Mechanics of Self-Similar Hierarchical Adhesive Structures Inspired by Gecko Feet - Structural Interfaces and Attachments in Biology - page 211

p

W ad

= pa

s cr ðy;'Þ¼ q

C½D 22 cos 2

(10.27)

ðy 'ÞþD 11 sin 2

sin ' ðy 'Þ

where

2

C ¼ ð 1 þ 4 e

Þ

(10.28)

4cosh 2 pe

Given material constants and the anisotropy direction y ,( 10.27 ) indicates that the
adhesion strength varies as a function of the pulling angle ' . To calculate the critical
(maximum and minimum) values as well as the corresponding directions, we solve
equation @s cr ðy;'Þ
@'

¼ 0 and obtain

s

E t ðE l n

2

1 þ D 22 =D 11
1 D 22 =D 11 cos

l E t Þ
E l ð 1 n

' ¼ cos ð 3 ' 2 yÞ; D 22 =D 11 ¼ (10.29)

2

t Þ

If the Young's modulus in the longitudinal direction (e.g., along a fiber array) is
much larger than that in the transverse direction (e.g., transverse to the fiber
direction), i.e., E l =E t 1, ( 10.29 ) has two roots

' 1 ¼ y;' 2 ¼ y= 2 þ p= 2 (10.30)

corresponding to the directions of the maximum and minimum pull-off stresses,
respectively. The adhesion releasability thus can be measured by the ratio of the
maximum to the minimum pull-off stresses:

1 = 2
¼ ð 1 þ cos yÞ
2 sin y

1 = 4

ðs cr Þ max

ðs cr Þ min ¼ ð 1 þ cos yÞ

D 11
D 22

E l ð 1 n

2

t Þ

(10.31)

2 sin y E t ðE l n

2

l E t Þ

For small Poisson's ratios, ( 10.31 ) suggests that the releasibility of adhesion
mainly depends on the stiffness ratio E l =E t and the anisotropy direction y . The
stronger the anisotropy, the higher the releasibility of adhesion. Assuming n t ¼ n l
¼ 0 3, y ¼ 30 , and E l =E t ¼ 10 4 , Fig. 10.9b plots the normalized pull-off stress as a

function of the pulling angle

:

. We can see that the elastic anisotropy causes about
an order of magnitude change in adhesion strength as the pulling angle varies.
A switch between attachment and detachment can thus be accomplished just by
shifting the pulling angle between these two directions. In contrast, the adhesion
strength for an isotropic material with E l ¼ E t and n l ¼ n t is much less sensitive to
the pulling direction. We conclude that strong elastic anisotropy can result in an
orientation-controlled switch between attachment and detachment. Similar
orientation-dependent behavior can also be seen from the adhesive contact between
a rigid sphere and an anisotropic elastic substrate [ 49 ].

'