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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.452 SNIP: 0.68 CiteScore™: 1.18

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

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2013005712
pages 369-387


Zhaoqian Xie
State Key Laboratory for Structural Analysis of Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, P. R. China
Hongwu Zhang
Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, P. R. China


A new multiscale finite element method is developed for mechanical analysis of periodic heterogeneous Cosserat materials. The main idea of the method is to numerically construct the multiscale base functions to capture the small-scale features of the coarse elements. Considering the existence of rotation in the Cosserat materials, specified boundary conditions of the base functions for extended multiscale finite element method (EMsFEM) are developed based on the relationship between transverse displacement and rotation (slope) of the two-node beam element, and the corresponding periodic boundary conditions are developed. By adopting both kinds of boundary conditions, the numerical base functions for displacement and rotation fields of Cosserat materials are constructed, respectively, to establish the relationship between the macroscopic deformation and the microscopic stress and strain. It is shown that the proposed method does not require the estimation of the overall material parameters of the heterogeneous Cosserat materials as the general homogenization methods. Numerical examples are carried out to verify the validity and efficiency of the developed multiscale finite element method.


  1. Aarnes, J. E., On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation. DOI: 10.1137/030600655

  2. Aarnes, J. E., Krogstad, S., and Lie, K. A., A hierarchical multiscale method for two-phase flow based upon mixed finite elements and nonuniform coarse grids. DOI: 10.1137/050634566

  3. Babuška, I., Homogenization Approach in Engineering.

  4. Babuška, I. and Osborn, E., Generalized finite element methods: Their performance and their relation to mixed methods. DOI: 10.1137/0720034

  5. Babuška, I., Caloz, G., and Osborn, E., Special finite element methods for a class of second order elliptic problems with rough coefficients. DOI: 10.1137/0731051

  6. Benssousan, A., Lions, J. L., and Papanicoulau, G., Asymptotic Analysis for Periodic Structures.

  7. Cosserat, E. and Cosserat, F., Theorie des Corps Deformables.

  8. de Borst, R., Simulation of strain localization: A reappraisal of the Cosserat continuum. DOI: 10.1108/eb023842

  9. Drugan, W. J. and Willis, J. R., A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. DOI: 10.1016/0022-5096(96)00007-5

  10. Efendiev, Y., Hou, T., and Ginting, V., Multiscale finite element methods for nonlinear problems and their applications.

  11. Efendiev, Y., Ginting, V., Hou, T. Y., Hou, T., and Ewing, R., Accurate multiscale finite element methods for two-phase flow simulations. DOI: 10.1016/

  12. Ellis, R. W. and Smith, C. W., A thin-plate analysis and experimental evaluation of couple-stress effects. DOI: 10.1007/BF02326308

  13. Fish, J., The s-version of the finite element method. DOI: 10.1016/0045-7949(92)90287-A

  14. Fish, J. and Guttal, R., The s-version of finite element method for laminated composites. DOI: 10.1002/(SICI)1097-0207(19961115)39:21<3641::AID-NME17>3.0.CO;2-P

  15. Fish, J. and Markolefas, S., The s-version of the finite element method for multilayer laminates. DOI: 10.1002/nme.1620330512

  16. Fish, J. and Yu, Q., Computational mechanics of fatigue and life predictions for composite materials and structures. DOI: 10.1016/S0045-7825(02)00401-2

  17. Forest, S., Barbe, F., and Cailletaud, G., Cosserat modelling of size effects in the mechanical behaviour of polycrystals and multi-phase materials. DOI: 10.1016/S0020-7683(99)00330-3

  18. Forest, S., Pradel, F., and Sab, K., Asymptotic analysis of heterogeneous Cosserat media. DOI: 10.1016/S0020-7683(00)00295-X

  19. Garikipati, K. and Hughes, T. J. R., A variational multiscale approach to strain localization-formulation for multidimensional problems. DOI: 10.1016/S0045-7825(99)00156-5

  20. Geers, M. G. D., Kouznetsova, V., and Brekelmans, W. A. M., Gradient-enhanced computational homogenization for the micro-macro scale transition. DOI: 10.1051/jp4:2001518

  21. Hughes, T. J. R., Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. DOI: 10.1016/0045-7825(95)00844-9

  22. Hughes, T. J. R., Feijoo, G. R., Mazzei, L., and Quincy, J.-B., The variational multiscale method-a paradigm for computational mechanics. DOI: 10.1016/S0045-7825(98)00079-6

  23. Hughes, T. J. R., Mazzei, L., and Jansen, K. E., Large eddy simulation and the variational multiscale method. DOI: 10.1007/s007910050051

  24. Hou, T. Y., Multiscale modelling and computation of fluid flow. DOI: 10.1002/fld.866

  25. Hou, T. Y. and Wu, X. H., A multiscale finite element method for elliptic problems in composite materials and porous media. DOI: 10.1006/jcph.1997.5682

