Continuum Limit and Homogenization of Stochastic and Periodic Discrete Systems – Fracture in Composite Materials
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- The limiting behaviour of a one‐dimensional discrete system is studied by means of Γ‐convergence. We consider a toy model of a chain of atoms. The interaction potentials are of Lennard‐Jones type and periodically or stochastically distributed. The energy of the system is considered in the discrete to continuum limit, i.e. as the number of atoms tends to infinity. During that limit, a homogenization process takes place. The limiting functional is discussed, especially with regard to fracture. Secondly, we consider a rescaled version of theThe limiting behaviour of a one‐dimensional discrete system is studied by means of Γ‐convergence. We consider a toy model of a chain of atoms. The interaction potentials are of Lennard‐Jones type and periodically or stochastically distributed. The energy of the system is considered in the discrete to continuum limit, i.e. as the number of atoms tends to infinity. During that limit, a homogenization process takes place. The limiting functional is discussed, especially with regard to fracture. Secondly, we consider a rescaled version of the problem, which yields a limiting energy of Griffith's type consisting of a quadratic integral term and a jump contribution. The periodic case can be found in [8], the stochastic case in [6,7].…
Autor(en): | Laura Lauerbach, Stefan Neukamm, Mathias Schäffner, Anja Schlömerkemper |
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URN: | urn:nbn:de:bvb:20-opus-211835 |
Dokumentart: | Artikel / Aufsatz in einer Zeitschrift |
Institute der Universität: | Fakultät für Mathematik und Informatik / Institut für Mathematik |
Sprache der Veröffentlichung: | Englisch |
Titel des übergeordneten Werkes / der Zeitschrift (Englisch): | Proceedings in Applied Mathematics & Mechanics |
Erscheinungsjahr: | 2019 |
Band / Jahrgang: | 19 |
Heft / Ausgabe: | 1 |
Seitenangabe: | e201900070 |
Originalveröffentlichung / Quelle: | Proceedings in Applied Mathematics & Mechanics 2019, 19(1):e201900070. DOI: 10.1002/pamm.201900070 |
DOI: | https://doi.org/10.1002/pamm.201900070 |
Allgemeine fachliche Zuordnung (DDC-Klassifikation): | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Freie Schlagwort(e): | continuum limit; discrete systems; homogenization |
Datum der Freischaltung: | 22.09.2020 |
Lizenz (Deutsch): | CC BY: Creative-Commons-Lizenz: Namensnennung 4.0 International |