Biomedical Engineering Reference
In-Depth Information
where
N
1,
N
2, and
N
3 are the interpolation functions as defined as
1
2
1
2
1
2
and A is the area determined by
1
{
}
(
)
(
)
(
)
N
= − +− +−
xyxy
y yxxxy
1
23
32
2
3
3
2
A
{
}
(
)
(
)
(
)
N
= − +− +−
x y
xy
y
y x
x
x
y
2
31
13
3
1
1
3
A
{
}
(
)
(
)
(
)
N
= − +− +−
xy
x y
y
y
x
x
x y
3
12
21
1
2
2
1
A
(5.70)
xy
= = −−++−
11
1
1
(
)
A
det 1
x y
xy
xy
x y
x y
x y
x y
2 2
12
13
21
23
31
32
2
2
1
xy
33
The element strain components are the derivatives of the displacement with respect
to the strain coordinate, given as
∂
== + +
∂
u
∂
N
∂
N
∂
N
1
2
3
ε
u
u
u
x
1
2
3
x
∂
x
∂
x
∂
x
N
∂
== + +
∂
v
∂
N
∂
N
∂
1
2
3
ε
v
v
v
y
1
2
3
y
∂
y
∂
y
∂
y
∂
N
∂
N
∂
N
∂
N
∂
N
∂
N
∂
∂
=+= + + + + +
∂∂∂ ∂ ∂ ∂ ∂ ∂
u
v
1
2
3
1
2
3
γ
u
u
u
v
v
v
xy
1
2
3
1
2
3
yx x
x
x
x
x
x
These are the three components of strain at a point, where
ε
and
ε
are strains in
the
x
- and
y
- coordinate respectively, and
x
γ
is the shear strain.
Similarly as presented for the 1D-case the [
B
] (strain-displacement) matrix is
y
−
y y
−
yy
−
y
0
0
0
2
3 3
11
2
1
[]
B
=
0
0
0
x
−
xx xx
−
−
x
3
21
32
1
2
A
x
−
x x
−
x x
−
xy
−
y y
−
yy
−
y
3
2 1
3 2
1 2
3 3
11
2
βββ
ααα
αααβββ
000
123
1
=
000
123
2
A
123123
Using Eqn. (5.32) for the elastic strain energy of the element gives
1
1
{ }
{ }
T
T
{ }
{ }
()
e
T
()
e
∫∫∫
∫∫∫
U
=
ε
[
D
]
ε
dV
=
δ
[
B DB
] [
][
]
δ
dV
(5.71)
2
2
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