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.

This paper is devoted to give a simplified proof of the trace theorem for functions of bounded deformation defined on bounded Lipschitz domains of R n . As a consequence, the existence of one-sided Lebesgue limits on countably H n−1 -rectifiable sets is also established.

This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution associated with the relaxed model. Firstly, a Γ-convergence analysis is performed with a surface energy density which does not provide weak compactness in the space of Special Functions of Bounded Variation. Then, the asymptotic analysis of the quasistatic crack evolution is presented in the case of bounded solutions that is with the simplifying assumption that every minimizing sequence is uniformly bounded in L ∞ .