Elasticity and Thermoelasticity

  • Julien YvonnetEmail author
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 258)


In this chapter, the definition and computation of effective properties in the context of linear elasticity are presented. First, the localization problem and the different types of boundary conditions are defined. Then, the definition of the effective elastic fourth-order tensor is introduced. The practical calculation of the effective elastic tensor with 2D and 3D FEM is detailed. An extension to thermoelasticity is described. Finally, reference solutions are provided for validation purpose.


  1. 1.
    Michel J-C, Moulinec H, Suquet P (1999) Effective properties of composite materials with periodic microstructure: a computational approach. Comput Methods Appl Mech Eng 172:109–143MathSciNetCrossRefGoogle Scholar
  2. 2.
    Huet C (1990) Application of variational concepts to size effects in elastic heterogeneous bodies. J Mech Phys Solids 38(6):813–841MathSciNetCrossRefGoogle Scholar
  3. 3.
    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):3647–3679CrossRefGoogle Scholar
  4. 4.
    Yvonnet J, Quang HL, He Q-C (2008) An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites. Comput Mech 42(1):119–131MathSciNetCrossRefGoogle Scholar
  5. 5.
    Nguyen TT, Yvonnet J, Bornert M, Chateau C, Bilteryst F, Steib E (2017) Large-scale simulations of quasi-brittle microcracking in realistic highly heterogeneous microstructures obtained from micro ct imaging. Extrem Mech Lett 17:50–55CrossRefGoogle Scholar
  6. 6.
  7. 7.
    Gmsh software, (2017)
  8. 8.
  9. 9.
    Hashin Z, Shtrikman S (1962) On some variational principles in anisotropic and nonhomogeneous elasticity. J Mech Phys Solids 10(4):335–342MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11(2):127–140MathSciNetCrossRefGoogle Scholar
  11. 11.
    Vaezi M, Seitz H, Yang S (2013) A review on 3D micro-additive manufacturing technologies. Int J Adv Manuf Technol 67(5–8):1721–1754CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.MSME LaboratoryUniversité Paris-Est Marne-la-ValléeMarne-la-Vallée Cedex2France

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