Olbermann, Heiner
[Universität Bonn]
Neukamm, Stefan
[Weiersrtasse Inst. fur Angewandte Analysis und Stochastik, Berlin]
We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of Gamma-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a "cylinder" or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions, using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.
- Friesecke Gero, James Richard D., Müller Stefan, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity : Geometric Rigidity and Plate Theory, 10.1002/cpa.10048
- Hornung Peter, Approximation of Flat W 2,2 Isometric Immersions by Smooth Ones, 10.1007/s00205-010-0374-y
- Hornung Peter, Fine Level Set Structure of Flat Isometric Immersions, 10.1007/s00205-010-0375-x
- Fonseca Irene, Kroemer Stefan, Multiple integrals under differential constraints: two-scale convergence and homogenization, 10.1512/iumj.2010.59.4249
- Anza Hafsa Omar, Mandallena Jean-Philippe, Homogenization of nonconvex integrals with convex growth, 10.1016/j.matpur.2011.03.003
- Hornung, P., Neukamm, S., Velčić, I.: Derivation of a homogenized nonlinear plate theory from 3d elasticity. Calc Var Partial Differ Equ 1–23 (2013)
- Hornung P., Velcic I.: Derivation of a homogenized von-Karman shell theory from 3d elasticity. arXiv preprint arXiv:1211.0045 (2012)
- Neukamm, S.: Homogenization, linearization and dimension reduction in elasticity with variational methods. PhD thesis, Technische Universität München (2010)
- Neukamm Stefan, Rigorous Derivation of a Homogenized Bending-Torsion Theory for Inextensible Rods from Three-Dimensional Elasticity, 10.1007/s00205-012-0539-y
- NEUKAMM STEFAN, VELČIĆ IGOR, DERIVATION OF A HOMOGENIZED VON-KÁRMÁN PLATE THEORY FROM 3D NONLINEAR ELASTICITY, 10.1142/s0218202513500449
- Velcic I.: A note on the derivation of homogenized bending plate model. arXiv preprint arXiv:1212.2594 (2012)
- Kirchheim B.: Geometry and rigidity of microstructures. Habilitation Thesis, Universität Leipzig (2001)
- Pakzad, M.R.: On the Sobolev space of isometric immersions. J. Differ. Geom. 66(1), 47–69 (2004)
- Müller Stefan, Pakzad Mohammad Reza, Regularity properties of isometric immersions, 10.1007/s00209-005-0804-y
- Hartman Philip, Nirenberg Louis, On Spherical Image Maps Whose Jacobians Do Not Change Sign, 10.2307/2372995
- Friesecke Gero, James Richard D., Müller Stefan, A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence, 10.1007/s00205-005-0400-7
- Nguetseng Gabriel, A General Convergence Result for a Functional Related to the Theory of Homogenization, 10.1137/0520043
- Allaire Grégoire, Homogenization and Two-Scale Convergence, 10.1137/0523084
- Visintin Augusto, Towards a two-scale calculus, 10.1051/cocv:2006012
- Visintin A., Two-scale convergence of some integral functionals, 10.1007/s00526-006-0068-3
- Attouch, H.: Variational convergence for functions and operators. Pitman (Advanced Publishing Program), Boston, MA, Applicable Mathematics Series (1984)
| Bibliographic reference |
Olbermann, Heiner ; Neukamm, Stefan. Homogenization of the nonlinear bending theory for plates. In: Calculus of Variations and Partial Differential Equations, Vol. 53, p. 719-753 (2015) |
| Permanent URL |
http://hdl.handle.net/2078/203024 |
