Towards uniformly Γ-equivalent theories for non-convex discrete systems

Scardia, L., Schloemerkemper, A. and Zanini, C. (2012) Towards uniformly Γ-equivalent theories for non-convex discrete systems. Discrete and Continuous Dynamical Systems: Series B, 17(2), pp. 661-686. (doi: 10.3934/dcdsb.2012.17.661)

Abstract

In this paper we consider a one-dimensional chain of atoms which interact with their nearest and next-to-nearest neighbours via a Lennard-Jones type potential. We are interested in a good approximation of the discrete energy of this system for a large number of atoms, i.e., in the continuum limit. We show that the canonical expansion by Gamma-convergence does not provide an accurate approximation of the discrete energy if the boundary conditions for the deformation are close to the threshold between elastic and fracture regimes. This suggests that a uniformly Gamma-equivalent approximation of the energy should be made, as introduced by Braides and Truskinovsky, to overcome the drawback of the lack of accuracy of the standard Gamma-expansion. In this spirit we provide a uniformly Gamma-equivalent approximation of the discrete energy at first order, which arises as the Gamma-limit of a suitably scaled functional.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Scardia, Dr Lucia
• Authors: Scardia, L., Schloemerkemper, A., and Zanini, C.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Discrete and Continuous Dynamical Systems: Series B
• ISSN: 1531-3492
• Published Online: 01 December 2011