Civil Engineering Reference
In-Depth Information
By using the divergence theorem, Equation (1.40) becomes
∂σ
1
i
∂x
1
+
d
V
∂σ
2
i
∂x
2
+
∂σ
3
i
∂x
3
F
v
i
=
(1.41)
V
Adding the component
X
i
of the body force per unit volume, the linearized Newton
equation for
V
may be written as
∂σ
1
i
∂x
1
+
d
V
=
0
∂x
3
+
X
i
−
ρ
∂
2
u
i
∂σ
2
i
∂x
2
+
∂σ
3
i
(1.42)
∂t
2
V
where
ρ
is the mass density of the material. This equation leads to the stress equations
of motion
ρ
∂
2
u
i
∂t
2
∂σ
1
i
∂x
1
+
∂σ
2
i
∂x
2
+
∂σ
3
i
∂x
3
+
X
i
−
=
i
=
1
,
2
,
3
0
(1.43)
With the aid of Equation (1.21) the equations of motion become
ρ
∂
2
u
i
∂t
2
λ
∂θ
2
µ
∂e
ii
2
µ
∂e
ji
=
∂x
i
+
∂x
i
+
∂x
j
+
X
i
i
=
1
,
2
,
3
(1.44)
j
=
i
Replacing
e
ji
by 1
/
2
(∂u
j
/∂x
i
+
∂u
i
/∂x
j
)
, Equations (1.44) can be written in terms
of displacement as
ρ
∂
2
u
i
∂t
2
=
(λ
+
µ)
∂
.
u
∂x
i
+
µ
∇
∇
2
u
i
+
X
i
i
=
1
,
2
,
3
(1.45)
is the Laplacian operator
∂
2
∂
2
∂x
2
+
∂
2
∂x
3
2
where
.
Using vector notation, Equations (1.45) can be written
∇
∂x
1
+
ρ
∂
2
u
∂t
2
2
u
+
X
=
(λ
+
µ)
∇∇
·
u
+
µ
∇
i
=
1
,
2
,
3
(1.46)
In this equation,
∇∇
·
u
is the gradient of the divergence
∇
·
u
of the vector field
u
,
and its components are
∂u
1
∂x
1
+
∂
∂x
i
∂u
2
∂x
2
+
∂u
3
∂x
3
i
=
1
,
2
,
3
(1.47)
2
u
is the Laplacian of the vector field
u
, having components
and the quantity
∇
∂
2
u
i
∂x
j
i
=
1
,
2
,
3
(1.48)
j
=
1
,
2
,
3
as indicated in Section 1.2.
