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Problems of polygonal inclusions in orthotropic materials with due consideration on the stresses at corners

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Abstract

Two-dimensional elastic Green’s functions in an orthotropic medium are obtained and expressed in a more explicit and detailed form than those shown in the preceding publications. The stress field associated with a line segment loaded by a uniform traction is found by a direct integration, and it has logarithmic singularities at both ends of the line segment. The basic solutions for the line segment are exploited to solve problems of polygonal inclusions. Stress distributions of circular, triangular, and square inclusions subjected to uniform dilatational eigenstrains in titanium single crystals are presented to show the effectiveness and accuracy of the present approach. The coefficients characterizing the strength of singularity for the stresses at corners are also numerically determined and discussed.

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References

  1. Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. A 241, 376–396 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  2. Mura, T.: Micromechanics of Defects in Solids. Martinus Nijhoff, Dordrecht (1982)

    Book  Google Scholar 

  3. Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogenous Materials. Elsevier, Amsterdam (1993)

    MATH  Google Scholar 

  4. Nozaki, H., Horibe, T., Taya, M.: Stress field caused by polygonal inclusion. JSME Int. J. Ser. A 44, 472–482 (2001)

    Article  Google Scholar 

  5. Dong, C.Y., Cheung, Y.K., Lo, S.H.: A regularized domain integral formulation for inclusion problems of various shapes by equivalent inclusion method. Comput. Method. Appl. Mech. Eng. 191, 3411–3421 (2002)

    Article  MATH  Google Scholar 

  6. Ru, C.Q.: Eshelby inclusion of arbitrary shape in an anisotropic plane or half-plane. Acta Mech. 160, 219–234 (2003)

    Article  MATH  Google Scholar 

  7. Pan, E.: Eshelby problems of polygonal inclusions in an anisotropic piezoelectric full- and half-planes. J. Mech. Phys. Solids. 52, 567–589 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gao, C.-F., Noda, N.: Faber series method for two-dimensional problems of an arbitrarily shaped inclusion in piezoelectric materials. Acta Mech. 171, 1–13 (2004)

    Article  MATH  Google Scholar 

  9. Lerma, J.D., Khraishi, T., Shen, Y.-L.: Elastic fields of 2D and 3D misfit particles in an infinite medium. Mech. Res. Commun. 34, 31–43 (2007)

    Article  MATH  Google Scholar 

  10. Zou, W., He, Q., Huang, M., Zheng, Q.: Eshelby’s problem of non-elliptical inclusions. J. Mech. Phys. Solids 58, 346–372 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhou, K., Keer, L.M., Wang, Q.J.: Semi-analytic solution for multiple interacting three-dimensional inhomogeneous inclusions of arbitrary shape in an infinite space. Int. J. Numer. Meth. Eng. 87, 617–638 (2011)

    Article  MATH  Google Scholar 

  12. Albrecht, J., Collatz, L., Hagedorn, P., Velte, W.: Numerical Treatment of Eigenvalue Problems, vol. 5. Birkhauser, Basel (1991)

    MATH  Google Scholar 

  13. Maz’ya, V., Nazarov, S., Plamenevskij, B.: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Birkhauser, Basel (2000)

    Book  Google Scholar 

  14. Lekhnitskii, S.G.: Anisotropic Plate. Gorden and Breach, New York (1963)

    MATH  Google Scholar 

  15. Michelitsch, T., Levin, V.M.: Green’s function for the infinite two-dimensional orthotropic medium. Int. J. Fract. 107, L33–L38 (2000)

    Article  Google Scholar 

  16. Kuznetsov, S.V.: Fundamental and singular solutions of Lamé equations for media with arbitrary elastic anisotropy. Q. Appl. Math. 63, 455–467 (2005)

    Article  Google Scholar 

  17. Chiang, C.R.: Eshelby’s tensor and its connection to ellipsoidal cavity problems with application to 2D transformation problems in orthotropic materials. Acta Mech. 226, 2631–2644 (2015). doi:10.1007/s00707-015-1343-1

    Article  MathSciNet  MATH  Google Scholar 

  18. Willis, J.R.: Anisotropic elastic inclusion problems. Q. J. Mech. Appl. Mech. 17, 157–174 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  19. Yang, H.C., Chou, Y.T.: Generalized plane problems of elastic inclusions in anisotropic solids. ASME J. Appl. Mech. 43, 424–430 (1976)

    Article  MATH  Google Scholar 

  20. Yang, H.C., Chou, Y.T.: Antiplane strain problems of an elliptic inclusion in an anisotropic medium. ASME J. Appl. Mech. 44, 437–441 (1977)

    Article  MATH  Google Scholar 

  21. Zeng, X., Rajapakse, R.K.N.D.: Eshelby tensor of piezoelectric inclusion and application to modeling of domain switching and evolution. Acta Mater. 51, 4121–4134 (2003)

    Article  Google Scholar 

  22. Reid, C.N.: Deformation Geometry for Materials Scientists. Pergamon, Oxford (1973)

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Power Mechanical Engineering, National Tsing Hua University, Hsin Chu, 30013, Taiwan

    Chun-Ron Chiang

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  1. Chun-Ron Chiang
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Correspondence to Chun-Ron Chiang.

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Chiang, CR. Problems of polygonal inclusions in orthotropic materials with due consideration on the stresses at corners. Arch Appl Mech 86, 769–785 (2016). https://doi.org/10.1007/s00419-015-1061-0

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  • DOI: https://doi.org/10.1007/s00419-015-1061-0

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