On the Passage from Atomistic Systems to Nonlinear Elasticity Theory

  • We derive continuum limits of atomistic models in the realm of nonlinear elasticity theory rigorously as the interatomic distances tend to zero. In particular we obtain an integral functional acting on the deformation gradient in the continuum theory which depends on the underlying atomistic interaction potentials and the lattice geometry. The interaction potentials to which our theory applies are general finite range models which in particular can also account for multi-pole interactions and bond-angle dependent contributions. Furthermore, we discuss the applicability of the Cauchy-Born rule. Our class of limiting energy densities consists of general quasiconvex functions and the class of linearized limiting energies consistent with the Cauchy-Born rule consists of general quadratic forms not restricted by the Cauchy relations.

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Metadaten
Author:Julian Braun, Bernd SchmidtGND
URN:urn:nbn:de:bvb:384-opus4-12756
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/1672
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2012-03)
Type:Preprint
Language:English
Publishing Institution:Universität Augsburg
Release Date:2012/03/06
Tag:Übergang von atomaren zu kontinuierlichen Systemen; Cauchy-Born Regel
nonlinear elasticity theory; gamma-convergence; discrete-to-continuum limits; Cauchy-Born rule
GND-Keyword:Gamma-Konvergenz; Nichtlineare Elastizitätstheorie; Variationsrechnung
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Nichtlineare Analysis
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):Deutsches Urheberrecht