Biomedical Engineering Reference
In-Depth Information
and thus finally
!
X
"
#
1
k
1
¼
1
;
k
2
¼
1
¼
X
N
N
o
k
2
ok
1
l
j
b
j
l
j
1
þ
2b
j
ð
3
:
413
Þ
j
¼
1
j
¼
1
Substituting (
3.413
)in(
3.411
) and further substituting in (
3.410
) leads to E
0
for
the O
GDEN
-H
ILL
model
2
P
þ
3
P
l
j
P
N
N
N
l
j
l
j
b
j
¼
o
j
¼
1
j
¼
1
j
¼
1
ok
1
P
I
1
j
k
1
¼
k
2
¼
1
¼
2
E
0
Ogden
Hill
ð
3
:
414
Þ
P
N
l
j
ð
1
þ
2b
j
Þ
j
¼
1
in consistence with elastic theory.
Similarly,
for
the
O
GDEN
model
for
slightly
compressible
materials
the
following expression for E
0
is obtained
3
P
N
D
j
P
N
l
j
¼
o
j
¼
1
j
¼
1
ok
1
P
I
1
j
k
1
¼
k
2
¼
1
¼
E
0
Ogden
:
ð
3
:
415
Þ
P
N
D
j
þ
6
P
N
l
j
j
¼
1
j
¼
1
In addition, the uniaxial load case may be used to establish a relation between
k
1
and k
2
as follows: With regard to k
2
6¼
0 and J
¼
k
1
k
2
(
3.408
)
2
reads
h
i
¼
0
X
¼
X
N
N
l
j
a
j
l
j
a
j
k
a
j
2
k
1
k
2
a
j
b
j
k
a
2
J
a
j
b
j
ð
3
:
416
Þ
j
¼
1
j
¼
1
For N [ 1, equation (
3.416
) cannot be expressed explicitly in terms of k
1
and k
2
respectively. However, it is satisfied if each of the N-terms vanishes. It is thus
sufficient to consider one term to establish the following relation between the
principal stretches k
1
and k
2
. For the j-th term in (
3.416
) the following expression
is found (Silber and Steinwender 2005)
b
j
1
þ
2b
j
k
2
¼
f k
ðÞ¼
k
:
ð
3
:
417
Þ
1
Equation (
3.417
) furthermore relates the b
j
, which determine the degree of
compressibility for each term of the strain-energy function, to the generalized
P
OISSON
'
S
ratio m
j
of each of the N terms. By utilizing G
2
¼
G
3
¼
mG
1
(in the
case N = 1) for uniaxial tensile loading and G
i
¼
ln k
i
(see (
3.396
)), the single m
j
can be expressed through m
j
¼
b
j
=ð
1
þ
2b
j
Þ
(see (
3.275
)). If b is equal to a con-
stant value for all N terms, a single Poisson's ratio m exists.