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Complex variable method for equivalence of the elliptical inhomogeneity to Eshelby’s elliptical inclusion under remote loading

  • Y. Z. ChenEmail author
Article
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Abstract

Two boundary value problems are formulated in this paper. Among them, one is for elliptical inhomogeneity embedded in an infinite matrix and other is for Eshelby’s elliptical eigenstrain inclusion in an infinite matrix. In both problems, the same remote loading is applied. Both problems are solved by using the complex variable method. In the first problem, the solution for stresses in the inhomogeneity depends on the elastic constants on two phases and the remote loading. In addition, the solution for stresses in the Eshelby’s elliptical eigenstrain inclusion depends on the elastic constants for the matrix or the inclusion and the remote loading. It is known that the stress components in the inclusion are uniform for two problems. Letting the stress components in the inclusion for two problems be the same, the equivalent eigenstrains are evaluated, which has a linear relation with respect to the remote loading.

Keywords

Inhomogeneity Eshelby’s inclusion Complex variable method Closed-form solution Numerical analysis 

Mathematics Subject Classification

74A20 74A40 74A60 74B05 74B10 

Notes

References

  1. 1.
    Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. R. Soc. Lond. A 241, 376–396 (1957)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Mura, T.: Micromechanics of Defects in Solids, 2nd edn. Martinus Nijhoff, Dordrecht (1987)CrossRefGoogle Scholar
  3. 3.
    Gong, S.X., Meguid, S.A.: On the elastic fields of an elliptical inhomogeneity under plane deformation. Proc. R. Soc. Lond. A 443, 457–471 (1993)CrossRefGoogle Scholar
  4. 4.
    Wang, X., Gao, X.L.: On the uniform stress state inside an inclusion of arbitrary shape in a three-phase composite. Z. Angew. Math. Phys. 62, 1101–1116 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Jin, X.Q., Wang, Z.J., Zhou, Q.H., Keer, L.M., Wang, Q.: On the solution of an elliptical inhomogeneity in plane elasticity by the equivalent inclusion method. J. Elast. 114, 1–18 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, Y.Z.: An innovative solution in closed form and numerical analysis for dissimilar elliptical inclusion in plane elasticity. Int. J. Appl. Mech. 6, 1450080 (2014). (16 pages)CrossRefGoogle Scholar
  7. 7.
    Wang, X., Shen, Y.P.: Two circular inclusions with inhomogeneous interfaces interacting with a circular Eshelby inclusion in anti-plane shear. Acta Mech. 158, 67–84 (2002)CrossRefGoogle Scholar
  8. 8.
    Li, Z., Sheng, Q., Sun, J.: A generally applicable approximate solution for mixed mode crack–inclusion interaction. Acta Mech. 187, 1–9 (2006)CrossRefGoogle Scholar
  9. 9.
    Dong, C.Y., Lee, K.Y.: A new integral equation formulation of two-dimensional inclusion–crack problems. Int. J. Solids Struct. 42, 5010–5020 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cao, C.K., Lu, L.M., Chen, C.K., Chen, F.M.: Analytic solution for a reinforcement layer bonded to an elliptic hole under a remote load. Int. J. Solids Struct. 46, 2959–2965 (2009)CrossRefGoogle Scholar
  11. 11.
    Zhou, L., Hoh, H.J., Wang, X., Keer, L.M., Pang, J.H.L., Song, B., Wang, Q.J.: A review of recent works on inclusions. Mech. Mater. 60, 144–158 (2013)CrossRefGoogle Scholar
  12. 12.
    Tian, L., Rajapakse, R.K.N.D.: Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity. Int. J. Solids Struct. 44, 7988–8005 (2007)CrossRefGoogle Scholar
  13. 13.
    Muskhelishvili, N.I.: Some Basic Problems of Mathematical Theory of Elasticity. Noordhoof, Groningen (1963)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Division of Engineering MechanicsJiangsu UniversityZhenjiangPeople’s Republic of China

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