Debonding from the substrate and subsequent blistering is a mechanism by which a  film can release compressive stresses. This happens for example upon cooling films deposited at high temperature on a substrate with a larger thermal expansion coefficient. Most studies of blistering are based on the Föppl-von Kármán theory of plates. Typical observed configurations exhibit folds perpendicular to the boundary, which coarsen in the interior of the debonded region. We explain this phenomenon through energy minimization. Indeed, we show that in order to achieve minimal energy the displacement field develops fine-scale oscillations. The oscillation period decreases approaching the boundary, corresponding to branching of the observed folds. We also show that, for this purpose, the Föppl-von Kármán model is equivalent to the full three-dimensional elasticity theory.
Branching of fine-scale oscillations is an ubiquitous feature in systems governed by a nonconvex energy. As an additional example, we discuss twin branching in austenite-martensite phase transitions.