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.