# American Institute of Mathematical Sciences

November  2015, 14(6): 2151-2168. doi: 10.3934/cpaa.2015.14.2151

## Homogenization of bending theory for plates; the case of oscillations in the direction of thickness

 1 University of Zagreb, Faculty of Science and Mathematics, Bijenička 30, 10 000 Zagreb, Croatia 2 University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10 000 Zagreb, Croatia

Received  October 2014 Revised  August 2015 Published  September 2015

In this paper we study the homogenization effects on the model of elastic plate in the bending regime, under the assumption that the energy density (material) oscillates in the direction of thickness. We study two different cases. First, we show, starting from 3D elasticity, by means of $\Gamma$-convergence and under general (not necessarily periodic) assumption, that the effective behavior of the limit is not influenced by oscillations in the direction of thickness. In the second case, we study periodic in-plane oscillations of the energy density coupled with periodic oscillations in the direction of thickness. In contrast to the first case we show that there are homogenization effects coming also from the oscillations in the direction of thickness.
Citation: Maroje Marohnić, Igor Velčić. Homogenization of bending theory for plates; the case of oscillations in the direction of thickness. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2151-2168. doi: 10.3934/cpaa.2015.14.2151
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar [2] J. M. Arrieta and M. C. Pereira., Homogenization in a thin domain with an oscillatory boundary,, J. Math. Pures Appl., 96 (2011), 29. doi: 10.1016/j.matpur.2011.02.003. Google Scholar [3] M. Bocea and I. Fonseca, Equi-integrability results for 3D-2D dimension reduction problems,, ESAIM Control Optim. Calc. Var., 7 (2002), 443. doi: 10.1051/cocv:2002063. Google Scholar [4] A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films,, Indiana Univ. Math. J., 49 (2000), 1367. doi: 10.1512/iumj.2000.49.1822. Google Scholar [5] A. Braides and C. I. Zeppieri, A note on equi-integrability in dimension reduction problems,, Calc. Var. Partial Differential Equations, 29 (2007), 231. doi: 10.1007/s00526-006-0065-6. Google Scholar [6] P. Courilleau and J. Mossino, Compensated compactness for nonlinear homogenization and reduction of dimension,, Calc. Var. Partial Differential Equations, 20 (2004), 65. doi: 10.1007/s00526-003-0228-7. Google Scholar [7] G. Friesecke, R. 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. doi: 10.1002/cpa.10048. Google Scholar [8] G. Friesecke, R. D. James, and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence,, Arch. Ration. Mech. Anal., 180 (2006), 183. doi: 10.1007/s00205-005-0400-7. Google Scholar [9] B. Gustafsson and J. Mossino, Compensated compactness for homogenization and reduction of dimension: the case of elastic laminates,, Asymptot. Anal., 47 (2006), 139. Google Scholar [10] P. Hornung, S. Neukamm and I. Velčić, Derivation of a homogenized nonlinear plate theory from 3d elasticity,, Calc. Var. Partial Differential Equations, 51 (2014), 677. doi: 10.1007/s00526-013-0691-8. Google Scholar [11] Peter Hornung, Approximation of flat $W^{2,2}$ isometric immersions by smooth ones,, Arch. Ration. Mech. Anal., 199 (2011), 1015. doi: 10.1007/s00205-010-0374-y. Google Scholar [12] P. Hornung and I. Velčić, Derivation of a homogenized von-Kármán shell theory from 3d elasticity,, accepted in Annales de l'Institut Henri Poincare (C) Non Linear Analysis., (). doi: 10.1016/j.anihpc.2014.05.003. Google Scholar [13] M. Jurak and Z. Tutek, A one-dimensional model of homogenized rod,, Glas. Mat. Ser. III, 24 (1989), 271. Google Scholar [14] H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity,, J. Math. Pures Appl., 74 (1995), 549. Google Scholar [15] M. Marohnić and I. Velčić, Non-periodic homogenization of bending-torsion theory for inextensible rods from 3d elasticity,, accepted in Annali di Matematica Pura ed Applicata., (). doi: 10.1007/s10231-015-0504-0. Google Scholar [16] S. Neukamm, Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods,, Phd thesis, (2010). Google Scholar [17] S. Neukamm, Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from three-dimensional elasticity,, Arch. Ration. Mech. Anal., 206 (2012), 645. doi: 10.1007/s00205-012-0539-y. Google Scholar [18] S. Neukamm and H. Olbermann, Homogenization of the nonlinear bending theory for plates,, Calc. Var. Partial Differential Equations, 53 (2015), 719. doi: 10.1007/s00526-014-0765-2. Google Scholar [19] S. Neukamm and I. Velčić, Derivation of a homogenized von Kármán plate theory from 3D elasticity,, M3AS, 23 (2013), 2701. doi: 10.1142/S0218202513500449. Google Scholar [20] B. Schmidt, Plate theory for stressed heterogeneous multilayers of finite bending energy,, J. Math. Pures Appl., 88 (2007), 107. doi: 10.1016/j.matpur.2007.04.011. Google Scholar [21] I. Velčić, On the general homogenization of von Kármán plate equations from 3d nonlinear elasticity,, accepted in Analysis and Applications., (). doi: 10.1142/S0219530515500244. Google Scholar [22] I. Velčić, Periodically wrinkled plate of Föppl von Kármán type,, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 12 (2013), 275. Google Scholar [23] I. Velčić, On the derivation of homogenized bending plate model,, Calc. Var. Partial Differential Equations, 53 (2015), 561. doi: 10.1007/s00526-014-0758-1. Google Scholar [24] A. Visintin, Towards a two-scale calculus,, ESAIM Control Optim. Calc. Var., 12 (2006), 371. doi: 10.1051/cocv:2006012. Google Scholar [25] A. Visintin, Two-scale convergence of some integral functionals,, Calc. Var. Partial Differential Equations, 29 (2007), 239. doi: 10.1007/s00526-006-0068-3. Google Scholar

