Library Subscription: Guest
Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections
International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2012002587
pages 281-293


Mohamed Ben Bettaieb
ArGEnCo Department, MS2F Division, University of Liege, Chemin des Chevreuils 1, 4000 Liege, Belgium
Olivier Debordes
LMA & ECM, IMT, Technopole Chateau-Gombert, F13383 Marseille Cedex 13, France
Abdelwaheb Dogui
LGM, ENIM, 5019 Monastir, Tunisia
Laurent Duchene
ArGEnCo Department, MS2F Division, University of Liege, Chemin des Chevreuils 1, 4000 Liege, Belgium


The main motivation of this paper consists of using the periodic homogenization theory to derive several relations between macroscopic Lagrangian (e.g., deformation gradient, Piola−Kirchhoff tensor) and Eulerian (e.g., velocity gradient, Cauchy stress) quantities. These relations demonstrate that these macroscopic quantities behave formally in the same way as their microscopic counterparts. We say therefore that these relations are stable with respect to the periodic homogenization. We also demonstrate the equivalence between the two forms of the macroscopic power density expressed in the Lagrangian and Eulerian formulations. Two simple examples illustrate these results, and indicate also that the Green−Lagrange strain tensor and the second Piola−Kirchhoff stress tensor are not stable with respect to periodic homogenization.


  1. Andia, P. C., Costanzo, F., and Gray, G. L., A Lagrangian-based continuum homogenization approach approach applicable to molecular dynamics simulations. DOI: 10.1016/j.ijsolstr.2005.05.027

  2. Ben Bettaieb, M., Modélisation numérique du comportement de matériaux polycristallins par homogénéisation périodique.

  3. Colby, C. and HyungJoo, K., Multi-scale unit cell analyses of textile composites.

  4. Débordes, O. and Dogui, A., Numerical homogenization of polycrystalline media.

  5. Duvaut, G., Méthodes variationnelles, initiation et applications.

  6. Elbououni, S., Simulation numérique d'anisotropie induite dans les matériaux polycristallins selon une approche micro-macro.

  7. Feyel, F. and Chaboche, J.-L., FE2 multiscale approach for modeling the elastoviscoplastic behavior of long fiber SiC/Ti composite materials. DOI: 10.1016/S0045-7825(99)00224-8

  8. Fish, J. and Fan, R., Mathematical homogenization of nonperiodic heterogeneous media subjected to large deformation transient loading. DOI: 10.1002/nme.2355

  9. Fish, J. and Kuznetsov, S., Computational continua. DOI: 10.1002/nme.2918

  10. Gurtin, M. E., An Introduction to Continuum Mechanics.

  11. Hill, R., Elastic properties of reinforced solids: Some theoretical principles. DOI: 10.1016/0022-5096(63)90036-X

  12. Matsui, K., Terada, K., and Yuge, K., Two-scale finite element analysis of heterogeneous solids with periodic microstructures. DOI: 10.1016/j.compstruc.2004.01.004

  13. Miehe, C. and Dettmar, J., A framework for micro–macro transitions in periodic particle aggregates of granular materials. DOI: 10.1016/j.cma.2003.10.004

  14. Miehe, C., Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy. DOI: 10.1016/S0045-7825(02)00564-9

  15. Nemat-Nasser, S., Averaging theorems in finite deformation plasticity. DOI: 10.1016/S0167-6636(98)00073-8

  16. Phillips, R., Crystals, Defects and Microstructures.

  17. Raviart, P. A. and Thomas, J. M., Introduction à l'analyse numérique des équations aux dérivées partielles.

  18. Suquet, P., Sur les équations de la plasticité: Existence et régularité des solutions.

  19. Taliercio, A. and Sagramoso, P., Uniaxial strength of polymeric-matrix fibrous composites predicted through a homogenization approach. DOI: 10.1016/0020-7683(94)00139-N