Stress-Deformation Analysis for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 694

The pores of a soil are comprised of a variety of shapes
and volumes. The pore-size distribution curve of a soil rep-
resents the volume and the shape of pores in the soil. The
pore-size distribution of a soil at any stress state provides
information related to the total volume of voids and the vol-
ume of water in the soil. The pore-size distribution curve
is changed when soil suction or mean net normal stress is
changed (Fig. 13.29). Water volume changes are related to
the pore-size distribution curve.
A comprehensive volume-mass constitutive model
for unsaturated soils should include (i) the selection of
appropriate stress state variables along with a specified
reference stress state; (ii) statements regarding the basic
assumptions for the response of a pore volume to changes in
net mean stress and soil suction; (iii) the formulation of the
stress-strain relationship for the soil structure surrounding
a group of pores, including changes to air-entry value,
water-entry value, yield stress, total volume, and water
volume; (iv) the determination of the compression (i.e.,
unloading/reloading indices) for groups of pores; and (v) an
accommodation of the hysteretic nature of the SWCC.

stress p can be assumed to control the mechanical behavior
of the soil structure surrounding the pore. Soil suction ψ
provides information as to when a pore is filled with water
or empty of water.
A mean net stress state of 1 kPa and zero soil suction
have been arbitrarily selected as the reference stress state of
the soil for the development of the proposed volume-mass
constitutive model which commences with a saturated
slurry soil (Pham, 2005). Soil suction plays the same role
as net mean stress (isotropic) when the pores in the soil
are filled with water (Alonso, 1993; Kohgo et al., 1993;
Wheeler and Sivakumar, 1995; Fredlund and Rahardjo,
1993b). The selected reference stress state assumes that the
air-entry value of the soil is greater than 1 kPa. The void
ratio e is selected to represent the overall volume changes
and gravimetric water content w is selected to represent the
amount of water in the soil.

13.6.1.1 Drying and Wetting Pore-Size Distribution
The pore-size distribution function provides basic infor-
mation for the development of a volume-mass constitutive
model for unsaturated soils. Numerous researchers have
described the meaning of the pore-size distribution (Brooks
and Cory, 1964; Fredlund and Xing, 1994; Vanapalli et al.,
1996a; M.D. Fredlund, 2000; Simms and Yanful, 2001; Cui
et al., 2002). The pore-size distribution has also been used
to model the hysteretic nature of the SWCC (Haines, 1930;
Mualem, 1973, 1974; Nimmo, 1992).
The pore-size distribution curve of a soil is generally plot-
ted using a semilogarithm scale for soil suction (M.D. Fred-
lund, 2000; Simms and Yanful, 2001) (Fig. 13.29). There
are two limiting pore-size distribution curves for a soil. The
pore-size distribution can be written as the ratio of pore vol-
ume per unit weight versus the “open pore diameters” (i.e.,
drying suction or air-entry value of the pore), referred to as
the drying pore - size distribution (DPD). The pore-size dis-
tribution can also be written as the ratio of pore volume per
unit weight versus the “body pore diameters” (i.e., wetting
suction or water-entry value of the pore), referred to as the
wetting pore - size distribution (WPD). The DPD and WPD
constitute the initial bounding drying curve and the bounding
wetting curve of the SWCCs for a soil (Fig. 13.30).
The pore-size distribution of a soil changes with soil
suction and net mean stress. Drying a soil from initially
slurry conditions to completely dry conditions (i.e., suction
of 10 6 kPa) gives the reference pore-size distribution curve
for the soil. A soil has two reference pore-size distribution
curves (i.e., one curve with respect to increasing soil suction
and one curve with respect to decreasing soil suction).
The reference DPD provides information regarding the
air-entry value and the volume distribution of the pores
(i.e., the initial bounding drying SWCC). The reference
WPD provides information regarding the water-entry value
and the volume distribution of the pores in the soil (i.e., the
bounding wetting SWCC). The reference DPD and WPD

13.6.1 Stress State and Concept of Pore
Volume Behavior
There are three stress state variables that can be used when
describing the mechanical behavior of a soil as it changes
from a saturated state to an unsaturated state. Under isotropic
loading conditions (i.e., σ x = σ y = σ z ), the state variables
are (i) net mean stress p = (σ 1 + σ 2 + σ 3 )/ 3 − u a , (ii) soil

suction, ψ = (u a − u w ) , and (iii) mean effective stress p ∗ =
(σ 1 + σ 2 + σ 3 )/ 3

− u w . Only two of the stress state vari-
ables are independent when considering any particular phys-
ical process even though there are three possible forms for
the stress state variables.
A pore volume in an unsaturated soil can be either filled
with water or essentially empty of water. The mean effective
stress p ∗ can be assumed to control the mechanical behavior
of the soil structure when a particular pore volume is filled
with water. Similarly, when the pore is empty, the net mean

Initial condition

After a change of
net mean stress or
soil suction

ψ

Log (open - body pore diameter or drying/wetting suction)

Figure 13.29 Changes in pore-size distribution for changes in net
mean stress or soil suction (from Pham, 2002).