Abo Bibliothek: Guest
Digitales Portal Digitale Bibliothek eBooks Zeitschriften Referenzen und Berichte Forschungssammlungen
International Journal for Multiscale Computational Engineering
Impact-faktor: 1.016 5-jähriger Impact-Faktor: 1.194 SJR: 0.554 SNIP: 0.82 CiteScore™: 2

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

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2013005817
pages 239-252


Erez Gal
Department of Structural Engineering, Ben-Gurion University, Beer-Sheva, 84105, Israel
E. Suday
Department of Structural Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel
Haim Waisman
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, New York 10027, USA


We study the homogenization of materials having three phases: matrix, inclusions, and inclusion coatings. To model the inclusions and their coatings we investigate two weakly discontinuous enrichment functions within the extended finite element method (XFEM) framework described by a single level set function. In both formulations the inclusion and coating shapes are independent of each other. The first approach, denoted as a V-type enrichment, combines several weak discontinuities by stacking the corresponding inclusion and coating enriched degrees of freedom in a single node (one on top of the other) while the second, denoted as a zigzag-type enrichment, only adds one additional degree of freedom per each direction. The XFEM approach is extremely efficient and avoids excessive remeshing compared to standard FEM, in particular when the coatings are very thin as in the case of aggregates surrounded by the interface transition zone (ITZ) in concrete materials. Comprehensive verification studies are presented including two-dimensional continuum problems and homogenization of concrete. Herein we mainly focus on the microscopic material response via homogenization of the unit cell. Nonetheless, once the homogenized material properties are obtained, the application to a full multiscale analysis is straightforward. While both enrichment types are different possible extensions to XFEM applied to such three-phase materials, both methods seem to work well and provide significant reduction in degrees of freedom and computation time as compared to standard FEM.


  1. Aboudi, J., Mechanics of Composite Materials—A Unified Micromechanical Approach.

  2. Aragón, A. M., Duarte, C. A., and Geubelle, P. H., Generalized finite element enrichment functions for discontinuous gradient field. DOI: 10.1002/nme.2772

  3. Babuška, I. and Melenk, J. M., The partition of unity finite element method: Basic theory and application. DOI: 10.1016/S0045-7825(96)01087-0

  4. Belytschko, T. and Black, T., Elastic crack growth in finite elements with minimal remeshing. DOI: 10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S

  5. Belytschko, T., Mo&euml;s, N., Usui, S., and Parimi, C., Arbitrary discontinuities in finite elements. DOI: 10.1002/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M

  6. Belytschko, T., Gracie, R., and Ventura, G., A review of extended/generalized finite element methods for material modeling. DOI: 10.1088/0965-0393/17/4/043001

  7. Benkemoun, N., Hautefeuille, M., Colliat, J. B., and Ibrahimbegovic, A., Failure of heterogeneous materials: 3D meso-scale FE models with embedded discontinuities. DOI: 10.1002/nme.2816

  8. Benkemoun, N., Ibrahimbegovic, A., and Colliat, J. B., Anisotropic constitutive model of plasticity capable of accounting for details of meso-structure of two-phase composite material. DOI: 10.1016/j.compstruc.2011.09.003

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

  10. Bentz, D. P., Stutzman, P. A., and Garboczi, E. J., Experimental and simulation studies of the interfacial zone in concrete. DOI: 10.1016/0008-8846(92)90113-A

  11. Bentz, D. P., Garboczi, E. J., and Stutzman, P. E., Computer modeling of the interfacial transition zone in concrete.

  12. Christensen, R. M., Mechanics of Composite Materials.

  13. Crouch, R. and Oskay, C., Symmetric meso-mechanical model for failure analysis of heterogeneous materials. DOI: 10.1615/IntJMultCompEng.v8.i5.20

  14. Cusatis, G. and Cedolin, L., Two-scale study of concrete fracturing behavior. DOI: 10.1016/j.engfracmech.2006.01.021

  15. Eshelby, J. D., The determination of the field of an ellipsoidal inclusion and related problems. DOI: 10.1098/rspa.1957.0133

  16. Fish, J. and Shek, K. L., Finite deformation plasticity of composite structures: Computational models and adaptive strategies. DOI: 10.1016/S0045-7825(98)00228-X

  17. Fish, J. and Yu, Q., Multiscale damage modeling for composite materials: Theory and computational framework. DOI: 10.1002/nme.276

  18. Fish, J., Shek, K., Pandheeradi, M., and Shephard, M. S., Computational plasticity for composite structures based on mathematical homogenization: Theory and practice. DOI: 10.1016/S0045-7825(97)00030-3

  19. Fries, T. P., A corrected XFEM approximation without problems in blending elements. DOI: 10.1002/nme.2259

  20. Gal, E. and Krivoruk, R., FRC material properties&mdash;a multi-scale approach.

  21. Gal, E. and Krivoruk, R., Meso-scale analysis of FRC using a two-step homogenization approach. DOI: 10.1016/j.compstruc.2011.02.006

  22. Gal, E., Ganz, A., Chadad, L., and Krivoruk, R., Development of a concrete unit cell. DOI: 10.1615/IntJMultCompEng.v6.i5.80

  23. Gitman, I. M., Askes, H., and Sluys, L. J., Representative volume: Existence and size determination engineering. DOI: 10.1016/j.engfracmech.2006.12.021

  24. Guedes, J. M. and Kikuchi, N., Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. DOI: 10.1016/0045-7825(90)90148-F

