Biomedical Engineering Reference
In-Depth Information
be referred to as memory part. In (
3.316
), P
I
0
must be defined using (
3.258
)
3
respectively (
3.269
)
3
.
Using partial integration,
=
C
can be rewritten
(
)
t
=
P
I
0
¼
P
ð
4
Þ
ð
4
Þ
ðÞ
P
I
0
ðÞ
1
ð Þ
e
t
t
0
t
ðÞ
P
I
0
t
ðÞ
P
s
t
0
¼
0
dt
0
ðÞ
Z
t
ð
4
Þ
d
dt
0
ð
4
Þ
t
ðÞ
P
I
0
1
ð Þ
e
t
t
0
t
ðÞ
þ
P
P
s
t
0
¼
0
ð
4
Þ
ð
4
Þ
ðÞ
P
I
0
ð
3
:
317
Þ
ðÞ
P
I
0
t
ðÞ
¼
P
ðÞ
1
ð
Þ
P
ð
4
Þ
ð
4
Þ
ðÞ
1
ð Þ
e
s
P
t
0
¼ ð Þ
P
I
0
t
0
¼ ð Þ
þ
P
dt
0
ðÞ
R
ð
4
Þ
t
ð
4
Þ
1
ð Þ
e
t
t
0
t
ðÞ
P
I
0
d
t
ðÞ
þ
P
dt
0
P
s
t
0
¼
0
Using the assumption of a stress-free primal state with P
II
0
ð
t
0
¼
0
Þ¼
0, the
ð
4
Þ
underlined term in (
3.317
) vanishes such that due to (
3.257
)
6
, the operator P
in the
double underlined term in (
3.317
) can be factored out and thus it follows
8
<
9
=
dt
0
0
ðÞþ
Z
t
=
P
I
0
¼
P
ð
4
Þ
ðÞ
cP
II
d
dt
0
ð
4
Þ
1
ð Þ
e
t
t
0
t
ðÞ
P
II
0
t
ðÞ
P
: ð
3
:
318
Þ
s
:
;
t
0
¼
0
With the identity (note (
3.257
)
6
and P
II
0
ð
t
0
¼
0
Þ¼
0)
ð
4
Þ
ð
4
Þ
ð
4
Þ
ðÞ
P
II
ðÞ
P
II
c P
0
ðÞ
P
ðÞ
c P
0
ðÞ
8
<
:
9
=
;
dt
0
ð
3
:
319
Þ
ðÞ
Z
t
ð
4
Þ
d
dt
0
ð
4
Þ
t
ðÞ
P
II
0
t
ðÞ
P
c
P
t
0
¼
0
and substitution in (
3.318
), finally, the form proposed in Simo (1987) alternative to
(
3.316
) follows:
ðÞ¼
Jp
0
C
1
þ
J
2
=
3
=
P
I
0
with
P
II
p
0
:
¼
df
0
ðÞ
dJ
dt
0
and
=
P
I
0
¼
P
ðÞ
R
ð
4
Þ
t
ð
4
Þ
ð
3
:
320
Þ
t
ðÞ
P
I
0
Kt
t
ð Þ
dt
0
P
t
ðÞ
t
0
¼
0
Kt
t
ð Þ
:
¼
c
þ
1
ð Þ
e
t
t
0
with
s
