Biomedical Engineering Reference
In-Depth Information
part A from double scalar multiplication with a second order tensor A (this
operation is sometimes referred to as dev
ðÞ
ð
4
Þ
ðÞ
by other authors) where
left and right double scalar multiplication of
ð
4
Þ
with I yields zero (a deviator is
trace-free):
ð
4
Þ
1
ð
4
Þ
ð
4
Þ
:
¼
I
1
3
II
with
I
1
A
¼
A
and
II
A
¼
tr
ð
I
A
devA
¼
A
A
1
3
ð
4
Þ
ð
3
:
261
Þ
p
tr
ð
I
I
ð
4
Þ
¼
ð
4
Þ
I
¼
0
ð
4
Þ
ð
4
Þ
¼
ð
4
Þ
:
with
and
Further, using trC
A
¼
trF
T
F
A
trF
A
F
T
and (
3.275
) and (
3.261
), the
following expression is obtained
F
T
¼
ð
4
Þ
F
A
F
T
ð
4
Þ
F
P
A
ð
3
:
262
Þ
which can be rewritten using (
3.260
)
2
and (
3.190
)to
0
1
F
T
¼
J
3
ð
4
Þ
F
P
II
F
T
¼
ð
4
Þ
J
3
F
ð
4
Þ
s
¼
J
3
F
P
P
II
@
P
II
J
3
F
T
A
|{z}
F
|{z}
F
T
ð
4
Þ
F
P
II
F
T
ð
3
:
263
Þ
Together with (
3.263
) and F
C
1
F
T
¼
I, the volumetric and deviatoric part
of the K
IRCHHOFF
stress tensor yield
p :
¼
o
f
ðÞ
oJ
s
J
¼
JpI
mit
ð
3
:
264
Þ
s :
¼
F
P
II
F
T
¼
2F
ow
ðÞ
ð
4
Þ
oC
F
T
:
s
¼
p
s
mit
Spectral Representation. Using (
3.183
), (
3.191
) and (
3.192
), the modified
right C
AUCHY
strain tensor reads
2
C
¼
J
2
=
3
C
¼
J
2
=
3
X
k
i
m
i
m
i
¼
X
m
i
m
i
¼
X
3
3
3
k
i
m
i
m
i
ð
3
:
265
Þ
J
1
=
3
k
i
i
¼
1
i
¼
1
i
¼
1
ðÞ¼
w k
1
ðÞ;
k
2
ðÞ;
k
3
where the deviatoric part of w can be written as w
ðÞ
(analogue to (
3.185
)). Using the chain rule (
3.266
) is obtained
oC
¼
X
3
ow
o
k
i
oC
:
ow
ðÞ
ðÞ
ok
i
ð
3
:
266
Þ
i
¼
1