Biomedical Engineering Reference
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Fig. 10.2 A fiber is brought into contact with a substrate. Depending upon the shape of the fiber
tip, the detachment process can occur either by ( a ) crack propagation ( singular shapes ) equivalent
to an infinite crack external to the contact area or by ( b ) uniform detachment ( optimal shapes )in
which the stress at pull-off is uniformly distributed and equal to the theoretical adhesion strength
s th . The difference between the adhesive strength of these two failure modes vanishes as the size of
the fibril is reduced to below a threshold R cr ¼
8 E Dg
, which is taken as the condition for flaw
tolerant adhesion
material that allows the contact to fail not by crack propagation, but always by
uniform detachment at the theoretical strength of adhesion, a concept termed as flaw
tolerance [ 7 , 25 , 26 ]. According to this concept, in an ideal flaw tolerant adhesion
system, there should be no crack propagation and coalescence as the contact
interface is pulled apart by uniform detachment.
For a single fiber on substrate, Gao and Yao [ 23 ] investigated the condition for
flaw tolerant adhesion from the point of view of variations in contact shape. It was
shown that there exist two extreme classes of contact shapes: one class (singular
shapes) gives rise to a singular stress field at pull-off similar to that of an external
crack (Fig. 10.2a ) and the other class (optimal shapes) leads to a uniform stress at
pull-off (Fig. 10.2b ). For singular shapes, the pull-off force can be calculated
according to the Griffith condition [ 10 ]as
1 = 2
E W ad
P crack ¼ pR 2
s Þ=E s 1 ,
E f ; E s ; n f ; n s being the Young's moduli and Poisson's ratios of the fiber and the
substrate, respectively. For a gecko sticking to a solid surface, we assume E s E f ,
therefore E E f 1 n
where W ad denotes the work of adhesion and E ¼½ð 1 n
f Þ=E f þð 1 n
f Þ . On the other hand, the pull-off force for optimal
contact shapes (Fig. 10.2b )is
P th ¼ pR 2
s th
where s th is the theoretical adhesion strength. Generally, P crack is much smaller than
P th . However, as the size of the fiber is reduced, the value of P crack increases towards
P th . At the critical size
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