  In this paper we introduce a variational model for the study of multilayer
films that allows for the treatment of both coherent and incoherent interfaces
between layers. The model is designed in the framework of the theory of Stress
Driven Rearrangement Instabilities, which are characterized by the competition
between elastic and surface energy effects. The surface of each film layer is
assumed to satisfy an ''exterior graph condition'', under which in particular
bulk cracks are allowed to be of non-graph type. By applying the direct method
of calculus of variations under a constraint on the number of connected
components of the cracks not connected to the surface of the film layers the
existence of energy minimizers is established in dimension 2. As a byproduct of
the analysis the state of art on the variational modeling of single-layered
films deposited on a fixed substrate is advanced by letting the substrate
surface free, by addressing the presence of multiple layers of various
materials, and by including the possibility of delamination between the various
film layers.
