Advertisement

Acta Mechanica

, Volume 230, Issue 10, pp 3543–3554 | Cite as

Derivation of a mesoscopic model for nonlinear particle-reinforced composites from a fully microscopic model

  • Asatur Zh. KhurshudyanEmail author
Original Paper
  • 30 Downloads

Abstract

Particle-reinforced composites (PRCs) are usually studied by some averaging or homogenization techniques. In this, the effective properties are derived by assuming that particles are dispersed within composites according to some given (probabilistic) distribution. Such approaches restrain the possibilities of studying the contribution of exact location and parameters of individual particles to the overall behavior of composites. In this paper, we attempt to fill this gap by deriving the mesoscopic model of such composites corresponding to a continuum with point inhomogeneities. We start from a fully microscopic model where the composite is regarded as a continuum with spherical inclusions. Letting the diameter of inclusions decrease to zero, material parameters of the composite are represented in terms of the Dirac distribution. The Mindlin–Reissner–von Kármán thick plate theory is considered as a particular case, and closed-form formulas are obtained for the plate stiffness coefficients. Numerical analysis of a thick composite plate reinforced over its mid-surface justifies the theoretical derivations.

Notes

Acknowledgements

The support of the State Administration of Foreign Expert Affairs of China is thankfully acknowledged. Valuable remarks of unknown reviewers allowed to improve the presentation of the results substantially.

