Biomedical Engineering Reference
In-Depth Information
To show convexity of the volumetric term of the O
GDEN
-H
ILL
Model, the
second derivative with regard to (
3.209
)
2
yields
J
a
j
b
j
2
0
dJ
2
¼
2
X
N
d
2
f
1
a
j
l
j
þ
b
j
ð
3
:
446
Þ
j
¼
1
which is true using the sufficient restrictions l
j
[ 0, a
j
[ 0 and b
j
[ 0. Other
sufficient case distinctions lead to negative values of b
j
.
The O
GDEN
model for slightly compressible materials (
3.208
)
2
, however, is not
a special case of (
3.443
) and can thus not be directly compared.
In the following, the B
AKER
-E
RICKSEN
inequalities (
3.434
) and (
3.436
) are
applied to the O
GDEN
-H
ILL
model (
3.274
), and the O
GDEN
model for slightly
compressible materials (
3.272
)
2
, to deduce implications on the material constants.
Using (
3.99
), (
3.253
)
1
and (
3.274
), the principle C
AUCHY
-stresses deduced from the
O
GDEN
-H
ILL
model read
S
ii
¼
2J
1
X
N
l
j
a
j
:
k
a
j
i
J
a
j
b
j
ð
3
:
447
Þ
j
¼
1
and in conjunction with (
3.434
) the first Baker-Ericksen inequality yields
2J
1
X
N
l
j
a
j
ð
k
a
j
i
k
a
k
Þð
k
i
k
k
Þ
[ 0
:
ð
3
:
448
Þ
j
¼
1
A sufficient condition regarding the choice of parameter l
j
in (
3.448
)is
l
j
[ 0
for
j
¼
1
;
...
;
N
ð
3
:
449
Þ
with no restrictions on parameter a
j
. Due to the form of the first B
AKER
-E
RICKSEN
inequalities no restriction can be deduced for parameter b
j
.
Furthermore, from (
3.436
) the following inequalities for the O
GDEN
-H
ILL
model
are obtained which restrict the choice of parameter a
j
and b
j
h
i
[ 0
:
2
X
N
l
j
a
j
ð
a
j
1
Þ
k
a
j
2
þð
a
j
b
j
þ
1
Þ
k
2
J
a
j
b
j
ð
3
:
450
Þ
i
i
j
¼
1
Sufficient conditions regarding the choice of parameters l
j
, a
j
and b
j
in (
3.450
)
are
l
j
[ 0
and
a
j
[ 1
and
b
j
[
1
for
j
¼
1
;
...
;
N
:
ð
3
:
451
Þ
The b
j
values are further restricted due to consistence with classical linear
theory, cf. (
3.432
). From comparison with (
3.445
) it can be seen that the restric-
tions vary only slightly.
