Environmental Engineering Reference
In-Depth Information
y
−
∂
x
+
∂
u
y
=
∂φ
∂
H
z
∂
H
x
∂
z
,
(4.15)
u
z
=
∂φ
∂
z
+
∂
H
y
x
−
∂
H
x
y
.
(4.16)
∂
∂
According to three components above, the corresponding stress
can bederived as
∂
u
x
∂
x
+
∂
u
y
∂
y
+
∂
u
z
+
2
μ
∂
u
x
T
xx
=
λ
∂
x
,
(4.17)
∂
z
∂
u
x
∂
x
+
∂
u
y
∂
y
+
∂
u
z
+
2
μ
∂
u
y
T
yy
=
λ
∂
y
,
(4.18)
∂
z
∂
u
x
∂
x
+
∂
u
y
∂
y
+
∂
u
z
+
2
μ
∂
u
z
T
zz
=
λ
∂
z
,
(4.19)
∂
z
∂
u
x
,
∂
y
+
∂
u
y
T
xy
=
T
yx
=
μ
(4.20)
∂
x
∂
,
∂
z
+
∂
u
y
u
z
∂
y
T
yz
=
T
zy
=
μ
(4.21)
∂
.
x
+
∂
u
z
∂
u
x
∂
T
xz
=
T
zx
=
μ
(4.22)
z
For a 2D system, at hand the displacement vector
U
is
independent of the
z
coordinate and one can take
∂/∂
z
=
∂
2
/∂
z
2
=
0,
Eqs. 4.14-4.16 can be written as
u
x
=
∂φ
∂
x
+
∂
H
z
∂
y
,
(4.23)
u
y
=
∂φ
∂
y
−
∂
H
z
∂
x
,
(4.24)
u
z
=
∂
H
y
∂
x
−
∂
H
x
∂
y
.
(4.25)
From Eqs. 4.23-4.25, one knows that a 2D system can support
two independent modes: the in-plane modes (
u
x
,
u
y
) and the out-
of-plane mode (
u
z
) (determined by Eqs. 4.23 and 4.24, and by
Eq. 4.25), respectively. Thus, it is convenient that they can be done
