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.

包含在固体中的 Eshelby 形式主义在力学和异质介质的其他领域具有重要的理论和实践意义。Eshelby 发现在无限各向同性介质中的孤立椭圆体包裹体中规定的均匀本征应变导致包裹体中的均匀弹性应变场，这导致推测椭球体是唯一具有所谓 Eshelby 均匀性特性的包裹体。此前，对于各向同性介质仅证明了该猜想的弱版本，而对于三维各向异性介质的猜想的一般有效性还有待探索。在这项工作中，首先，我们证明了具有立方、横向各向同性、正交各向异性的各向异性介质的广义 Eshelby 猜想的弱版本，在具有这些对称性的无限介质中的孤立包裹体中，所有均匀的本征应变都转化为均匀的弹性应变场。其次，我们证明了在这些各向异性介质中，存在着可以将特定偶次多项式本征应变转化为其中偶数次多项式弹性应变场的非椭球体包裹体。这些结果在强烈意义上构成了广义高阶 Eshelby 猜想（Eshelby 多项式守恒定理的逆问题）的反例) 用于各向异性介质和各向同性介质中的多项式本征应变（仅二次本征应变）。此外，我们还表明，对于这些各向异性介质中均匀本征应变的广义 Eshelby 猜想的强版本存在反例。为这些反例包含的存在提供了充分条件。这些发现揭示了本征应变之间的均匀性与标准椭圆包裹体之外的各向异性介质中包裹体中相应诱导的弹性应变之间的惊人丰富性。由于嵌入和固有各向异性量子点晶体中的应变场是量子点发射光子质量的有效调谐旋钮，因此结果可能对量子信息技术产生影响，