Biomedical Engineering Reference
In-Depth Information
Fig. 7
Planar fiber model. Notation
The actual shape f
ð
x
;
e
F
Þ¼
r
ð
x
;
e
F
Þ
n at the along-the-chord strain level e
F
is
determined by solving the following differential problem, resulting by applying the
Principle of Virtual Works [
59
]:
r
ð
x
;
e
F
Þ¼
E
eq
ð
e
F
Þ
E
f
ð
e
f
Þ
½
a
1
ð
x
;
e
F
Þ
n
þ
a
2
ð
x
;
e
F
Þ
t
e
F
;
ð
16
Þ
with
a
1
ð
x
;
e
F
Þ¼
xa
ð
e
F
Þþ
Z
x
0
df
sin a
A
F
I
F
f f
cos a
ð
17
Þ
Z
L
x
A
F
I
F
f
cos a
df
;
x
df
a
2
ð
x
;
e
F
Þ¼
f
ð
x
;
e
F
Þ
a
ð
e
F
Þþ
Z
x
0
cos a
þ
A
F
I
F
f
2
cos a
ð
18
Þ
Z
L
þ
f
ð
x
;
e
F
Þ
A
F
I
F
f
cos a
df
;
x
:
1
2 cos a
i
2
A
F
I
F
a
ð
e
F
Þ¼
h
sin 2a
f
hi
ð
19
Þ
h
i
In order to show effectiveness and soundness of the proposed microscale
description, numerical results obtained for an isolated collagen crimped fiber are
discussed, addressing the fiber along-the-chord response. The reference centerline
curve f
ð
x
;
0
Þ
is defined as the sum of two sinusoidal waves depending on the
parameters x
2
N (x
0) and v
2
R:
f
ð
x
;
0
Þ¼
H
o
sin 2px
=
L
o
ð
Þ
vH
o
sin 2xpx
=
L
o
ð
Þ
ð
20
Þ
and, as a notation rule, H
max
;
o
¼
max
f
f
ð
x
;
0
Þg
. Moreover, nanoscale parameters
defining the fiber's material behavior are assumed to be equal to
'
p
¼
14
:
5 nm,
'
c
¼
287 nm, E
o
¼
1 GPa, E
¼
100 GPa, g
¼
10, e
o
¼
0
:
35
;'
kinks
¼
22 nm,
A
m
¼
1
:
41 nm
2
, T
¼
310
:
15 K, l
¼
1 and kk
cl
¼
10 pN/nm.
By integrating Eq. (
16
), reference and actual (at e
F
¼
0
:
1) shapes of curvilinear
fibers with different centerline curves are shown in Fig.
8
. The typical stiffening
Search WWH ::
Custom Search