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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 |
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. |
show all references
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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