A constrained proof of the strong version of the Eshelby conjecture for three-dimensional isotropic media 三维各向同性介质中Eshelby强猜想的受限证明

Yuan，Tianyu1,2; Huang，Kefu3; Wang，Jianxiang2,4

2023-07-01

10.1007/s10409-023-22064-x

Acta Mechanica Sinica/Lixue Xuebao   Impact Factor & Quartile

0567-7718

1614-3116

Eshelby’s seminal work on the ellipsoidal inclusion problem leads to the conjecture that the ellipsoid is the only inclusion possessing the uniformity property that a uniform eigenstrain is transformed into a uniform elastic strain. For the three-dimensional isotropic medium, the weak version of the Eshelby conjecture has been substantiated. The previous work (Ammari et al., 2010) substantiates the strong version of the Eshelby conjecture for the cases when the three eigenvalues of the eigenstress are distinct or all the same, whereas the case where two of the eigenvalues of the eigenstress are identical and the other one is distinct remains a difficult problem. In this work, we study the latter case. To this end, firstly, we present and prove a necessary condition for an inclusion being capable of transforming a uniform eigenstress into a uniform elastic stress field. Since the necessary condition is not enough to determine the shape of the inclusion, secondly, we introduce a constraint that is concerned with the material parameters, and by introducing the concept of dissimilar media we prove that there exist combinations of uniform eigenstresses and the elastic tensors of dissimilar isotropic media such that only an ellipsoid can have the Eshelby uniformity property for these combinations simultaneously. Finally, we provide a more specifically constrained proof of the conjecture by proving that for the uniform strain fields constrained to those induced by an ellipsoid from a set of specified uniform eigenstresses, the strong version of the Eshelby conjecture is true for a set of isotropic elastic tensors which are associated with the specified uniform eigenstresses. This work makes some progress towards the complete solution of the intriguing and longstanding Eshelby conjecture for three-dimensional isotropic media.

Eigenstrain Elasticity Eshelby conjecture Inclusion problem Isotropic medium

SCI

Chinese

Corresponding

National Natural Science Foundation of China[11521202]

Engineering ; Mechanics

Engineering, Mechanical ; Mechanics

WOS:001024361600001

SPRINGER HEIDELBERG

ENGINEERING

2-s2.0-85165220393

Scopus

1.Institute for Advanced Study,Chengdu University,Chengdu,610106,China
2.State Key Laboratory for Turbulence and Complex System,Department of Mechanics and Engineering Science,College of Engineering,Peking University,Beijing,100871,China
3.Department of Mechanics and Aerospace Engineering,Southern University of Science and Technology,Shenzhen,518055,China
4.CAPT-HEDPS,and IFSA Collaborative Innovation Center of MoE,College of Engineering,Peking University,Beijing,100871,China

Yuan，Tianyu,Huang，Kefu,Wang，Jianxiang. A constrained proof of the strong version of the Eshelby conjecture for three-dimensional isotropic media 三维各向同性介质中Eshelby强猜想的受限证明[J]. Acta Mechanica Sinica/Lixue Xuebao,2023,39(7).

Yuan，Tianyu,Huang，Kefu,&Wang，Jianxiang.(2023).A constrained proof of the strong version of the Eshelby conjecture for three-dimensional isotropic media 三维各向同性介质中Eshelby强猜想的受限证明.Acta Mechanica Sinica/Lixue Xuebao,39(7).

Yuan，Tianyu,et al."A constrained proof of the strong version of the Eshelby conjecture for three-dimensional isotropic media 三维各向同性介质中Eshelby强猜想的受限证明".Acta Mechanica Sinica/Lixue Xuebao 39.7(2023).