Biomedical Engineering Reference
In-Depth Information
At the lowest level of hierarchy, the failure process is governed by the van der
Waals interaction between the smallest fibers (ultrastructure) and a solid surface.
In this case, the maximum fiber radius ensuring flaw tolerant adhesion is given by
DgE
f
pð
1
n
8
R
1
¼
(10.8)
2
2
th
f
Þs
where the work of adhesion is simply equal to the surface energy
Dg
due to van der
Waals interaction and
s
th
is the theoretical strength of van der Waals forces.
In light of the anti-bunching condition of (
10.7
), the maximum fiber length of the
bottom level can be expressed as a function of the area fraction
'
1
of this level as
1
=
3
1
=
2
p
'
max
='
1
E
f
R
1
g
f
L
1
ð'
1
Þ¼R
1
a
1
(10.9)
With these parameters, the work of adhesion associated with the next (second)
level is given by
ð'
1
Þ¼
s
th
L
1
W
ad
2
2
E
f
þ Dg
'
1
(10.10)
which is a function of the area fraction
'
1
. This function exhibits a maximum at a
specific value of
'
1
due to the opposing trends of variation of the parameters
L
1
and
'
1
: denser fibers with larger
'
1
require smaller
L
1
for stability against bunching.
Therefore, we can choose the fiber area fraction
'
1
to maximize the work of
adhesion at the next level according to (
10.10
). After
'
1
is calculated, the fiber
length
L
1
is immediately determined by (
10.9
). In this way, all the structural
parameters characterizing the first level
R
1
,
L
1
,
'
1
have been determined. Then,
by using (
10.10
), the work of adhesion for the second level
W
a
2
is obtained as well.
We now advance further to design the second (next) level. The fiber radius is
again chosen to ensure flaw tolerant adhesion,
8
W
a
2
E
f
pð
1
n
8
W
a
2
E
f
pð
1
n
R
2
¼
2
¼
(10.11)
2
2
2
f
ÞðS
2
Þ
f
Þð'
1
s
th
Þ
where
S
2
¼ '
1
s
th
is the effective adhesion strength of the second level. Similarly,
the anti-bunching condition allows the fiber length to be determined as a function of
the area fraction
'
2
as
1
=
3
1
=
2
p
'
max
='
2
E
f
R
2
g
f
L
2
ð'
2
Þ¼R
2
a
1
(10.12)
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