Comptes Rendus
no. 1
Boundary conditions for the high order homogenized equation: laminated rods, plates and composites
[Conditions aux limites pour l'équation homogénéisée d'ordre élévé : barres, plaques et composites stratifiés]

Présenté par : Évariste Sanchez-Palencia

Grigory Panasenko1
1 LaMUSE EA 3989, University of Saint Étienne, 23, rue P. Michelon, 42023 Saint Étienne, France
Comptes Rendus. Mécanique, Volume 337 (2009) no. 1, pp. 8-14.

La technique d'homogénéisation d'ordre élévé mène à l'équation homogénéisée d'ordre élévé. Ses coefficients ont été largement discutés dans la literature de la mécanique des composites parce qu'ils sont liés aux théories des gradients des déformations d'ordre élévé. Neanmoins, la nature mathématique de cette équation n'a pas été complètement clarifiée et les conditions aux limites asymptotiquement exactes n'ont pas été définies. Dans cette Note nous donnons la formulation variationnelle de l'équation homogénéisée d'ordre élévé. Cette formulation est dérivée par la projection du problème initial sur l'espace du dévéloppement asymptotique. Elle engendre les conditions aux limites appropriées pour l'équation homogénéisée d'ordre élévé. L'estimation de la difference entre la solution exacte et la solution approchée est obtenue.

The high order homogenization technique generates the so called infinite order homogenized equation. Its coefficients were widely discussed in composite mechanics literature because they are closely related to the so called high order strain gradients theories. However, it was not clear what is the correct mathematical setting for this equation and what are the asymptotically exact boundary conditions. In the present Note we give a variational formulation for the high order homogenized equation by the projection of the initial problem on the “ansatz subspace”. This formulation generates the appropriate boundary conditions for the high order homogenized equation. The error estimates for the solution of the original problem and the homogenized one are obtained.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2008.10.008
Keywords: Homogenization, High order homogenized equation, High order boundary conditions, Strain gradient theories
Mot clés : Homogénéisation, Equation homogénéisée d'ordre élévé, Conditions aux limites d'ordre élévé, Théories des gradients des déformations
Grigory Panasenko 1

1 LaMUSE EA 3989, University of Saint Étienne, 23, rue P. Michelon, 42023 Saint Étienne, France 
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Grigory Panasenko. Boundary conditions for the high order homogenized equation: laminated rods, plates and composites. Comptes Rendus. Mécanique, Volume 337 (2009) no. 1, pp. 8-14. doi : 10.1016/j.crme.2008.10.008. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2008.10.008/

[1] E. Cosserat; F. Cosserat Théorie des Corps Déformables, Hermanns, Paris, 1909

[2] R.D. Mindlin Microstructure in linear elasticity, Arch. Ration. Mech. Anal., Volume 16 (1965), pp. 51-78

[3] R.A. Tupin Elastic materials with couple stresses, Arch. Ration. Mech. Anal., Volume 11 (1962), pp. 385-414

[4] S.A. Nazarov; A.S. Zorin Two-term asymptotics of the problem on longitudinal deformation of a plate with clamped edge, Computational Mechanics of Solids, Volume 2 (1991), pp. 10-21 (in Russian)

[5] S.A. Nazarov On the accuracy of asymptotic approximations for longitudinal deformation of a thin plate, Math. Modelling Numer. Anal., Volume 30 (1996) no. 2, pp. 185-213

[6] N.S. Bakhvalov Averaging partial differential equations with rapidly oscillating coefficients, Dokl. Acad. Nauk SSSR, Volume 221 (1975), pp. 516-519

[7] N.S. Bakhvalov; G.P. Panasenko Homogenization: Averaging Processes in Periodic Media, Mathematics and Its Applications (Soviet Series), vol. 36, Nauka, Moscow, 1984 (in Russian). English translation in:, 1989, Kluwer Academic Publishers, Dordrecht

[8] G.P. Panasenko Boundary layer in homogenization problems for non-homogeneous media, BAIL 4 (S.K. Godunov; J.J.H. Miller; V.A. Novikov, eds.), Boole Press, Dublin (1986), pp. 398-402

[9] G.P. Panasenko Multi-Scale Modelling for Structures and Composites, Springer, 2005

[10] N.S. Bakhvalov; M.E. Eglit Variational properties of averaged equations for periodic media I, Trudy Mat. Inst. Steklova, Volume 192 (1990), pp. 3-18 (English version: Proc. Steklov Inst. Math., 192, 1992)

[11] N.S. Bakhvalov; M.E. Eglit Investigation of the effective equations with dispersion for wave propagation in stratified media and thin plates, Dokl. RAN, Volume 383 (2002) no. 6

[12] N.S. Bakhvalov; K.Yu. Bogachev; M.E. Eglit Investigation of the effective equations with dispersion for wave propagation in heterogeneous thin rods, Dokl. RAN, Volume 387 (2002) no. 6, pp. 749-753

[13] M.B. Panfilov, Macroscopic models of flows in stronglyheterogeneous porous media, Doctor Sci. Thesis, 1992

[14] V.P. Smyshlyaev; K.D. Cherednichenko On derivation of “strain gradient” effects in the overall behaviour of periodic heterogeneous media, J. Mech. Phys. Solids, Volume 48 (2000), pp. 1325-1357

[15] C. Boutin Microstructural effects in elastic composites, Int. J. Solids Structures, Volume 33 (1996), pp. 1023-1051

[16] G.P. Panasenko On the existence and uniqueness of the solution of some elliptic equations, set in Rn (V.A. Morozov, ed.), Proc. of the Conference of young researchers of the Dept. of Numerical Analysis and Informatics of Moscow State University, Moscow University Publ., Moscow, 1976, pp. 53-61 (in Russian)

[17] G.P. Panasenko Asymptotic of higher order of solutions of equations with rapidly oscillating coefficients, Dokl. Akad. Nauk SSSR, Volume 240 (1978) no. 6, pp. 1293-1296 (in Russian). English translation in Soviet Math. Dokl., 1979

[18] G.P. Panasenko High order asymptotics of solutions of problems on the contact of periodic structures, Math. Sb., Volume 110 (152) (1979) no. 4, pp. 505-538 (in Russian). English translation in Math. USSR Sb., 38, 4, 1981, pp. 465-494

[19] E. Sanchez-Palencia Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, New York, 1980

[20] J.L. Lions Some Methods in Mathematical Analysis of Systems and their Control, Science Press, Gordon and Breach, Beijing, New York, 1981

[21] G.P. Panasenko The partial homogenization: continuous and semi-discretized versions, Math. Models Methods Appl. Sci., Volume 8 (2007) no. 17, pp. 1183-1209

[22] E.M. Landis; G.P. Panasenko A theorem on the asymptotics of solutions of elliptic equations with coefficients periodic in all variables except one, Dokl. Akad. Nauk SSSR, Volume 235 (1977) no. 6, pp. 1140-1143 (in Russian). English translation in Soviet Math. Dokl., 18, 4, 1977

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