Environmental Engineering Reference
In-Depth Information
The equilibrium analysis of a one-phase solid can be writ-
ten based upon the conservation of linear momentum. The
conservation of linear momentum in the y- direction can be
written as
∂τ xy
∂x
The stress tensor shown in Eq. 3.17 can be used as the
building block for a science for any one-phase material (e.g.,
steel, concrete). It can also be viewed as the total stress or
overall stress tensor for a multiphase material.
ρg dx dy dz
ρ Dv y
Dt
∂σ y
∂y
∂τ zy
∂z
3.3.2 Equilibrium Equations for Multiphase System
An equilibrium analysis can also be applied to a cubical
element of a multiphase system such as a saturated soil or
an unsaturated soil. Newton's second law is applied to the
soil element by summing forces in each direction (i.e., x -,
y -, and z -directions). A saturated soil can be considered
as a two-phase system and equilibrium equations can be
written for each of the phases. Equilibrium equations can
also be written for each phase of the four phases of an
unsaturated soil element (i.e., air, water, contractile skin,
and soil particles). Each phase is assumed to behave as an
independent, linear, continuous, and coincident stress field
in each of the three Cartesian coordinate directions.
Equilibrium equations can also be overlain or summed
using the principle of superposition. However, this may not
give rise to equilibrium equations with stresses that can be
measured. For example, the interparticle stresses between the
soil solids cannot be directly measured. The force equilibrium
equations for the air phase, water phase, and contractile skin
together with the total equilibrium equation for the soil ele-
ment can be used in formulating the equilibrium equation for
the soil structure (Fredlund, 1973b). It is necessary to com-
bine the independent phases in such a way that measurable
stresses appear in the equilibrium equation for the soil solids.
The total number of independent equilibrium equations
will be equal to 3 times the number of phases in a multiphase
system. The stress variables that control the equilibrium of
the soil structure also control the equilibrium of the contrac-
tile skin (Fredlund, 1973b).
+
+
+
=
(3.13)
where:
τ xy
=
shear stress on the x -plane in the y -direction,
σ y
=
total normal stress on the y -plane,
τ zy
=
shear stress on the z -plane in the y -direction,
ρ
=
total density of the soil,
g
=
gravitational acceleration,
dx, dy, dz
=
dimension of the element in the x -, y -, and
z -directions, respectively,
Dv y
Dt =
v y
∂t +
v y
∂y
∂y
∂t =
acceleration
in
the
y -
direction, and
v y
=
velocity in the y -direction.
Let us assume that the REV does not undergo acceleration.
Then the right-hand side of the above equation becomes zero
and is referred to as the Newtonian equation of equilibrium:
∂τ xy
∂x
ρg dx dy dz
∂σ y
∂y
∂τ zy
∂z
+
+
+
=
0
(3.14)
Similarly, equilibrium equations can be written for the
x -direction,
∂σ x
∂x
dx dy dz
∂τ yx
∂y
∂τ zx
∂z
+
+
=
0
(3.15)
and the z -direction,
∂τ xz
∂x +
3.3.3 Independent Phase Equilibrium
The soil particles and the contractile skin are assumed to
behave as solids in an unsaturated soil. In other words, it
is assumed that these phases come to equilibrium under
applied stress gradients. The arrangement of soil particles
is referred to as the soil structure. The water phase and the
air phase qualify as fluids since they flow under an applied
stress gradient. Each phase is assumed to have an indepen-
dent, linear, continuous, and coincident stress field in each
direction. The principle of superposition can be applied to
the equilibrium equations for each of the phases because the
stress fields are linear. The sum of the equilibrium equations
for each phase is equal to the total or overall equilibrium
of the soil element (Westergaard, 1952; Green and Naghdi,
1965; Truesdell, 1966; Faizullaev, 1969).
The equilibrium equations for the water phase, air phase,
and contractile skin can be written independently. The three
individual phase equilibrium equations together with the
total equilibrium equation form the basis for formulating
the equilibrium equation for the soil structure.
dx dy dz
∂τ yz
∂y +
∂σ z
∂z
=
0
(3.16)
Once the equilibrium equations have been written for each
of the Cartesian coordinate directions, it is possible to extract
the surface tractions and place them in the form of a tensor
(i.e., a 3
3 matrix). The tensor represents the stress state
at a point for a one-phase solid:
×
σ x
τ yx
τ zx
τ xy
σ y
τ zy
(3.17)
τ xz
τ yz
σ z
The stresses in the uppermost row in the matrix (i.e., σ x , τ yx ,
τ zx ) were extracted from the x- direction equilibrium equation.
The middle row of stresses was from the y- direction equilib-
rium equation and the lowermost row of stresses was from the
z- direction equilibrium equation. The 3
3 matrix is related
to the three Cartesian coordinate directions commonly used to
formulate engineering problems and is called a stress tensor.
×
 
Search WWH ::




Custom Search