Compressibility and Pore Pressure Parameters - Unsaturated Soil Mechanics in Engineering Practice - page 796

V w = volume of water, can be computed by multiplying the compressibility of the

air-water mixture, C a w , by the pore fluid volume V v and the
total stress increment dσ (Fig. 15.12a):
dV ν
V 0

m 1

= coefficient of water volume change with re-

spect to a change in net normal stress, and

m 2 = coefficient of water volume change with re-

spect to a change in matric suction.

0 = C aw ndσ

(15.29)

The continuity requirement for a referential element can
be expressed as

where:

( dV ν /V 0 ) 0 = volume change during undrained compres-

sion resulting from the compression of the
pore fluids referenced to the initial total vol-
ume and

dV v
V 0 =

dV a
V 0 +

dV w
V 0

(15.26)

The following conditions must be satisfied when consid-
ering volumetric continuity:

n = porosity (i.e., V ν /V can be assumed to be

equal to V ν /V 0 for small changes in stress
state variables and small volume changes).

m 1 = m 1 + m 1

(15.27)

m 2 = m 2 + m 2 (15.28)

Volume change can also be written in terms of the
stress state variable changes as defined by the constitutive

The above constitutive relations can be used to compute
volume changes that occur during undrained compression.
The changes in volume that occur between the initial and the
final conditions can be calculated using a marching-forward
technique with finite changes in each of the stress state vari-
ables [i.e., d(σ − u a ) and d(u a − u w ) ]. The volume of the
soil, water, and air must be accounted for after each incre-
ment of total stress.
Although three constitutive relationships (i.e., soil struc-
ture, air, and water) have been presented, only two of the
relationships are required when deriving the pore pressure
parameter equations. The constitutive equations for the soil
structure and the air phase will be selected for the derivation
of the pore pressure parameters.

15.5 DRAINED AND UNDRAINED LOADING

The constitutive relationships obtained from drained loading
can be applied to undrained loading conditions. The applica-
tion of an all-around positive (i.e., compressive) total stress,
dσ, either under drained or undrained loading conditions,
causes a change in volume. In drained loading, air and water
are allowed to drain from the soil subsequent to the applica-
tion of the total stress increment. The stress state variables
in the soil are altered and volume of the soil changes. The
volume change can be computed from the stress state vari-
able changes in accordance with the constitutive relationship
for the soil structure.
In undrained loading, the air and water are not allowed
to drain from the soil. The total stress increase causes the
pore-air and pore-water pressures to increase, and conse-
quently the stress state variables also change. An increase
in the pore fluid pressures occurs in response to a com-
pression of the pore fluid (i.e., mainly air). Volume changes
during undrained loading can be regarded as the volume
change equivalent to the pore fluid compression. The vol-
ume change equivalent to the pore fluid compression, dV ν ,

Figure 15.12 Volume changes in unsaturated soil during loading:
(a) volume change with respect to change in σ − u a or σ ;(b)

volume change with respect to change in u a − u w .