References

  1. 1.
    German, R.M.: Particulate Composites: Fundamentals and Applications. Springer, Basel (2016)CrossRefGoogle Scholar
  2. 2.
    Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New York (2002)CrossRefGoogle Scholar
  3. 3.
    Lewiński, T., Telega, J.J.: Plates, Laminates and Shells: Asymptotic Analysis and Homogenization. World Scientific Publishing, Singapore (2000)CrossRefGoogle Scholar
  4. 4.
    Kamiński, M.: Computational Mechanics of Composite Materials. Sensitivity, Randomness and Multiscale Behaviour. Springer, London (2005)Google Scholar
  5. 5.
    Böhm, H.J., Eckschlager, A., Han, W.: Multi-inclusion unit cell models for metal matrix composites with randomly oriented discontinuous reinforcements. Computat. Mater. Sci. 25(1–2), 42–53 (2002)CrossRefGoogle Scholar
  6. 6.
    Nazarenko, L., Stolarski, H.: Energy-based definition of equivalent inhomogeneity for various interphase models and analysis of effective properties of particulate composites. Compos. Part B 94, 82–94 (2016)CrossRefGoogle Scholar
  7. 7.
    Nazarenko, L., Bargmann, S., Stolarski, H.: Closed-form formulas for the effective properties of random particulate nanocomposites with complete Gurtin–Murdoch model of material surfaces. Contin. Mech. Thermodyn. 29(1), 77–96 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Nazarenko, L., Stolarski, H., Altenbach, H.: Effective properties of particulate composites with surface-varying interphases. Compos. Part B 149, 268–284 (2018)CrossRefGoogle Scholar
  9. 9.
    Nazarenko, L., Stolarski, H., Khoroshun, L., Altenbach, H.: Effective thermo-elastic properties of random composites with orthotropic components and aligned ellipsoidal inhomogeneities. Int. J. Solids Struct. 136–137, 220–240 (2018)CrossRefGoogle Scholar
  10. 10.
    Picu, C.R., Sorohan, S., Soare, M.A., Constantinescu, D.M.: Designing particulate composites: the effect of variability of filler properties and filler spatial distribution. In: Trovalusci, P. (ed.) Materials with Internal Structure Multiscale and Multifield Modeling and Simulation, pp. 89–108. Springer, Berlin (2016)Google Scholar
  11. 11.
    Nazarenko, L., Chirkov, A.Y., Stolarski, H., Altenbach, H.: On modeling of carbon nanotubes reinforced materials and on influence of carbon nanotubes spatial distribution on mechanical behavior of structural elements. Int. J. Eng. Sci. 143, 1–13 (2019)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Berger, H., et al.: An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites. Int. J. Solids Struct. 42(21–22), 5692–5714 (2005)CrossRefGoogle Scholar
  13. 13.
    Lin, Ch-H, Muliana, A.: Micromechanics models for the effective nonlinear electro-mechanical responses of piezoelectric composites. Acta Mech. 224(7), 1471–1492 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ju, J.W., Chen, T.M.: Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities. Acta Mech. 103, 103–121 (1994)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Tong, J., Nan, C.-W., Fu, J., Guan, X.: Effect of inclusion shape on the effective elastic moduli for composites with imperfect interface. Acta Mech. 146(3–4), 127–134 (2001)CrossRefGoogle Scholar
  16. 16.
    Nazarenko, L., Stolarski, H., Altenbach, H.: A definition of equivalent inhomogeneity applicable to various interphase models and various shapes of inhomogeneity. Procedia IUTAM 21, 63–70 (2017)CrossRefGoogle Scholar
  17. 17.
    Kushnevsky, V., Morachkovsky, O., Altenbach, H.: Identification of effective properties of particle reinforced composite materials. Comput. Mech. 22(4), 317–325 (1998)CrossRefGoogle Scholar
  18. 18.
    Altenbach, H.: Modelling of anisotropic behavior in fiber and particle reinforced composites. In: Sadowski, T. (ed.) Multiscale Modelling of Damage and Fracture Processes in Composite Materials. CISM International Centre for Mechanical Sciences (Courses and Lectures), vol. 474. Springer, Vienna (2005)Google Scholar
  19. 19.
    Altenbach, H., Altenbach, J., Kissing, W.: Mechanics of Composite Structural Elements, 2nd edn. Springer, Singapore (2018)CrossRefGoogle Scholar
  20. 20.
    Muc, A., Barski, M.: Design of particulate-reinforced composite materials. Materials 11, 234 (2018)CrossRefGoogle Scholar
  21. 21.
    Kamiński, M.: Deterministic and probabilistic homogenization limits for particle-reinforced composites with nearly incompressible components. Compos. Struct. 187, 36–47 (2018)CrossRefGoogle Scholar
  22. 22.
    Wriggers, P., Hain, M.: Micro-meso-macro modelling of composite materials. In: Oñate, E., Owen, R. (eds.) Computational Plasticity. Springer, Berlin (2007)Google Scholar
  23. 23.
    Kamiński, M.: Homogenization of particulate and fibrous composites with some non-Gaussian material uncertainties. Compos. Struct. 210, 778–786 (2019)CrossRefGoogle Scholar
  24. 24.
    Sokolovski, D., Kamiński, M.: Homogenization of carbon/polymer composites with anisotropic distribution of particles and stochastic interface defects. Acta Mech. 229, 3727–3765 (2018)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Ju, J.W., Yanase, K.: Micromechanics and effective elastic moduli of particle-reinforced composites with near-field particle interactions. Acta Mech. 215, 135–153 (2010)CrossRefGoogle Scholar
  26. 26.
    Tartar, L.: The General Theory of Homogenization. A Personalized Introduction. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  27. 27.
    Nordmann, J., Aßmus, M., Altenbach, H.: Visualising elastic anisotropy—theoretical background and computational implementation. Contin. Mech. Thermodyn. 30(4), 689–708 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Sakata, S., Ashida, F.: Hierarchical stochastic homogenization analysis of a particle reinforced composite material considering non-uniform distribution of microscopic random quantities. Comput. Mech. 48, 529–540 (2011)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations. Springer, Cham (2016)CrossRefGoogle Scholar
  30. 30.
    Mikhlin, S.G.: Error Analysis in Numerical Processes. Wiley, Chichester (1991)zbMATHGoogle Scholar
  31. 31.
    Pica, A., Wood, R.D., Hinton, E.: Finite element analysis of geometrically nonlinear plate behaviour using a Mindlin formulation. Comput. Struct. 11, 203–215 (1980)MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department on Dynamics of Deformable Systems and Coupled Fields, Institute of MechanicsNational Academy of Sciences of ArmeniaYerevanArmenia
  2. 2.Institute of Natural SciencesShanghai Jiao Tong UniversityShanghaiChina

Personalised recommendations