Biomedical Engineering Reference
In-Depth Information
oC
¼
X
3
o
w
ðÞ
oC
¼
o
w k
1
;
k
2
;
k
3
ð
Þ
¼
o
w
ok
1
ok
1
oC
þ
o
w
ok
2
oC
þ
o
w
ok
3
o
w
ok
i
o
k
i
oC
ð
3
:
250
Þ
oC
ok
2
ok
3
i
¼
1
where the partial derivations qk/qC can be derived using the G
ÂTEAUX
variation to
(Silber and Steinwender 2005)
o
k
i
ðÞ
oC
¼
1
2k
i
ðÞ
m
i
m
i
ð
i
¼
1
;
2
;
3
Þ
i not summed!
ð
Þ
ð
3
:
251
Þ
and, substituting (
3.251
)in(
3.250
) finally yields
oC
¼
X
3
o
w
ðÞ
1
2k
i
ðÞ
o
w
ok
i
m
i
m
i
:
ð
3
:
252
Þ
i
¼
1
Together with (
3.252
) and (
3.245
), the C
AUCHY
stress tensor and the first and
second P
IOLA
-K
IRCHHOFF
stress tensors derive to
S
¼
J
1
X
3
k
i
o
w
ok
i
n
i
n
i
mit
J
¼
k
1
k
2
k
3
i
¼
1
ð
3
:
253
Þ
P
I
¼
X
ok
i
m
i
n
i
;
P
II
¼
X
3
3
o
w
1
k
i
o
w
ok
i
m
i
m
i
:
i
¼
1
i
¼
1
Split into Deviatoric (Isochoric) and Volumetric Parts. With respect to the
theory outlined in
Sect. 3.2.6.5
regarding (linear-) viscoelastic materials valid for
finite strains, a split into deviatoric and volumetric parts is presented using the
example of the second P
IOLA
-K
IRCHHOFF
stress tensor P
II
and the Kirchhoff stress
tensor s, respectively (a transformation to other stress measures can be done using
(
3.96
), (
3.98
) and (
3.99
). Note, however, that splitting the material equations may
lead to nonphysical effects as outlined in Eipper (1998), especially when using the
example of uniaxial compressing testing at higher volumetric strain!
The strain energy function w according to (
3.194
) is divided into a deviatoric
part w
ðÞ
and a volumetric part f(J). According to (
3.245
)
3
and using the chain
rule, the split of the second P
IOLA
-K
IRCHHOFF
stress tensor yields
P
II
¼
P
I
J
þ
P
II
ð
3
:
254
Þ
with the volumetric and the deviatoric parts
o C
o
C
P
I
J
¼
2
of
ðÞ
oC
¼
2
o
f
ðÞ
oJ
oC
;
P
II
¼
2
ow
oC
¼
2
ow
ðÞ
ðÞ
oC
: ð
3
:
255
Þ
oJ
=
oC
¼
C o J
2
=
3
=
oC
þ
J
2
=
3
o C
=
oC
¼
o J
2
=
3
C
From
(
3.192
)
1
results
ð
4
Þ
ð
4
Þ
oC
=
oC such that using J
ðÞ¼
C
1
=
2
III
is the fourth order
identity tensor, together with (
3.247
)
3
and the G
ÂTEAUX
differential, the following
derivatives from (
3.255
) are derived
and oC
=
oC
¼
I
1
where I
1
