An atomistic derivation of von-Kármán plate theory

Julian Braun; Bernd Schmidt

doi:10.3934/nhm.2022019

An atomistic derivation of von-Kármán plate theory

Julian Braun^*, Bernd Schmidt

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

We derive von-Kármán plate theory from three dimensional, purely atomistic models with classical particle interaction. This derivation is established as a Γ-limit when considering the limit where the interatomic distance ε as well as the thickness of the plate h tend to zero. In particular, our analysis includes the ultrathin case where ε ∼ h, leading to a new von-Kármán plate theory for finitely many layers.

Original language	English
Pages (from-to)	613-644
Number of pages	32
Journal	Networks and Heterogeneous Media
Volume	17
Issue number	4
Early online date	1 Apr 2022
DOIs	https://doi.org/10.3934/nhm.2022019
Publication status	Published - Aug 2022

Keywords

dimension reduction
Discrete to continuum
elastic plates
gamma-convergence
von-Kármán plate theory

ASJC Scopus subject areas

Statistics and Probability
General Engineering
Computer Science Applications
Applied Mathematics

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10.3934/nhm.2022019

https://www.aimsciences.org/article/doi/10.3934/nhm.2022019

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title = "An atomistic derivation of von-K{\'a}rm{\'a}n plate theory",

abstract = "We derive von-K{\'a}rm{\'a}n plate theory from three dimensional, purely atomistic models with classical particle interaction. This derivation is established as a Γ-limit when considering the limit where the interatomic distance ε as well as the thickness of the plate h tend to zero. In particular, our analysis includes the ultrathin case where ε ∼ h, leading to a new von-K{\'a}rm{\'a}n plate theory for finitely many layers.",

keywords = "dimension reduction, Discrete to continuum, elastic plates, gamma-convergence, von-K{\'a}rm{\'a}n plate theory",

author = "Julian Braun and Bernd Schmidt",

note = "Funding Information: Acknowledgments. This work was supported by the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) under project number 285722765, as well as the Engineering and Physical Sciences Research Council (EPSRC) under the grant EP/R043612/1. Publisher Copyright: {\textcopyright} 2022, American Institute of Mathematical Sciences. All rights reserved.",

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month = aug,

doi = "10.3934/nhm.2022019",

language = "English",

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AU - Schmidt, Bernd

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N2 - We derive von-Kármán plate theory from three dimensional, purely atomistic models with classical particle interaction. This derivation is established as a Γ-limit when considering the limit where the interatomic distance ε as well as the thickness of the plate h tend to zero. In particular, our analysis includes the ultrathin case where ε ∼ h, leading to a new von-Kármán plate theory for finitely many layers.

AB - We derive von-Kármán plate theory from three dimensional, purely atomistic models with classical particle interaction. This derivation is established as a Γ-limit when considering the limit where the interatomic distance ε as well as the thickness of the plate h tend to zero. In particular, our analysis includes the ultrathin case where ε ∼ h, leading to a new von-Kármán plate theory for finitely many layers.

KW - dimension reduction

KW - Discrete to continuum

KW - elastic plates

KW - gamma-convergence

KW - von-Kármán plate theory

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JO - Networks and Heterogeneous Media

JF - Networks and Heterogeneous Media

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An atomistic derivation of von-Kármán plate theory

Abstract

Keywords

ASJC Scopus subject areas

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