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Technical Brief

On the Displacement of a Two-Dimensional Eshelby Inclusion of Elliptic Cylindrical Shape

[+] Author and Article Information
Xiaoqing Jin

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

Xiangning Zhang, Pu Li, Zheng Xu, Yumei Hu

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

Leon M. Keer

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

1Corresponding author.

2Jin et al., 2011, “A Closed-Form Solution for the Eshelby Tensor and the Elastic Field Outside an Elliptic Cylindrical Inclusion,” ASME J. Appl. Mech., 78(3), p. 031009.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 24, 2017; final manuscript received May 20, 2017; published online June 7, 2017. Assoc. Editor: M Taher A Saif.

J. Appl. Mech 84(7), 074501 (Jun 07, 2017) (6 pages) Paper No: JAM-17-1217; doi: 10.1115/1.4036820 History: Received April 24, 2017; Revised May 20, 2017

In a companion paper,ff2 we have obtained the closed-form solutions to the stress and strain fields of a two-dimensional Eshelby inclusion. The current work is concerned with the complementary formulation of the displacement. All the formulae are derived in explicit closed-form, based on the degenerate case of a three-dimensional (3D) ellipsoidal inclusion. A benchmark example is provided to validate the present analytical solutions. In conjunction with our previous study, a complete elasticity solution to the classical elliptic cylindrical inclusion is hence documented in Cartesian coordinates for the convenience of engineering applications.

FIGURES IN THIS ARTICLE
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Copyright © 2017 by ASME
Topics: Tensors , Displacement
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References

Eshelby, J. D. , 1957, “ The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems,” Proc. R. Soc. London, Ser. A, 241(1226), pp. 376–396. [CrossRef]
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Mura, T. , Shodja, H. M. , and Hirose, Y. , 1996, “ Inclusion Problems,” ASME Appl. Mech. Rev., 49(10S), pp. S118–S127. [CrossRef]
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Zhou, K. , Hoh, H. J. , Wang, X. , Keer, L. M. , Pang, J. H. L. , Song, B. , and Wang, Q. J. , 2013, “ A Review of Recent Works on Inclusions,” Mech. Mater., 60, pp. 144–158. [CrossRef]
Chiu, Y. P. , 1980, “ On the Internal Stresses in a Half Plane and a Layer Containing Localized Inelastic Strains or Inclusions,” ASME J. Appl. Mech., 47(2), pp. 313–318. [CrossRef]
Nozaki, H. , and Taya, M. , 1997, “ Elastic Fields in a Polygon-Shaped Inclusion With Uniform Eigenstrains,” ASME J. Appl. Mech., 64(3), pp. 495–502. [CrossRef]
Muskhelishvili, N. I. , 1953, Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff, Groningen, The Netherlands.
Ru, C. Q. , 1999, “ Analytic Solution for Eshelby's Problem of an Inclusion of Arbitrary Shape in a Plane or Half-Plane,” ASME J. Appl. Mech., 66(2), pp. 315–322. [CrossRef]
Jin, X. , Keer, L. M. , and Wang, Q. , 2009, “ New Green's Function for Stress Field and a Note of Its Application in Quantum-Wire Structures,” Int. J. Solids Struct., 46(21), pp. 3788–3798. [CrossRef]
Eshelby, J. , 1959, “ The Elastic Field Outside an Ellipsoidal Inclusion,” Proc. R. Soc. London, Ser. A, 252(1271), pp. 561–569. [CrossRef]
Jin, X. , Lyu, D. , Zhang, X. , Zhou, Q. , Wang, Q. , and Keer, L. M. , 2016, “ Explicit Analytical Solutions for a Complete Set of the Eshelby Tensors of an Ellipsoidal Inclusion,” ASME J. Appl. Mech., 83(12), p. 121010. [CrossRef]
Jin, X. , Keer, L. M. , and Wang, Q. , 2011, “ A Closed-Form Solution for the Eshelby Tensor and the Elastic Field Outside an Elliptic Cylindrical Inclusion,” ASME J. Appl. Mech., 78(3), p. 031009. [CrossRef]
Ju, J. W. , and Sun, L. Z. , 1999, “ A Novel Formulation for the Exterior-Point Eshelby's Tensor of an Ellipsoidal Inclusion,” ASME J. Appl. Mech., 66(2), pp. 570–574. [CrossRef]
Jin, X. , Wang, Z. , Zhou, Q. , Keer, L. M. , and Wang, Q. , 2014, “ On the Solution of an Elliptical Inhomogeneity in Plane Elasticity by the Equivalent Inclusion Method,” J. Elasticity, 114(1), pp. 1–18. [CrossRef]
Maugis, D. , 2000, Contact, Adhesion and Rupture of Elastic Solids, Springer, Berlin.

Figures

Grahic Jump Location
Fig. 1

Schematic illustration of a confocal imaginary ellipsoid with its outward unit normal vector at x denoted by n⇀

Grahic Jump Location
Fig. 2

A plane strain elliptical cavity subjected to a remote uniform loading

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