Solutions to the generalized Eshelby conjecture for anisotropic media: Proofs of the weak version and counter-examples to the high-order and the strong versions

Yuan，Tianyu1,2; Huang，Kefu3; Wang，Jianxiang1,2

2022

10.1016/j.jmps.2021.104648

JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS   影响因子和分区

0022-5096

The Eshelby formalism for an inclusion in a solid is of significant theoretical and practical implications in mechanics and other fields of heterogeneous media. Eshelby's finding that a uniform eigenstrain prescribed in a solitary ellipsoidal inclusion in an infinite isotropic medium results in a uniform elastic strain field in the inclusion leads to the conjecture that the ellipsoid is the only inclusion that possesses the so-called Eshelby uniformity property. Previously, only the weak version of the conjecture has been proved for the isotropic medium, whereas the general validity of the conjecture for anisotropic media in three dimensions is yet to be explored. In this work, firstly, we present proofs of the weak version of the generalized Eshelby conjecture for anisotropic media that possess cubic, transversely isotropic, orthotropic, and monoclinic symmetries, which substantiates that only the ellipsoidal shape can transform all uniform eigenstrains into uniform elastic strain fields in a solitary inclusion in infinite media possessing these symmetries. Secondly, we prove that in these anisotropic media, there exist non-ellipsoidal inclusions that can transform particular polynomial eigenstrains of even degrees into polynomial elastic strain fields of the same even degrees in them. These results constitute counter-examples, in the strong sense, to the generalized high-order Eshelby conjecture (inverse problem of Eshelby's polynomial conservation theorem) for polynomial eigenstrains in both anisotropic media and the isotropic medium (quadratic eigenstrain only). In addition, we also show that there are counter-examples to the strong version of the generalized Eshelby conjecture for uniform eigenstrains in these anisotropic media. A sufficient condition for the existence of those counter-example inclusions is provided. These findings reveal striking richness of the uniformity between the eigenstrains and the correspondingly induced elastic strains in inclusions in anisotropic media beyond the canonical ellipsoidal inclusions. Since the strain fields in embedded and inherently anisotropic quantum dot crystals are effective tuning knobs of the quality of the emitted photons by the quantum dots, the results may have implications in the technology of quantum information, in addition to in mechanics and materials science.

Anisotropic media Eigenstrain Elasticity Eshelby conjecture Inclusion problems

SCI ; EI

英语

通讯

WOS:000708676700003

20214211020089

Anisotropic media ; Anisotropy ; Inverse problems ; Nanocrystals ; Polynomials ; Quantum optics ; Strain

Semiconductor Devices and Integrated Circuits:714.2 ; Light/Optics:741.1 ; Nanotechnology:761 ; Algebra:921.1 ; Physical Properties of Gases, Liquids and Solids:931.2 ; Quantum Theory; Quantum Mechanics:931.4 ; Crystalline Solids:933.1 ; Materials Science:951

ENGINEERING

2-s2.0-85116866390

Scopus

1.State Key Laboratory for Turbulence and Complex System,Department of Mechanics and Engineering Science,College of Engineering,Peking University,Beijing,100871,China
2.CAPT-HEDPS,and IFSA Collaborative Innovation Center of MoE,College of Engineering,Peking University,Beijing,100871,China
3.Department of Mechanics and Aerospace Engineering,Southern University of Science and Technology,Shenzhen,Guangdong,518055,China

Yuan，Tianyu,Huang，Kefu,Wang，Jianxiang. Solutions to the generalized Eshelby conjecture for anisotropic media: Proofs of the weak version and counter-examples to the high-order and the strong versions[J]. JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS,2022,158.

Yuan，Tianyu,Huang，Kefu,&Wang，Jianxiang.(2022).Solutions to the generalized Eshelby conjecture for anisotropic media: Proofs of the weak version and counter-examples to the high-order and the strong versions.JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS,158.

Yuan，Tianyu,et al."Solutions to the generalized Eshelby conjecture for anisotropic media: Proofs of the weak version and counter-examples to the high-order and the strong versions".JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 158(2022).