Biomedical Engineering Reference
In-Depth Information
where the E
i
are the moduli of elasticity and the G
ij
are the shear moduli and the m
ij
are the P
OISSON
ratios. Similarly, this approach applies for arbitrary anisotropies
such as monotropy, transversal isotropy etc.
In the case of plane state stress where w.l.o.g, both orthonormal basis vectors e
1
and e
2
are situated in-plane and a change in thickness is neglected e
33
¼
e
23
¼
e
13
¼
0
;
(
3.290
) degenerates to
r
11
¼
u
1
e
11
þ
u
2
e
22
;
r
22
¼
u
2
e
11
þ
u
4
e
22
;
r
12
¼
r
21
¼
u
9
e
12
ð
3
:
294
Þ
with the four material coefficients resulting from (
3.290
) (the moduli of elasticity
and the bulk moduli and the P
OISSON
ratios with index 3 vanish in this process):
u
1
E
1
u
2
m
12
E
1
E
2
u
4
E
1
E
2
;
;
;
u
9
2G
12
N
N
N
N :
¼
E
1
m
12
E
2
u
3
¼
u
5
¼
u
6
¼
u
7
¼
u
8
¼
0
;
ð
3
:
295
Þ
Anisotropic Materials - H
OLZAPFEL
-G
ASSER
-O
GDEN
Model. Based on the
strain energy function of the H
OLZAPFEL
-G
ASSER
-O
GDEN
model (
3.230
)
2
, the vol-
umetric part of the K
IRCHHOFF
stress tensors (
3.264
)
1
derives to
s
J
¼
1
D
I
:
J
2
1
ð
3
:
296
Þ
Considering the partial derivative derived using (
3.230
)
1
-(
3.232
)
Þ
k
1
X
N
o
w
o C
¼
U
1
I
þ
j 1
3j
U
2
i
K
0i
ð
i
¼
1
with
e
k
2
E
i
ð
3
:
297
Þ
C
;
H
ð Þ
:
¼
c
1
þ
j
2
k
1
X
N
C
I
3
þ
1
3j
j
U
1
¼
U
1
C
IV
i
1
ð
Þ
i
¼
1
e
k
2
E
i
C
;
H
ð Þ
:
¼
C
I
3
þ
1
3j
j
U
2
i
¼
U
2
i
C
IV
i
1
ð
Þ
and using (
3.264
)
3
, the deviatoric part of the K
IRCHHOFF
stress tensors is obtained:
"
#
Þ
k
1
X
N
s
¼
2
ð
4
Þ
U
1
B
þ
j 1
3j
U
2
i
F
K
0i
F
T
ð
ð
3
:
298
Þ
i
¼
1
where B :
¼
F
F
T
is the modified left C
AUCHY
strain tensor.
In the isotropic case with j
¼
1
=
3 and considering the definitions (
3.232
) and
(
3.297
),
(
3.298
)
transforms
into
the
following
constitutive
equation,
which
depends only on B (note that C
I
¼
B
I
)
