  Contact unit size reduction is a widely studied mechanism as a means to
improve adhesion in natural fibrillar systems, such as those observed in
beetles or geckos. However, these animals also display complex structural
features in the way the contact is subdivided in a hierarchical manner. Here,
we study the influence of hierarchical fibrillar architectures on the load
distribution over the contact elements of the adhesive system, and the
corresponding delamination behaviour. We present an analytical model to derive
the load distribution in a fibrillar system, including hierarchical splitting
of contacts, i.e. a "hierarchical shear-lag" model that generalizes the
well-known shear-lag model used in mechanics. The influence on the detachment
process is investigated introducing a numerical procedure that allows the
derivation of the maximum delamination force as a function of the considered
geometry, including statistical variability of local adhesive energy. Our study
suggests that contact splitting generates improved adhesion only in the ideal
case of infinitely compliant contacts. In real cases, to produce efficient
adhesive performance, contact splitting needs to be coupled with hierarchical
architectures to counterbalance high load concentrations resulting from contact
unit size reduction, generating multiple delamination fronts and helping to
avoid detrimental non-uniform load distributions. We show that these results
can be summarized in a generalized adhesion scaling scheme for hierarchical
structures, proving the beneficial effect of multiple hierarchical levels. The
model can thus be used to predict the adhesive performance of hierarchical
adhesive structures, as well as the mechanical behaviour of composite materials
with hierarchical reinforcements.
