was lost from the
soil specimens.

elasticity form, the E , H , μ , E w , and H w parameters are
required to define the volumetric deformations associated
with the soil structure and water phase constitutive equations.
In the compressibility form, the soil structure and water phase
constitutive surfaces make use of m 1 , m 2 , m 1 , and m 2 vol-
umetric coefficients. These coefficients of volume change
can be expressed in terms of the above elastic parameters,
as shown in Table 13.1. In soil mechanics terminology, a t ,
a m , b t , and b m coefficients are used to define the slope of the
void ratio and water content relations versus the respective
stress states.
This section discusses the relationships among the various
volumetric deformation coefficients. Theoretical and experi-
mental methods that can be used to obtain these relationships
are briefly discussed. Emphasis is given to the coefficients
used in the compressibility and soil mechanics forms of the
constitutive surfaces (i.e., m 1 , m 2 , m 1 , m 2 and a t , a m , b t ,
b m ). These coefficients can, in general, be obtained from
various laboratory tests. In some cases modifications must
be made to the laboratory equipment in order to properly
perform the tests.

13.4.6 Verification of Constitutive Surfaces Using
Large Stress State Variable Changes
Matyas and Radhakrishna (1968) experimentally tested the
void ratio and degree-of-saturation constitutive surfaces for
uniqueness under conditions of large stress state changes.
Soil specimens consisting of 80% flint powder and 20%
kaolin were prepared using static compaction with the same
compaction effort. The initial water contents and dry densi-
ties were also the same. The specimens were tested under
isotropic and K 0 loading with controlled total, pore-air, and
pore-water pressures. The tests were performed using a mod-
ified triaxial apparatus.
The constitutive surfaces obtained from isotropic loading
are shown in Fig. 13.1 and the results for K 0 loading were
shown in Fig. 13.2. It was found that the change in void
ratio between any two state points was independent of the
deformation-stress path followed for monotonic deforma-
tion. Monotonic deformation was obtained by following paths
with increasing degrees of saturation where the specimens
were not allowed to swell. The degree-of-saturation constitu-
tive surface, however, did not indicate complete uniqueness.
The constitutive surfaces obtained from K 0 loading are
presented in Fig. 13.2. All test paths appeared to form a sin-
gle warped void ratio surface (Fig. 13.2a). Collapsing soil
behavior was again observed. The degree of saturation was
found to be more sensitive to stress state changes than was
the void ratio constitutive surface (Fig. 13.2b). The unique-
ness of both constitutive surfaces was again restricted to
monotonic deformation (Matyas and Radhakrishna, 1968).
Barden et al., (1969a) studied the volume change charac-
teristics of unsaturated Westwater and Derwent clays under
K 0 loading conditions. The constitutive surface was traced
following different test paths through various combinations
of stress state variables σ y − u a and u a − u w , as shown in
Fig. 13.3. The results were earlier presented in Figs. 13.4
and 13.5. The results exhibited uniqueness or loading path
independence as long as deformations were monotonic. Hys-
teresis associated with wetting and drying processes was
considered to be the major cause of loading path depen-
dence. The degree-of-saturation constitutive surface showed
uniqueness provided the degree-of-saturation change was
monotonic.

13.5.1 Relationship of Volumetric Deformation
Coefficients for Void Ratio and Water Content Surfaces
The void ratio and water content surfaces for an unsaturated
soil are illustrated in Fig. 13.15. The orientation of each
point on the void ratio surface can be defined in terms of two
slope angles. The first slope angle is referenced to net normal
stress and is defined by the coefficient of compressibility, a t :

∂e
∂ σ − u a

a t = (13.62)

The second slope angle is referenced to matric suction and
is defined by the coefficient of compressibility a m :

∂e
∂ u a − u w

a m = (13.63)

Similarly, a point on the water content surface has two
slopes, which can be defined as b t and b m with respect to
the net normal stress and matric suction, respectively:

∂ w
∂ σ − u a

b t = (13.64)

∂ w
∂ u a − u w

b m = (13.65)

Both the void ratio and water content surfaces are gener-
ally nonlinear. Therefore, the four coefficients a t , a m , b t , and
b m vary over the constitutive surfaces. In other words, these
coefficients are a function of the state point on the surface.
However, the slopes on the constitutive surface have some
limiting conditions as observed from experimental results.

13.5 RELATIONSHIP AMONG VOLUMETRIC
DEFORMATION COEFFICIENTS

The constitutive equations or surfaces for unsaturated soils
have been formulated and presented in three forms. In the