Continuum limits of discrete thin films with superlinear growth densities

We provide a rigorous derivation by Γ -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 ω, we p...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2008-11, Vol.33 (3), p.267-297
Main Authors: Alicandro, Roberto, Braides, Andrea, Cicalese, Marco
Format: Article
Language:English
Subjects:
Analysis
Calculus of variations
Calculus of Variations and Optimal Control
Optimization
Continuums
Control
Density
Derivation
Homogenizing
Mathematical analysis
Mathematical and Computational Physics
Mathematical models
Mathematics
Mathematics and Statistics
Reproduction
Systems Theory
Theoretical
Thin films
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Summary:We provide a rigorous derivation by Γ -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 ω, 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.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-008-0159-4