An atomistic derivation of von-Kármán plate theory

Julian Braun; Bernd Schmidt; Julian Braun; Bernd Schmidt

doi:10.3934/nhm.2022019

Networks and Heterogeneous Media

2022, Volume 17, Issue 4: 613-644. doi: 10.3934/nhm.2022019

Previous Article Next Article

An atomistic derivation of von-Kármán plate theory

Julian Braun ^{1
,
,},
Bernd Schmidt ^{2
,}

1.
Heriot-Watt University, United Kingdom
2.
Universität Augsburg, Germany

Received: 01 July 2021 Revised: 01 February 2022 Published: 13 April 2022
Primary: 49J45, 70C20, 74K20

We derive von-Kármán plate theory from three dimensional, purely atomistic models with classical particle interaction. This derivation is established as a $ \Gamma $-limit when considering the limit where the interatomic distance $ \varepsilon $ as well as the thickness of the plate $ h $ tend to zero. In particular, our analysis includes the ultrathin case where $ \varepsilon \sim h $, leading to a new von-Kármán plate theory for finitely many layers.
- Discrete to continuum,
- gamma-convergence,
- dimension reduction,
- elastic plates,
- von-Kármán plate theory
Citation: Julian Braun, Bernd Schmidt. An atomistic derivation of von-Kármán plate theory[J]. Networks and Heterogeneous Media, 2022, 17(4): 613-644. doi: 10.3934/nhm.2022019

Related Papers:

Abstract

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

References

[1]	A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. (2004) 36: 1-37.
[2]	Numerical solution of a Föppl–von Kármán model. SIAM J. Numer. Anal. (2017) 55: 1505-1524.
[3]	From molecular models to continuum mechanics. Arch. Ration. Mech. Anal. (2002) 164: 341-381.
[4]	Connecting atomistic and continuous models of elastodynamics. Arch. Ration. Mech. Anal. (2017) 224: 907-953.
[5]	J. Braun and B. Schmidt, Existence and convergence of solutions of the boundary value problem in atomistic and continuum nonlinear elasticity theory, Calc. Var. Partial Differential Equations, 55 (2016), 36pp.
[6]	On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with $p$-growth. Netw. Heterog. Media (2013) 8: 879-912.
[7]	Confining thin elastic sheets and folding paper. Arch. Ration. Mech. Anal. (2008) 187: 1-48.
[8]	Energy minimizing configurations of pre-strained multilayers. J. Elasticity (2020) 140: 303-335.
[9]	M. de Benito Delgado and B. Schmidt, A hierarchy of multilayered plate models, ESAIM Control Optim. Calc. Var., 27 (2021), 35pp.
[10]	Cauchy-Born rule and the stability of crystalline solids: Static problems. Arch. Ration. Mech. Anal. (2007) 183: 241-297.
[11]	A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods. J. Mech. Phys. Solids (2000) 48: 1519-1540.
[12]	The Föppl-von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity. C. R. Math. Acad. Sci. Paris (2002) 335: 201-206.
[13]	A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence. Arch. Ration. Mech. Anal. (2006) 180: 183-236.
[14]	A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. (2002) 55: 1461-1506.
[15]	The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. (9) (1995) 74: 549-578.
[16]	Interpenetration of matter in plate theories obtained as Γ-limits. ESAIM Control Optim. Calc. Var. (2017) 23: 119-136.
[17]	Justification of the Cauchy-Born approximation of elastodynamics. Arch. Ration. Mech. Anal. (2013) 207: 1025-1073.
[18]	A derivation of continuum nonlinear plate theory from atomistic models. Multiscale Model. Simul. (2006) 5: 664-694.
[19]	On the derivation of linear elasticity from atomistic models. Netw. Heterog. Media (2009) 4: 789-812.
[20]	On the passage from atomic to continuum theory for thin films. Arch. Ration. Mech. Anal. (2008) 190: 1-55.
[21]	Qualitative properties of a continuum theory for thin films. Ann. Inst. H. Poincaré C Anal. Non Linéaire (2008) 25: 43-75.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)