The analysis of periodic composites with randomly damaged constituents

  • Michael RyvkinEmail author
  • Jacob Aboudi
Original Paper


A new method is offered for the modeling of periodic composites in which some damaged constituents are randomly distributed. The representative volume element for such composites consists of many repetitive cells, and its direct analysis by standard methods is an expensive computational task. The proposed approach allows to diminish significantly the volume of calculations. It is based on the representative cell method and the higher-order theory. In the former, the problem of the periodic composite with randomly located damages is reduced, in conjunction with the discrete Fourier transform, to a single cell problem, formulated in the transform domain. The solution of this boundary value problem is obtained by the latter one. The inversion of the transform, in conjunction with an iterative procedure, establishes the elastic field at any point of the damaged composite. The method is employed for the evaluation of the effective properties and field distribution for a specified loading of unidirectional fiber-reinforced composites with numerous randomly located defective fibers such as lost, missing, and damaged fibers. In addition, porous materials with random distributions of pores and unidirectional composites with fibers that are randomly located within the matrix can be considered. Furthermore, random distributions of matrix cracks in composites can be simulated, and the resulting behavior can be predicted. Finally, broken layers and interfacial cracks in periodically layered composites in which the cracks are randomly located can be similarly analyzed.



The first author gratefully acknowledges support by the Israel Science Foundation (ISF) (Grant No. 1494/16).


