Biomedical Engineering Reference
In-Depth Information
c
¼
0
;
k
1
¼
1
;
k
2
¼
1
c
¼
0
;
k
1
¼
1
;
k
2
¼
1
c
¼
0
;
k
1
¼
1
;
k
2
¼
1
oc
X
3
2
w
oc
2
l
0
¼
o
P
I
21
¼
o
o
w
ok
i
o
k
i
oc
¼
o
: ð
3
:
425
Þ
oc
i
¼
1
In conjunction with (
3.421
) the following derivatives arise
2
;
o
2
k
1
oc
2
j
c
¼
0
¼
o
2
k
2
oc
j
c
¼
0
¼
o
2
J
o
k
1
oc
j
c
¼
0
¼
o
k
2
oc
j
c
¼
0
¼
1
oc
2
j
c
¼
0
¼
1
4
;
oJ
oc
2
j
c
¼
0
¼
0
:
ð
3
:
426
Þ
Considering the O
GDEN
-H
ILL
model (
3.209
) together with (
3.421
), initial tan-
gent shear modulus l
0
thus derives to
oc
2
j
c
¼
0
;
k
1
¼
k
2
¼
1
¼
X
N
2
w
¼
o
l
0
Ogden
Hill
l
j
:
ð
3
:
427
Þ
j
¼
1
In the same manner, for the O
GDEN
model for slightly compressible materials,
l
0
is found identical to (
3.427
)
oc
2
j
c
¼
0
;
k
1
¼
k
2
¼
1
¼
X
2
w
N
¼
o
l
0
Ogden
l
j
:
ð
3
:
428
Þ
j
¼
1
Volumetric Deformation: The initial tangent bulk modulus K
0
at infinitesimal
strains can be related to the material constants of the O
GDEN
-H
ILL
model. The bulk
modulus is defined as (Ryder 2007)
K
0
:
¼
V
o
p
oV
p
¼
o
w
oJ
with
ð
3
:
429
Þ
where p represents the volumetric stress and hydrostatic pressure (see (
3.258
) and
(
3.264
)), respectively. Since the principal stretches k
j
are identical under volu-
metric deformation, the volume ratio (in the case of homogeneous deformation)
becomes J
¼
VV
0
¼
k
1
k
2
k
3
¼
k
v
. The initial bulk modulus thus derives using
the derivative of the first P
IOLA
-K
IRCHHOFF
stress measure with respect to J with
J
!
1 to account for the initial stress-free state to
"
*
+
#
X
N
2
w
oJ
2
j
J
¼
1
¼
J
¼
J
o
dJ
2
d
l
j
a
j
J
a
j
=
3
J
a
j
b
j
K
0
Ogden
Hill
j
J
¼
1
J
j
¼
1
:
¼
2
X
N
1
3
þ
b
j
l
j
ð
3
:
430
Þ
j
¼
1
Similarly,
for
the
Ogden
model
for
slightly
compressible
materials
the
following expression for K
0
is obtained
