Research Papers

Explicit Analytical Solutions for the Complete Elastic Field Produced by an Ellipsoidal Thermal Inclusion in a Semi-Infinite Space

[+] Author and Article Information
Ding Lyu, Xiangning Zhang, Pu Li, Dahui Luo, Yumei Hu

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

Xiaoqing Jin

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

Liying Zhang

Department of Biomedical Engineering,
Wayne State University,
Detroit, MI 48201

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 January 9, 2018; final manuscript received February 13, 2018; published online March 7, 2018. Assoc. Editor: Thomas Siegmund.

J. Appl. Mech 85(5), 051005 (Mar 07, 2018) (8 pages) Paper No: JAM-18-1022; doi: 10.1115/1.4039373 History: Received January 09, 2018; Revised February 13, 2018

Thermal inclusion in an elastic half-space is a classical micromechanical model for describing localized heating near a surface. This paper presents explicit analytical solutions for the complete elastic fields, including displacements, strains, and stresses, produced by an ellipsoidal thermal inclusion in a three-dimensional semi-infinite space. Unlike the famous Eshelby solution corresponding to the infinite space case, the present work demonstrates that the interior strain and stress components are no longer uniform and appear to be much more complex. Nevertheless, the results can be represented in a more compact and geometrically meaningful form by constructing auxiliary confocal ellipsoids. The derived explicit solution indicates that the shear components of the stress and strain may be represented in closed-form. The jump conditions are examined and proven to be exactly identical to the infinite space case. A purposely selected benchmark example is studied to illustrate the free boundary surface effects. The degenerate case of a spherical thermal inclusion may be derived in a closed form, and is verified by the well-known Mindlin solution.

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Grahic Jump Location
Fig. 1

Geometric configuration of a half-space with an ellipsoidal inclusion

Grahic Jump Location
Fig. 2

Schematic of the original, mirror image, and imaginary confocal ellipsoidal inclusions. The imaginary confocal ellipsoid is constructed passing an exterior points x, where the outward unit normal vector is denoted by n⇀.

Grahic Jump Location
Fig. 3

Variation of σ33 along the x3-axis

Grahic Jump Location
Fig. 4

Variations of displacement, strain, and stress components along a vertical target line: (a) geometric configuration; (b) variation of displacements; (c) variation of normal strains; (d) variation of shear strains; (e) variation of normal stresses; and (f) variation of shear stresses




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