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Fig. 10.3 Work of adhesion of a hairy surface. ( a ) Schematic of a hairy surface containing arrays
of fibrillar protrusions contacting a substrate. ( b ) Effective stress-separation law for the hairy
surface on substrate vs. that for two smooth surfaces
E f W ad
ð 1 n
R cr ¼
f Þs
the pull-off force for the singular shapes predicted by the Griffith condition in ( 10.1 )
reaches that of the optimal shapes in ( 10.2 ). Alternative derivations based on partial
contact [ 7 ] or perfectly bonded contact [ 24 ] lead to similar, but more relaxed,
conditions on the fiber size. In this chapter, we shall adopt ( 10.3 ) as the basic
flaw tolerant condition for adhesion of a single fiber.
10.3.2 Energy Dissipation in Fibrillar Structures
It can be seen from ( 10.3 ) that R cr is proportional to the work of adhesion W ad which
is commonly taken as the differential surface energy
Dg ¼ g f þ g s g fs , where g f , g s ,
g fs denote the surface energies of fiber, substrate, and fiber-substrate interface,
respectively. However, this interpretation is correct only in the absence of other
dissipation mechanisms. For slender elastic hairs in strong, flaw tolerant adhesion
with a solid surface, additional energy dissipation terms should be taken into
Consider the adhesion between a larger fiber with a hairy tip surface in contact
with a substrate, as shown in Fig. 10.3a . Compared to the case shown in Fig. 10.2 ,
the larger fiber in Fig. 10.3a consists of a number of finer fibrils on its tip, resulting
in a two-leveled structure: an array of smaller fibrils on the tip surface of a larger
fiber. For this structure, the effective work of adhesion for the larger fiber is no
longer equal to
Dg even though the small fibrils interact with the substrate only via
van der Waals forces. To estimate the work of adhesion of the large fiber, we
assume that the fibrils are thin enough to meet the condition for flaw tolerant
adhesion. Figure 10.3b plots the effective stress-separation relationship for the
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