August  2022, 17(4): 613-644. doi: 10.3934/nhm.2022019

An atomistic derivation of von-Kármán plate theory

1. 

Heriot-Watt University, United Kingdom

2. 

Universität Augsburg, Germany

*Corresponding author: Julian Braun

Received  July 2021 Revised  February 2022 Published  August 2022 Early access  April 2022

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.

Citation: Julian Braun, Bernd Schmidt. An atomistic derivation of von-Kármán plate theory. Networks and Heterogeneous Media, 2022, 17 (4) : 613-644. doi: 10.3934/nhm.2022019
References:
[1]

R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth, SIAM J. Math. Anal., 36 (2004), 1-37.  doi: 10.1137/S0036141003426471.

[2]

S. Bartels, Numerical solution of a Föppl–von Kármán model, SIAM J. Numer. Anal., 55 (2017), 1505-1524.  doi: 10.1137/16M1069791.

[3]

X. BlancC. Le Bris and P.-L. Lions, From molecular models to continuum mechanics, Arch. Ration. Mech. Anal., 164 (2002), 341-381.  doi: 10.1007/s00205-002-0218-5.

[4]

J. Braun, Connecting atomistic and continuous models of elastodynamics, Arch. Ration. Mech. Anal., 224 (2017), 907-953.  doi: 10.1007/s00205-017-1091-6.

[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. doi: 10.1007/s00526-016-1048-x.

[6]

J. Braun and B. Schmidt, On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with $p$-growth, Netw. Heterog. Media, 8 (2013), 879-912.  doi: 10.3934/nhm.2013.8.879.

[7]

S. Conti and F. Maggi, Confining thin elastic sheets and folding paper, Arch. Ration. Mech. Anal., 187 (2008), 1-48.  doi: 10.1007/s00205-007-0076-2.

[8]

M. de Benito Delgado and B. Schmidt, Energy minimizing configurations of pre-strained multilayers, J. Elasticity, 140 (2020), 303-335.  doi: 10.1007/s10659-020-09771-y.

[9]

M. de Benito Delgado and B. Schmidt, A hierarchy of multilayered plate models, ESAIM Control Optim. Calc. Var., 27 (2021), 35pp. doi: 10.1051/cocv/2020067.

[10]

W. E and P. Ming, Cauchy-Born rule and the stability of crystalline solids: Static problems, Arch. Ration. Mech. Anal., 183 (2007), 241-297.  doi: 10.1007/s00205-006-0031-7.

[11]

G. Friesecke and R. D. James, A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods, J. Mech. Phys. Solids, 48 (2000), 1519-1540.  doi: 10.1016/S0022-5096(99)00091-5.

[12]

G. FrieseckeR. D. James and S. Müller, The Föppl-von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity, C. R. Math. Acad. Sci. Paris, 335 (2002), 201-206.  doi: 10.1016/S1631-073X(02)02388-9.

[13]

G. FrieseckeR. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal., 180 (2006), 183-236.  doi: 10.1007/s00205-005-0400-7.

[14]

G. FrieseckeR. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002), 1461-1506.  doi: 10.1002/cpa.10048.

[15]

H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9), 74 (1995), 549-578. 

[16]

H. Olbermann and E. Runa, Interpenetration of matter in plate theories obtained as Γ-limits, ESAIM Control Optim. Calc. Var., 23 (2017), 119-136.  doi: 10.1051/cocv/2015042.

[17]

C. Ortner and F. Theil, Justification of the Cauchy-Born approximation of elastodynamics, Arch. Ration. Mech. Anal., 207 (2013), 1025-1073.  doi: 10.1007/s00205-012-0592-6.

[18]

B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models, Multiscale Model. Simul., 5 (2006), 664-694.  doi: 10.1137/050646251.

[19]

B. Schmidt, On the derivation of linear elasticity from atomistic models, Netw. Heterog. Media, 4 (2009), 789-812.  doi: 10.3934/nhm.2009.4.789.

[20]

B. Schmidt, On the passage from atomic to continuum theory for thin films, Arch. Ration. Mech. Anal., 190 (2008), 1-55.  doi: 10.1007/s00205-008-0138-0.

