Conductivity of Fibre Composites: Analytical Homogenization Approach

  • Igor V. AndrianovEmail author
  • Jan AwrejcewiczEmail author
  • Vladyslav V. DanishevskyyEmail author
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 77)


Application of the multi-scale asymptotic homogenization method allowed us to separate global and local components of the solution and to reduce the input boundary value problem in a multi-connected domain to a recurrent sequence of local problems, considered within a representative unit cell of the composite structure.


  1. 1.
    Bakhvalov, N., and G. Panasenko. 1989. Averaging processes in periodic media. Mathematical Problems in Mechanics of Composite Materials. Dordrecht: Kluwer.Google Scholar
  2. 2.
    Perrins, W.T., D.R. McKenzie, and R.C. McPhedran. 1979. Transport properties of regular arrays of cylinders. Proceedings of the Royal Society London A 369: 207–225.Google Scholar
  3. 3.
    Perrins, W.T., and R.C. McPhedran. 2010. Metamaterials and the homogenization of composite materials. Metamaterials 4: 24–31.Google Scholar
  4. 4.
    Vanin, G.A. 1985. Micromechanics of composite materials. Kyiv (in Russian): Naukova Dumka.Google Scholar
  5. 5.
    Jiang, C.P., Y.L. Xu, Y.K. Cheung, and S.H. Lo. 2004. A rigorous analytical method for doubly periodic cylindrical inclusions under longitudinal shear and its application. Mechanics of Materials 36: 225–237.Google Scholar
  6. 6.
    Hashin, Z. 2002. Thin interphase/imperfect interface in elasticity with application to coated fiber composites. Journal of the Mechanics and Physics of Solids 50: 2509–2537.Google Scholar
  7. 7.
    Keller, J.B. 1964. A theorem on the conductivity of a composite medium. Journal of Mathematical Physics 5: 548–549.Google Scholar
  8. 8.
    Kozlov, G.M. 1989. Geometrical aspects of averaging. Russian Mathematical Surveys 44 (2): 91–144.Google Scholar
  9. 9.
    Berlyand, L.V., and V. Mityushev. 2001. Generalized Clausius-Mossotti formula for random composite with circular fibres. Journal of Statistical Physics 102: 115–145.Google Scholar
  10. 10.
    Berlyand, L., and V. Mityushev. 2005. Increase and decrease of the effective conductivity of two phase composites due to polydispersity. Journal of Statistical Physics 118: 481–509.Google Scholar
  11. 11.
    Keller, J.B., L.A. Rubenfeld, and L.A. Molyneux. 1967. Extremum principles for slow viscous flows with applications to suspensions. Journal of Fluid Mechanics. 30: 97–125.Google Scholar
  12. 12.
    Rubenfeld, L.A., and J.B. Keller. 1969. Bounds on the elastic moduli of composite media. SIAM Journal on Applied Mathematics 17: 495–510.Google Scholar
  13. 13.
    Torquato, S. 2002. Random heterogeneous materials. Microstructure and macroscopic properties. New York: Springer.Google Scholar
  14. 14.
    Torquato, S., and J. Rubinshtein. 1991. Improved bounds on the effective conductivity of high-contrast suspensions. Journal of Applied Physics 69: 7118–7125.Google Scholar
  15. 15.
    Hashin, Z., and S. Shtrikman. 1962. A variational approach to the theory of the effective magnetic permeability of multiphase materials. Journal of Applied Physics 33: 1514–1517.Google Scholar
  16. 16.
    Hashin, Z., and S. Shtrikman. 1963. A variational approach to the theory of the elastic behaviour of multiphase materials. Journal of the Mechanics Physics of Solids 11: 127–140.Google Scholar
  17. 17.
    Dykhne, A.M. 1970. Conductivity of a two-dimensional system. Soviet Physics JETP 32: 63–65.Google Scholar
  18. 18.
    Christensen, R.M. 2005. Mechanics of composite materials. Mineola, New York: Dover Publications.Google Scholar
  19. 19.
    Lenci, S. 1999. Bonded joints with nonhomogeneous adhesives. Journal of Elasticity 53: 23–35.Google Scholar
  20. 20.
    Mbanefo, U., and R.A. Westmann. 1990. Axisymmetric stress analysis of a broken, debonded fiber. Journal of Applied Mechanics 57: 654–660.Google Scholar
  21. 21.
    Abramowitz, M., and I.A. Stegun (eds.). 1965. Handbook of mathematical functions, with formulas, graphs, and mathematical tables. New York: Dover Publications.Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institut für Allgemeine MechanikRWTH Aachen UniversityAachenGermany
  2. 2.Automation, Biomechanics and MechatronicsLodz University of TechnologyŁódźPoland
  3. 3.School of Computing and MathematicsKeele UniversityKeeleUK

Personalised recommendations