Civil Engineering Reference
In-Depth Information
Q
′
P
′
u
′
x
3
u
Q
(x
1
+
∆
x
1
, x
2
+
∆
x
2
,x
3
+
∆
x
3
)
P
(x
1
, x
2
, x
3
)
O
x
2
x
1
The displacement of
P
and
Q
to
P
and
Q
in a deformable medium.
Figure 1.1
A rotation vector
(
1
,
2
,
3
)
and a 3
×
3 strain tensor
e
canbedefinedat
P
by
the following equations:
∂u
3
∂x
2
−
,
∂u
1
∂x
3
−
1
2
∂u
2
∂x
3
1
2
∂u
3
∂x
1
1
=
2
=
∂u
2
∂x
1
−
(1.10)
∂u
1
∂x
2
1
2
3
=
∂u
i
∂x
j
+
1
2
∂u
j
∂x
i
e
ij
=
(1.11)
The displacement components of
Q
can be rewritten as
u
1
=
u
1
+
(
2
x
3
−
3
x
2
)
+
(e
11
x
1
+
e
12
x
2
+
e
13
x
3
)
u
2
=
u
2
+
(
3
x
1
−
1
x
3
)
+
(e
21
x
1
+
e
22
x
2
+
e
23
x
3
)
(1.12)
u
3
=
u
3
+
(
1
x
2
−
2
x
1
)
+
(e
31
x
1
+
e
32
x
2
+
e
33
x
3
)
The terms in the first parenthesis of each equation are associated with rotations around
P
, while those in the second parenthesis are related to deformations. The three compo-
nents
e
11
,
e
22
and
e
33
, which are equal to
∂u
1
∂x
1
,
∂u
2
∂x
2
,
∂u
3
∂x
3
e
11
=
e
22
=
e
33
=
(1.13)
are an estimation of the extensions parallel to the axes.
The cubical dilatation
θ
is the limit of the ratio of the change in the volume to the
initial volume when the dimensions of the initial volume approach zero. Hence,
lim
(x
1
+
e
11
x
1
)(x
2
+
e
22
x
2
)(x
3
+
e
33
x
3
)
−
x
1
x
2
x
3
θ
=
(1.14)
x
1
x
2
x
3
and is equal to the divergence of
u
:
∂u
1
∂x
1
+
∂u
2
∂x
2
+
∂u
3
∂x
3
=
θ
=
∇
·
u
=
e
11
+
e
22
+
e
33
(1.15)
