Derivation of a homogenized von-Kármán shell theory from 3D elasticity
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 5, p. 1039-1070
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

We derive homogenized von Kármán shell theories starting from three dimensional nonlinear elasticity. The original three dimensional model contains two small parameters: the period of oscillation ε of the material properties and the thickness h of the shell. Depending on the asymptotic ratio of these two parameters, we obtain different asymptotic theories. In the case hϵ we identify two different asymptotic theories, depending on the ratio of h and ϵ 2 . In the case of convex shells we obtain a complete picture in the whole regime hϵ.

DOI : https://doi.org/10.1016/j.anihpc.2014.05.003
Keywords: Elasticity, Dimension reduction, Homogenization, Shell theory, Two-scale convergence
@article{AIHPC_2015__32_5_1039_0,
     author = {Hornung, Peter and Vel\v ci\'c, Igor},
     title = {Derivation of a homogenized von-K\'arm\'an shell theory from 3D elasticity},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {32},
     number = {5},
     year = {2015},
     pages = {1039-1070},
     doi = {10.1016/j.anihpc.2014.05.003},
     zbl = {1329.74178},
     mrnumber = {3400441},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2015__32_5_1039_0}
}
Hornung, Peter; Velčić, Igor. Derivation of a homogenized von-Kármán shell theory from 3D elasticity. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 5, pp. 1039-1070. doi : 10.1016/j.anihpc.2014.05.003. http://www.numdam.org/item/AIHPC_2015__32_5_1039_0/

[1] I. Aganović, M. Jurak, E. Marušić-Paloka, Z. Tutek, Moderately wrinkled plate, Asymptot. Anal. 16 no. 3–4 (1998), 273 -297 | MR 1612817 | Zbl 0944.74052

[2] Grégoire Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 no. 6 (1992), 1482 -1518 | MR 1185639 | Zbl 0770.35005

[3] I. Aganović, E. Marušić-Paloka, Z. Tutek, Slightly wrinkled plate, Asymptot. Anal. 13 no. 1 (1996), 1 -29 | MR 1406165 | Zbl 0855.73034

[4] José M. Arrieta, Marcone C. Pereira, Homogenization in a thin domain with an oscillatory boundary, J. Math. Pures Appl. (9) 96 no. 1 (2011), 29 -57 | MR 2812711 | Zbl 1223.35039

[5] Andrea Braides, Irene Fonseca, Gilles Francfort, 3D–2D asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J. 49 no. 4 (2000), 1367 -1404 | MR 1836533 | Zbl 0987.35020

[6] Andrea Braides, Homogenization of some almost periodic coercive functional, Rend. Accad. Naz. Sci. Detta Accad. XL, Parte I, Mem. Mat. (5) 9 no. 1 (1985), 313 -321 | MR 899255 | Zbl 0582.49014

[7] Philippe G. Ciarlet, Mathematical Elasticity, vol. III, Stud. Math. Appl. vol. 29 , North-Holland Publishing Co., Amsterdam (2000) | MR 1757535 | Zbl 0953.74004

[8] P. Courilleau, J. Mossino, Compensated compactness for nonlinear homogenization and reduction of dimension, Calc. Var. Partial Differ. Equ. 20 no. 1 (2004), 65 -91 | MR 2047146 | Zbl 1072.35028

[9] Gero Friesecke, Richard D. James, Stefan Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity, Commun. Pure Appl. Math. 55 no. 11 (2002), 1461 -1506 | MR 1916989 | Zbl 1021.74024

[10] Gero Friesecke, Richard D. James, Stefan Müller, A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence, Arch. Ration. Mech. Anal. 180 no. 2 (2006), 183 -236 | MR 2210909 | Zbl 1100.74039

[11] Gero Friesecke, Richard D. James, Maria Giovanna Mora, Stefan Müller, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence, C. R. Math. Acad. Sci. Paris 336 no. 8 (2003), 697 -702 | MR 1988135 | Zbl 1140.74481

[12] Marius Ghergu, Georges Griso, Houari Mechkour, Bernadette Miara, Homogenization of thin piezoelectric perforated shells, M2AN Math. Model. Numer. Anal. 41 no. 5 (2007), 875 -895 | Numdam | MR 2363887 | Zbl 1138.74039

[13] Björn Gustafsson, Jacqueline Mossino, Compensated compactness for homogenization and reduction of dimension: the case of elastic laminates, Asymptot. Anal. 47 no. 1–2 (2006), 139 -169 | MR 2224410 | Zbl 1113.35018

