Computational Mechanics

, Volume 40, Issue 2, pp 281–298 | Cite as

A numerical method for homogenization in non-linear elasticity

Original Paper

Abstract

In this work, homogenization of heterogeneous materials in the context of elasticity is addressed, where the effective constitutive behavior of a heterogeneous material is sought. Both linear and non-linear elastic regimes are considered. Central to the homogenization process is the identification of a statistically representative volume element (RVE) for the heterogeneous material. In the linear regime, aspects of this identification is investigated and a numerical scheme is introduced to determine the RVE size. The approach followed in the linear regime is extended to the non-linear regime by introducing stress–strain state characterization parameters. Next, the concept of a material map, where one identifies the constitutive behavior of a material in a discrete sense, is discussed together with its implementation in the finite element method. The homogenization of the non-linearly elastic heterogeneous material is then realized through the computation of its effective material map using a numerically identified RVE. It is shown that the use of material maps for the macroscopic analysis of heterogeneous structures leads to significant reductions in computation time.

Keywords

Homogenization Elasticity Micromechanics 

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References

  1. 1.
    Aboudi J (1991) Mechanics of composite materials: a unified micromechanical approach. Elsevier, AmsterdamMATHGoogle Scholar
  2. 2.
    Brieu M, Devries F (1999) Micro-mechanical approach and algorithm for the study of damage appearance in elastomer composites. Comp Struct 46:309–319CrossRefGoogle Scholar
  3. 3.
    Castañeda PP (1989) The overall constitutive behavior of nonlinearly elastic composites. Proc R Soc Lond A 422(1862):147–171MATHGoogle Scholar
  4. 4.
    Christensen RM (1991) Mechanics of composite materials. Krieger, New YorkGoogle Scholar
  5. 5.
    Ciarlet PG (1993) Mathematical elasticity: three dimensional elasticity. North-Holland, AmsterdamGoogle Scholar
  6. 6.
    Cioranescu D, Donato P (1998) An introduction to homogenization. Oxford University Press, New YorkGoogle Scholar
  7. 7.
    Fung YC (1993) Biomechanics: mechanical properties of living tissues, 2nd edn. Springer, Berlin Heidelberg New YorkGoogle Scholar
  8. 8.
    Hashin Z (1983) Analysis of composite materials. J Appl Mech 50:481–505MATHCrossRefGoogle Scholar
  9. 9.
    Hill R (1972) On constitutive macro-variables for heterogeneous solids at finite strain. Proc R Soc Lond A 326(1565):131–147MATHCrossRefGoogle Scholar
  10. 10.
    Hill R, Rice JR (1973) Elastic potentials and the structure of inelastic constitutive laws. SIAM J Appl Math 25(3):448–461CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Huet C (1990) Application of variational concepts to size effects in elastic heterogeneous bodies. J Mech Phys Solids 38(6):813–841CrossRefMathSciNetGoogle Scholar
  12. 12.
    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 40:3647–3679MATHCrossRefGoogle Scholar
  13. 13.
    Kouznetsova V, Brekelmans WAM, Baaijens FPT (2001) An approach to micro–macro modeling of heterogeneous materials. Comput Mech 27:37–48MATHCrossRefGoogle Scholar
  14. 14.
    Liu I-S (2002) Continuum mechanics. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  15. 15.
    Miehe C, Schotte J, Schröder J (1999) Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Comput Materials Sci 16:372–382CrossRefGoogle Scholar
  16. 16.
    Miehe C, Schröder J, Becker M (2002) Computational homogenization analysis in finite elasticity: material and structural instabilities on the micro- and macro-scales of periodic composites and their interaction. Comput Methods Appl Mech Eng 191:4971–5005MATHCrossRefGoogle Scholar
  17. 17.
    Müller S (1987) Homogenization of nonconvex integral functionals and cellular elastic materials. Arch Ration Mech Anal 25:189–212Google Scholar
  18. 18.
    Mura T (1987) Micromechanics of defects in solids. Martinus Nijhoff, The HagueGoogle Scholar
  19. 19.
    Nemat-Nasser S (2004) Plasticity: a treatise on finite deformation of heterogeneous inelastic materials. Cambridge Press, Cambridge.MATHGoogle Scholar
  20. 20.
    Nemat-Nasser S, Hori M (1999) Micromechanics: overall properties of heterogeneous materials, 2nd edn. North-Holland, AmsterdamGoogle Scholar
  21. 21.
    Ogden RW (1978) Extremum principles in non-linear elasticity and their application to composites–i. Int J Solids Struct 14:265–282MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Saiki I, Terada K, Ikeda K, Hori M (2002) Appropriate number of unit cells in a representative volume element for micro-structural bifurcation encountered in a multi-scale modeling. Comput Methods Appl Mech Eng 191:2561–2585MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Stroeven M, Askes H, Sluys LJ (2004) Numerical determination of representative volumes for granular materials. Comput Methods Appl Mech Eng 193:3221–3238MATHCrossRefGoogle Scholar
  24. 24.
    Sugimoto T (2001) Monodispersed particles. Elsevier, AmsterdamGoogle Scholar
  25. 25.
    Talbot DRS, Willis JR (1985) Variational principles for inhomogeneous non-linear media. IMA J Appl Math 35:39–54MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Torquato S (2002) Random heterogeneous materials: microstructure and macroscopic properties. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  27. 27.
    Willis JR (1994) Upper and lower bounds for non-linear composite behavior. Materials Sci Eng A175:7–14CrossRefGoogle Scholar
  28. 28.
    Zohdi TI, Wriggers P (2005) Introduction to computational micromechanics. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  29. 29.
    Zohdi T, Wriggers P, Huet C (2001) A method of substructuring large-scale computational micromechanical problems. Comput Methods Appl Mech Eng 190:5636–5656Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of CaliforniaBerkeleyUSA

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