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Портал Begell Электронная Бибилиотека e-Книги Журналы Справочники и Сборники статей Коллекции
International Journal for Multiscale Computational Engineering
Импакт фактор: 1.016 5-летний Импакт фактор: 1.194 SJR: 0.554 SNIP: 0.68 CiteScore™: 1.18

ISSN Печать: 1543-1649
ISSN Онлайн: 1940-4352

Том 17, 2019 Том 16, 2018 Том 15, 2017 Том 14, 2016 Том 13, 2015 Том 12, 2014 Том 11, 2013 Том 10, 2012 Том 9, 2011 Том 8, 2010 Том 7, 2009 Том 6, 2008 Том 5, 2007 Том 4, 2006 Том 3, 2005 Том 2, 2004 Том 1, 2003

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2012002059
pages 213-227


Sandra Klinge
Institute of Mechanics, Ruhr-University Bochum, D-44780 Bochum, Germany
Klaus Hackl
Ruhr University Bochum, Bochum, Germany

Краткое описание

In this contribution the properties and application of the multiscale finite element program MSFEAP are presented. This code is developed on basis of coupling the homogenization theory with the finite element method. According to this concept, the investigation of an appropriately chosen representative volume element yields the material parameters needed for the simulation of a macroscopic body. The connection of scales is based on the principle of volume averaging and the Hill-Mandel macrohomogeneity condition. The latter leads to the determination of different types of boundary conditions for the representative volume element and in this way to the postulation of a well-posed problem at this level. The numerical examples presented in the contribution investigate the effective material behavior of microporous media. An isotropic and a transversally anisotropic microstructure are simulated by choosing an appropriate orientation and geometry of the representative volume element in each Gauss point. The results are verified by comparing them with Hashin-Shtrikman's analytic bounds. However, the chosen examples should be understood as simply an illustration of the program application, while its main feature is a modular structure suitable for further development.


  1. Bathe, K. J., Finite Element Procedures.

  2. Budianski, B., On the elastic moduli of some heterogeneous materials. DOI: 10.1016/0022-5096(65)90011-6

  3. Castañeda, P. P., The effective mechanical properties of nonlinear isotropic composites. DOI: 10.1016/0022-5096(91)90030-R

  4. Castañeda, P. P., New variational principles in plasticity and their application to composite materials. DOI: 10.1016/0022-5096(92)90050-C

  5. Castañeda, P. P., Second-order homogenization estimates for nonlinear composites incorporating field fluctuations. DOI: 10.1016/S0022-5096(01)00099-0

  6. Chaterjee, A., An introduction to the proper orthogonal decomposition.

  7. deBotton, G., Hariton, I., and Socolsky, E. A., Neo-Hookean fibre reinforced composites in finite elasticity. DOI: 10.1016/j.jmps.2005.10.001

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

  9. Feyel, F., A multilevel finite element method (FE2) to describe the response of highly-nonlinear structures using generalized continua. DOI: 10.1016/S0045-7825(03)00348-7

  10. Gilbert, P. P. and Panachenko, A., Effective acoustic equations for a two-phase medium with microstructure. DOI: 10.1016/j.mcm.2004.07.002

  11. Gilbert, P. P., Panachenko, A., and Xie, X., Homogenization of viscoelastic matrix in linear frictional contact. DOI: 10.1002/mma.570

  12. Hashin, Z. and Shtrikman, S., On some variational principles in anisotropic and nonhomogeneous elasticity. DOI: 10.1016/0022-5096(62)90004-2

  13. Hashin, Z. and Shtrikman, S., A variational approach to the theory of the elastic behaviour of polycrystals. DOI: 10.1016/0022-5096(62)90005-4

  14. Hashin, Z. and Shtrikman, S., A variational approach to the theory of the elastic behaviour of multiphase materials. DOI: 10.1016/0022-5096(63)90060-7

  15. Hazanov, S., Hill condition and overall properties of composites. DOI: 10.1007/s004190050173

  16. Hazanov, S.,, On apparent properties of nonlinear heterogeneous bodies smaller than the representative volume element.