  26. Hou, T. Y., Wu, X. H., and Cai, Z. Q., Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. DOI: 10.1090/S0025-5718-99-01077-7

  27. Kanoute, P., Boso, D. P., Chaboche, J. L., and Schrefler, B. A., Multiscale methods for composites: A review. DOI: 10.1007/s11831-008-9028-8

  28. Kouznetsova, V., Computational Homogenization for the Multiscale Analysis of Multi-Phase Materials.

  29. Kouznetsova, V., Geers, M. G. D., and Brekelmans, W. A. M., Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. DOI: 10.1002/nme.541

  30. Lakes, R. S., Strongly Cosserat elastic lattice and foam materials for enhanced toughness.

  31. Li, X. K., Liu, Q. P., and Zhang, J. B., A micro-macro homogenization approach for discrete particle assembly-Cosserat continuum modeling of granular materials. DOI: 10.1016/j.ijsolstr.2009.09.033

  32. Miehe, C. and Bayreuther, C. G., On multiscale FE analyses of heterogeneous structures: From homogenization to multigrid solvers. DOI: 10.1002/nme.1972

  33. Neff, P. and Forest, S., A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure: Modelling, existence of minimizers, identification of moduli and computational results. DOI: 10.1007/s10659-007-9106-4

  34. Ostoja-Starzewski, M., Boccara, S. D., and Jasiuk, I., Couple-stress moduli and characteristic length of a two-phase composite. DOI: 10.1016/S0093-6413(99)00039-7

  35. Paumelle, P., Hassim, F., and Lene, F., Microstress analysis in woven composite structures.

  36. Pecullan, S., Gibiansky, L. V., and Torquato, S., Scale effects on the elastic behavior of periodic andhierarchical two-dimensional composites. DOI: 10.1016/S0022-5096(98)00111-2

  37. Providas, E. and Kattis, M. A., Finite element method in plane Cosserat elasticity. DOI: 10.1016/S0045-7949(02)00262-6

  38. Schijve, J., Note on couple stresses. DOI: 10.1016/0022-5096(66)90042-1

  39. Smyshlyaev, V. P. and Cherednichenko, K. D., On rigorous derivation of strain gradient effects in the overall behavior of periodic heterogeneous media. DOI: 10.1016/S0022-5096(99)00090-3

  40. Smyshlyaev, V. P. and Fleck, N. A., Bounds and estimates for linear composites with strain gradient effects. DOI: 10.1016/0022-5096(94)90016-7

  41. Tekoglu, C. and Onck, P. R., A comparison of discrete and Cosserat continuum analyses for cellular materials.

  42. Tekoglu, C. and Onck, P. R., Identification of Cosserat constant for cellular materials.

  43. Triantafyllidis, N. and Bardenhagen, S., The influence of scale size on the stability of periodic solids and the role of associated higher order gradient continuum models. DOI: 10.1016/0022-5096(96)00047-6

  44. Weinan, E. and Engquist, B., The heterogeneous multi-scale method for homogenization problems. DOI: 10.1007/3-540-26444-2_4

  45. Weinan, E., Engquist, B., Li, X., Ren., W., and Vanden-Eijnden, E., The heterogeneous multiscale method: A review. DOI: 10.1017/S0962492912000025

  46. Xia, Z. H., Zhou, C. W., Yong, Q. L., and Wang, X., On selection of repeated unit cell model and application of unified periodic boundary conditions in micro-mechanical analysis of composites. DOI: 10.1016/j.ijsolstr.2005.03.055

  47. Yang, J. F. C. and Lakes, R. S., Experimental study of micropolar and couple stress elasticity in compact bone in bending. DOI: 10.1016/0021-9290(82)90040-9

  48. Yuan, Z. and Fish, J., Toward realization of computational homogenization in practice. DOI: 10.1002/nme.2074

  49. Zhang, H. W., Zhang, S., Bi, J. Y., and Schrefler, B. A., Thermo-mechanical analysis of periodic multiphase materials by a multiscale asymptotic homogenization approach. DOI: 10.1002/nme.1757

  50. Zhang, H. W., Fu, Z. D., and Wu, J. K., Coupling multiscale finite element method for consolidation analysis of heterogeneous saturated porous media. DOI: 10.1016/j.advwatres.2008.11.002

  51. Zhang, H. W., Wu, J. K., and Fu, Z. D., Extended multiscale finite element method for elasto-plastic analysis of 2D periodic lattice truss materials. DOI: 10.1007/s00466-010-0475-3

  52. Zhang, H. W., Wu, J. K., L&uuml;, J., and Fu, Z. D., Extended multiscale finite element method for mechanical analysis of heterogeneous materials. DOI: 10.1007/s10409-010-0393-9

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