三维各向同性介质中Eshelby强猜想的受限证明  

A constrained proof of the strong version of the Eshelby conjecture for three-dimensional isotropic media

作　　者：袁天宇 黄克服 王建祥 Tianyu Yuan;Kefu Huang;Jianxiang Wang(Institute for Advanced Study,Chengdu University,Chengdu 610106,China;State Key Laboratory for Turbulence and Complex System,Department of Mechanics and Engineering Science,College of Engineering,Peking University,Beijing 100871,China;Department of Mechanics and Aerospace Engineering,Southern University of Science and Technology,Shenzhen 518055,China;CAPT-HEDPS,and IFSA Collaborative Innovation Center of MoE,College of Engineering,Peking University,Beijing 100871,China)

机构地区：[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

出　　处：《Acta Mechanica Sinica》2023年第6期67-79,共13页力学学报（英文版）

基　　金：supported by the National Natural Science Foundation of China(Grant No.11521202).

摘　　要：Eshelby在椭球夹杂问题的研究中提出过一个著名的猜想——Eshelby猜想,即椭球是唯一一种能将均匀本征应变转化为均匀弹性应变的夹杂构型,且该类性质被称为Eshelby均匀特性.在后来的研究中,Eshelby猜想被分为强弱两个版本,在三维各向同性介质中,Eshelby弱猜想已经得到证明,但Eshelby强猜想仍然悬而未决,其中Ammari等人(2010)仅针对本征应力有三个相同或三个不同特征值的情形证明了强猜想成立.在这项工作中,我们考虑尚未被证明的情形,即本征应力仅有两个相同特征值的情形.首先,我们提出了具有Eshelby均匀特性的夹杂须满足的必要条件.接着,由于该必要条件不足以帮我们判别夹杂构型是否只能为椭球,因此我们额外引入一个与材料常数相关的约束条件,继而基于非相似材料的概念,证明了,在两个非相似各向同性介质中都具有Eshelby均匀特性的夹杂构型只能为椭球。最后,我们提出了一个更具体的强猜想的受限证明,我们约束具有Eshelby均匀特性的夹杂其内部应变场与椭球在相同本征应变下产生的内部应变场相同,继而证明了,对特定本征应力及与本征应力相关的弹性张量的组合,Eshelby强猜想成立.综上,该工作在完全证明三维各向同性介质Eshelby强猜想的探索道路上又向前迈进了一步.Eshelby’s seminal work on the ellipsoidal inclusion problem leads to the conjecture that the ellipsoid is the only inclusion pos-sessing 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 eigen-stresses.This work makes some progress towards the complete solution of the intriguing and longstanding Eshelby conjecture for three-dimensional isotropic media.

关 键 词：各向同性介质 本征应力 本征应变 材料常数 弹性应变 特征值 相似材料 非相似 

分 类 号：O753.3[理学—晶体学]