Periodically wrinkled plate model of the Föppl-von Kármán type

Velčić, Igor

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 2, pp. 275-307.

In this paper we derive, by means of $\Gamma$-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 $(x_1,x_2)\to (x_1,x_2,h^2\theta (\frac{x_1}{h},\frac{x_2}{h}))$, where $\theta$ is ${[0,1]}^2$ periodic function. We also assume that the strain energy of the plate has the order $h^8={\left(h^2\right)}^4$, 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.

Classification : 74K20, 74B20

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

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

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

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

[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 | Zbl

[6] A. Braides, “$\Gamma$-convergence for Beginners”, Oxford University Press, Oxford, 2002. | MR | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[14] 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. | MR | Zbl

[15] G. Friesecke, R. D. James and S. Müller, The Föppl-von Kármán plate theory as a low energy $\Gamma$-limit of nonlinear elasticity, C. R. Math. Acad. Sci. Paris 335 (2002), 201–206. | MR | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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