Continuum Mechanics and Thermodynamics

, Volume 31, Issue 1, pp 71–77 | Cite as

Anti-plane eigenstrain problem of an inclusion of arbitrary shape in an anisotropic bimaterial with a semi-infinite interface crack

  • Xu Wang
  • Peter SchiavoneEmail author
Original Article


We consider an Eshelby inclusion of arbitrary shape with uniform anti-plane eigenstrains embedded in one of two bonded dissimilar anisotropic half planes containing a semi-infinite interface crack situated along the negative real axis. Using two consecutive conformal mappings, the upper and lower halves of the physical plane are first mapped onto two separate quarters of the image plane. The corresponding boundary value problem is then analyzed in this image plane rather than in the original physical plane. Corresponding analytic functions in all three phases of the composite are derived via the construction of an auxiliary function and repeated application of analytic continuation across the real and imaginary axes in the image plane. As a result, the local stress intensity factor is then obtained explicitly. Perhaps most interestingly, we find that the satisfaction of a particular condition makes the inclusion (stress) invisible to the crack.


Eshelby inclusion Interface crack Anisotropic bimaterial Conformal mapping Analytic continuation 


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This work is supported by the National Natural Science Foundation of China (Grant No. 11272121) and through a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No. RGPIN-2017-03716115112).


  1. 1.
    Zhang, H.T., Chou, Y.T.: Antiplane eigenstrain problem of an elliptic inclusion in a two-phase anisotropic medium. ASME J. Appl. Mech. 52, 87–90 (1985)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Yu, H.Y., Sanday, S.C.: Elastic field in jointed semi-infinite solids with an inclusion. Proc. R. Soc. Lond. A 434, 521–530 (1991)ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Yu, H.Y., Sanday, S.C., Rath, B.B., Chang, C.I.: Elastic fields due to defects in transversely isotropic bimaterials. Proc. R. Soc. Lond. A 449, 1–30 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Yu, H.Y.: Two-dimensional elastic defects in orthotropic bicrystals. J. Mech. Phys. Solids 49, 261–287 (2001)ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Ru, C.Q., Schiavone, P., Mioduchowski, A.: Elastic fields in two jointed half-planes with an inclusion of arbitrary shape. Z Angew. Math. Phys. 52, 18–32 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ru, C.Q.: A two-dimensional Eshelby problem for two bonded piezoelectric half-planes. Proc. R. Soc. Lond. A 457, 865–883 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Wang, X., Sudak, L.J., Ru, C.Q.: Elastic fields in two imperfectly bonded half-planes with a thermal inclusion of arbitrary shape. Z Angew. Math. Phys. 58, 488–509 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wang, X., Pan, E.: Two-dimensional Eshelby’s problem for two imperfectly bonded piezoelectric half-planes. Int. J. Solids Struct. 47, 148–160 (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    Kuvshinov, B.N.: Elastic and piezoelectric fields due to polyhedral inclusions. Int. J. Solids Struct. 45, 1352–1384 (2008)CrossRefzbMATHGoogle Scholar
  10. 10.
    Ting, T.C.T.: Anisotropic Elasticity-Theory and Applications. Oxford University Press, New York (1996)CrossRefzbMATHGoogle Scholar
  11. 11.
    Milne-Thomson, L.M.: Antiplane Elastic Systems. Springer, Berlin (2012)zbMATHGoogle Scholar
  12. 12.
    Ting, T.C.T.: Common errors on mapping of nonelliptic curves in anisotropic elasticity. ASME J. Appl. Mech. 67, 655–657 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Savin, G.N.: Stress Concentration Around Holes. Pergamon Press, London (1961)zbMATHGoogle Scholar
  14. 14.
    England, A.H.: Complex Variable Methods in Elasticity. Wiley, London (1971)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mechanical and Power EngineeringEast China University of Science and TechnologyShanghaiChina
  2. 2.Department of Mechanical Engineering, 10-203 Donadeo Innovation Centre for Engineering EdmontonUniversity of AlbertaEdmontonCanada

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