Biomedical Engineering Reference
In-Depth Information
"
*
+
#
oJ
X
N
2
w
¼
J
o
oJ
2
j
J
¼
1
¼
J
o
2j
D
j
j
J
¼
1
¼
2
J
ð Þ
2j
1
K
0
Ogden
ð
3
:
431
Þ
D
1
j
¼
1
where all terms of the derivative of the expression in angle brackets with j [ 1
vanish.
If the elastic modulus E
0
, the initial shear modulus l
0
and/or the initial bulk K
0
modulus are determined from experiments, they define dependencies among the
material parameters which can be used in the parameter optimization process.
Furthermore, from equations (
3.427
) and (
3.428
) as well as (
3.430
) and (
3.431
), a
stress-free reference state is satisfied if l
0
and K
0
(at small strains and under
assumption of isotropy, E
0
is clearly related to l
0
and K
0
) are positive values
which lead to sufficient restrictions on the parameters l
j
, b
j
and D
1
in terms of the
following inequalities
b
j
[
1
3
Ogden
Hill model (highly comp. Mat.):
l
j
[ 0
and
; ð
3
:
432
Þ
Ogden model (slightly comp. Mat.):
l
j
[ 0
and
D
1
[ 0
:
ð
3
:
433
Þ
Using
(
3.275
)
3
,
inequality
(
3.432
)
2
implies m
j
[
1
as
a
lower
bound
restriction.
3.4.11 Constitutive Inequalities
As already introduced with the concept of the D
RUCKER
stability criterion,
restrictions on the strain-energy function may be paraphrased with the concept of
'stress increase is accompanied with strain increase', i.e. physically reasonable
results are to be ensured. On the basis of this concept Baker and Ericksen (1954)
intuitively proposed inequalities, the so-called B
AKER
-E
RICKSEN
inequalities, which
were based on the principle that under the assumption of isotropy the largest
principal stress should always occur in the direction of the largest principal stretch.
The difference of pairs of C
AUCHY
stress components must thus have the same sign
as the difference of their conjugate deformation measures. This can reasonably be
formulated as
S
i
S
k
k
i
k
k
ð
S
i
S
k
Þ
k
i
k
k
ð
Þ
[ 0
,
[ 0
for
k
i
6¼
k
k
ð
3
:
434
Þ
where S
i
is a principal C
AUCHY
stress component and k
i
is the conjugate principal
stretch component. Using the stress coordinates of the C
AUCHY
stress tensor
(
3.253
)
1
S
i
¼
k
i
J
1
ow
=
ok
i
, the inequality of (
3.434
) can be written as
k
i
o
w
=
o
k
i
k
k
o
w
=
o
k
k
k
i
k
k
[ 0
for
k
i
6¼
k
k
ð
3
:
435
Þ
