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
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
1 n
8 W a 2 E f
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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