Biomedical Engineering Reference
In-Depth Information
S
¼
u
1
e
11
þ
u
2
e
22
þ
u
3
e
33
ð
Þ
e
1
e
1
þ
u
2
e
11
þ
u
4
e
22
þ
u
5
e
33
ð
Þ
e
2
e
2
þ
u
3
e
11
þ
u
5
e
22
þ
u
6
e
33
ð
Þ
e
3
e
3
ð
3
:
289
Þ
þ
u
7
e
12
e
1
e
2
þ
e
2
e
1
ð
Þþ
u
8
e
13
e
1
e
3
þ
e
3
e
1
ð
Þþ
u
9
e
23
e
2
e
3
þ
e
3
e
2
ð
Þ:
and from (
3.289
) by comparison of coordinates, the six equations of H
OOKE
'
S
law
for orthotropic materials in coordinate notation follow:
r
11
¼
u
1
e
11
þ
u
2
e
22
þ
u
3
e
33
r
22
¼
u
2
e
11
þ
u
4
e
22
þ
u
5
e
33
r
33
¼
u
3
e
11
þ
u
5
e
22
þ
u
6
e
33
r
12
¼
u
9
e
12
;
ð
3
:
290
Þ
r
13
¼
u
8
e
13
;
r
23
¼
u
7
e
23
:
Transforming (
3.290
) into the form r
¼
P
e with the V
OIGT
6
1 vectors
e :
¼ð
e
11
;
e
22
;
e
33
;
e
23
;
e
13
;
e
12
Þ
and r :
¼ð
r
11
;
r
22
;
r
33
;
r
23
;
r
13
;
r
12
Þ
as well as the
symmetric stiffness matrix P
2
3
u
1
u
2
u
3
000
u
2
u
4
u
5
000
u
3
u
5
u
6
000
000u
7
00
0000u
8
0
00000u
9
4
5
P :
¼
;
ð
3
:
291
Þ
inverting
the flexibility
matrix
for
the
orthotropic
case
(cf.
(Altenbach
and
Altenbach 1994))
2
4
3
5
1
=
E
1
m
12
=
E
1
m
13
=
E
1
0
0
0
m
12
=
E
1
1
=
E
2
m
23
=
E
2
0
0
0
m
13
=
E
1
m
23
=
E
2
1
=
E
3
0
0
0
P
1
:
¼
ð
3
:
292
Þ
0
0
0
2
=
G
23
0
0
0
0
0
0
2
=
G
31
0
0
0
0
0
0
2
=
G
12
and comparing then the coordinates with (
3.291
) leads to the following nine
independent material coefficients
u
1
E
1
E
2
m
23
E
3
u
2
E
1
E
2
m
12
E
2
þ
m
13
m
23
E
3
ð
Þ
u
3
E
1
E
2
E
3
m
13
þ
m
12
m
23
ð
Þ
;
;
N
N
N
u
4
E
2
E
1
m
13
E
3
u
5
E
2
E
3
m
23
E
1
þ
m
12
m
13
E
2
ð
Þ
u
6
E
2
E
3
E
1
m
12
E
2
;
;
N
N
N
u
7
2G
12
;
u
8
2G
31
;
u
9
2G
23
þ
m
13
E
2
E
3
m
13
þ
m
12
m
23
N
E
2
E
1
m
12
E
2
ð
Þþ
m
23
E
3
m
23
E
1
þ
m
12
m
13
E
2
ð
Þ
ð
3
:
293
Þ
