Mechanics of Self-Similar Hierarchical Adhesive Structures Inspired by Gecko Feet - Structural Interfaces and Attachments in Biology - page 194

between elastic and surface energies similar to Griffith's [ 10 ] criterion for crack
growth in an elastic solid. The JKR theory introduces into the Hertz solution an
additional crack-like singular term which satisfies the Griffith condition near the
contact edge. While the JKR theory is quite appropriate for modeling contact
between large and soft materials, the assumption of a crack-like singular field
becomes increasingly inaccurate for small and stiff materials, in which case
different assumptions on the elastic deformation of contacting objects have led to
the models of DMT (Derjaguin-Muller-Toporov) [ 11 ] and Bradley [ 12 ]. Maugis
[ 13 ] generalized the Dugdale model of a crack in a plastic sheet [ 14 ] to adhesive
contact and developed a more general model (Maugis-Dugdale model) that includes
the JKR and DMT models as two limiting cases. More recent studies have further
extended these theories to viscoelastic materials [ 15 , 16 ], coupled normal and shear
loads [ 17 ] and biological attachments [ 4 , 5 , 18 - 21 ].
For contact between single smooth asperities, one can define adhesion strength
as the tensile force per unit contact area at pull-off. It has been shown that the
adhesion strength can be enhanced up to the theoretical adhesion strength via size
reduction [ 7 , 22 - 24 ]. In this respect, it is interesting to note that the existing contact
mechanics theories, including JKR, DMT, and Maugis-Dugdale models, all
predicted infinite adhesion strength as the size of contacting objects is reduced to
zero. This behavior seems contradictory to the physics that adhesion strength
would never exceed the theoretical strength of adhesive interaction. The fact that
this behavior also occurs in the Maugis-Dugdale model is especially surprising
since the original Dugdale model correctly predicted that the fracture strength is
bounded by the yield strength of the material. Gao et al. [ 7 ] found that the root of
this illogical behavior of the classical contact models can be attributed to the
original Hertz approximation of contact surfaces as parabolas, which is strictly
valid only if the size of the contact area is much smaller than the overall dimension
of the contacting objects; the lack of strength saturation in these models is thus
explained from the fact that the parabolic approximation fails in the limit of very
small contacting bodies. As an example, Gao et al. [ 7 ] showed that, if the exact
geometry of a sphere is taken into account, the adhesion strength indeed saturates
at the theoretical strength as the diameter of the sphere is reduced to zero. On the
other hand, Gao and Yao [ 23 ] showed that the adhesion strength can in principle
approach the theoretical strength for any contact size via shape optimization.
In practice, interfacial crack-like flaws due to surface roughness or contaminants
inevitably weaken the actual adhesion strength. Gao et al. [ 7 ] performed finite
element calculations to show that the adhesion strength of a flat-ended cylindrical
punch in partial contact with a rigid substrate saturates at the theoretical strength
below a critical radius around 200 nm for the van der Waals interaction. Similar
discussions of strength saturation for small contacting objects have been made by
Persson [ 22 ] for a rigid cylindrical punch on an elastic half-space and by Glass-
maker et al. [ 24 ] for an elastic cylindrical punch in perfect bonding with a rigid
substrate. Gao and Yao [ 25 ] showed that the theoretical strength can be achieved
by either optimizing the shape of the contact surfaces or by reducing the size of the
contact area; the smaller the size, the less important the shape. A shape-insensitive