Skip to main content
SpringerLink
Log in

Effective Field Method in the Theory of Heterogeneous Media

  • Chapter
  • First Online:
Effective Properties of Heterogeneous Materials

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 193))

Abstract

The work is devoted to the effective field method and its application in the theory of heterogenous materials. For many years, various versions of the method have been used for the calculation of effective physical and mechanical properties of composite materials (the homogenization problem). In the historical survey, the most important steps of the development of the method are indicated starting from nineteenth century. The main attention is focused on the combination of the effective field and numerical methods that yields efficient numerical algorithms for the calculation of effective properties and detailed fields in periodic and random composite materials. Examples of the application of the method to prediction of conductive, elastic, and elasto-plastic properties of composites are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Benveniste Y (1977) A new approach to the application of the Mori-Tanaka theory in composite materials. Mech Mater 6:147–157

    Google Scholar 

  2. Benveniste Y, Dvorak G, Chen T (1991) On diagonal and elastic symmetry of the approximate effective stiffness tensor of heterogeneous media. J Mech Phys Solids 39:927–946

    Article  MathSciNet  MATH  Google Scholar 

  3. Berriman J, Berge P (1996) Critique of two explicit schemes for estimating elastic properties of multiphase composites. Mech Mater 22:149–164

    Article  Google Scholar 

  4. Eshelby J (1957) The determination of the elastic field of an elliptical inclusion, and related problems. Proc R Soc Lond A241:376–396

    MathSciNet  Google Scholar 

  5. Faraday M (1838) Experimental researches on electricity. Philos Trans R Soc Lond II Ser. 1ff

    Google Scholar 

  6. Ferrary M (1991) Asymmetry and the high concentration limit of the mori-tanaka effective medium theory. Mech Mater 11:251–256

    Article  Google Scholar 

  7. Foldy L (1945) The multiple scattering of waves. Phys Rev 67:107–119

    Article  MathSciNet  MATH  Google Scholar 

  8. Fricke H (1924) A mathematical treatment of the electric conductivity and capacity of disperse systems. I. The electric conductivity and capacity of disperse systems. Phys Rev 24:575–587

    Google Scholar 

  9. Gonzalez C, Segurado J, Llorca J (2004) Numerical simulation of elasto-plastic deformation of composites: evolution of stress microfields and implications for homogenization models. J Mech Phys Solids 52(7):1573–1593

    Article  MATH  Google Scholar 

  10. Gusev A (1999) Representative volume element size for elastic composites: a numerical study. J Mech Phys Solids 45:1449–1459

    Article  Google Scholar 

  11. Golub G, Van Loan C (1993) Matrix computations. Johns Hopkins University Press, Baltimore

    Google Scholar 

  12. Hartree DR (1957) The calculation of atomic structures. Wiley, Newyork

    MATH  Google Scholar 

  13. Kachanov LM (2004) Fundamentals of the theory of plasticity. Dover Publications, New York

    Google Scholar 

  14. Kanaun S (1975) The method of self-consistent field in the problem of effective properties of composites. J Appl Mech Tech Phys N4:191–203

    Google Scholar 

  15. Kanaun S (1977) The approximation of self-consistent field for composite elastic media. J Appl Mech Tech Phys N2:160–169

    Google Scholar 

  16. Kanaun S. (2009) Fast calculation of elastic fields in a homogeneous medium with isolated heterogeneous inclusions. Int J Multiscale Comput Eng 7(4):263–276

    Google Scholar 

  17. Kanaun S (2011) An efficient numerical method for calculation of elastic and thermo-elastic fields in a homogeneous media with several heterogeneous inclusions. World J Mech 1(2):31–43

    Article  Google Scholar 

  18. Kanaun S (2011) Calculation of electro and thermo static fields in matrix composite materials of regular or random microstructures. Int J Eng Sci 49:41–60

    Article  MathSciNet  Google Scholar 

  19. Kanaun S (2012) An efficient homogenization method for composite materials with elasto-plastic components. J Eng Sci 57:36–49

    Article  MathSciNet  Google Scholar 

  20. Kanaun S, Jeulin D (2001) Elastic properties of hybrid composites by the effective field approach. J Mech Phys Solids 49:2339–2367

    Article  MATH  Google Scholar 

  21. Kanaun S, Levin V (2008a) Self-consistent methods for composites. V.I, Static problems. Springer, Dortrecht

    Google Scholar 

  22. Kanaun S, Levin V (2008b) Self-consistent methods for composites. V.II, Wave propagation in heterogeneous materials. Springer, Dortrecht

    Google Scholar 

  23. Kanaun S, Martinez R (2012) Numerical solution of the integral equations of elasto-plasticity for a homogeneous medium with several heterogeneous inclusions. Comput Mater Sci 55:147–156

    Article  Google Scholar 

  24. Kanaun S, Pervago E (2011) Combining self-consistent and numerical methods for the calculation of elastic fields and effective properties of 3D-matrix composites with periodic and random microstructures. Int J Eng Sci 49(5):420–442

