On Lennard-Jones systems with finite range interactions and their asymptotic analysis

Please always quote using this URN: urn:nbn:de:bvb:20-opus-228428
  • The aim of this work is to provide further insight into the qualitative behavior of mechanical systems that are well described by Lennard-Jones type interactions on an atomistic scale. By means of Gamma-convergence techniques, we study the continuum limit of one-dimensional chains of atoms with finite range interactions of Lennard-Jones type, including the classical Lennard-Jones potentials. So far, explicit formula for the continuum limit were only available for the case of nearest and next-to-nearest neighbour interactions. In this work, weThe aim of this work is to provide further insight into the qualitative behavior of mechanical systems that are well described by Lennard-Jones type interactions on an atomistic scale. By means of Gamma-convergence techniques, we study the continuum limit of one-dimensional chains of atoms with finite range interactions of Lennard-Jones type, including the classical Lennard-Jones potentials. So far, explicit formula for the continuum limit were only available for the case of nearest and next-to-nearest neighbour interactions. In this work, we provide an explicit expression for the continuum limit in the case of finite range interactions. The obtained homogenization formula is given by the convexification of a Cauchy-Born energy density. Furthermore, we study rescaled energies in which bulk and surface contributions scale in the same way. The related discrete-to-continuum limit yields a rigorous derivation of a one-dimensional version of Griffith' fracture energy and thus generalizes earlier derivations for nearest and next-to-nearest neighbors to the case of finite range interactions. A crucial ingredient to our proofs is a novel decomposition of the energy that allows for re fined estimates.show moreshow less

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Metadaten
Author: M. Schäffner, A. Schlömerkemper
URN:urn:nbn:de:bvb:20-opus-228428
Document Type:Journal article
Faculties:Fakultät für Mathematik und Informatik / Institut für Mathematik
Language:English
Parent Title (English):Networks and Heterogeneous Media
Year of Completion:2018
Volume:13
Issue:1
Pagenumber:95-118
Source:Networks and Heterogeneous Media, 13(1): 95-118.
DOI:https://doi.org/10.3934/nhm.2018005
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Tag:Cauchy-Born rule; Discrete-to-continuum limits; Gamma-convergence; atomistic models; variational fracture
Release Date:2023/01/25
Licence (German):License LogoCC BY-NC-ND: Creative-Commons-Lizenz: Namensnennung, Nicht kommerziell, Keine Bearbeitungen 4.0 International