Biomedical Engineering Reference
In-Depth Information
h
i
¼
1
2
C
I
:
¼
trC
I
C
;
C
II
:
¼
1
2
I
ð Þ
2
C
C
C
I
trC
2
ð
3
:
180
Þ
C
III
:
¼
det C
1
3
C
I
C
II
C
I
trC
2
þ
trC
3
such that the sef in the isotropic case is a function of the three invariants C
i
of the
right C
AUCHY
strain tensor
w
ðÞ¼
wC
I
;
C
II
;
C
III
ð
Þ:
ð
3
:
181
Þ
Strain energy Functions as Functions of Stretch. Commonly, in the literature
and in FE-software, strain energy functions w are given as functions of the prin-
cipal stretches of the right stretch tensor U, such that the derived constitutive
equations result in spectral form. In this process, the right stretch tensor U is
transformed into spectral form using (
3.60
) and the particular eigen-value problem
(analogue to the principal axis transformation of the C
AUCHY
stress tensor S, cf.
''Principal Stresses'' in
Sect. 3.2.4.4
)
2
3
k
1
00
0 k
2
0
00k
3
U
¼
X
3
4
5
m
i
m
h i:
k
i
m
i
m
i
and
½¼
ð
3
:
182
Þ
i
¼
1
In (
3.182
), the k
i
are the eigen-values or principal stretches and the m
i
are the
eigen-vectors or principal directions of U (in the ICFG). Using (
3.66
) and (
3.182
)
and considering m
i
m
j
¼
d
ij
;
the corresponding spectral representation of the
right C
AUCHY
strain tensor yields
2
4
3
5
m
i
m
h i: ð
3
:
183
Þ
k
1
00
0 k
2
0
00k
3
C
¼
X
3
j
i
¼
k
i
j
i
m
i
m
i
with
and
½¼
i
¼
1
It can be seen that C has the same eigen-vectors m
i
as does U, the eigen-values,
however, are the squared values of that of U! The three eigen-vectors m
i
and the
eigen-values j
i
of C are derived based on the eigen-value problem C
j
i
ð Þ
m
i
¼
0 (i = 1, 2, 3). Using (
3.183
), the three basic invariants of C (
3.180
) derive
as functions of the three principal stretches:
¼
k
1
k
2
þ
k
2
k
3
þ
k
1
k
3
C
I
¼
trC
¼
k
1
þ
k
2
þ
k
3
;
C
I
trC
2
C
II
¼
2
C
III
¼
det C
¼
k
1
k
2
k
3
:
ð
3
:
184
Þ
According to (
3.184
), C
i
¼
C
i
ð
k
1
;
k
2
;
k
3
Þ
for (i = I, II, III) such that the strain
energy function w (
3.181
) can also be formulated in terms of functions of the
principal stretches:
w
ðÞ¼
wC
I
k
1
;
k
2
;
k
3
½
ð
Þ;
C
II
k
1
;
k
2
;
k
3
ð
Þ;
C
III
k
1
;
k
2
;
k
3
ð
Þ
¼
w k
1
;
k
2
;
k
3
ð
Þ
ð
3
:
185
Þ
¼
w k
1
ðÞ;
k
2
ðÞ;
k
3
ðÞ
½
:
