Mechanics of Self-Similar Hierarchical Adhesive Structures Inspired by Gecko Feet - Structural Interfaces and Attachments in Biology - page 198

hairy surface, assuming Lennard-Jones (e.g., [ 30 ]) or Dugdale [ 14 ] interaction law.
While the stress-separation curves for two smooth surfaces are described by the van
der Waals or Dugdale interaction laws at the atomic scale, the separation at the level
of the larger fiber is strongly influenced by the elastic properties and geometry of
the fibrils. For sufficiently long fibrils, the elastic deformation of the fibrils will
make significant contributions to the separation process and adhesion failure occurs
by an abrupt drop in stress near the theoretical strength of surface interaction. In this
way, the strain energy stored in the fiber becomes part of the cohesive energy to be
dissipated through dynamic snapping of the thin fibrils. In other words, the thin
fibrils behave effectively as cohesive bonds for the larger fiber. The work of
adhesion for the large fiber should therefore include the elastic energy stored in
the fibrils when they are stretched to failure, i.e.,

2

W ad ¼ðDg þ s

th L=

2 E f Þ' (10.4)

where L is the length of the fibrils and is the area fraction of the fibril array. The first

term within the bracket represents the original van der Waals interaction energy and
the second term is the elastic energy lost during dynamic snapping of the fibrils as
they are detached from the substrate near the theoretical strength of van der Waals
interaction. Equation (4) also shows why it is important to optimize the strength of
the lower level fibril structure via size reduction: the strength of the lower scale
fibrils directly contributes to the work of adhesion of the larger scale fiber. Taking
Dg ¼ 0

'

:

01 J/m 2 , s th ¼ 20 MPa, L ¼ 100

m

m, E f ¼ 1 GPa,

' ¼ 0 :

5, the work of
adhesion for the hairy tipped fiber is calculated to be W ad 10 J/m 2 , a value much
larger than

Dg . Such enhancement in work of adhesion by fibrillar structures has also
been reported or discussed by Jagota and Bennison [ 31 ], Persson [ 20 ], Gao et al.
[ 25 ], and Tang et al. [ 32 ]. Hence, slender hairs with large aspect ratios can signifi-
cantly increase the work of adhesion and contribute to the robustness of adhesion at
larger scales. However, the length of the fibrils cannot be too long, as there is an
instability leading to fiber bunching as the aspect ratio of the fibrils increases.

10.3.3 Anti-Bunching Condition of Fibrillar Array

In an array of slender hairs planted on a solid surface, the van der Waals interaction
between neighboring fibers will cause them to bundle together [ 7 , 20 , 33 - 36 ].
The anti-bunching condition is an important factor in the design of hairy adhesion
structures. The exact form of the anti-bunching condition depends on the geometry
of the fiber. For example, the anti-bunching condition for fibers of square cross
section has been discussed by Hui et al. [ 35 ] and Gao et al. [ 7 ]. Here, we focus on
cylindrical fibers that have been considered by Glassmaker et al. [ 33 ].
Consider two neighboring identical cylindrical fibers with circular cross
sections. When the separation 2 w becomes small, the surface adhesive forces may