Skip to main content
SpringerLink
Log in
Book cover

Computational Homogenization of Heterogeneous Materials with Finite Elements pp 29–51Cite as

Conduction Properties

  • Chapter
  • First Online:

  • 1403 Accesses

Part of the book series: Solid Mechanics and Its Applications ((SMIA,volume 258))

Abstract

The objective of this chapter is to present the different basic concepts of computational homogenization through the simplest problem: defining the effective conductivity of a heterogeneous medium in steady-state regime. First, the notion of RVE is introduced. Then, the localization problems and the effective quantities are defined, and the numerical procedures using FEM to compute the effective conductivity tensor are presented.

This is a preview of subscription content, 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   159.99
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

Learn about institutional subscriptions

References

  1. Torquato S (2001) Random heterogeneous materials: microstructure and macroscopic properties. Springer, Berlin

    MATH  Google Scholar 

  2. Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40(13–14):3647–3679

    Article  Google Scholar 

  3. Gitman IM, Askes H, Sluys LJ (2007) Representative volume: existence and size determination. Eng Fract Mech 74(16):2518–2534

    Article  Google Scholar 

  4. Monetto I, Drugan WJ (2009) A micromechanics-based non local constitutive equation and minimum RVE size estimates for random elastic composites containing aligned spheroidal heterogeneities. J Mech Phys Solids 57:1578–1595

    Article  MathSciNet  Google Scholar 

  5. Pensée V, He Q-C (2007) Generalized self-consistent estimation of the apparent isotropic elastic moduli and minimum representative volume element size of heterogeneous media. Int J Solids Struct 44(7):2225–2243

    Article  Google Scholar 

  6. Drugan WJ, Willis JR (1996) A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. J Mech Phys Solids 44(4):497–524

    Article  MathSciNet  Google Scholar 

  7. Bulsara V, Talreja R, Qu J (1999) Damage initiation under transverse loading of unidirectional composites with arbitrarily distributed fibers. Compos Sci Technol 59:673–682

    Article  Google Scholar 

  8. Gonzalez C, LLorca J, (2007) Mechanical behavior of unidirectional fiber-reinforced polymers under transverse compression: microscopic mechanisms and modeling. Compos Sci Technol 67(13):2795–2806

    Article  Google Scholar 

  9. Rakow JF, Waas AM (2004) Size effects in metal foam cores for sandwich structures. AIAA J 42:7

    Article  Google Scholar 

  10. Swaminathan S, Ghosh S (2006) Statistically equivalent representative volume elements for unidirectional composite microstructures: II. with interfacial debonding. J Compos Mater 49:605–621

    Article  Google Scholar 

  11. Swaminathan S, Ghosh S, Pagano NJ (2006) Statistically equivalent representative volume elements for unidirectional composite microstructures: I. without damage. J Compos Mater 40:583–604

    Article  Google Scholar 

  12. Ranganathan IS, Ostoja-Starzewski M (2008) Scaling function, anisotropy and the size of RVE in elastic random polycrystals. J Mech Phys Solids 56:2773–2791

    Article  MathSciNet  Google Scholar 

  13. Elvin A, Sunder SS (1996) Microcracking due to grain boundary sliding in polycrystalline ice under uniaxial compression. Acta Mater 44(1):43–56

    Article  Google Scholar 

  14. Gusev AA (1997) Representative volume element size for elastic composites: a numerical study. J Mech Phys Solids 45:1449–1459

    Article  Google Scholar 

  15. Ostoja-Starzewski M (2006) Material spatial randomness: from statistical to representative volume element. Probabilistic Eng Mech 21:112–132

    Article  Google Scholar 

  16. Ren ZY, Zheng QS (2002) A quantitative study of minimum sizes of representative volume elements of cubic polycrystals - numerical experiments. J Mech Phys Solids 50:881–893

    Article  Google Scholar 

  17. Zhodi TI, Wriggers P (2000) On the sensitivity of homogenized material responses at infinitesimal and finite strains. Commun Numer Methods Eng 16:657–670

    Article  Google Scholar 

  18. Du X, Ostoja-Starzewski M (2006) On the size of representative volume element for darcy law in random media. Proc R Soc A 462:2949–2963

    Article  MathSciNet  Google Scholar 

  19. Grimal Q, Raum K, Gerisch A, Laugier P (2011) A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties. Biomech Model Mechanobiol 10:925–937

    Article  Google Scholar 

  20. Valavala PK, Odegard GM, Aifantis EC (2009) Influence of representative volume element size on predicted elastic properties of polymer materials. Model Simul Mater Sci Eng 17:045004

    Article  Google Scholar 

  21. Salmi M, Auslender F, Bornert M, Fogli M (2012) Various estimates of representative volume element sizes based on a statistical analysis of the apparent behavior of random linear composites. Comptes Rendus de Mécanique 340:230–246

    Article  Google Scholar 

  22. Soize C (2008) Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size. Probabilistic Eng Mech 23:307–323

    Article  Google Scholar 

  23. Sab K, Nedjar B (2005) Periodization of random media and representative volume element size for linear composites. Comptes Rendus de Mécanique 333:187–195

    Article  Google Scholar 

  24. Hoang TH, Guerich M, Yvonnet J (2016) Determining the size of rve for nonlinear random composites in an incremental computational homogenization framework. J Eng Mech 142(5):04016018

    Article  Google Scholar 

  25. Huet C (1990) Application of variational concepts to size effects in elastic heterogeneous bodies. J Mech Phys Solids 38(6):813–841

    Article  MathSciNet  Google Scholar 

  26. Yvonnet J, He Q-C, Zhu Q-Z, Shao J-F (2011) A general and efficient computational procedure for modelling the Kapitza thermal resistance based on XFEM. Comput Mater Sci 50(4):1220–1224

    Article  Google Scholar 

  27. Yvonnet J, He Q-C, Toulemonde C (2008) Numerical modelling of the effective conductivities of composites with arbitrarily shaped inclusions and highly conducting interface. Compos Sci Technol 68(13):2818–2825

    Article  Google Scholar 

  28. Le-Quang H, Bonnet G, He Q-C (2010) Size-dependent Eshelby tensor fields and effective conductivity of composites made of anisotropic phases with highly conducting imperfect interfaces. Phys Rev B 81(6): 064203

    Google Scholar 

  29. Bertsekas PD (2014) Constrained optimization and Lagrange multiplier methods. Academic Press, New York

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. MSME Laboratory, Université Paris-Est Marne-la-Vallée, Marne-la-Vallée Cedex2, France

    Julien Yvonnet

Authors
  1. Julien Yvonnet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julien Yvonnet .

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Yvonnet, J. (2019). Conduction Properties. In: Computational Homogenization of Heterogeneous Materials with Finite Elements. Solid Mechanics and Its Applications, vol 258. Springer, Cham. https://doi.org/10.1007/978-3-030-18383-7_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-18383-7_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-18382-0

  • Online ISBN: 978-3-030-18383-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation