Biomedical Engineering Reference
In-Depth Information
Step 4
With
u
=
u
x
e
x
+
u
y
e
y
and
x
=
x
e
x
+
y
e
y
the strain components may be
written as
⎡
⎣
⎤
⎦
⎡
⎤
∂
u
x
∂
x
ε
xx
⎣
⎦
∂
u
y
∂
y
∼
=
ε
yy
=
.
(18.42)
+
∂
u
y
∂
x
∂
u
x
∂
y
2
ε
xy
This is frequently rewritten as
∼
=
B
∼
,
(18.43)
with
B
an operator defined by
⎡
⎤
∂
∂
x
0
⎣
⎦
∂
∂
y
B
0
=
,
(18.44)
∂
∂
y
∂
∂
x
while
∼
represents the displacement field:
u
x
u
y
.
∼
=
(18.45)
Step 5
Within each element the displacement field is interpolated according to
n
T
∼
xe
u
x
|
e
=
N
i
(
x
,
y
)
u
xei
=
∼
i
=
1
n
u
y
e
=
T
∼
ye
,
N
i
(
x
,
y
)
u
yei
=
∼
(18.46)
i
=
1
where
∼
xe
and
∼
ye
denote the nodal displacements of element
e
in the
x
- and
y
-direction, respectively. Using this discretization the strain within an element can
be written as
⎡
⎣
⎤
⎦
∂
N
i
∂
x
u
xei
n
∂
N
i
∂
y
u
yei
∼
=
.
(18.47)
i
=
1
∂
N
i
∂
y
u
xei
+
∂
N
i
∂
x
u
yei
