We study fracture and delamination of a thin stiff film bonded on a rigid substrate through a thin compliant bonding layer. Starting from the three-dimensional system, upon a scaling hypothesis, we provide an asymptotic analysis of the three-dimensional variational fracture problem as the thickness goes to zero, using Γ-convergence. We deduce a two-dimensional limit model consisting of a brittle membrane on a brittle elastic foundation. The fracture sets are naturally discriminated between transverse cracks in the film (curves in 2D) and debonded surfaces (two-dimensional planar regions). We introduce the vectorial plane-elasticity case, applying the rigorous results established for scalar displacement fields, in order to numerically investigate the typical cracking scenarios encountered in applications. To this end, we formulate a reduced-dimension, rate-independent, irreversible evolution law for transverse fracture and debonding of thin film systems. Finally, we propose a numerical implementation based on a regularized formulation of the fracture problem via a gradient damage functional. We provide an illustration of the capabilities of the formulation exploring complex crack patterns in one and two dimensions, showing a qualitative comparison with geometrically involved real life examples.

We study multifissuration and debonding phenomena of a thin film bonded to a stiff substrate using the variational approach to fracture mechanics. We consider a reduced one-dimensional membrane model where the loading is introduced through uniform inelastic (e. g. thermal) strains in the film or imposed displacements of the substrate. Fracture phenomena are accounted for by adopting a Griffith model for debonding and transverse fracture. On the basis of energy minimization arguments, we recover the key qualitative properties of the experimental evidences, like the periodicity of transverse cracks and the peripheral debonding of each regular segment. Phase diagrams relate the maximum number of transverse cracks that may be created before debonding takes place, as a function of the material properties and the sample's geometry. The theoretical results are illustrated with numerical simulations obtained through a finite element discretization and a regularized variational formulation of the Ambrosio-Tortorelli type, which is suited to further extensions in two-dimensional settings.