    Article  MathSciNet  MATH  Google Scholar 

  25. Kunin I (1965) Methods of tensor analysis in the theory of dislocation. US Department of Commerce, Clearing House for Fed. Sci. Techn. Inform, Springfield, VA 22151

    Google Scholar 

  26. Kunin I (1983) The theory of elastic media with microstructure II. Springer, Berlin

    Book  Google Scholar 

  27. Kushch VI (1997) Conductivity of a periodic particle composite with transversely isotropic phases. Proc R Soc Lond A 453:65–76

    Article  MATH  Google Scholar 

  28. Kuster G, Toksőz M (1974) Velocity and attenuation of seismic waves in two-phase media. I. Theoretical formulations. II. Experimental results. Geophysics 39:587–618

    Google Scholar 

  29. Lax M (1952) Multiple scattering of waves. II. The effective field in dense systems. Rev Mod Phys 85:621-629

    Google Scholar 

  30. Landauer R (1978) Electrical conductivity in inhomogeneous media, In: Proceedings of conference on electrical transport and optical properties of inhomogeneous media (ETOPIM 1), pp 2–46

    Google Scholar 

  31. Levin V (1976) On the determination of elastic and thermoelastic constants of composite materials. Proc Acad Sci USSR Mech Solids N1:88–93

    Google Scholar 

  32. Levin V (1977) On the stress concentration on inclusions in composite materials. Appl Math Mech (PMM) 41:136–145

    Article  Google Scholar 

  33. Lorenz L (1880) Űber die Refractionskonstante. Ann Phys Chem 11:70ff

    Google Scholar 

  34. Markov K (2001) Elementary micromechanics of heterogeneous media. In: Markov K, Preziosi L (eds) Heterogeneous media. Micromechanics modeling methods and simulations. Birkhauser, Boston, pp 1–162

    Google Scholar 

  35. Maz’ya V, Schmidt G (2007) Approximate approximation, mathematical surveys and monographs vol 141. American Mathematical Society, Providence

    Google Scholar 

  36. Maxwell J (1954) A treatise on electricity and magnetism, 3rd edn. Dover, New York (Republication of 1891)

    Google Scholar 

  37. McPhedran RC, McKenzie DR (1978) The conductivity of lattice of spheres. I. The simple cubic lattice. Proc R Soc Lond A 359:45–63

    Google Scholar 

  38. Mikhlin S (1965) Multidimensional singular integrals and integral equations. Pergamon Press, Oxford

    Google Scholar 

  39. Mori T, Tanaka K (1973) Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 21:571–574

    Google Scholar 

  40. Mossotti O (1850) Discussione analitica sul’influenza che l’azione di un mezzo dieléttrrico he sulla distribuzione dell’elettricitá alla superficie di piú corpi elettrici dissemenati in eso. Mem Mat Fis dilla Soc Ital di Sci Modena 24:49–74

    Google Scholar 

  41. Peterson AF, Ray SL, Mittra R (1997) Computational methods for electromagnetics. IEEE Press, New York

    Google Scholar 

  42. Poisson S (1824) Mémoire sur la théorie du magnétisme. Mem de l’Acad R de France V:247–338

    Google Scholar 

  43. Press W, Flannery B, Teukolsky S, Vetterling W (1992) Numerical recipes in FORTRAN: the art of scientific computing, 2nd edn. Cambridge University Press, Newyork

    Google Scholar 

  44. Prochorov Yu, Rosanov Yu (1969) Probability theory: basic concepts, limit theorems, random processes. Springer, Berlin

    Google Scholar 

  45. Qui Y, Weng G (1990) On the application of the Mori-Tanaka theory involving transversely isotropic spheroidal inclusions. Int J Eng Sci 28:1121–1137

    Google Scholar 

  46. Segurado J, Llorca J (2002) A numerical approximation to the elastic properties of sphere-reinforced composites. J Mech Phys Solids 50:2107–2121

    Google Scholar 

  47. Slater JC (1974) The self-consistent fields for molecules and solids. McGrow-Hill, New York

    Google Scholar 

  48. Stanley HE (1971) Introduction to phase transition and critical phenomena. Oxford University Press, Oxford

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technological Institute of Higher Education of Monterrey, State Mexico Campus, México, USA

    S. Kanaun

  2. Mexican Oil Institute, México DF, México, USA

    V. Levin

Authors
  1. S. Kanaun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. Levin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Kanaun .

Editor information

Editors and Affiliations

  1. , Department of Mechanical Engineering, Tufts University, College Ave 200, Medford, 02155, Massachusetts, USA

    Mark Kachanov

  2. College of Engineering, Dept. of Mechanical & Aerospace Eng., New Mexico State University, Las Cruces, 88003-8001, New Mexico, USA

    Igor Sevostianov

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Kanaun, S., Levin, V. (2013). Effective Field Method in the Theory of Heterogeneous Media. In: Kachanov, M., Sevostianov, I. (eds) Effective Properties of Heterogeneous Materials. Solid Mechanics and Its Applications, vol 193. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5715-8_3

Download citation

  • DOI: https://doi.org/10.1007/978-94-007-5715-8_3

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-5714-1

  • Online ISBN: 978-94-007-5715-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation