Biomedical Engineering Reference
In-Depth Information
eigenvalue problem has a set of non-vanishing solutions j
1
;
j
2
;
j
3
if the deter-
minant of the coefficients vanishes, i.e.
P
ð
j
i
Þ¼ð
1
j
i
Þ½ð
1
j
i
Þð
1
j
i
þ
c
2
Þ
c
2
¼
0
: ð
3
:
419
Þ
det
ð
C
j
i
I
Þ¼
0
;
Equation (
3.419
)
2
is a cubic equation in j
i
and its roots k
i
j
i
(i = 1,2,3)
represent the squared principal stretches of the right stretch tensor U (see (
3.182
)
in
Sect. 3.2.6.1
). The solution of (
3.419
)
2
leads to the following set of principal
stretches as functions of the shear gradient c
s
1
þ
c
2
r
1
þ
c
2
2
k
1
;
2
ðÞ¼
p
j
1
;
2
¼
2
c
ð
3
:
420
Þ
k
3
¼
p
j
3
¼
1
:
According to (
3.420
) the stretches (eigenvalues) k
i
depend solely on the shear
deformation c such that according to (
3.185
), the strain energy function is also only
a function of c:
w
ð
C
Þ¼
w k
1
ðÞ;
k
2
ðÞ;
k
3
ðÞ
½
¼
w
ðÞ¼
w c
ð½ :
ð
3
:
421
Þ
The initial shear modulus may be generated as follows: from (
3.242
) it follows
D
¼
F
T
C
F
1
=
2 such that inserting in (
3.150
) and rearranging and solving
for w and using (
3.96
) leads to the following equality
w
¼
JS
D
¼
P
I
F
:
ð
3
:
422
Þ
Together with (
3.421
) and the time derivative of the deformation gradient
F
¼
ce
1
e
2
following from (
3.417
)
2
as well the representation of the first P
IOLA
-
K
IRCHHOFF
stress tensors with respect to an OBS, P
I
¼
P
ij
e
i
e
j
results from the RHS
of (
3.422
)to
c
¼
P
I
21
c
: ð
3
:
423
Þ
w
d
dt
g¼
dw
dc
c
¼
P
I
ce
1
e
ð Þ
P
I
e
1
e
2
f
w c
ð
t
½
For arbitrary c and by comparison of the underlined terms in (
3.423
) the cor-
relation between the shear coordinate P
I
21
of the first P
IOLA
-K
IRCHHOFF
stress
tensors and the shear deformation c follows to
g
¼
X
3
P
I
21
¼
o
w
oc
¼
o
o
w
ok
i
o
k
i
oc
w k
1
ðÞ;
k
2
ðÞ;
k
3
ðÞ
½
ð
3
:
424
Þ
f
oc
i
¼
1
Analogue
to
the
definition
of
the
initial
Y
OUNG
'
S
modulus
E
0
:
¼
oP
I
11
=
ok
1
k
1
¼
1
;
k
2
¼
1
, the initial tangent shear modulus can be defined as l
0
:
¼
oP
I
21
=
oc
c
¼
0
;
k
1
¼
1
;
k
2
¼
1
. Using (
3.424
) the initial tangent shear modulus yields
