Biomedical Engineering Reference
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upon which the work of adhesion for the third level can be determined,
!
!
2
2
ð'
2
Þ¼ W
a
2
þ
ð
S
2
Þ
L
2
2
E
f
W
a
2
þ
ð'
1
s
th
Þ
L
2
W
ad
3
'
2
¼
'
2
(10.13)
2
E
f
Next, the area fraction
'
2
is determined by maximizing
W
a
3
ð'
2
Þ
. Once
'
2
is
known, the fiber length
L
2
is determined from (
10.12
). Hence all the structural
parameters,
R
2
,
L
2
,
'
2
, for the second hierarchical level, as well as the work of
adhesion
W
a
3
for the third level, have been determined.
An iterative procedure can now be described to determine the structural parameters
at all hierarchical levels, starting from the lowest level. Assuming we have completed
the design from the first to (
n
1)-th levels so that
R
i
,
L
i
,
'
i
,
W
ad
(
i ¼
1
;
2
; ; n
1) as
i
well as
W
ad
n
have been determined, for the
n
-th level (
n >
1), the (maximum) fiber
radius ensuring flaw tolerant adhesion is given by
8
W
a
n
E
f
ð
1
n
8
W
a
n
E
f
R
n
¼
2
¼
(10.14)
2
2
2
f
ÞpðS
n
Þ
ð
1
n
f
Þpðs
th
F
n
1
Þ
where
S
n
¼ s
th
F
n
1
; F
n
1
¼ '
1
'
2
'
n
1
¼
Y
n
1
i¼
1
'
i
(10.15)
is the effective adhesion strength of the
n
-th level. The (maximum allowable) fiber
length of the
n
-th level can then be expressed, according to the ant-bunching
condition, as a function of the area fraction
'
n
,
1
=
3
1
=
2
p
'
max
='
n
E
f
R
n
g
f
L
n
ð'
n
Þ¼aR
n
1
(10.16)
The work of adhesion for the (
n
+ 1)-th level is
!
!
2
2
nþ
1
ð'
n
Þ¼ W
a
n
þ
ð
S
n
Þ
L
n
2
E
f
W
a
n
þ
ðs
th
F
n
1
Þ
L
n
W
ad
'
n
¼
'
n
(10.17)
2
E
f
The area fraction for the
n
-th level
'
n
can now be determined by maximizing
W
ad
nþ
1
ð'
n
Þ
, upon which
L
n
and
W
ad
nþ
1
can be readily calculated. This iterative, bottom-
up design procedure can be repeated until the desired size scale for flaw tolerant
adhesion is reached. Upon the knowledge of the fiber radius and area fraction of
each level, we can calculate the number of fibrils on the tip of a fiber at the next
higher level,
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