This paper is concerned with the asymptotic analysis of plates with periodically rapidly varying heterogeneities. The formal asymptotic procedure is performed when both the periods of changes of the material properties and the thickness of the plate are of the same orders of magnitude. Our approach is based on a sequence of recursive minimization problems. We consider a plate made of Ciarlet—Geymonat type materials. Depending on the order of magnitude of the applied loads, we obtain a nonlinear membrane model and a nonlinear membrane inextensional bending model as announced by Pruchnicki.
Ciarlet, P.G.Mathematical Elasticity, Vol. II. Theory of Plates, North-Holland , Amsterdam, 1997.
2.
Lewiński, T. and Telega, J.J.Plates, Laminates and Shells: Asymptotic Analysis and Homogenization, World Scientific, Singapore, 2000.
3.
Pruchnicki, E.Non linearly elastic membrane model for heterogeneous shells by using a new double scale variational formulation: A formal asymptotic approach. J. Elasticity, 84, 245-280 (2006).
4.
Pruchnicki, E.Two-dimensional nonlinear models for heterogeneous plates. Comptes Rendus de l’Académie des sciences, Paris, 337(5), 297-302 (2009).
5.
Pruchnicki, E.Nonlinearly elastic membrane model for heterogeneous plates : a formal asymptotic approach by using a new double scale variationnal formulation. International Journal of Engineering Science , 40, 2183-2202 (2002).
6.
Ciarlet, P.G. and Geymonat, G.Sur les lois de comportement en élasticité non linéaire compressible. Comptes Rendus de l’Académie des sciences, Paris, Series II, 295, 423-426 (1982).
7.
Ball, J.M.Convexity condition and existence theorems in nonlinear elasticity. Archive for Rational Mechanics and Analysis , 63(4), 337-403 (1976).
Panasenko, G.Muticomponent homogenization of the vibration problem for incompressible media with heavy and rigid inclusions. Comptes Rendus de l’Académie des sciences, Paris, Series I, 321, 1109-1114 (1995).
10.
Pantz, O.Quelques problèmes de modélisation en élasticité non linéaire, PhD Thesis, University of ParisVI, 2001.
11.
Pantz, O.Dérivation des modèles de plaques membranaires non linéaires à partir de l’élasticité tri-dimensionnelle. Comptes Rendus de l’Académie des sciences, Paris, Series I, 331, 171-174 (2000).
12.
Meunier, N.Recursive derivation of one-dimensional models from three-dimensional nonlinear elasticity. Mathematics and Mechanics of Solids, 13(2), 172-194 (2008).
13.
Trabelsi, K.Nonlinear thin plate models for a family of Ogden materials. Comptes Rendus de l’Académie des sciences, Paris, Series I, 337, 819-824 (2003).
14.
Giroud, P.Analyse asymptotique de coques inhomogènes en élasticité linéarisé anisotrope. Comptes Rendus de l’Académie des sciences, Paris, Series I, 327, 1011-1014 ( 1998).
15.
Caillerie, D. and Sanchez-Palencia , E.Elastic thin shells: Asymptotic theory in the anisotropic and heterogeneous cases. Mathematical Models and Methods in Applied Science, 8(4), 473-496 (1995).
16.
Trabelsi, K.Nonlinearly elastic thin plate models for a class of Ogden materials: II. The bending model. Analysis and Applications (Singapore), 3, 271-283 (2005).
17.
Nečas, J.Les méthodes directes en théorie des équations elliptiques, Masson. Paris, 1967.
18.
Caillerie, D.Thin elastic and periodic plates. Mathematical Models and Methods in Applied Science, 6, 159-191 ( 1984).
19.
Lewiński, T.Effective models of composite periodic plates I. Asymptotic solution. International Journal of Solids and Structures, 27, 1173-1174 (1991).
20.
Ciarlet, P.G. and Lods, V.Asymptotic analysis of linearly elastic shells I. Justification of membrane shell equations. Archive for Rational Mechanics and Analysis, 136, 119-161 (1996).
