# 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 and 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), 1482-1518. doi: 10.1137/0523084. [2] J. M. Arrieta and M. C. Pereira., Homogenization in a thin domain with an oscillatory boundary, J. Math. Pures Appl., 96 (2011), 29-57. doi: 10.1016/j.matpur.2011.02.003. [3] M. Bocea and I. Fonseca, Equi-integrability results for 3D-2D dimension reduction problems, ESAIM Control Optim. Calc. Var., 7 (2002), 443-470. doi: 10.1051/cocv:2002063. [4] A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J., 49 (2000), 1367-1404. doi: 10.1512/iumj.2000.49.1822. [5] A. Braides and C. I. Zeppieri, A note on equi-integrability in dimension reduction problems, Calc. Var. Partial Differential Equations, 29 (2007), 231-238. doi: 10.1007/s00526-006-0065-6. [6] P. Courilleau and J. Mossino, Compensated compactness for nonlinear homogenization and reduction of dimension, Calc. Var. Partial Differential Equations, 20 (2004), 65-91. doi: 10.1007/s00526-003-0228-7. [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), 1461-1506. doi: 10.1002/cpa.10048. [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), 183-236. doi: 10.1007/s00205-005-0400-7. [9] B. Gustafsson and J. Mossino, Compensated compactness for homogenization and reduction of dimension: the case of elastic laminates, Asymptot. Anal., 47 (2006), 139-169. [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), 677-699. doi: 10.1007/s00526-013-0691-8. [11] Peter Hornung, Approximation of flat $W^{2,2}$ isometric immersions by smooth ones, Arch. Ration. Mech. Anal., 199 (2011), 1015-1067. doi: 10.1007/s00205-010-0374-y. [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. [13] M. Jurak and Z. Tutek, A one-dimensional model of homogenized rod, Glas. Mat. Ser. III, 24 (1989), 271-290. [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), 549-578. [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. [16] S. Neukamm, Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods, Phd thesis, Tecnische Universität München, 2010. [17] S. Neukamm, Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from three-dimensional elasticity, Arch. Ration. Mech. Anal., 206 (2012), 645-706. doi: 10.1007/s00205-012-0539-y. [18] S. Neukamm and H. Olbermann, Homogenization of the nonlinear bending theory for plates, Calc. Var. Partial Differential Equations, 53 (2015), 719-753. doi: 10.1007/s00526-014-0765-2. [19] S. Neukamm and I. Velčić, Derivation of a homogenized von Kármán plate theory from 3D elasticity, M3AS, 23 (2013), 2701-2748. doi: 10.1142/S0218202513500449. [20] B. Schmidt, Plate theory for stressed heterogeneous multilayers of finite bending energy, J. Math. Pures Appl., 88 (2007), 107-122. doi: 10.1016/j.matpur.2007.04.011. [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. [22] I. Velčić, Periodically wrinkled plate of Föppl von Kármán type, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 12 (2013), 275-307. [23] I. Velčić, On the derivation of homogenized bending plate model, Calc. Var. Partial Differential Equations, 53 (2015), 561-586. doi: 10.1007/s00526-014-0758-1. [24] A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var., 12 (2006), 371-397 (electronic). doi: 10.1051/cocv:2006012. [25] A. Visintin, Two-scale convergence of some integral functionals, Calc. Var. Partial Differential Equations, 29 (2007), 239-265. doi: 10.1007/s00526-006-0068-3.

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [2] J. M. Arrieta and M. C. Pereira., Homogenization in a thin domain with an oscillatory boundary, J. Math. Pures Appl., 96 (2011), 29-57. doi: 10.1016/j.matpur.2011.02.003. [3] M. Bocea and I. Fonseca, Equi-integrability results for 3D-2D dimension reduction problems, ESAIM Control Optim. Calc. Var., 7 (2002), 443-470. doi: 10.1051/cocv:2002063. [4] A. Braides, I. Fonseca and G. Francfort, 3D-2D asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J., 49 (2000), 1367-1404. doi: 10.1512/iumj.2000.49.1822. [5] A. Braides and C. I. Zeppieri, A note on equi-integrability in dimension reduction problems, Calc. Var. Partial Differential Equations, 29 (2007), 231-238. doi: 10.1007/s00526-006-0065-6. [6] P. Courilleau and J. Mossino, Compensated compactness for nonlinear homogenization and reduction of dimension, Calc. Var. Partial Differential Equations, 20 (2004), 65-91. doi: 10.1007/s00526-003-0228-7. [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), 1461-1506. doi: 10.1002/cpa.10048. [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), 183-236. doi: 10.1007/s00205-005-0400-7. [9] B. Gustafsson and J. Mossino, Compensated compactness for homogenization and reduction of dimension: the case of elastic laminates, Asymptot. Anal., 47 (2006), 139-169. [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), 677-699. doi: 10.1007/s00526-013-0691-8. [11] Peter Hornung, Approximation of flat $W^{2,2}$ isometric immersions by smooth ones, Arch. Ration. Mech. Anal., 199 (2011), 1015-1067. doi: 10.1007/s00205-010-0374-y. [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. [13] M. Jurak and Z. Tutek, A one-dimensional model of homogenized rod, Glas. Mat. Ser. III, 24 (1989), 271-290. [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), 549-578. [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. [16] S. Neukamm, Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods, Phd thesis, Tecnische Universität München, 2010. [17] S. Neukamm, Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from three-dimensional elasticity, Arch. Ration. Mech. Anal., 206 (2012), 645-706. doi: 10.1007/s00205-012-0539-y. [18] S. Neukamm and H. Olbermann, Homogenization of the nonlinear bending theory for plates, Calc. Var. Partial Differential Equations, 53 (2015), 719-753. doi: 10.1007/s00526-014-0765-2. [19] S. Neukamm and I. Velčić, Derivation of a homogenized von Kármán plate theory from 3D elasticity, M3AS, 23 (2013), 2701-2748. doi: 10.1142/S0218202513500449. [20] B. Schmidt, Plate theory for stressed heterogeneous multilayers of finite bending energy, J. Math. Pures Appl., 88 (2007), 107-122. doi: 10.1016/j.matpur.2007.04.011. [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. [22] I. Velčić, Periodically wrinkled plate of Föppl von Kármán type, Ann. Sc. Norm. Super. Pisa Cl. Sci.(5), 12 (2013), 275-307. [23] I. Velčić, On the derivation of homogenized bending plate model, Calc. Var. Partial Differential Equations, 53 (2015), 561-586. doi: 10.1007/s00526-014-0758-1. [24] A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var., 12 (2006), 371-397 (electronic). doi: 10.1051/cocv:2006012. [25] A. Visintin, Two-scale convergence of some integral functionals, Calc. Var. Partial Differential Equations, 29 (2007), 239-265. doi: 10.1007/s00526-006-0068-3.
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