Environmental Engineering Reference
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does not affect the soil structure surrounding the pore. The
pore-size distribution function of the soil provides infor-
mation regarding the maximum stress acting on the soil
structure surrounding each pore during the drying process.
Similarly, the maximum stress that acts on the soil struc-
ture surrounding each pore during a wetting process is the
water-entry value of the pore.
Assumption 3. Each water-filled pore in a soil has
two indices: (i) a virgin compression index C c and (ii) an
unloading-reloading compression index C s . The compres-
sion indices of a saturated soil are defined as the change in
void ratio of the soil with respect to a change in logarithm of
mean effective stress under saturated conditions (Pham et al.,
2005). The virgin compression and unloading-reloading
indices of a water-filled pore are defined as follows:
w
w u
Boundary wetting curve
Boundary drying curve
y
Logarithmic soil suction (kPa)
Wetting pore-size
distribution, WPD
v b
v a
V s ζ
dv p
V s d log (p ) =
C c
(p > yield stress)
=
Drying pore-size
distribution, DPD
(13.86)
v b
v a
V s ζ
dv p
V s d log (p ) =
C s
(p < yield stress)
=
(13.87)
where:
y
Log (open/body pore diameter or drying/wetting suction)
C c
=
virgin compression index of the water-filled pore,
Figure 13.30 Change in pore-size distribution resulting from
changes in net mean stress or soil suction (from Pham, 2002).
C s
=
unloading-reloading index of a water-filled pore,
v b
=
volume of the pore prior to an increase in load,
v a
=
volume of the pore after an increase in load, and
ζ
=
infinitesimal increment of logarithmic mean effec-
tive stress.
curves form the basic building units for the development of
a volume-mass constitutive model.
Assumption 4. There are two types of pores: (i) col-
lapsible pores and (ii) noncollapsible pores. Collapsible
pores are relatively large pores while noncollapsible pores
are relatively small interconnected pores (Fig. 13.31).
Noncollapsible pores are assumed to be incompressible.
This assumption is similar to the macro- and microstruc-
tures described by Alonso et al. (1994) and Wheeler and
Sivakumar (1995). Measured pore-size distribution curves for
soils under various loading and compaction conditions show
that micropores do not appear to change significantly when
the volume of the soil changes (Sridharan et al., 1971; Ahmed
et al., 1974; Delage and Graham, 1995; Al-Mukhtar, 1995;
Alonso et al., 1995; Wan et al., 1995; Lloret et al., 2003).
Data showing the effect of net mean stress on the pore-size
distribution of montmorillonite clay are shown in Fig. 13.32.
The assumption related to collapsible and noncollapsible
pores is consistent with the hysteretic nature of SWCCs.
The two types of pores explain the hysteretic phenomena
associated with the SWCC (Fig. 13.33). The drying and
wetting processes associated with the two types of pores
along with the hysteretic SWCCs are shown in Fig. 13.34.
It is assumed that water in the soil exists primarily in
noncollapsible pores once the soil suction exceeds residual
13.6.2 Basic Assumptions for Volume-Mass
Constitutive Model
The Pham and Fredlund (2011) volume-mass constitutive
model is based on several assumptions. These assumptions
are based on the findings from several research studies.
Assumption 1. A particular pore in an unsaturated soil
has one of two possible states: (i) the pore can be filled with
water or (ii) the pore can be empty or essentially dry. This
assumption 1 has been widely accepted in the development
of hysteresis models for SWCCs (Neel, 1942, 1943; Poulo-
vassilis, 1962; Mualem, 1973, 1974; Pham et al., 2005). This
assumption allows pore volumes to be assumed to be either
empty or filled at a designated suction.
Assumption 2. Soil suction affects only the water-filled
pores and does not directly affect the dry-filled pores while
net mean stress has an effect on all pores in the soil. This
assumption is consistent with the mechanical unsaturated
soil behavior suggested by Alonso et al. (1994) and Wheeler
and Sivakumar (1995). The maximum stress acting on the
soil structure surrounding a pore is equal to the air-entry
value of the pore during the drying process under zero net
mean stress. Therefore, the pore is empty and soil suction
 
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