Compressibility and Pore Pressure Parameters - Unsaturated Soil Mechanics in Engineering Practice - page 797

relationship for the soil structure. The increase in total
stress results in an increase in net normal stress, d(σ − u a ) ,
and a decrease in matric suction, d(u a − u w ) . The increase
in net normal stress causes a decrease in volume:
dV ν
V 0

also extremely small because of the low compressibility of
the water.
The pore voids in a dry soil are primarily filled with air,
which is much more compressible than the soil structure. The
pore-air pressure is low and the isothermal compressibility
of air is equal to the inverse of the absolute air pressure. A
total stress increase dσ during undrained loading is almost
entirely transferred to the soil structure. The pore pressure
remains practically constant or stated another way, the pore
pressure parameter approaches zero. The volume change can

1 = m 1 d(σ − u a )

(15.30)

where:

be computed from the change in net normal stress σ − u a ,
in accordance with the constitutive relationship for the soil
structure. Volume changes associated with a change in matric
suction are negligible for a dry soil. The compression of the
air during undrained loading follows the compression of the
soil structure. A substantial volume change is required to
produce a significant pore-air pressure since the air phase
has a high compressibility.

( dV ν /V 0 ) 1 = volume change due to the change in net

normal stress, d(σ − u a ) , referenced to the
initial total volume.

The decrease in matric suction generally causes an
increase in soil volume (Fig. 15.12b):
dV ν
V 0

2 = m 2 d(u a − u w )

(15.31)

where:

15.5.1 Total Stress and Soil Anisotropy
The pore pressure parameters can be derived for various
conditions of loading and soil anisotropy. Possible loading
conditions are similar to those outlined by Lambe and Whit-
man (1979) for saturated soils. These loading conditions are
summarized in Fig. 15.13, where increments in the major,
intermediate, and minor principal stresses are denoted as
dσ 1 ,dσ 2 , and dσ 3 , respectively. For K 0 loading, the soil can
only change volume in one direction (i.e., the y -direction).
For the other loading conditions, the soil can change vol-
ume three dimensionally (i.e., x -, y -, and z -directions). The
loading conditions differ in the magnitude and direction in
which total stresses are applied.
An isotropic soil is defined as a soil with compressibility
values that are constant with respect to different directions.
That is to say, the magnitudes of m 1 and m 1 are constant in
the x -, y -, and z -directions. Only the constitutive equations
for the soil structure and the air phase have been selected
for the derivation of the pore pressure parameters.
Soil anisotropy is herein defined as the condition where
the soil compressibility with respect to a change in each
stress state variable varies with direction. The number of
independent modulus values required depends on whether
the longitudinal strain in each direction is to be predicted
independently or whether three-dimensional volume changes
are to be predicted. As a result, the compressibility m 1 varies
in the x -, y -, and z -directions and is designated as m 11 ,m 12 ,
and m 13 , respectively. Similarly, the air phase has compress-
ibility values m 11 ,m 12 ,m 13 in the x -, y -, and z -directions,
respectively. However, the compressibility with respect to a
change in matric suction can be assumed to produce singular
compressibility values (i.e., m 2 and m 2 ).
The following sections present the derivations of pore
pressure parameters for an isotropic soil under one-
dimensional loading (i.e., K 0 loading). The volume change
modulus for K 0 loading will be given the subscript k (i.e.,

( dV ν /V 0 ) 2 = volume change due to a change in matric

suction, d(u a − u w ), referenced to the ini-
tial total volume.

The matric suction decrease yields a volume increase or
swelling. The m 2 value for the unloading path must be used
in Eq. 15.31. The total volume change obtained from the
constitutive relations for the soil structure can be written as
the sum of Eqs. 15.30 and 15.31:
dV v
V 0

2 = m 1 d(σ − u a ) + m 2 d u a − u w
(15.32)
The total volume change obtained from the constitutive
relationship can be equated to the volume change due to
pore fluid compression:
dV ν
V 0

dV v
V 0

1 +

1 +

dV ν
V 0

dV ν
V 0

2 =

(15.33)

0

or

m 1 d σ − u a + m 2 d u a − u w = C aw ndσ

(15.34)

In a saturated soil, the pore voids are filled with water.
The pore fluid compressibility is equal to the compressibility
of water, which is far less than the compressibility of the soil
structure. A total stress increase dσ in undrained loading is
almost entirely transferred to the water phase (i.e., du w ≈
dσ ), or stated another way, the pore-water pressure parame-
ter approaches 1.0. The effective stress in undrained loading

remains essentially unchanged [i.e., d σ − u w ≈

0 . 0]. As
a result, the volume change computed from the constitutive
relationship for the soil structure is extremely small. The soil
volume change obtained from the pore fluid compression is