Geoscience Reference
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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
 
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