Environmental Engineering Reference
In-Depth Information
chosen for a referential element and its motion is traced. The
description is generally called the Lagrangian description
when the reference configuration is the initial configura-
tion. The time variable disappears when equilibrium condi-
tions are achieved. The referential description is commonly
used in problems involving the elasticity of solids where
the initial geometry, boundary, and loading conditions are
specified.
The position of each particle is described as a function
of its current position and time for a spatial description of
deformation. A fixed region in space is chosen instead of an
element of mass. The spatial description is generally used in
fluid mechanics and is referred to as the Eulerian description.
The time variable vanishes under steady-state flow condi-
tions. The Lagrangian and the Eulerian descriptions give
the same results for cases with infinitesimal deformation.
The continuity requirement can be rewritten as follows for
the case where the contractile skin volume change is internal
to the element:
V
v
V
0
V
w
V
0
V
a
V
0
=
+
(13.2)
The above continuity requirement for an unsaturated soil
shows that the volume changes associated with any two of
the three volume variables must be measured while the third
volume change can be computed. In practice, the overall
and water volume changes are usually measured while the
air volume change is calculated. Suitable deformation state
variables can now be defined that are consistent with the
“continuity requirement.”
13.2.2 Overall Volume Change
The overall or total volume change of a soil refers to the vol-
ume change of the soil structure. Consider a two-dimensional
representation of a referential element of unsaturated soil as
shown in Fig. 13.6. The element is referenced to a fixed mass
of soil particles. The element has infinitesimal dimensions of
dx, dy
, and
dz
in the
x-, y-,
and
z-
directions, respectively.
Only the
x-
and
y-
directions are shown in Fig. 13.6.
The soil element is assumed to undergo translations of
u,
v,
and
w
from their original
x-, y-,
and
z-
coordinate posi-
tions, respectively. The final position of the element becomes
x
+
u
,
y
+
13.2.1 Continuity Requirements
A saturated soil is visualized as a fluid-solid multiphase. The
soil particles form a structure with voids filled with water.
The soil structure deforms and changes volume under an
applied stress gradient. The soil structure volume change
represents the overall volume change of the soil. Overall
volume change must be equal to the sum of changes in vol-
umes associated with the solid phase (i.e., soil particles) and
the fluid phase (i.e., water). The equality concept associated
with a multiphase system is referred to as the “continuity
requirement” (Fredlund, 1973a).
The continuity requirement is a volumetric restriction
that prevents “gaps” from forming between the phases of a
deformed multiphase system. The volumetric requirement
also ensures that conservation of mass is maintained. Volume
changes in a saturated soil are primarily the result of water
flowing in or out of the soil since the particles are essentially
incompressible. An unsaturated soil can be visualized as a
mixture with two phases that come to equilibrium under
applied stress gradients (i.e., soil particles and contractile
skin) and two phases that flow under applied stress gradients
(i.e., air and water). Consider an element of soil that
deforms under an applied stress gradient. The total volume
change of the soil element must be equal to the sum of the
volume changes associated with each phase. The continuity
requirement for the unsaturated soil can be stated as follows
for the case where the soil particles are assumed to be
incompressible:
V
v
V
0
w
. The element is assumed to deform
in response to an applied stress gradient. The deformation
consists of a change in length and a rotation of the element
sides, as illustrated in Fig. 13.6. The changes in length in
the
x-, y-,
and
z-
directions can be written as
(∂u/∂x) dx
,
(∂
v
/∂y) dy
, and
(∂
w
/∂z) dz
. Defining normal strain
ε
as a
change in length per unit length, the normal strains of the soil
structure in the
x-, y-,
and
z-
directions can be expressed as
v
, and
z
+
∂u
∂x
ε
x
=
(13.3)
∂
v
∂y
ε
y
=
(13.4)
∂
w
∂z
ε
z
=
(13.5)
where:
ε
x
=
normal strain in the
x-
direction,
ε
y
=
normal strain in the
y-
direction, and
V
w
V
0
V
a
V
0
V
c
V
0
ε
z
=
normal strain in the
z-
direction.
=
+
+
(13.1)
The above normal strains are positive for an increase in
length and negative for a decrease in length. The distortion
of the element is expressed in terms of shear strains, which
correspond to two orthogonal directions. Shear strain
γ
is
defined as the change in the original right angle between two
axes (Chou and Pagano, 1967a). The angle is measured in
radians. A positive shear strain indicates that the right angle
where:
V
0
=
initial overall volume of an unsaturated soil element,
V
v
=
volume of soil voids,
V
w
=
volume of water,
V
a
=
volume of air, and
V
c
=
volume of contractile skin.
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