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
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