  17. Hill, R., The elastic behaviour of a crystalline aggregate. DOI: 10.1088/0370-1298/65/5/307

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

  19. Hill, R., On constitutive macro-variables for heterogeneous solids at finite strain. DOI: 10.1098/rspa.1972.0001

  20. Huet, C., Universal conditions for assimilation of a heterogeneous material to an effective medium.

  21. Huet, C., On the definition and experimental determination of effective constitutive equations for assimilating heterogeneous materials. DOI: 10.1016/0093-6413(84)90064-8

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

  23. Hughes, T. J. R., The Finite Element Method.

  24. Ilic, S., Application of the Multiscale FEM to the Modeling of Composite Materials.

  25. Ilic, S., Hackl, K., and Gilbert, R. P., Application of the multiscale FEM to the modelling of cancellous bone. DOI: 10.1007/s10237-009-0161-6

  26. Ilic, S., User manual for the multiscale FE program MSFEAP.

  27. Kröner, E., Elastic moduli of perfectly disordered composite materials. DOI: 10.1016/0022-5096(67)90026-9

  28. Lebensohn, R. A., N-site modeling of a 3D viscoplastic polycrystal using fast Fourier Transfrom.

  29. Lopez-Pamies, O. and Castañeda, P. P.,, Second-order estimated for the macroscopic response and loss of ellipticity in porous rubbers at large deformations. DOI: 10.1007/s10659-005-1405-z

  30. Miehe, C., Schotte, J., and Lambrecht, M., Homogenisation of inelastic solid materials at finite strains based on incremental minimization principles. DOI: 10.1016/S0022-5096(02)00016-9

  31. Michel, J. C. and Suquet, P., Nonuniform transformation field analysis. DOI: 10.1016/S0020-7683(03)00346-9

  32. Michel, J. C. and Suquet, P., Computational analysis of nonlinear composite structures using the nonuniform transformation fields analysis. DOI: 10.1016/j.cma.2003.12.071

  33. Mori, T. and Tanaka, K., Average stress in matrix and average elastic energy of materials with misfitting inclusions. DOI: 10.1016/0001-6160(73)90064-3

  34. Moulinec, H. and Suquet, P., Intraphase strain heterogeneity in nonlinear composites: A computational approach. DOI: 10.1016/S0997-7538(03)00079-2

  35. Oden, J. T. and Zohdi, T. I., Analysis and adaptive modeling of highly heterogeneous elastic structures. DOI: 10.1016/S0045-7825(97)00032-7

  36. Schröder, J., Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Stabilitäts Problemen.

  37. Simo, J. C. and Hughes, T. J. R., Computational Inelasticity.

  38. Suquet, P., Effective Properties of Nonlinear Composites.

  39. Talbot, D. R. S. and Willis, J. R., Variational principles for inhomogeneous non-linear media. DOI: 10.1093/imamat/35.1.39

  40. Taylor, R. L., Feap Usear Manual.

  41. Terada, K. and Kikuchi, N., A class of general algorithms for multi-scale analysis of heterogeneous media. DOI: 10.1016/S0045-7825(01)00179-7

  42. Willis, J. R., Bounds and self-consistent estimates for the overall properties of anisotropic composites. DOI: 10.1016/0022-5096(77)90022-9

  43. Yvonet, J. and He, Q.-C., The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. DOI: 10.1016/j.jcp.2006.09.019

  44. Zienkiewicz, O. C. and Taylor, R. L., The Finite Element Method.

  45. Zohdi, T. I., Oden, J. T., and Rodin, G. J., Hierarchical modeling of heterogeneous bodies. DOI: 10.1016/S0045-7825(96)01106-1

  46. Zohdi, T. I. and Wriggers, P., A domain decomposition method for bodies with heterogeneous microstructure based on the material regularization. DOI: 10.1016/S0020-7683(98)00124-3

  47. Zohdi, T. I., Wriggers, P., and Huet, C., A method of substructuring large-scale computational micromechanical problems.