  1. 1.
    Aboudi, J., Ryvkin, M.: The effect of localized damage on the behavior of composites with periodic microstructure. Int. J. Eng. Sci. 52, 41–55 (2012)CrossRefGoogle Scholar
  2. 2.
    Aboudi, J., Ryvkin, M.: The analysis of localized effects in composites with periodic microstructure. Philos. Trans. R. Soc. A 371, 20120373 (2013)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aboudi, J., Arnold, S.M., Bednarcyk, B.A.: Micromechanics of Composite Materials: A Generalized Multiscale Analysis Approach. Elsevier, Oxford (2013)Google Scholar
  4. 4.
    Arabnejad, S., Burnett Johnston, R., Pura, J.A., Singh, B., Tanzer, M., Pasini, D.: High-strength porous biomaterials for bone replacement: a strategy to assess the interplay between cell morphology, mechanical properties, bone ingrowth and manufacturing constraints. Acta Biomater. 30, 345–356 (2016)CrossRefGoogle Scholar
  5. 5.
    Astrom, J., Alava, M., Timonen, J.: Crack dynamics and crack surfaces in elastic beam lattices. Phys. Rev. E 57(2), R1259–R1262 (1998)CrossRefGoogle Scholar
  6. 6.
    Bagheri, Z., Melancon, D., Liu, L., Johnston, R., Pasini, D.: Compensation strategy to reduce geometry and mechanics mismatches in porous biomaterials built with Selective Laser Melting. J. Mech. Behav. Biomed. Mater. 70, 17–27 (2017)CrossRefGoogle Scholar
  7. 7.
    Bulsara, V., Talreja, R., Qu, J.: Damage initiation under transverse loading of unidirectional composites with arbitrary distributed fibers. Compos. Sci. Technol. 59, 673–682 (1999)CrossRefGoogle Scholar
  8. 8.
    Chen, C., Lu, T.J., Fleck, N.A.: Effect of imperfections on the yielding of two-dimensional foams. J. Mech. Phys. Solids 47, 2235–2272 (1999)CrossRefGoogle Scholar
  9. 9.
    Cherkaev, A., Ryvkin, M.: Damage propagation in 2d beam lattices: 1. Uncertainty and assumptions. Arch. Appl. Mech. 89, 485–501 (2019)CrossRefGoogle Scholar
  10. 10.
    Cui, X., Xue, Z., Pei, Y., Fang, D.: Preliminary study on ductile fracture of imperfect lattice materials. Int. J. Solids Struct. 48, 3453–3461 (2011)CrossRefGoogle Scholar
  11. 11.
    Ghosh, S., Lee, K., Raghavan, P.: A multi-level computational model for multi-scale damage analysis in composite and porous materials. Int. J. Solids Struct. 38, 2335–2385 (2001)CrossRefGoogle Scholar
  12. 12.
    Gotlib, V.A., Sato, T., Beltzer, A.: Neural computing of effective properties of random composite materials. Comput. Struct. 79, 1–6 (2001)CrossRefGoogle Scholar
  13. 13.
    Grenestedt, J.: Influence of imperfections on effective properties of cellular solids. In: Materials Research Society Symposium-Proceedings, vol. 521, pp. 3–13 (1998)Google Scholar
  14. 14.
    Grenestedt, J.L.: On interactions between imperfections in cellular solids. J. Mater. Sci. 40, 5853–5857 (2005)CrossRefGoogle Scholar
  15. 15.
    Grigorovitch, M., Gal, E.: The local response in structures using the embedded unit cell approach. Comput. Struct. 157, 189–200 (2015)CrossRefGoogle Scholar
  16. 16.
    Grigorovitch, M., Gal, E.: Homogeniztion of non-periodic zones in periodic domains using the embedded unit cell approach. Comput. Struct. 179, 95–108 (2017)CrossRefGoogle Scholar
  17. 17.
    Guinovart-Daz, R., Rodriguez-Ramos, R., Bravo-Castillero, J., Lopez-Realpozo, J.C., Sabina, F.J., Sevostianov, I.: Effective elastic properties of a periodic fiber reinforced composite with parallelogram-like arrangement of fibers and imperfect contact between matrix and fibers. Int. J. Solids Struct. 50, 2022–2032 (2013)CrossRefGoogle Scholar
  18. 18.
    Herrmann, H., Hansen, A., Roux, S.: Fracture of disordered, elastic lattices in two dimensions. Phys. Rev. B 39, 637–648 (1989)CrossRefGoogle Scholar
  19. 19.
    Jiang, W., Zhong, R., Qin, H.Q., Tong, Y.: Homogenized finite element analysis on effective elastoplastic mechanical behaviors of composite with imperfect interfaces. Int. J. Mol. Sci. 15, 23389–23407 (2014)CrossRefGoogle Scholar
  20. 20.
    Laws, N.: Composite materials: theory vs. experiment. J. Compos. Mater. 22, 396–400 (1988)CrossRefGoogle Scholar
  21. 21.
    Liang, S., Wei, Y., Wu, Z.: Multiscale modeling elastic properties of cement-based materials considering imperfect interface effect. Constr. Build. Mater. 154, 567–579 (2017)CrossRefGoogle Scholar
  22. 22.
    Liu, L., Kamm, P., Garcia-Moreno, F., Banhart, J., Pasini, D.: Elastic and failure response of imperfect three-dimensional metallic lattices: the role of geometric defects induced by Selective Laser Melting. J. Mech. Phys. Solids 107, 160–184 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Milton, G.W.: The Theory of Composites. Cambridge University Press, New York (2002)CrossRefGoogle Scholar
  24. 24.
    Nie, S., Basaran, C.: A micromechanical model for effective elastic properties of particulate composites with imperfect interfacial bonds. Int. J. Solids Struct. 42, 4179–4191 (2005)CrossRefGoogle Scholar
  25. 25.
    Postma, G.W.: Wave propagation in stratified medium. Geophysics 20, 780–806 (1955)CrossRefGoogle Scholar
  26. 26.
    Romijn, N., Fleck, N.A.: The fracture toughness of planar lattices: imperfection sensitivity. J. Mech. Phys. Solids 55, 2538–2564 (2007)CrossRefGoogle Scholar
  27. 27.
    Ryvkin, M.: Employing the discrete Fourier transform in the analysis of multiscale problems. Int. J. Multiscale Comput. Eng. 6, 435–449 (2008)CrossRefGoogle Scholar
  28. 28.
    Ryvkin, M., Aboudi, J.: Stress redistribution due to cracking in periodically layered composites. Eng. Fract. Mech. 93, 225238 (2012)CrossRefGoogle Scholar
  29. 29.
    Ryvkin, M., Hadar, O.: Employing the discrete Fourier transform for evaluation of crack-tip field in periodic materials. Int. J. Eng. Sci. 86, 10–19 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Ryvkin, M., Nuller, B.: Solution of quasi-periodic fracture problems by representative cell method. Comput. Mech. 20, 145–149 (1997)CrossRefGoogle Scholar
  31. 31.
    Schmidt, I., Fleck, N.A.: Ductile fracture of two-dimensional cellular structures. Int. J. Fract. 111, 327–342 (2001)CrossRefGoogle Scholar
  32. 32.
    Sevostianov, I., Rodriguez-Ramos, R., Guinovart-Diaz, R., Bravo-Castillero, J., Sabina, F.J.: Connections between different models describing imperfect interfaces in periodic fiber-reinforced composites. Int. J. Solids Struct. 49, 1518–1525 (2012)CrossRefGoogle Scholar
  33. 33.
    Symons, D.D., Fleck, N.A.: The imperfection sensitivity of isotropic two-dimensional elastic lattices. J. Appl. Mech. 75, 051011-1-8 (2008)CrossRefGoogle Scholar
  34. 34.
    Talreja, R., Singh, C.V.: Damage and Failure in Composite Materials. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  35. 35.
    Tandon, G.P., Weng, G.J.: Average stress in the matrix and effective moduli of randomly oriented composites. Compos. Sci. Technol. 27, 111–132 (1986)CrossRefGoogle Scholar
  36. 36.
    Taya, M., Arsenault, R.J.: Metal Matrix Composites. Pergamon, Oxford (1989)Google Scholar
  37. 37.
    Zeman, J., Sejnhola, M.: Homogenization of balanced plain weave composites with imperfect microstructure: part I. Theoretical formulation. Int. J. Solids Struct. 41, 6549–6571 (2004)CrossRefGoogle Scholar
  38. 38.
    Zohdi, T.I., Kachanov, M., Sevostianov, I.: On perfectly plastic flow in porous material. Int. J. Plast. 18, 1649–1659 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of Engineering, School of Mechanical EngineeringTel Aviv UniversityRamat AvivIsrael

Personalised recommendations