Civil Engineering Reference
In-Depth Information
If
x
denotes the vector having components
x
1
,
x
2
and
x
3
, after a rotation
characterized by the rotation vector
, the initial vector becomes
x
related to
x
by
x
−
x
=
∧
x
(1.16)
The rotation vector
, in vector notation, is
1
2
=
curl u
(1.17)
1.4
Stress in a deformable medium
Two kinds of forces may act on a body, body forces and surface forces. Surface forces
act across the surface, including its boundary. Consider a volume
V
in a deformable
medium as represented in Figure 1.2.
Let
S
be the surface limiting
V
and
S
an element of
S
around a point
P
that lies
on
S
. The side of
S
which is outside
V
is called (
+
) while the other is called (
−
). The
force exerted on
V
across
S
is denoted by
F
. A stress vector at
P
is defined by
0
F
T
(P)
=
lim
S
(1.18)
S
→
The stress vector
T
(
P
) depends on
P
and on the direction of the positive outward unit
normal
n
to the surface
S
at
P
. The stress vectors can be obtained from
T
1
(σ
11
,σ
12
,σ
13
)
,
T
2
(σ
21
,σ
22
,σ
23
)
,and
T
3
(σ
31
,σ
32
,σ
33
)
corresponding to surfaces with normal
n
parallel
to the
x
1
,
x
2
and
x
3
axes, respectively.
The components
T
1
,
T
2
,
T
3
of
T
can be expressed in the general case as
T
1
=
σ
11
n
1
+
σ
21
n
2
+
σ
31
n
3
T
2
=
σ
12
n
1
+
σ
22
n
2
+
σ
32
n
3
T
3
=
σ
13
n
1
+
σ
23
n
2
+
σ
33
n
3
(1.19)
In these equations
n
1
,
n
2
and
n
3
are the direction cosines of the positive normal
n
to
S
at
P
. The quantities
σ
ij
are the nine components of the stress tensor at
P
.These
components are symmetrical, i.e.
σ
ij
σ
ji
, like the components
e
ij
. An illustration is
given in Figure 1.3 for a cube with faces of unit area parallel to the coordinate planes.
=
X
3
n
∆
S
P
V
S
O
X
2
X
1
Figure 1.2
A volume
V
in a deformable medium, with an element
S
belonging to
the surface
S
limiting
V
.
