Biomedical Engineering Reference
In-Depth Information
where, according to (
3.195
), the following properties must be satisfied
C
¼
I
C
¼
I
;
C
¼
I
f
0
J
¼ ð Þ¼
0
;
w
0
ð
Þ
0
w
d
ð
Þ¼
0
:
ð
3
:
306
Þ
In (
3.305
), w
0
ð
C
;
J
Þ
is the initial strain energy function with the deviatoric (or
isochoric) elastic part w
0
ðÞ
and the volumetric elastic part f
0
ð
J
Þ
in the equilib-
rium state of the material and w
d
ð
C
;
C
Þ
is the dissipative part of the strain energy
function w, extended for viscoelastic materials. Considering (
3.305
)
1
, equation
(
3.304
) transforms into
P
II
¼
2
o
w C
; ð Þ
oC
and
o
w C
; ð Þ
oC
C
0
:
ð
3
:
307
Þ
Note: In particular the split into deviatoric and volumetric parts is analogous to
that regarding hyperelastic materials, starting with equation (
3.254
). The entities
regarding equilibrium elasticity introduced in the 'hyperelastic materials-' section
are additionally indexed with ''0''!
Considering (
3.305
) and following the outlines for the one-dimensional form
(
3.301
)
1
, the stress tensor (
3.307
)
1
can be divided into an equilibrium elasticity
part P
I
0
and a dissipative part P
I
d
where P
I
0
can be further divided into a volumetric
and isochoric part P
I
0J
and P
I
0
such that employing the chain rule
ow
0
ð
C
;
J
Þ=
oC
¼
o w
0
ð
C
Þ=
oC
þ
of
0
ð
J
Þ=
o w
0
ð
C
Þ
¼½
ow
0
ð
C
Þ=
oC
½
oC
=
oC
þ½
of
0
ð
J
Þ=
oJ
ð
oJ
=
oC
Þ
all parts can be written as follows
P
II
¼
P
I
0
þ
P
I
d
¼
P
I
0J
þ
P
I
0
þ
P
I
d
ð
3
:
308
Þ
with
P
I
0
¼
2
ow
0
C
; ð Þ
P
I
d
¼
2
ow
d
C
; ð Þ
:
¼
P
I
0J
þ
P
I
0
ð
3
:
309
Þ
and
oC
oC
and
P
I
0
¼
2
ow
0
o C
o
C
P
I
0J
¼
2
o
f
0
ðÞ
oC
¼
2
o
f
0
ðÞ
oJ
oC
;
ðÞ
oC
¼
2
ow
0
ðÞ
oJ
oC
ð
3
:
310
Þ
P
I
d
¼
o
C
C
ð Þ
¼
o
C
C
ð Þ
o
C
½
½
oC
:
oC
oC
With the expression
o
C
C
ð Þ
½
¼
C
ð
3
:
311
Þ
o C
as well as (
3.256
) and (
3.257
), the relations (
3.310
) and thus all three terms of
constitutive equation (
3.308
), transform to