Science China Technological Sciences

, Volume 60, Issue 3, pp 452–458 | Cite as

An anisotropic micromechanical model for calculation of effective elastic moduli of Ni-based single crystal superalloys



An anisotropic micromechanical model based on Mori-Tanaka method is developed to calculate the effective elastic moduli of Ni-based single crystal superalloys. In the micromechanical model, the γ' precipitate with very high volume fraction is regarded as matrix, γ phase is divided into three parts as three different kinds of inclusions, and the actual cubic structure and orthogonal anisotropy properties of γ phase and γ′ precipitate are taken into account. Based on this anisotropic micromechanical model, the effective elastic moduli of Ni-based single crystal superalloys composite materials is obtained, and the influences of volume fraction and elastic constants of γ′ precipitate on the effective elastic moduli are also discussed. The results provide useful information for understanding mechanical behavior of composite materials in Ni-based single crystal superalloys and other anisotropic polygonal inclusion problem.


Ni-based single crystal superalloys effective elastic moduli Mori-Tanaka method anisotropic inclusion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Hershey A V. The elasticity of an isotropic aggregate of anisotropic cubic crystals. J Appl Mech, 1954, 21: 236–241MATHGoogle Scholar
  2. 2.
    Hill R. Continuum micro-mechanics of elastoplastic polycrystals. J Mech Phys Solids, 1965, 13: 89–101CrossRefMATHGoogle Scholar
  3. 3.
    Hill R. A self-consistent mechanics of composite materials. J Mech Phys Solids, 1965, 13: 213–222CrossRefGoogle Scholar
  4. 4.
    Hutchinson J W. Elastic-plastic behaviour of polycrystalline metals and composites. Proc R Soc A-Math Phys Eng Sci, 1970, 319: 247–272CrossRefGoogle Scholar
  5. 5.
    Budiansky B. On the elastic moduli of some heterogeneous materials. J Mech Phys Solids, 1965, 13: 223–227CrossRefGoogle Scholar
  6. 6.
    Weng G J. Some elastic properties of reinforced solids, with special reference to isotropic ones containing spherical inclusions. Int J Eng Sci, 1984, 22: 845–856CrossRefMATHGoogle Scholar
  7. 7.
    Benveniste Y. A new approach to the application of Mori-Tanaka’s theory in composite materials. Mech Mater, 1987, 6: 147–157CrossRefGoogle Scholar
  8. 8.
    Mori T, Tanaka K. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall, 1973, 21: 571–574CrossRefGoogle Scholar
  9. 9.
    Eshelby J D. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc R Soc A-Math Phys Eng Sci, 1957, 241: 376–396MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Eshelby J D. Elastic inclusions and inhomogeneities. Prog Solid Mech, 1961, 2: 89–140MathSciNetGoogle Scholar
  11. 11.
    Wang H, Li Q. Prediction of elastic modulus and Poisson’s ratio for unsaturated concrete. Int J Solids Struct, 2007, 44: 1370–1379CrossRefMATHGoogle Scholar
  12. 12.
    Gong S, Li Z, Zhao Y Y. An extended Mori-Tanaka model for the elastic moduli of porous materials of finite size. Acta Mater, 2011, 59: 6820–6830CrossRefGoogle Scholar
  13. 13.
    Puchi-Cabrera E S, Staia M H, Iost A. A description of the composite elastic modulus of multilayer coated systems. Thin Solid Films, 2015, 583: 177–193CrossRefGoogle Scholar
  14. 14.
    Zhu Y, Dui G. Micromechanical modeling of the stress-induced superelastic strain in magnetic shape memory alloy. Mech Mater, 2007, 39: 1025–1034CrossRefGoogle Scholar
  15. 15.
    Zhu Y, Yu K. A model considering mechanical anisotropy of magnetic- field-induced superelastic strain in magnetic shape memory alloys. J Alloys Compd, 2013, 550: 308–313CrossRefGoogle Scholar
  16. 16.
    Zhu Y, Dui G. Model for field-induced reorientation strain in magnetic shape memory alloy with tensile and compressive loads. J Alloys Compd, 2008, 459: 55–60CrossRefGoogle Scholar
  17. 17.
    Chang J C, Allen S M. Elstic energy changes accompanying gamma-prime rafting in nickel-base superalloys. J Mater Res, 1991, 6: 1843–1855CrossRefGoogle Scholar
  18. 18.
    Miyazaki T, Nakamura K, Mori H. Experimental and theoretical investigations on morphological changes of ?' precipitates in Ni-Al single crystals during uniaxial stress-annealing. J Mater Sci, 1979, 14: 1827–1837CrossRefGoogle Scholar
  19. 19.
    Ratel N, Bruno G, Bastie P, et al. Plastic strain-induced rafting of precipitates in Ni superalloys: Elasticity analysis. Acta Mater, 2006, 54: 5087–5093CrossRefGoogle Scholar
  20. 20.
    Wu W P, Guo Y F, Dui G S, et al. A micromechanical model for predicting the directional coarsening behavior in Ni-based superalloys. Comp Mater Sci, 2008, 44: 259–264CrossRefGoogle Scholar
  21. 21.
    Li S Y, Wu W P, Chen M X. An anisotropic micromechanics model for predicting the rafting direction in Ni-based single crystal superalloys. Acta Mech Sin, 2016, 32: 135–143MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Zhou L, Li S X, Chen C R, et al. Three-dimensional finite element analysis of stresses and energy density distributions around ?' before coarsening loaded in the [110]-direction in Ni-based superalloy. Mater Sci Eng-A, 2003, 352: 300–307CrossRefGoogle Scholar
  23. 23.
    Siebörger D, Knake H, Glatzel U. Temperature dependence of the elastic moduli of the nickel-base superalloy CMSX-4 and its isolated phases. Mater Sci Eng-A, 2001, 298: 26–33CrossRefGoogle Scholar
  24. 24.
    Fahrmann M, Hermann W, Fahrmann E, et al. Determination of matrix and precipitate elastic constants in Ni-base model alloys, and their relevance to rafting. Mater Sci Eng-A, 1999, 260: 212–221CrossRefGoogle Scholar
  25. 25.
    Mura T. Micromechanics of Defects in Solids. 2nd Edition. Dordrecht: Kluwer Academic Publishers, 1987CrossRefMATHGoogle Scholar
  26. 26.
    Onaka S, Kobayashi N, Kato M. Two-dimensional analysis on elastic strain energy due to a uniformly eigenstrained supercircular inclusion in an elastically anisotropic material. Mech Mater, 2002, 34: 117–125CrossRefGoogle Scholar
  27. 27.
    Pan E. Eshelby problem of polygonal inclusions in anisotropic piezoelectric full- and half-planes. J Mech Phys Solids, 2004, 52: 567–589MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Ting T C T. Anisotropic Elasticity: Theory and Applications. Oxford: Oxford University Press, 1996MATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • ShuangYu Li
    • 1
  • WenPing Wu
    • 1
    • 2
  • YunLi Li
    • 1
  • MingXiang Chen
    • 1
  1. 1.Department of Engineering Mechanics, School of Civil EngineeringWuhan UniversityWuhanChina
  2. 2.State Key Laboratory of Water Resources & Hydropower Engineering ScienceWuhan UniversityWuhanChina

Personalised recommendations