Biomedical Engineering Reference
In-Depth Information
p
0
:
¼
o
f
0
ðÞ
oJ
P
I
0J
¼
Jp
0
C
1
:
¼
2
o
w
0
ð
4
Þ
ðÞ
o C
P
I
0
P
I
0
¼
J
2
=
3
P
I
0
P
ð
3
:
312
Þ
with
P
I
d
¼
o
C
C
ð Þ
ð
4
Þ
¼
J
2
=
3
P
C
oC
Substituting (
3.312
)in(
3.308
)
1
, it yields
:
ð
4
Þ
P
II
¼
P
I
0J
þ
P
I
0
þ
P
I
d
¼
Jp
0
C
1
þ
J
2
=
3
P
I
0
C
ð
3
:
313
Þ
P
In order to determine the inner variable C(t)in(
3.313
), in (Simo 1987), the
following evolution equation is introduced
C
þ
1
s
C
¼
1
c
ð
4
Þ
P
I
0
P
with
C t
¼ ð Þ ¼
0
ð
3
:
314
Þ
s
where s is the relaxation time and c is a constant (for c = 0 and c = 1, respec-
tively, the special cases of a M
AXWELL
fluid and an elastic solid ensue).
Based on the previous, it is assumed that the viscous behaviour of the material
and the inner variable C characterizing the dissipative effects are determined
exclusively
function w
0
ðÞ
(cf.
(
3.312
)
3
). The solution of the inhomogenous first order differential equation for
C(t) in the (current) time (
3.314
) can be found by variation of constants using the
principle of determinism (consideration of all the states including the remote past
to the present time, cf.
Sect. 3.2.6.1
) as follows
by the
deviatoric
part
of
the
strain
energy
C
ðÞ¼
Z
t
1
c
s
ð
4
Þ
e
t
t
0
t
ðÞ
P
I
0
t
ðÞ
dt
0
P
ð
3
:
315
Þ
s
t
0
¼
0
Substituting (
3.315
)in(
3.313
) leads to the material equation for the second
P
IOLA
-K
IRCHHOFF
stress tensor for linear-viscoelastic materials at finite strains
ðÞ¼
Jp
0
C
1
þ
J
2
=
3
=
P
I
0
with
p
0
:
¼
df
0
ðÞ
dJ
P
II
2
3
ðÞ
P
I
0
ðÞ
Z
t
and
=
P
I
0
:
¼
P
ð
4
Þ
1
c
s
ð
4
Þ
t
ðÞ
P
I
0
t
ðÞ
dt
0
e
t
t
0
4
5
P
s
t
0
¼
0
ð
3
:
316
Þ
where the first term of the functional
=
C
defined in (
3.316
)
3
represents the
spontaneously elastic part (stress of the spring in parallel) and the second term can
