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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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]
Mura, T. , 1982, Micromechanics of Defects in Solids, Springer, Dordrecht, The Netherlands. [CrossRef]
Li, S. , and Wang, G. , 2008, Introduction to Micromechanics and Nanomechanics, World Scientific, Singapore. [CrossRef]
Yu, H. Y. , and Sanday, S. C. , 1990, “ Axisymmetric Inclusion in a Half Space,” ASME J. Appl. Mech., 57(1), pp. 74–77. [CrossRef]
Davies, J. H. , 2003, “ Elastic Field in a Semi-Infinite Solid Due to Thermal Expansion or a Coherently Misfitting Inclusion,” ASME J. Appl. Mech., 70(5), pp. 655–660. [CrossRef]
Liu, S. , Jin, X. , Wang, Z. , Keer, L. M. , and Wang, Q. , 2012, “ Analytical Solution for Elastic Fields Caused by Eigenstrains in a Half-Space and Numerical Implementation Based on FFT,” Int. J. Plast., 35, pp. 135–154. [CrossRef]
Mindlin, R. D. , and Cheng, D. H. , 1950, “ Thermoelastic Stress in the Semi-Infinite Solid,” J. Appl. Phys., 21(9), pp. 931–933. [CrossRef]
Chiu, Y. P. , 1978, “ On the Stress Field and Surface Deformation in a Half Space With a Cuboidal Zone in Which Initial Strains are Uniform,” ASME J. Appl. Mech., 45(2), pp. 302–306. [CrossRef]
Seo, K. , and Mura, T. , 1979, “ The Elastic Field in a Half Space Due to Ellipsoidal Inclusions With Uniform Dilatational Eigenstrains,” ASME J. Appl. Mech., 46(3), pp. 568–572. [CrossRef]
Kuvshinov, B. N. , 2008, “ Elastic and Piezoelectric Fields Due to Polyhedral Inclusions,” Int. J. Solids Struct., 45(5), pp. 1352–1384. [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. , 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]
Jin, X. , Zhang, X. , Li, P. , Xu, Z. , Hu, Y. , and Keer, L. M. , 2017, “ On the Displacement of a Two-Dimensional Eshelby Inclusion of Elliptic Cylindrical Shape,” ASME J. Appl. Mech., 84(7), p. 074501. [CrossRef]
Ferrers, N. M. , 1877, “ On the Potentials of Ellipsoids, Ellipsoidal Shells, Elliptic Laminae and Elliptic Rings of Variable Densities,” Q. J. Pure Appl. Math., 14(1), pp. 1–22.
Dyson, F. W. , 1891, “ The Potentials of Ellipsoids of Variable Densities,” Q. J. Pure Appl. Math., 25, pp. 259–288.
Mindlin, R. D. , and Cheng, D. H. , 1950, “ Nuclei of Strain in the Semi-Infinite Solid,” J. Appl. Phys., 21(9), pp. 926–930. [CrossRef]
Liu, S. , and Wang, Q. , 2005, “ Elastic Fields Due to Eigenstrains in a Half-Space,” ASME J. Appl. Mech., 72(6), pp. 871–878. [CrossRef]
Mindlin, R. D. , 1936, “ Force at a Point in the Interior of a Semi-Infinite Solid,” Physics, 7(5), pp. 195–202. [CrossRef]


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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