Civil Engineering Reference
In-Depth Information
When operating on a vector field
v
with components
(υ
1
,υ
2
,υ
3
)
, the vector operator
∇
yields the divergence of
v
∂υ
1
∂x
1
+
∂υ
2
∂x
2
+
∂υ
3
∂x
3
div
v
=
∇
·
v
=
(1.3)
The Laplacian of
ϕ
is:
∂
2
ϕ
∂x
1
+
∂
2
ϕ
∂x
2
+
∂
2
ϕ
∂x
3
2
ϕ
=
div
grad
ϕ
=
∇
·
∇
ϕ
=
∇
(1.4)
When operating on the vector
v
, the Laplacian operator yields a vector field whose
components are the Laplacians of
υ
1
,υ
2
and
υ
3
∂
2
υ
i
∂ϕ
1
+
∂
2
υ
i
∂ϕ
2
+
∂
2
υ
i
∂ϕ
3
2
v
)
i
=
(
∇
(1.5)
The gradient of the divergence of a vector
v
is a vector of components
∂υ
1
∂x
1
+
∂
∂x
i
∂υ
2
∂x
2
+
∂υ
3
∂x
3
(
∇∇
·
v
)
i
=
(1.6)
The vector
curl
is denoted by
curl v
=
∇
∧
v
(1.7)
and is equal to
curl v
=
i
1
∂υ
3
+
i
2
∂υ
1
+
i
3
∂υ
2
∂υ
2
∂x
3
∂υ
3
∂x
1
∂υ
3
∂x
2
∂x
2
−
∂x
3
−
∂x
1
−
(1.8)
1.3
Strain in a deformable medium
Let us consider the coordinates of the two points
P
and
Q
in a deformable medium
before and after deformation. The two points
P
and
Q
are represented in Figure 1.1.
The coordinates of
P
are
(x
1
,x
2
,x
3
)
and become
(x
1
+
u
3
)
after
deformation. The quantities
(u
1
,u
2
,u
3
)
are then the components of the displacement
vector
u
of
P
. The components of the displacement vector for the neighbouring point
Q
,
having initial coordinates
(x
1
u
1
,x
2
+
u
2
,x
3
+
+
x
1
,x
2
+
x
2
,x
3
+
x
3
)
, are to a first-order approxi-
mation
∂u
1
∂x
1
x
1
∂u
1
∂x
2
x
2
∂u
1
∂x
3
x
3
u
1
=
u
1
+
+
+
∂u
2
∂x
1
x
1
∂u
2
∂x
2
x
2
∂u
3
∂x
3
x
3
u
2
=
u
2
+
+
+
(1.9)
∂u
3
∂x
1
x
1
+
∂u
3
∂x
2
x
2
+
∂u
3
∂x
3
x
3
u
3
=
u
3
+