[21]

B. Schmidt, Qualitative properties of a continuum theory for thin films, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 25 (2008), 43-75.  doi: 10.1016/j.anihpc.2006.09.001.

show all references

References:
[1]

R. Alicandro and M. Cicalese, A general integral representation result for continuum limits of discrete energies with superlinear growth, SIAM J. Math. Anal., 36 (2004), 1-37.  doi: 10.1137/S0036141003426471.

[2]

S. Bartels, Numerical solution of a Föppl–von Kármán model, SIAM J. Numer. Anal., 55 (2017), 1505-1524.  doi: 10.1137/16M1069791.

[3]

X. BlancC. Le Bris and P.-L. Lions, From molecular models to continuum mechanics, Arch. Ration. Mech. Anal., 164 (2002), 341-381.  doi: 10.1007/s00205-002-0218-5.

[4]

J. Braun, Connecting atomistic and continuous models of elastodynamics, Arch. Ration. Mech. Anal., 224 (2017), 907-953.  doi: 10.1007/s00205-017-1091-6.

[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. doi: 10.1007/s00526-016-1048-x.

[6]

J. Braun and B. Schmidt, On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with $p$-growth, Netw. Heterog. Media, 8 (2013), 879-912.  doi: 10.3934/nhm.2013.8.879.

[7]

S. Conti and F. Maggi, Confining thin elastic sheets and folding paper, Arch. Ration. Mech. Anal., 187 (2008), 1-48.  doi: 10.1007/s00205-007-0076-2.

[8]

M. de Benito Delgado and B. Schmidt, Energy minimizing configurations of pre-strained multilayers, J. Elasticity, 140 (2020), 303-335.  doi: 10.1007/s10659-020-09771-y.

[9]

M. de Benito Delgado and B. Schmidt, A hierarchy of multilayered plate models, ESAIM Control Optim. Calc. Var., 27 (2021), 35pp. doi: 10.1051/cocv/2020067.

[10]

W. E and P. Ming, Cauchy-Born rule and the stability of crystalline solids: Static problems, Arch. Ration. Mech. Anal., 183 (2007), 241-297.  doi: 10.1007/s00205-006-0031-7.

[11]

G. Friesecke and R. D. James, A scheme for the passage from atomic to continuum theory for thin films, nanotubes and nanorods, J. Mech. Phys. Solids, 48 (2000), 1519-1540.  doi: 10.1016/S0022-5096(99)00091-5.

[12]

G. FrieseckeR. D. James and S. Müller, The Föppl-von Kármán plate theory as a low energy Γ-limit of nonlinear elasticity, C. R. Math. Acad. Sci. Paris, 335 (2002), 201-206.  doi: 10.1016/S1631-073X(02)02388-9.

[13]

G. FrieseckeR. D. James and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal., 180 (2006), 183-236.  doi: 10.1007/s00205-005-0400-7.

[14]

G. FrieseckeR. D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Comm. Pure Appl. Math., 55 (2002), 1461-1506.  doi: 10.1002/cpa.10048.

[15]

H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9), 74 (1995), 549-578. 

[16]

H. Olbermann and E. Runa, Interpenetration of matter in plate theories obtained as Γ-limits, ESAIM Control Optim. Calc. Var., 23 (2017), 119-136.  doi: 10.1051/cocv/2015042.

[17]

C. Ortner and F. Theil, Justification of the Cauchy-Born approximation of elastodynamics, Arch. Ration. Mech. Anal., 207 (2013), 1025-1073.  doi: 10.1007/s00205-012-0592-6.

[18]

B. Schmidt, A derivation of continuum nonlinear plate theory from atomistic models, Multiscale Model. Simul., 5 (2006), 664-694.  doi: 10.1137/050646251.

[19]

B. Schmidt, On the derivation of linear elasticity from atomistic models, Netw. Heterog. Media, 4 (2009), 789-812.  doi: 10.3934/nhm.2009.4.789.

[20]

B. Schmidt, On the passage from atomic to continuum theory for thin films, Arch. Ration. Mech. Anal., 190 (2008), 1-55.  doi: 10.1007/s00205-008-0138-0.

[21]

B. Schmidt, Qualitative properties of a continuum theory for thin films, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 25 (2008), 43-75.  doi: 10.1016/j.anihpc.2006.09.001.

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