A spherical Eshelby inclusion with a Steigmann-Ogden interface in a finite domain

doi:10.1007/s10483-026-3403-9

Abstract

Abstract:

We derive closed-form solutions to the three-dimensional Eshelby’s problem of a spherical Eshelby inclusion undergoing uniform deviatoric eigenstrains concentrically embedded in an isotropic elastic finite spherical domain with a traction-free or rigidly clamped boundary. The interface between the inclusion and its surrounding domain is assumed to be of Steigmann-Ogden type. Our solutions indicate that the stresses and strains within the spherical inclusion are generally nonuniform because of the effects of the finite spherical domain and the Steigmann-Ogden imperfect interface. The internal elastic field of stresses and strains is uniform within the spherical inclusion when a condition that relates the single interface parameter to the geometric parameter and Poisson’s ratio of the finite domain is satisfied. When the spherical edge is rigidly clamped, a Gurtin-Murdoch interface is found to be sufficient to achieve this interior uniformity property. In contrast, when the spherical edge is traction-free, a Steigmann-Ogden interface with nonzero and positive bending stiffness parameters must be used to achieve the interior uniformity property.

Key words: spherical inclusion, spherical domain, Steigmann-Ogden interface, Gurtin-Murdoch interface, Neumann problem, Dirichlet problem, uniformity property

2010 MSC Number:

O343.3

Xu WANG, P. SCHIAVONE. A spherical Eshelby inclusion with a Steigmann-Ogden interface in a finite domain. Applied Mathematics and Mechanics (English Edition), 2026, 47(7): 1569-1580.

TrendMD

Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks

URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3403-9

https://www.amm.shu.edu.cn/EN/Y2026/V47/I7/1569

Figures/Tables 7

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

[1]	LI, S., SAUER, R., and WANG, G. A circular inclusion in a finite domain I: the Dirichlet-Eshelby problem. Acta Mechanica, 179(1), 67–90 (2005)
[2]	WANG, G., LI, S., and SAUER, R. A circular inclusion in a finite domain II, the Neumann-Eshelby problem. Acta Mechanica, 179(1), 91–110 (2005)
[3]	LI, S. F., SAUER, R. A., and WANG, G. The Eshelby tensors in a finite spherical domain: part I: theoretical formulations. Journal of Applied Mechanics, 74(4), 770–783 (2007)
[4]	LI, S. F., WANG, G., and SAUER, R. A. The Eshelby tensors in a finite spherical domain: part II: applications to homogenization. Journal of Applied Mechanics, 74(4), 784–797 (2007)
[5]	ZOU, W. N., HE, Q. C., and ZHENG, Q. S. Inclusions in a finite elastic body. International Journal of Solids and Structures, 49(13), 1627–1636 (2012)
[6]	ZOU, W. N. and HE, Q. C. Eshelby’s problem of a spherical inclusion eccentrically embedded in a finite spherical body. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 473(2198), 20160808 (2017)
[7]	ZOU, W. N. and LEE, Y. Completely explicit solutions of Eshelby’s problems of smooth inclusions embedded in a circular disk, full- and half-planes. Acta Mechanica, 229(5), 1911–1926 (2018)
[8]	ZOU, W. N., LEE, Y. G., and HE, Q. C. Inclusions inside a bounded elastic body undergoing anti-plane shear. Mathematics and Mechanics of Solids, 23(4), 588–605 (2018)
[9]	PAN, C. L. and YU, Q. Inclusion problem of a two-dimensional finite domain: the shape effect of matrix. Mechanics of Materials, 77, 86–97 (2014)
[10]	PAN, C. L. and YU, Q. Investigation of an arbitrarily shaped inclusion embedded in a two-dimensional finite domain. International Journal of Mechanical Sciences, 126, 142–150 (2017)
[11]	KRASNITCKII, S. A., SMIRNOV, A. M., and GUTKIN, M. Y. Misfit stress and energy in composite nanowire with polygonal core. International Journal of Engineering Science, 193, 103959 (2023)
[12]	WANG, X. and SCHIAVONE, P. A circular Eshelby inclusion in a finite domain possessing an n-fold axis of rotational symmetry. European Journal of Mechanics-A/Solids, 119, 106148 (2026)
[13]	SHARMA, P. and GANTI, S. Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies. Journal of Applied Mechanics, 71(5), 663–671 (2004)
[14]	STEIGMANN, D. J. and OGDEN, R. W. Plane deformations of elastic solids with intrinsic boundary elasticity. Proceedings: Mathematical, Physical and Engineering Sciences, 453(1959), 853–877 (1997)
[15]	STEIGMANN, D. J. and OGDEN, R. W. Elastic surface: substrate interactions. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 455(1982), 437–474 (1999)
[16]	GURTIN, M. E. and IAN MURDOCH, A. A continuum theory of elastic material surfaces. Archive for Rational Mechanics and Analysis, 57(4), 291–323 (1975)
[17]	GURTIN, M. E. and IAN MURDOCH, A. Surface stress in solids. International Journal of Solids and Structures, 14(6), 431–440 (1978)
[18]	GURTIN, M. E., WEISSMÜLLER, J., and LARCHÉ, F. A general theory of curved deformable interfaces in solids at equilibrium. Philosophical Magazine A, 78(5), 1093–1109 (1998)
[19]	CHRISTENSEN, R. M. and LO, K. H. Solutions for effective shear properties in three phase sphere and cylinder models. Journal of the Mechanics and Physics of Solids, 27(4), 315–330 (1979)
[20]	ZEMLYANOVA, A. Y. and MOGILEVSKAYA, S. G. On spherical inhomogeneity with Steigmann-Ogden interface. Journal of Applied Mechanics, 85(12), 121009 (2018)
[21]	RU, C. Q. Analytic solution for Eshelby’s problem of an inclusion of arbitrary shape in a plane or half-plane. Journal of Applied Mechanics, 66(2), 315–523 (1999)