State Variables for Unsaturated Soils - Unsaturated Soil Mechanics in Engineering Practice - page 84

volume changes) are related to directional changes in the
soil structure (i.e., arrangement of the soil solids), form-
ing an REV. Volume changes and distortions can be written
in the form of a strain tensor in a manner consistent with
continuum mechanics (or theory of elasticity or plasticity):

3.1.5 Constitutive Relations Stage
Once the stress state variables and the deformation state
variables are defined for a multiphase system (e.g., an
unsaturated soil), the stage is set for proposing constitutive
relationships that can be used to represent changes in inde-
pendent state variables. Constitutive relations require the
incorporation of soil properties. The types of geotechnical
engineering problems that are addressed will dictate the
number of required constitutive relations. The constitutive
relations stage of development of a science brings in the
need to test the “uniqueness” of any proposed constitutive
relation on a variety of soil types. In other words, there is
no need to test for uniqueness at the state variable stage;
rather, tests for uniqueness are a characteristic associated
with the constitutive stage of the development of a science.
Constitutive relations are largely empirical in nature. Con-
stitutive relations can also be thought of as the stage in the
development of a science where the geotechnical engineer
becomes involved in establishing workable empirical laws
for solving practical engineering problems.
A variety of empirical constitutive models can be pro-
posed for the relationship between the stress and deformation
state variables. In each case assumptions need to be made
with respect to anticipated soil behavior and limitations in
the type of soil behavior being represented. Volume-mass
models can range from those consistent with historical soil
mechanics properties (e.g., coefficient of compressibility and
coefficient of volume change) to incremental elasticity mod-
els (i.e., Young's modulus and Poisson's ratio) (Fredlund and
Rahardjo, 1993a). There can also be elastoplastic constitutive
models that use other soil properties (e.g., λ and κ ) to repre-
sent initial compression and rebound-reloading compression,
respectively, on a semilog scale (Alonso et al, 1990; Wheeler
and Sivakumar 1995, 2003; Blatz and Graham, 2003). A com-
plete critical state model must also embrace shear strength
behavior.
Shear strength models may take the form of an extension
of a Mohr-Coulomb representation (Fredlund et al., 1978)
or the representation of shear strength at critical state con-
ditions (Wheeler and Sivakumar, 1995; Sheng et al., 2009).

⎡ ⎤

ε xx γ yx γ zx

⎣ ⎦

γ xy ε yy γ zy

(3.6)

γ xz γ yz ε zz

where:

ε xx ,ε yy ,ε zz = longitudinal strain in the x -,

y -, and z -directions, respec-
tively, and

γ yx ,γ zx ,γ xy ,γ zy ,γ xz ,γ yz = shear strains on the x-, y -,

and z -planes.

The trace of the strain tensor (i.e., ε xx + ε yy + ε zz ) yields

volumetric strain ε v :

ε v = ε xx + ε yy + ε zz (3.7)

The amount of air and water in the same REV can be
described on a volumetric basis, θ a (volumetric air con-
tent) and θ (volumetric water content). The overall volume
change ε v of a REV must equal changes in the volumes
representing air and water:

ε v = dθ + dθ a (3.8)

Equation 3.8 can be referred to as a volumetric requirement
for a referential element. Volume changes associated with the
air and water phases refer to changes in the volume of the
referential element that is represented by air and the volume
that is represented by water. In other words, when reference is
made to a change in water volume, dθ , in the REV, reference
is being made to a change in the portion of the REV that is
comprised of water. The same logic is true for the air phase.
Even though the air phase is highly compressible, a change
in the volumetric air content, dθ a , simply refers to a change in
the portion of the REV that is comprised of air. A change in
the volume of air in the REV, for example, could be the result
of overall volume changes in the REV or the result of flow
of either water or air from the element.
The overall volume change of a REV can be due to strains
in the x-, y-, and z- directions; however, the terms dθ and
dθ a should not be considered as strains. It is for this reason
that the term deformation state variables is used rather than
strain state variables when referring to the water and air
phases.

3.2 BASIS FOR STRESS STATE VARIABLES

Let us further examine the justification for the proposed
stress state variables. There does not appear to be any theory
which can be used to prove that a particular set of stress state
variables must be used as the basis for formulating a sci-
ence for unsaturated soils. The selection of the most suitable
stress state variables should be judged based on the follow-
ing criteria: (1) variables that can be experimentally tested
in the laboratory, (2) variables that can be theoretically jus-
tified using equilibrium considerations, (3) variables where
the component stresses can be measured in engineering prac-
tice, and (4) variables that meet the requirements set out by
the definition of state variables in continuum mechanics.