, Volume 53, Issue 10, pp 2659–2667 | Cite as

Two-dimensional Eshelby’s problem for piezoelectric materials with a parabolic boundary

  • Xu Wang
  • Peter Schiavone


We use complex variable techniques to obtain analytic solutions of Eshelby’s problem consisting of an inclusion of arbitrary shape in an anisotropic piezoelectric plane with a parabolic boundary. The region of the physical plane below the parabola is mapped onto the lower half of the image plane. The problem is then more conveniently studied in the image plane rather than in the physical plane. The critical step in our approach lies in the construction of certain auxiliary functions in the image plane which allow for the technique of analytic continuation to be applied to an inclusion of arbitrary shape.


Eshelby inclusion Piezoelectric material Parabolic boundary Stroh octet formalism Analytic solution 



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).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Wang B (1992) Three-dimensional analysis of an ellipsoidal inclusion in a piezoelectric material. Int J Solids Struct 29:293–308CrossRefzbMATHGoogle Scholar
  2. 2.
    Chung MY, Ting TCT (1996) Piezoelectric solid with an elliptic inclusion or hole. Int J Solids Struct 33:3343–3361MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dunn ML, Wienecke HA (1997) Inclusions and inhomogeneities in transversely isotropic piezoelectric solids. Int J Solids Struct 34:3571–3582CrossRefzbMATHGoogle Scholar
  4. 4.
    Ru CQ (2000) Eshelby’s problem for two-dimensional piezoelectric inclusions of arbitrary shape. Proc R Soc Lond A 456:1051–1068MathSciNetCrossRefzbMATHADSGoogle Scholar
  5. 5.
    Ru CQ (2001) A two-dimensional Eshelby problem for two bonded piezoelectric half-planes. Proc R Soc Lond A 457:865–883MathSciNetCrossRefzbMATHADSGoogle Scholar
  6. 6.
    Wang X, Pan E (2010) Two-dimensional Eshelby’s problem for two imperfectly bonded piezoelectric half-planes. Int J Solids Struct 47:148–160CrossRefzbMATHGoogle Scholar
  7. 7.
    Ting TCT, Hu Y, Kirchner HOK (2001) Anisotropic elastic materials with a parabolic or hyperbolic boundary: a classical problem revisited. ASME J Appl Mech 68:537–542MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Suo Z, Kuo CM, Barnett DM, Willis JR (1992) Fracture mechanics for piezoelectric ceramics. J Mech Phys Solids 40:739–765MathSciNetCrossRefzbMATHADSGoogle Scholar
  9. 9.
    Wang X (1994) Trial discussions on the mathematical structure of inclusion, dislocation and crack. Dissertation, Xi’an Jiaotong UniversityGoogle Scholar
  10. 10.
    Ting TCT (1996) Anisotropic elasticity-theory and applications. Oxford University Press, New YorkzbMATHGoogle Scholar
  11. 11.
    Savin GN (1961) Stress concentration around holes. Pergamon Press, LondonzbMATHGoogle Scholar
  12. 12.
    England AH (1971) Complex variable methods in elasticity. Wiley, LondonzbMATHGoogle Scholar
  13. 13.
    Wang X, Chen L, Schiavone P (2017) Eshelby inclusion of arbitrary shape in isotropic elastic materials with a parabolic boundary. J Mech Mater Struct (in press) Google Scholar

Copyright information

© Springer Science+Business Media B.V., 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 EngineeringUniversity of AlbertaEdmontonCanada

Personalised recommendations