Mechanics of Self-Similar Hierarchical Adhesive Structures Inspired by Gecko Feet - Structural Interfaces and Attachments in Biology - page 213

Fig. 10.10 Releasable adhesion in an attachment pad. ( a ) Geometry of the attachment pad used in
FEM calculations. ( b ) Variation of normalized pull-off force with the pulling angle. Periodic B.C.:
periodic boundary condition

s

D n
d cn

2
þ

2

D t
d ct

l ¼ (10.36)

D t the tangential separation; d cn and
d ct are the corresponding critical separations. The force function

Here, D n denotes the normal separation and

' is taken to be

trilinear,

8
< s max l=L 1

s max
s max ð 1 lÞ=ð 1 L 2 Þ;

ðl <L 1 Þ
ðL 1 <l<L 2 Þ
ðL 2 <l<

'ðlÞ¼ (10.37)

:

1 Þ

where L 2 define the values of l at which the cohesive force reaches the peak.

Taking the partial derivatives of the potential with respect to the normal and
tangential separations gives the normal and tangential tractions as

L 1 and

T n ¼ @F

@D n ¼ D n d cn 'ðlÞ

(10.38)

l

T t ¼ @F

@D t ¼ d cn d ct D t d ct 'ðlÞ

(10.39)

l

Clearly, Tvergaard-Hutchison law takes into account both normal and tangential
tractions with a constant work of adhesion

W ad ¼ 1 = 2 ð 1 þ L 2 L 1 Þs max d cn (10.40)

The simulation system consists of a plane-strain anisotropic (transversely iso-
tropic) elastic pad adhering to a rigid substrate with a crack situated at the central
region of the contact interface (representing an adhesion flaw due to surface