Explicit analytical solutions for a complete set of the eshelby tensors of an ellipsoidal inclusion

Xiaoqing Jin*, Ding Lyu, Xiangning Zhang, Qinghua Zhou, Q Jane Wang, Leon M Keer

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

25 Scopus citations


The celebrated solution of the Eshelby ellipsoidal inclusion has laid the cornerstone for many fundamental aspects of micromechanics. A well-known difficulty of this classical solution is to determine the elastic field outside the ellipsoidal inclusion. In this paper, we first analytically present the full displacement field of an ellipsoidal inclusion subjected to uniform eigenstrain. It is demonstrated that the displacements inside inclusion are linearly related to the coordinates and continuous across the interface of inclusion and matrix. The exterior displacement, which is less detailed in existing literatures, may be expressed in a more compact, explicit, and simpler form through utilizing the outward unit normal vector of an auxiliary confocal ellipsoid. Other than many practical applications in geological engineering, the displacement solution can be a convenient starting point to derive the deformation gradient, and subsequently in a straightforward manner to accomplish the full-field solutions of the strain and stress. Following Eshelby's definition, a complete set of the Eshelby tensors corresponding to the displacement, deformation gradient, strain, and stress are expressed in explicit analytical form. Furthermore, the jump conditions to quantify the discontinuities across the interface are discussed and a benchmark problem is provided to validate the present formulation.

Original languageEnglish (US)
Article number121010
JournalJournal of Applied Mechanics, Transactions ASME
Issue number12
StatePublished - Dec 1 2016


  • Displacement solution
  • Eigenstrain
  • Ellipsoidal inclusion
  • Eshelby tensor
  • Exterior field

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering


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