[14] Giuseppe Geymonat, Enrique Sánchez-Palencia, On the rigidity of certain surfaces with folds and applications to shell theory, Arch. Ration. Mech. Anal. 129 no. 1 (1995), 11 -45 | MR 1328470 | Zbl 0830.73040

[15] Peter Hornung, Stefan Neukamm, Igor Velčić, Derivation of the homogenized bending plate model from 3D nonlinear elasticity, Calc. Var. Partial Differ. Equ. (2014), http://dx.doi.org/10.1007/s00526-013-0691-8 | Zbl 1327.35018

[16] Peter Hornung, Continuation of infinitesimal bendings on developable surfaces and equilibrium equations for nonlinear bending theory of plates, Commun. Partial Differ. Equ. (2014) | MR 3169749 | Zbl 06219503

[17] Peter Hornung, The Willmore functional on isometric immersions, 2012, MIS MPG preprint.

[18] Jürgen Jost, Riemannian Geometry and Geometric Analysis, Universitext , Springer, Heidelberg (2011) | MR 2829653 | Zbl 1227.53001

[19] M. Jurak, Z. Tutek, A one-dimensional model of homogenized rod, Glas. Mat. 24(44) no. 2–3 (1989), 271 -290 | MR 1074871 | Zbl 0709.73003

[20] Hervé Le Dret, Annie Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9) 74 no. 6 (1995), 549 -578 | MR 1365259 | Zbl 0847.73025

[21] H. Le Dret, A. Raoult, The membrane shell model in nonlinear elasticity: a variational asymptotic derivation, J. Nonlinear Sci. 6 no. 1 (1996), 59 -84 | MR 1375820 | Zbl 0989.74048

[22] Marta Lewicka, Maria Giovanna Mora, Mohammad Reza Pakzad, Shell theories arising as low energy Γ-limit of 3d nonlinear elasticity, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 9 no. 2 (2010), 253 -295 | Numdam | MR 2731157 | Zbl 05791996

[23] Marta Lewicka, Maria Giovanna Mora, Mohammad Reza Pakzad, The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells, Arch. Ration. Mech. Anal. 200 no. 3 (2011), 1023 -1050 | MR 2796137 | Zbl 1291.74130

[24] T. Lewiński, J.J. Telega, Asymptotic analysis and homogenization, Plates, Laminates and Shells, Ser. Adv. Math. Appl. Sci. vol. 52 , World Scientific Publishing Co. Inc., River Edge, NJ (2000) | MR 1758600 | Zbl 0965.74003

[25] Adam Lutoborski, Homogenization of thin elastic shell, J. Elast. 15 no. 1 (1985), 69 -87 | MR 788838 | Zbl 0558.73055

[26] Stefan Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Ration. Mech. Anal. 99 no. 3 (1987), 189 -212 | MR 888450 | Zbl 0629.73009

[27] Stefan Neukamm, Homogenization, linearization and dimension reduction in elasticity with variational methods, Tecnische Universität München (2010)

[28] Stefan Neukamm, Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from three-dimensional elasticity, Arch. Ration. Mech. Anal. 206 no. 2 (2012), 645 -706 | MR 2980530 | Zbl 1295.74049

[29] Stefan Neukamm, Igor Velčić, Derivation of a homogenized von Kármán plate theory from 3D elasticity, Math. Models Methods Appl. Sci. 23 no. 14 (2013), 2701 -2748 | MR 3119636 | Zbl 1282.35039

[30] Bernd Schmidt, Plate theory for stressed heterogeneous multilayers of finite bending energy, J. Math. Pures Appl. (9) 88 no. 1 (2007), 107 -122 | MR 2334775 | Zbl 1116.74034

[31] Igor Velčić, A note on the derivation of homogenized bending plate model, http://www.mis.mpg.de/publications/preprints/2013/prepr2013-34.html | MR 3347471 | Zbl 1329.35050

[32] Igor Velčić, On the general homogenization and γ-closure for the equations of von kármán plate, http://www.mis.mpg.de/preprints/2013/preprint2013_61.pdf | Zbl 06667859

[33] Igor Velčić, Periodically wrinkled plate of Föppl von Kármán type, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 12 no. 2 (2013), 275 -307 | MR 3114006 | Zbl 1271.74290

[34] Augusto Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var. 12 no. 3 (2006), 371 -397 | Numdam | MR 2224819 | Zbl 1110.35009

[35] Augusto Visintin, Two-scale convergence of some integral functionals, Calc. Var. Partial Differ. Equ. 29 no. 2 (2007), 239 -265 | MR 2307775 | Zbl 1129.35011