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International Journal for Multiscale Computational Engineering
Fator do impacto: 1.016 FI de cinco anos: 1.194 SJR: 0.554 SNIP: 0.82 CiteScore™: 2

ISSN Imprimir: 1543-1649
ISSN On-line: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2011002047
pages 503-513


Karam Sab
Université Paris-Est, Laboratoire Navier, Ecole des Ponts ParisTech, IFSTTAR, CNRS


The homogenization of elastic periodic plates is as follows: The three-dimensional (3D) heterogeneous body is replaced by a homogeneous Love-Kirchhoff plate whose stiffness constants are computed by solving an auxiliary boundary problem on a 3D unit cell that generates the plate by periodicity in the in-plane directions. In the present study, a generalization of the above-mentioned approach is presented for the case of a plate cut from a block of linear elastic composite material considered to be statistically uniform random in the in-plane directions. The homogenized bending stiffness and the moduli for in-plane deformation of the random plate are defined in four equivalent manners: (1) the first definition considers statistically invariant stress and strain fields in the infinite plate. In the other definitions, a finite representative volume element of the plate is submitted on its lateral boundary to suitable (2) kinematically uniform conditions, (3) statically uniform conditions, and (4) periodic conditions. The relationships between these four definitions are studied and hierarchical bounds are derived.


  1. Caillerie, D., Thin elastic and periodic plates. DOI: 10.1002/mma.1670060112

  2. Cecchi, A. and Sab, K., Out of plane model for heterogeneous periodic materials: The case of masonry. DOI: 10.1016/S0997-7538(02)01243-3

  3. Cecchi, A. and Sab, K., A comparison between a 3D discrete model and two homogenized plate models for periodic elastic brickwork. DOI: 10.1016/j.ijsolstr.2003.12.020

  4. Cecchi, A. and Sab, K., A homogenized Reissner–Mindlin model for orthotropic periodic plates. Application to Brickwork panels. DOI: 10.1016/j.ijsolstr.2007.02.009

  5. Cecchi, A. and Sab, K., A homogenized Love–Kirchhoff model for out-of-plane loaded Random 2D lattices: Application to “quasiperiodic” Brickwork panels. DOI: 10.1016/j.ijsolstr.2009.03.021

  6. Gusev, A. A., Representative volume element size for elastic composites: A numerical study. DOI: 10.1016/S0022-5096(97)00016-1

  7. Huet, C., Application of variational concepts to size effects in elastic heterogeneous bodies. DOI: 10.1016/0022-5096(90)90041-2

  8. Jeulin, D. and Ostoja-Starzewski, M., CISM Courses and Lectures 430.

  9. Kanit, T., Forest, S., Galliet, I., Mounoury, V., and Jeulin, D., Determination of the size of the representative volume element for random composites: Statistical and numerical approach. DOI: 10.1016/S0020-7683(03)00143-4

  10. Kohn, R. and Vogelius, M., A new model for thin plates with rapidly varying thickness. DOI: 10.1016/0020-7683(84)90044-1

  11. Kolpakov, A. G., Variational principles for stiffnesses of a non-homogeneous plate. DOI: 10.1016/S0022-5096(99)00010-1

  12. Kolpakov, A. G. and Sheremet, I. G., The stiffnesses of non-homogeneous plates. DOI: 10.1016/S0021-8928(99)00078-7

  13. Kozlov, S. M., Averaging of random operators. DOI: 10.1070/SM1980v037n02ABEH001948

  14. Kozlov, S. M., Olenik, O., and Zhikov, V., Homogenization of Differential Operators.

  15. Lewinski, T. and Telega, J. J., Plates Laminates and Shells. Asymptotic Analysis and Homogenization.

  16. Nguyen, T. K., Sab, K., and Bonnet, G., Shear correction factors for functionally graded plates. DOI: 10.1080/15376490701672575

  17. Nguyen, T. K., Sab, K., and Bonnet, G., Green's operator for a periodic medium with traction-free boundary conditions and computation of the effective properties of thin plates. DOI: 10.1016/j.ijsolstr.2008.08.015

  18. Nguyen, T. K., Sab, K., and Bonnet, G., First-order shear deformation plate model for functionally graded materials. DOI: 10.1016/j.compstruct.2007.03.004

  19. Nguyen, T. K., Sab, K., and Bonnet, G., Bounds for the effective properties of heterogeneous plates. DOI: 10.1016/j.euromechsol.2009.05.006

  20. Ostoja-Starzewski, M., Material spatial randomness: From statistical to representative volume element. DOI: 10.1016/j.probengmech.2005.07.007

  21. Sab, K., Hill's principle and homogenization of random materials.

  22. Sab, K., On the homogenization and simulation of random materials.

  23. Sab, K., Homogenization of nonlinear random media by a duality method. Application to plasticity. DOI: 10.3233/ASY-1994-9402

  24. Sab, K. and Nedjar, B., Periodization of random media and representative volume element size for linear composites. DOI: 10.1016/j.crme.2004.10.003

  25. Terada, K., Ito, T., and Kikuchi, N., Characterization of the mechanical behaviors of solid–fluid mixture by the homogenization method. DOI: 10.1016/S0045-7825(97)00071-6

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