Compressibility and Pore Pressure Parameters - Unsaturated Soil Mechanics in Engineering Practice - page 807

15.7 RHEOLOGICAL MODEL TO REPRESENT
RELATIVE COMPRESSIBILITIES OF
UNSATURATED SOIL

Another linear rheological element is used to model the
effect of the solid phase and the air phase (i.e., σ − u a ).
The two linear elements are connected in series and
represent the two independent stress state variables.
The Hookean constants c w and c a account for the com-
pressibility of water and air, respectively. The fluids in the
dashpots are assumed to be incompressible, but the escape
or outflow of air and water is allowed. The Hookean spring
constants κ w and κ a account for the compressibility of soil
structure with respect to each of the stress state variables
σ − u w and σ − u a , respectively, where σ is the total stress
applied, u w is the pore-water pressure, and u a

The primary value of rheological models lies in their role as
an aid in visualizing soil behavior. For example, a rheolog-
ical model can illustrate how a load applied to a saturated
soil is initially carried by the water phase and is slowly
transferred to the soil structure as the pore-water pressure
dissipates.
Two linear rheological elements connected in series, one
for the water phase and the other for the air phase, can
be used to simulate the behavior of a layer of unsaturated
soil (Fig. 15.27). The unsaturated soil stratum is simulated
by interconnecting two rheological elements in series: one
element represents the air portion and another element rep-
resents the water portion of the soil.
Fredlund and Morgenstern (1977) showed that it was
possible to use any two of the three possible stress state
variables (i.e., σ − u w ,σ − u a , and u a − u w ) when describ-
ing soil behavior. It appears that the σ − u w and σ − u a
combination has advantages over other combinations for
the visualization of drained and undrained processes in an
unsaturated soil. Consequently, the σ − u w and σ − u a
combination of stress state variables will be used for the
development of a rheological model to simulate unsaturated
soil behavior.
One linear rheological element is used to model the
effect of the solid phase and the water phase (i.e., σ − u w ).

is the pore-air

pressure.
The hydraulic conductivity properties of the soil with
respect to water and air are represented by the viscosity
constants η w and η a , respectively. The total strain of a unit
infinitesimal layer is denoted by ε . The strain of the linear
rheological elements for the water phase and the air phase
are denoted by ε w and ε a , respectively.

15.7.1 Effect of Relative Compressibility
Values on Initial Pore Pressures Generated upon
Loading
A portion of the load applied to an unsaturated soil will be
carried by each of the linear rheological models. In other
words, there will be a sharing of the applied load between
the air phase and water phase rheological elements. The
amount of load carried by the rheological model for the
air phase and the water phase will depend upon the rela-
tive soil compressibility values associated with the σ − u w
and σ − u a stress state variables as well as the compress-
ibility values associated with the air and water phases. The
compressibility of water is low and the compressibility of
air is high and highly nonlinear. The nonlinearity of the
air phase is not taken into consideration in the rheological
models.
Let us suppose that a pressure of 100 kPa represents a
total stress σ which is instantaneously applied to the rheo-
logical models that are connected in series. The total applied
stress will immediately be shared by the rheological model
associated with the air and water phases. The sharing of
the total stress will depend upon the magnitude of the four
relative compressibility values. Since the compressibility of
the air phase is much greater than the compressibility of
the water phase, the water phase rheological model will
experience the greatest initial change in pore pressure. The
instantaneous change in pore-air pressure (i.e., excess pore
air pressure) and the instantaneous change in pore-water
pressure (i.e., excess pore-water pressure) will then dissi-
pate over time. The dissipation of pore-air and pore-water
pressures with time represents independent processes that

s

Maxwell
portion
for water

η w

Hookean
portion for
s - u w

κ w

c w

Maxwell
portion
for air

η a

Hookean
portion for
s

κ a

- u a

c a

s

Figure 15.27 Use of two linear rheological elements in series
to represent the behavior of the σ − u w and σ − u a

stress state

variables for unsaturated soil.