Geoscience Reference
In-Depth Information
t
zz
+ (
∂t
zz
/
∂
z) (
d
z/ 2 )
t
zx
+ (
∂t
zx
/
∂
z) (
d
z/ 2 )
t
xz
+ (
∂t
xz
/
∂
x) (
d
x/ 2 )
t
xx
+ (
∂t
xx
/
∂
x) (
d
x/ 2 )
t
zy
+ (
∂t
zy
/
∂
z) (
d
z/ 2 )
t
yz
+ (
∂t
yz
/
∂
y) (
d
y/ 2 )
t
xy
+ (
∂t
xy
/
∂
x) (
d
x/ 2 )
t
yx
+ (
∂t
yx
/
∂
x) (
d
x/ 2 )
t
yy
+ (
∂t
yy
/
∂
y) (
d
y/ 2 )
)
τ
zy
−
(
∂
t
zy
/
∂
z) (
d
z/ 2 )
t
zx
−
(
∂t
zx
/
∂
z) (
d
z/ 2 )
z
t
zz
−
(
∂t
zz
/
∂
z) (
d
z/ 2 )
x
y
Fig. 8.33 Some of the effective or intergranular stress components acting on an elementary cube of porous
material with dimensions (
δ
x
δ
y
δ
z
), when the stresses at the center are
τ
xx
,τ
xy
,τ
xz
,τ
yx
,τ
yy
,
etc.; the first subscript indicates the direction of the surface on which the stress component acts,
and the second subscript is the direction of the stress component itself.
Stresses
The total stress related to the aforementioned displacements,
u
,
w
, and
v
, consists of three
parts, corresponding to the forces acting on each of three phases at a point in space. It
can be written as (Biot, 1955; Brutsaert, 1964)
τ
xx
+
τ
w
+
τ
a
,τ
xy
,τ
xz
,τ
yx
,τ
yy
+
τ
w
+
τ
a
,τ
yz
,τ
zx
,τ
zy
,
and
τ
zz
+
τ
w
+
τ
a
. The
τ
xx
,τ
xy
,
etc., components, namely those with two
subscripts, represent the stress tensor acting on the solid part of the material, also known as
the intergranular or effective stress. A few of the components of the effective stress tensor
are illustrated in Figure 8.33, as they act on the sides of a small cubic element with volume
(
δ
x
δ
y
δ
z
). The first subscript of each effective stress component indicates the direction of
the surface on which it acts, and the second subscript is the direction of the stress component
itself. A component is taken as positive, when it acts in a positive direction on a positive
surface, or in a negative direction on a negative surface. The part of the total stress tensor
applied to the water is
τ
w
,
given by
p
w
τ
w
=−
χ
(8.64)
where
p
w
is the incremental fluid pressure over and above the initial hydrostatic pressure
under equilibrium conditions prior to the displacement. This initial pressure is denoted by
p
wi
and the total pressure is
p
w
=
p
wi
+
p
w
. The term
χ
is the effective stress function, a concept
introduced for partly saturated soils by Bishop (1961) in 1955. It is generally accepted that
χ
equals zero when
S
=
0 and that it is close to unity when
S
=
1 at saturation, in accordance
with Terzaghi's (1925, 1943) effective stress concept. It should be added that Terzaghi's
concept (i.e.
χ
=
1 for
S
=
1) is valid only for granular materials, if the grains are in point
contact with each other, and if (see Bishop and Blight, 1963) the grains are incompressible.
It would seem that besides the effect of partial occupancy of the pores by water,
χ
is also
