Geoscience Reference
In-Depth Information
δ
z
(
∂
u
x
/
∂
z)
C
′
G
D
′
F
δ
z/ 2
u
z
+ (
∂
u
z
/
∂
z)
δ
x
(
∂
u
z
/
∂
x)
H
u
z
C
D
B
′
δ
x
E
A
′
δ
z
A
B
u
x
z
δ
δ
u
x
−
(
∂
u
x
/
∂
x)
x/ 2
δ
x+ (
∂
u
x
/
∂
x)
x
δ
x/ 2
u
x
+ (
∂
u
x
/
∂
x)
x
Fig. 8.32 Some of the displacements of a porous material subjected to deformation. Before the
deformation the solid mass
ρ
s
(1
−
n
0
)
δ∀
occupies the volume element
δ∀
=
(
δ
x
δ
y
δ
z
)at
ABCD, and after the displacement this same solid mass has moved to A
B
C
D
. The center of
the mass has moved from (
x
0
,
y
0
,
z
0
)to(
x
0
+
u
x
,
y
0
+
u
y
,
z
0
+
u
z
). The figure is shown in
two-dimensions for clarity; the third coordinate
y
can be imagined as pointing out of the plane
of the drawing.
are defined such that, when multiplied by the total or bulk cross-sectional area of porous
material, they produce the displaced water and air volume, respectively.
The strain components of the solid are defined as
e
xx
=
∂
u
x
/∂
x
,
e
xy
=
(
∂
u
x
/∂
y
+
∂
2, etc. Their physical significance is illustrated in Figure 8.32. The position of
the center of the cube shown in the figure is at (
x
0
,
u
y
/∂
x
)
/
y
0
,
z
0
) prior to the deformation, and the
displacement components (
u
x
,
u
z
) refer to the displacement of the center of the cube.
After the deformation the position of the point H is at
u
y
,
∂
x
2
δ
2
x
H
=
x
0
−
δ
2
+
u
x
−
∂
x
u
x
∂
x
δ
2
+
∂
x
2
u
x
x
2
1
2
−···
and
z
H
=
z
0
+
u
z
−
∂
u
z
∂
x
∂
x
2
δ
x
2
2
u
z
δ
x
2
+
∂
1
2
−···
2
The position of the point F is
∂
x
2
δ
x
2
x
F
=
x
0
+
δ
x
2
+
u
x
+
∂
u
x
δ
x
2
+
∂
2
u
x
1
2
+···
∂
x
2
and
∂
x
2
δ
x
2
u
z
+
∂
u
z
∂
x
δ
x
2
+
∂
2
u
z
1
2
+···
z
F
=
z
0
+
2