Periodically wrinkled plate model of the Föppl-von Kármán type
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 2, p. 275-307

In this paper we derive, by means of Γ-convergence, the periodically wrinkled plate model starting from three dimensional nonlinear elasticity. We assume that the thickness of the plate is h 2 and that the mid-surface of the plate is given by (x 1 ,x 2 )(x 1 ,x 2 ,h 2 θ(x 1 h,x 2 h)), where θ is [0,1] 2 periodic function. We also assume that the strain energy of the plate has the order h 8 =(h 2 ) 4 , which corresponds to the Föppl-von Kármán model in the case of the ordinary plate. The obtained model mixes the bending part of the energy with the stretching part.

Published online : 2019-02-21
Classification:  74K20,  74B20
@article{ASNSP_2013_5_12_2_275_0,
     author = {Vel\v ci\'c, Igor},
     title = {Periodically wrinkled plate model of the F\"oppl-von K\'arm\'an type},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {2},
     year = {2013},
     pages = {275-307},
     zbl = {1271.74290},
     mrnumber = {3114006},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2013_5_12_2_275_0}
}
Velčić, Igor. Periodically wrinkled plate model of the Föppl-von Kármán type. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 2, pp. 275-307. http://www.numdam.org/item/ASNSP_2013_5_12_2_275_0/

[1] R.A. Adams, “Sobolev Spaces”, Academic press, New York, 1975. | MR 450957 | Zbl 1098.46001

[2] I. Aganović, E. Marušić-Paloka and Z. Tutek, Slightly Wrinkled Plate, Asymptotic Anal. 13 (1996), 1–29. | MR 1406165 | Zbl 0855.73034

[3] I. Aganović, E. Marušić-Paloka, and Z. Tutek, Moderately Wrinkled Plate, Asymptot. Anal. 16 (1998), 273–297. | MR 1612817 | Zbl 0944.74052

[4] G. Allaire, Homogenization and two scale convergence, SIAM J. Math. Anal. 23 (6) (1992), 1482–1518. | MR 1185639 | Zbl 0770.35005

[5] J. F. Babadjian and M. Baía, 3D-2D analysis of a thin film with periodic microstructure, Proc. Roy. Soc. Edinburgh, Section A 136 (2006), 223–243. | MR 2218151 | Zbl 1125.49035

[6] A. Braides, “Γ-convergence for Beginners”, Oxford University Press, Oxford, 2002. | MR 1968440 | Zbl 1198.49001

[7] A. Braides, I. Fonseca and G. Francfort, 3D-2D Asymptotic analysis for inhomogeneous thin films, Indiana Univ. Math. J. 49 (2000), 1367–1404. | MR 1836533 | Zbl 0987.35020

[8] P. G. Ciarlet, “Mathematical Elasticity. Vol. I, Three-dimensional Elasticity”, North-Holland Publishing Co., Amsterdam, 1988. | MR 936420 | Zbl 0953.74004

[9] P. G. Ciarlet, “Mathematical elasticity. Vol. II. Theory of plates. Studies in Mathematics and its Applications”, 27. North-Holland Publishing Co., Amsterdam 1997. | MR 1477663 | Zbl 0953.74004

[10] P. G. Ciarlet, “Mathematical elasticity. Vol. III. Theory of shells. Studies in Mathematics and its Applications”, 29. North-Holland Publishing Co., Amsterdam 2000. | MR 1757535 | Zbl 0953.74004

[11] G. Dal Maso, “An Introduction to Γ-convergence, Progress in Nonlinear Differential Equations and Their Applications”, Birkäuser, Basel 1993. | MR 1201152 | Zbl 0816.49001

[12] L. C. Evans, “Partial Differential Equations”, Second Edition, American Mathematical Society, Providence, Rhode Island, 1998.

[13] 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. | MR 1916989 | Zbl 1021.74024

[14] G. Friesecke, R. D. James and S. Müller, A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Γ-Convergence, Arch. Ration. Mech. Anal. 180 (2006), 183–236. | MR 2210909 | Zbl 1100.74039

[15] G. Friesecke, R. 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. | MR 1920020 | Zbl 1041.74043

[16] G. Friesecke,, R. James, M. G. Mora and S. Müller, Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Γ-convergence, C. R. Math. Acad. Sci. Paris 336 (2003), 697–702. | MR 1988135 | Zbl 1140.74481

[17] H. Le Dret and A. Raoult, The nonlinear membrane model as a variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. 74 (1995), 549–578. | MR 1365259 | Zbl 0847.73025

[18] H. Le Dret and A. Raoult, The membrane shell model in nonlinear elasticity: a variational asymptotic derivation, J. Nonlinear Sci. 6 (1996), 59–84. | MR 1375820 | Zbl 0844.73045

[19] M. Lecumberry and S. Müller, Stability of slender bodies under compression and validity of the von Kármán theory, Arch. Ration. Mech. Anal. 193 (2009), 255–310. | MR 2525119 | Zbl 1200.74060

[20] M. Lewicka, M. G. Mora and M. Pakzad, Shell theories arising as low energy Γ-limit of 3d nonlinear elasticity, Ann. Sc. Norm. Super. Pisa Cl. Sci. 9 (2010), 1–43. | Numdam | MR 2731157

[21] M. Lewicka and M. Pakzad, The infinite hierarchy of elastic shell models: some recent results and a conjecture, In: “Infinite dimensional dynamical systems”, Fields Inst. Commun., Vol. 64, Springer, New York, 2013, 407–420. | MR 2986945 | Zbl 1263.74035

[22] M. Lewicka, M. R. Packzad and L. Mahadevan, The Föppl-von Kármán equations for plates with incompatible strains, Proc. R. Soc. London, Ser. A 467 (2011) 402–426. | MR 2748099 | Zbl 1219.74027

[23] S. Neukamm, Homogenization, linearization and dimension reduction in elasticity with variational methods, Dissertation Technische Universität München, 2010.

[24] 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. | MR 2980530 | Zbl 1295.74049

[25] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), 608–625. | MR 990867 | Zbl 0688.35007

[26] G. A. Pavliotis and A. M. Stuart, “Multiscale Methods: Averaging and Homogenization”, Springer, New York, 2008. | MR 2382139 | Zbl 1160.35006

[27] R. Temam, “Navier Stokes Equations and Nonlinear Functional Analysis”, Second Edition, SIAM Society for Industrial and Applied Mathematics, Philadelphia, 1995. | MR 1318914 | Zbl 0833.35110

[28] I. Velčić, Shallow shell models by Γ-convergence, Math. Mech. Solids 17 (2012), 781–802. | MR 3179426

[29] I. Velčić, Nonlinear weakly curved rod by Γ-convergence, J. Elasticity 108 (2012), 125–150. | MR 2948053 | Zbl 1243.74108