  25. Haffner, S., Eckardt, S., Luther, T., and Koknke, C., Mesoscale modeling of concrete: Geometry and numerics. DOI: 10.1016/j.compstruc.2005.10.003

  26. Hashin, Z., The elastic moduli of heterogeneous materials. DOI: 10.1115/1.3636446

  27. He, H., Guo, Z., Stroeven, P., Stroeven, M., and Sluys, L. H., Influence of particle packing on elastic properties of concrete. DOI: 10.1680/macr.10.00163

  28. Hill, R., A self-consistent mechanics of composite materials. DOI: 10.1016/0022-5096(65)90010-4

  29. Hiriyur, B., Waisman, H., and Deodatis, G., Uncertainty quantification in homogenization of heterogeneous microstructures modeled by extended finite element method. DOI: 10.1002/nme.3174

  30. Ibrahimbegovic, A. and Markovic, D., Strong coupling methods in multiphase and multiscale modeling of inelastic behavior of heterogeneous structures. DOI: 10.1016/S0045-7825(03)00342-6

  31. Ibrahimbegovic, A. and Melnyk, S., Embedded discontinuity finite element method for modeling of localized failure in heterogeneous materials with structured mesh: An alternative to extended finite element method. DOI: 10.1007/s00466-006-0091-4

  32. Ibrahimbegovic, A., Gre&scaron;ovnik, I., Markovic, D., Melnyk, S., and Rodic, T., Shape optimization of two-phase material with microstructure. DOI: 10.1108/02644400510603032

  33. Karihaloo, B. L. and Xiao, Q. Z., Modeling of stationary and growing cracks in FE framework without remeshing: A state-of-the-art review. DOI: 10.1016/S0045-7949(02)00431-5

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

  35. Mang, H. A., Lackner, R., Meschke, G., and Mosler, J., Computational modeling of concrete structures.

  36. 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

  37. Mo&euml;s, N., Dolbow, J., and Belytschko, T., A finite element method for crack growth without remeshing. DOI: 10.1002/(SICI)1097-0207(19990910)46:1<131::AID-NME726>3.0.CO;2-J

  38. Mohammadi, S., Extended Finite Element Method for Fracture Analysis of Structures.

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

  40. Nilsen, A. U. and Monteiro, P. J. M., Concrete: A three-phase material. DOI: 10.1016/0008-8846(93)90145-Y

  41. Oskay, C. and Fish, J., Eigendeformation-based reduced order homogenization for failure analysis of heterogeneous material. DOI: 10.1016/j.cma.2006.08.015

  42. Prokopski, G. and Halbiniak, J., Interfacial transition zone in cementitious materials. DOI: 10.1016/S0008-8846(00)00210-6

  43. Sanchez-Palencia, E., Non-Homogeneous Media and Vibration Theory: Lecture Notes in Physics.

  44. Scrivener, K. L. and Nemati, K. M., The percolation of pore space in the cement paste/aggregate interfacial zone of concrete. DOI: 10.1016/0008-8846(95)00185-9

  45. Scrivener, K. L. and Gariner, E. M., Microstructural gradients in cement paste around aggregate particles. DOI: 10.1557/PROC-114-77

  46. Simeonov, P. and Ahmad, S., Effect of transition zone on the elastic behavior of cement-based composites. DOI: 10.1016/0008-8846(94)00124-H

  47. Sukumar, N., Chopp, D. L., Mo&euml;s, N., and Belytschko, T., Modeling holes and inclusions by level sets in the extended finite-element method. DOI: 10.1016/S0045-7825(01)00215-8

  48. Sun, Z., Garboczi, E. J., and Shah, S. P., Modeling the elastic properties of concrete composites: Experiment, differential effective medium theory and numerical simulation. DOI: 10.1016/j.cemconcomp.2006.07.020

  49. Terada, K. and Kikuchi, N., Nonlinear homogenization method for practical applications.

  50. 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

  51. Tran, A. B., Yvonnet, J., He, Q.-C., Toulemonde, C., and Sanahuja, J., A multiple level set approach to prevent numerical artefacts in complex microstructures with nearby inclusions within XFEM. DOI: 10.1002/nme.3025

  52. Van Der Meer, F. P., Mo&euml;s, N., and Sluys, L. J., A level set model for delamination&mdash;Modeling crack growth without cohesive zone or stress singularity. DOI: 10.1016/j.engfracmech.2011.10.013

  53. Wriggers, P. and Moftah, S. O., Mesoscale models for concrete: Homogenization and damage behavior. DOI: 10.1016/j.finel.2005.11.008

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

  55. Zohdi, T. I. and Wriggers, P., Introduction to Computational Micromechanics.

Articles with similar content:

Toward Two-Scale Adaptive FEM Modeling of Nonlinear Heterogeneous Materials
International Journal for Multiscale Computational Engineering, Vol.8, 2010, issue 3
Marta Serafin, Witold Cecot
Effects of Shape and Size of Crystal Grains on the Strengths of Polycrystalline Metals
International Journal for Multiscale Computational Engineering, Vol.4, 2006, issue 4
Kenjiro Terada, Masayoshi Akiyama, Ikumu Watanabe
International Journal for Multiscale Computational Engineering, Vol.10, 2012, issue 4
Haim Waisman, Xia Liu, Jacob Fish
International Journal for Multiscale Computational Engineering, Vol.9, 2011, issue 4
F. Ashida, Sei-ichiro Sakata, K. Enya
International Journal for Multiscale Computational Engineering, Vol.11, 2013, issue 6
Mirmohammadreza Kabiri, Franck J. Vernerey