Biomedical Engineering Reference
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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 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 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.
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