Research Papers

Explicit Analytical Solutions for a Complete Set of the Eshelby Tensors of an Ellipsoidal Inclusion

[+] Author and Article Information
Xiaoqing Jin

State Key Laboratory
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
e-mail: jinxq@cqu.edu.cn

Ding Lyu, Xiangning Zhang

State Key Laboratory
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China

Qinghua Zhou

School of Aeronautics and Astronautics,
Sichuan University,
Chengdu 610065, China

Qian Wang, Leon M. Keer

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 10, 2016; final manuscript received September 9, 2016; published online October 5, 2016. Assoc. Editor: Shaoxing Qu.

J. Appl. Mech 83(12), 121010 (Oct 05, 2016) (12 pages) Paper No: JAM-16-1347; doi: 10.1115/1.4034705 History: Received July 10, 2016; Revised September 09, 2016

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.

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Grahic Jump Location
Fig. 1

Schematic of an ellipsoidal inclusion. For any exterior point x, an imaginary ellipsoid is constructed, where the outward unit normal vector at x is denoted by n⇀.

Grahic Jump Location
Fig. 2

Verification of the present analytical solution with Ref. [15] solution. Results of displacements, strains, and stress are compared along a vertical target line at x1 = x2 = 0.25a1, parallel to the x3 axis. (a) Schematic illustration of benchmark example, (b) displacement components, (c) normal strain components, (d) shear strain components, (e) normal stress components, (f) shear stress components.



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