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482. doi: 10.1137/0523084. Google Scholar [2] J. M. Arrieta and M. C. Pereira., Homogenization in a thin domain with an oscillatory boundary,, J. Math. Pures Appl., 96 (2011), 29. doi: 10.1016/j.matpur.2011.02.003. Google Scholar [3] M. Bocea and I. Fonseca, Equi-integrability results for 3D-2D dimension reduction problems,, ESAIM Control Optim. Calc. Var., 7 (2002), 443. doi: 10.1051/cocv:2002063. Google Scholar [4] A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films,, Indiana Univ. Math. J., 49 (2000), 1367. doi: 10.1512/iumj.2000.49.1822. Google Scholar [5] A. Braides and C. I. Zeppieri, A note on equi-integrability in dimension reduction problems,, Calc. Var. Partial Differential Equations, 29 (2007), 231. doi: 10.1007/s00526-006-0065-6. Google Scholar [6] P. Courilleau and J. Mossino, Compensated compactness for nonlinear homogenization and reduction of dimension,, Calc. Var. Partial Differential Equations, 20 (2004), 65. doi: 10.1007/s00526-003-0228-7. Google Scholar [7] G. Friesecke, R. 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. doi: 10.1002/cpa.10048. Google Scholar [8] G. Friesecke, R. D. James, and S. Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence,, Arch. Ration. Mech. Anal., 180 (2006), 183. doi: 10.1007/s00205-005-0400-7. Google Scholar [9] B. Gustafsson and J. Mossino, Compensated compactness for homogenization and reduction of dimension: the case of elastic laminates,, Asymptot. Anal., 47 (2006), 139. Google Scholar [10] P. Hornung, S. Neukamm and I. Velčić, Derivation of a homogenized nonlinear plate theory from 3d elasticity,, Calc. Var. Partial Differential Equations, 51 (2014), 677. doi: 10.1007/s00526-013-0691-8. Google Scholar [11] Peter Hornung, Approximation of flat $W^{2,2}$ isometric immersions by smooth ones,, Arch. Ration. Mech. Anal., 199 (2011), 1015. doi: 10.1007/s00205-010-0374-y. Google Scholar [12] P. Hornung and I. Velčić, Derivation of a homogenized von-Kármán shell theory from 3d elasticity,, accepted in Annales de l'Institut Henri Poincare (C) Non Linear Analysis., (). doi: 10.1016/j.anihpc.2014.05.003. Google Scholar [13] M. Jurak and Z. Tutek, A one-dimensional model of homogenized rod,, Glas. Mat. Ser. III, 24 (1989), 271. Google Scholar [14] H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity,, J. Math. Pures Appl., 74 (1995), 549. Google Scholar [15] M. Marohnić and I. Velčić, Non-periodic homogenization of bending-torsion theory for inextensible rods from 3d elasticity,, accepted in Annali di Matematica Pura ed Applicata., (). doi: 10.1007/s10231-015-0504-0. Google Scholar [16] S. Neukamm, Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods,, Phd thesis, (2010). Google Scholar [17] S. Neukamm, Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from three-dimensional elasticity,, Arch. Ration. Mech. Anal., 206 (2012), 645. doi: 10.1007/s00205-012-0539-y. Google Scholar [18] S. Neukamm and H. Olbermann, Homogenization of the nonlinear bending theory for plates,, Calc. Var. Partial Differential Equations, 53 (2015), 719. doi: 10.1007/s00526-014-0765-2. Google Scholar [19] S. Neukamm and I. Velčić, Derivation of a homogenized von Kármán plate theory from 3D elasticity,, M3AS, 23 (2013), 2701. doi: 10.1142/S0218202513500449. Google Scholar [20] B. Schmidt, Plate theory for stressed heterogeneous multilayers of finite bending energy,, J. Math. Pures Appl., 88 (2007), 107. doi: 10.1016/j.matpur.2007.04.011. Google Scholar [21] I. Velčić, On the general homogenization of von Kármán plate equations from 3d nonlinear elasticity,, accepted in Analysis and Applications., (). doi: 10.1142/S0219530515500244. Google Scholar [22] I. Velčić, Periodically wrinkled plate of Föppl von Kármán type,, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 12 (2013), 275. Google Scholar [23] I. Velčić, On the derivation of homogenized bending plate model,, Calc. Var. Partial Differential Equations, 53 (2015), 561. doi: 10.1007/s00526-014-0758-1. Google Scholar [24] A. Visintin, Towards a two-scale calculus,, ESAIM Control Optim. Calc. Var., 12 (2006), 371. doi: 10.1051/cocv:2006012. Google Scholar [25] A. Visintin, Two-scale convergence of some integral functionals,, Calc. Var. Partial Differential Equations, 29 (2007), 239. doi: 10.1007/s00526-006-0068-3. Google Scholar
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