Continuum limits of discrete thin films with superlinear growth densities

We provide a rigorous derivation by Gamma-convergence of an effective theory of thin films in hyperelastic regime in the so-called discrete to continuous framework. By considering a discrete thin film obtained piling up at microscopic distance a finite number M of copies of a discrete monolayer omega, we provide a continuum description analogous to that in the dimension-reduction theories for continuum thin films. Our energetic description of the continuum limit model accounts for microscopic effects and in particular depends in a non-trivial way on M. We also consider the problem of homogenization and discuss several cases of interest when an explicit formula for the homogenized energy density can be obtained, with an interpretation in terms of the Cauchy-Born rule.