  Adhesion hysteresis can be caused by elastic instabilities that are triggered
by surface roughness or chemical heterogeneity. However, the role of these
instabilities in adhesion hysteresis remains poorly understood because we lack
theoretical and numerical models accounting for realistic roughness. Our work
focuses on the adhesion of soft elastic spheres with low roughness or weak
heterogeneity, where the indentation process can be described as a
Griffith-like propagation of a nearly circular external crack. We discuss how
to describe the contact of spheres with chemical heterogeneity that leads to
fluctuations in the local work of adhesion. We introduce a variational
first-order crack-perturbation model and validate our approach using
boundary-element simulations. The crack-perturbation model faithfully predicts
contact shapes and hysteretic force-penetration curves, provided that the
contact perimeter remains close to a circle and the contact area is simply
connected. Computationally, the crack-perturbation model is orders of magnitude
more efficient than the corresponding boundary element formulation, allowing
for realistic heterogeneity fields. Furthermore, our crack-front formulation
clarifies the connection of adhesion hysteresis to classic theories on pinning
of elastic lines.
