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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December 2016
Research-Article
Explicit Analytical Solutions for a Complete Set of the Eshelby Tensors of an Ellipsoidal Inclusion
Xiaoqing Jin,
Xiaoqing Jin
State Key Laboratory
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
e-mail: jinxq@cqu.edu.cn
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
e-mail: jinxq@cqu.edu.cn
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Ding Lyu,
Ding Lyu
State Key Laboratory
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
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Xiangning Zhang,
Xiangning Zhang
State Key Laboratory
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
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Qinghua Zhou,
Qinghua Zhou
School of Aeronautics and Astronautics,
Sichuan University,
Chengdu 610065, China
Sichuan University,
Chengdu 610065, China
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Qian Wang,
Qian Wang
Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
Northwestern University,
Evanston, IL 60208
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Leon M. Keer
Leon M. Keer
Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
Northwestern University,
Evanston, IL 60208
Search for other works by this author on:
Xiaoqing Jin
State Key Laboratory
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
e-mail: jinxq@cqu.edu.cn
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
e-mail: jinxq@cqu.edu.cn
Ding Lyu
State Key Laboratory
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
Xiangning Zhang
State Key Laboratory
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
of Mechanical Transmission,
Chongqing University,
Chongqing 400030, China
Qinghua Zhou
School of Aeronautics and Astronautics,
Sichuan University,
Chengdu 610065, China
Sichuan University,
Chengdu 610065, China
Qian Wang
Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
Northwestern University,
Evanston, IL 60208
Leon M. Keer
Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
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. Dec 2016, 83(12): 121010 (12 pages)
Published Online: October 5, 2016
Article history
Received:
July 10, 2016
Revised:
September 9, 2016
Citation
Jin, X., Lyu, D., Zhang, X., Zhou, Q., Wang, Q., and Keer, L. M. (October 5, 2016). "Explicit Analytical Solutions for a Complete Set of the Eshelby Tensors of an Ellipsoidal Inclusion." ASME. J. Appl. Mech. December 2016; 83(12): 121010. https://doi.org/10.1115/1.4034705
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