Biomedical Engineering Reference
In-Depth Information
o
C
oC
¼
J
2
=
3
P
T
ð
4
Þ
oC
¼
1
oJ
2
JC
1
;
:
ð
3
:
256
Þ
ð
4
Þ
is a (fourth order) material deviator operator which generates the
material deviatoric part
In (
3.256
)
3
P
A from double scalar multiplication with a second order
ð
4
Þ
tensor A (this operation is sometimes referred to as DEV
ðÞ
P
ðÞ
by other
ð
4
Þ
authors). Since left double scalar multiplication of P
with C and right double
scalar multiplication with C
1
yields zero (a deviator is trace-free), the following
relations are obtained
ð
4
Þ
ð
4
Þ
1
ð
4
Þ
3
C
1
C
C
1
C
A
¼
trC
A
Þ
C
1
P
:
¼
I
1
with
I
1
A
¼
A
and
ð
ð
4
Þ
A
DEVA
¼
A
A
1
3
Þ
C
1
P
ð
trC
A
ð
4
Þ
ð
4
Þ
ð
4
Þ
ð
4
Þ
ð
4
Þ
C
1
¼
0
with
C
P
¼
P
and
P
P
¼
P
:
ð
3
:
257
Þ
ð
4
Þ
ð
4
Þ
A
T
for arbitrary sec-
ond order tensors A yields the following terms of the constitutive equation (
3.254
)
for the second P
IOLA
-K
IRCHHOFF
stress tensor
Substituting (
3.256
)in(
3.255
) and using A
P
T
¼
P
P
I
J
¼
2
o
f
ðÞ
p :
¼
o
f
ðÞ
oJ
oC
¼
JpC
1
with
ð
3
:
258
Þ
P
II
¼
2
ow
:
¼
2
o
w
ðÞ
oC
¼
J
2
=
3
ð
4
Þ
ðÞ
o C
P
II
P
II
P
with
where the quantity P
II
in (
3.258
)
3
is referred to as a fictitious second P
IOLA
-
K
IRCHHOFF
stress tensor. The respective parts of the Kirchhoff stress tensor follow
from substitution of (
3.254
) and (
3.258
)in(
3.99
) (to distinguish between material
and spatial deviator, the deviator of s has been modified using a circumflex!)
which preliminarily leads to
F
T
¼
s
J
þ
s
s
¼
F
P
II
F
T
¼
F
P
I
J
þ
P
II
ð
3
:
259
Þ
with
s
J
¼
F
P
I
J
F
T
¼
JpF
C
1
F
T
F
T
:
ð
3
:
260
Þ
ð
4
Þ
s
¼
F
P
II
F
T
¼
J
2
=
3
F
P
P
II
For further advancement, it is reasonable to introduce a fourth order spatial
deviator operator
ð
4
Þ
(analogue to (
3.257
)) which generates the spatial deviatoric
