Biomedical Engineering Reference
In-Depth Information
The second product term in (
3.266
) is given by (analogue to (
3.251
))
ok
i
ðÞ
oC
¼
1
2k
i
ðÞ
m
i
m
i
ð
i
¼
1
;
2
;
3
Þ
i not summed!
Þ:
ð
3
:
267
Þ
and, substituting (
3.267
)in(
3.266
) and further in (
3.258
) leads to the fictitious
second P
IOLA
-K
IRCHHOFF
stress tensor
:
¼
2
ow
oC
¼
X
3
ðÞ
1
k
i
o
w
ok
i
P
II
m
i
m
i
:
ð
3
:
268
Þ
i
¼
1
Substitution of (
3.268
)in(
3.258
) as well as in (
3.264
) and considering (
3.183
)
finally leads to the volumetric and deviatoric part of the material equation of the
second P
IOLA
-K
IRCHHOFF
stress tensor in spectral form
P
I
J
¼
Jp
X
3
k
2
i
p :
¼
o
f
ðÞ
oJ
m
i
m
i
mit
i
¼
1
ð
3
:
269
Þ
ð
4
Þ
P
II
¼
P
3
P
II
¼
J
2
=
3
P
II
k
i
ow
1
P
mit
ok
i
m
i
m
i
i
¼
1
and the K
IRCHHOFF
stress tensor (note that I
¼
n
i
n
i
)
p :
¼
o
f
ðÞ
oJ
s
J
¼
Jpn
i
n
i
mit
ð
3
:
270
Þ
s
¼
P
3
ð
4
Þ
k
i
o
w
s
¼
p
s
mit
ok
i
n
i
n
i
i
¼
1
Slightly Compressible Materials. With regard to (
3.208
), (
3.261
), (
3.270
),
I
¼
n
i
n
i
and I
n
i
n
i
¼
n
i
n
i
¼
1, finally, the following constitutive equations in
the form of the volumetric and deviatoric part of the K
IRCHHOFF
stress tensor for
slightly compressible hyperelastic materials are obtained
s
J
¼
2
X
X
D
k
JJ
ð Þ
2k
1
n
i
n
i
¼
2
X
3
N
N
k
k
D
k
JJ
ð Þ
2k
1
I
i
¼
1
k
¼
1
k
¼
1
s
¼
2
X
3
X
N
l
k
a
k
s
¼
ð
4
Þ
s
k
a
i
n
i
n
i
with
or
i
¼
1
k
¼
1
!
!
s
¼
2
X
X
X
n
i
n
i
2
X
X
X
X
3
N
3
3
N
N
3
l
k
a
k
k
a
k
i
1
3
l
k
a
k
k
a
i
n
i
n
i
2
3
l
k
a
k
k
a
k
j
k
a
j
I
i
¼
1
k
¼
1
j
¼
1
i
¼
1
k
¼
1
k
¼
1
j
¼
1
ð
3
:
271
Þ
respectively.
