Mechanics of Self-Similar Hierarchical Adhesive Structures Inspired by Gecko Feet - Structural Interfaces and Attachments in Biology - page 196

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
ps

, which is taken as the condition for flaw 2

th

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

r 8

p

1 = 2

E W ad
R

P crack ¼ pR 2 (10.1)

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 =ð 1 n

where W ad denotes the work of adhesion and E ¼½ð 1 n

2 2

f Þ=E f þð 1 n

2

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 (10.2)

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