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Continuum limits of discrete thin films with superlinear growth densities. (English) Zbl 1148.49010
Summary: 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.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 74K35 Thin films 74B20 Nonlinear elasticity 82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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