21.
Miara, B.Nonlinearly elastic shell models: a formal asymptotic approach I. The membrane model. Archive for Rational Mechanics and Analysis, 142, 331-353 (1998).
22.
Collard, C. and Miara, B.Asymptotic analysis of the stresses in thin elastic shells. Archive for Rational Mechanics and Analysis , 148, 233-264 (1999).
23.
Mardare, C.Asymptotic analysis of linearly elastic shells: error estimates in the membrane case. Asymptotic Analysis , 17, 31-51 (1998).
24.
Lions, J.L.Some Methods in the Mathematical Analysis of Systems and their Control, Gordon and Breach Science Publishers, New York, 1981.
25.
Dauge, M. and Gruais, I.Développement asymptotique d’ordre arbitraire pour une élastique mince encastrée. Comptes Rendus de l’Académie des sciences, Paris, Series I, 321, 375-380 (1995).
26.
Dauge, M. and Gruais, I.Asymptotics of arbitrary order for a thin elastic clamped plate. I: Optimal error estimates. Asymptotic Analysis, 13, 167-197 (1996).
27.
Pruchnicki, E.Hyperelastic homogenized law for reinforced elastomer at finite strain with edge effects. Acta Mechanica , 129, 139-162 (1998).
28.
Pruchnicki, E.Overall properties of thin hyperelastic plate at finite strain with edge effects using asymptotic method. International Journal of Engineering Science, 36, 973-1000 ( 1998).
29.
Bakhvalov, N.S. and Panasenko G.Homogenisation: Averaging Processes in Periodic Media (Mathematics and its Applications Soviet Series), Kluwer Academic, Dordrecht (1989 ).
30.
Muller, S.Homogenization of nonconvex cellular elastic materials. Archive for Rational Mechanics and Analysis, 99, 189-212 (1987).
31.
Geymonat, G., Muller, S. and Triantafyllidis , N.Homogenization of non linearly elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity. Archive for Rational Mechanics and Analysis , 122, 231- 290 (1993).
32.
Michel, J.C. , Lopez-Pamies, O., Ponte-Castañeda , P. and Triantafyllidis, N.Microscopic and macroscopic instabilities in finitely strained porous elastomers. Journal of the Mechanics and Physics of Solids, 55, 900-938 ( 2007).
33.
Trabelsi, K.Nonlinearly elastic thin plate models for a class of Ogden materials: I The membrane model. Analysis and Applications (Singapore), 3, 195-221 ( 2005).
34.
Friesecke, G., James, R.D. and Muller, S.A herarchy of plate models derived from nonlinear elasticity by gamma-convergence. Archive for Rational Mechanics and Analysis, 180, 183-236 (2006).
35.
Ciarlet, P.G. and Lods, V.Asymptotic analysis of linearly elastic shells. III. Justification of Koiter’s shell equations. Archive for Rational Mechanics and Analysis, 136, 191-200 (1996).
36.
Ciarlet P.G. and Roquefort, A.Justification of a two-dimensional nonlineat shell model of Koiter’s type. Chinese Annals of Mathematics , 22B, 129-144 (2001).
37.
Koiter, W.T.On the nonlinear theory of thin elastic shells. Proceedings of the Academic Congress, Netherlands, Wetensch, B69, 1-54 ( 1966).
38.
Ciarlet, P.G. , Lods, V. and Miara, B.Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations. Archive for Rational Mechanics and Analysis, 136, 163-190 (1996).
39.
Lods, V. and Miara, B.Nonlinearly elastic shell models: A formal asymptotic approach II. The flexural model. Archive for Rational Mechanics and Analysis, 142, 355-374 (1998).
40.
Steigmann, D.J.Two-dimensional models for the combined stretching of plates and shells on three-dimensional linear elasticity. International Journal of Engineering Science, 46, 654-676 ( 2008).
