Continuum limits of discrete thin films with superlinear growth densities

Alicandro, Roberto; Braides, Andrea; Cicalese, Marco
November 2008
Calculus of Variations & Partial Differential Equations;Nov2008, Vol. 33 Issue 3, p267
Academic Journal
We provide a rigorous derivation by G-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 ?, 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.


Related Articles

  • Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media. Caffarelli, Luis; Souganidis, Panagiotis // Inventiones Mathematicae;May2010, Vol. 180 Issue 2, p301 

    We establish a logarithmic-type rate of convergence for the homogenization of fully nonlinear uniformly elliptic second-order pde in strongly mixing media with similar, i.e., logarithmic, decorrelation rate. The proof consists of two major steps. The first, which is actually the only place in...

  • Maximum Principle in the Optimal Design of Plates with Stratified Thickness. Roub�&cek, Tom� // Applied Mathematics & Optimization;Mar/Apr2005, Vol. 51 Issue 2, p183 

    An optimal design problem for a plate governed by a linear, elliptic equation with bounded thickness varying only in a single prescribed direction and with unilateral isoperimetrical-type constraints is considered. Using Murat-Tartar�s homogenization theory for stratified plates and...

  • Compensated compactness for nonlinear homogenization and reduction of dimension. P. Courilleau; J. Mossino // Calculus of Variations & Partial Differential Equations;May2004, Vol. 20 Issue 1, p65 

    We study the limit behaviour of some nonlinear monotone equations, such as: $-div(A^\epsilon \varphi (B^\epsilon \nabla U^\epsilon)) = F^\epsilon$ , in a domain $\Omega^\epsilon$ which is thin in some directions (e.g. $\Omega^\epsilon$ is a plate or a thin cylinder). After rescaling to a fixed...

  • Homogenization of the Prager model in one-dimensional plasticity. Schweizer, Ben // Continuum Mechanics & Thermodynamics;Apr2009, Vol. 20 Issue 8, p459 

    We propose a new method for the homogenization of hysteresis models of plasticity. For the one-dimensional wave equation with an elasto-plastic stress-strain relation we derive averaged equations and perform the homogenization limit for stochastic material parameters. This generalizes results of...

  • Framework for deformation induced anisotropy in glassy polymers. Harrysson, Magnus; Ristinmaa, Matti; Wallin, Mathias; Menzel, Andreas // Acta Mechanica;May2010, Vol. 211 Issue 3/4, p195 

    In this paper a constitutive model for glassy polymers is developed. Glassy polymers consist of a number of polymer chains that at a microscopic level form a network. If the distribution of the polymer chains shows some preferred direction, the mechanical response at a global macroscopic level...

  • Homogenization and Multigrid. Neuss, N.; J�ger, W.; Wittum, G. // Computing;2001, Vol. 66 Issue 1, p1 

    For elliptic partial differential equations with periodically oscillating coefficients which may have large jumps, we prove robust convergence of a two-grid algorithm using a prolongation motivated by the theory of homogenization. The corresponding Galerkin operator on the coarse grid turns out...

  • Homogenization of First-Order Equations with $$(u/\varepsilon)$$ -Periodic Hamiltonians. Part I: Local Equations. Imbert, Cyril; Monneau, R�gis // Archive for Rational Mechanics & Analysis;Jan2008, Vol. 187 Issue 1, p49 

    In this paper, we present a result of homogenization of first-order Hamilton�Jacobi equations with ( $$u/\varepsilon$$ )-periodic Hamiltonians. On the one hand, under a coercivity assumption on the Hamiltonian (and some natural regularity assumptions), we prove an ergodicity property of...

  • Multiscale Young Measures in almost Periodic Homogenization and Applications. Ambrosio, Luigi; Frid, Hermano // Archive for Rational Mechanics & Analysis;Apr2009, Vol. 192 Issue 1, p37 

    We prove the existence of multiscale Young measures associated with almost periodic homogenization. We give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure, such as scalar conservation laws, nonlinear transport...

  • Homogenization of the Three-dimensional Hall Effect and Change of Sign of the Hall Coefficient. BRIANE, MARC; MILTON, GRAEME W. // Archive for Rational Mechanics & Analysis;Sep2009, Vol. 193 Issue 3, p715 

    The notion of a Hall matrix associated with a possibly anisotropic conducting material in the presence of a small magnetic field is introduced. Then, for any material having a microstructure we prove a general homogenization result satisfied by the Hall matrix in the framework of the...


Read the Article


Sign out of this library

Other Topics