Since Eshelby’s (1957) result (Eshelby, JD. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc R Soc London A 1957; 421: 379–396) that ellipsoids in an infinite matrix have uniform localization tensors, all attempts to find other finite domain shapes sharing that same property have failed and valuable proofs were provided that none could exist. Since that “uniformity property” also applies to infinite cylinders and layers as limits of prolate and oblate spheroids, we examine the cases of hyperboloidal domains of which infinite cylinders and platelets also are the limits. As members of the quadric surface family, hyperboloids expectably also have uniform Eshelby tensor and Green operator when embedded in infinite media, with specific features expectable too from unboundedness and not convex curvatures. Using the Radon transform method applied by the author to various inclusion shapes, as well as to finite and infinite patterns, since the uniformity of a shape function (inverse Radon transform of the domain indicator function) implies the uniformity of the related Green operator and Eshelby tensor, we examine the shape functions of axially symmetric hyperboloids. We establish that those of the two-sheet types are uniform and those of the one-sheet types are not, an additional neck-related term carrying the non-uniformity. The Green operators are next examined in considering an isotropic embedding medium with elastic- (including dielectric-) like properties. The results regarding the operator (non) uniformity correspond to those concerning the shape functions. The established operator uniformity characteristics imply validity of all Eshelby-derived ellipsoid properties. Yet, determining the Green operator solution calls for overcoming the issue of the infinite hyperbolic planar sections (the operator finiteness), with also attention being paid to positive definiteness. Options are compared from which an obtained satisfying solution with regard to both issues raises questioning theoretical and practical points on mathematical and mechanical grounds. While further studies are in progress, some application tracks are indicated.

中文翻译：

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无限介质中双曲面域的 Green 算子和 Eshelby 张量的一致性

由于 Eshelby (1957) 的结果 (Eshelby, JD. 椭圆体包含弹性场的确定和相关问题。Proc R Soc London A 1957; 421: 379–396) 无限矩阵中的椭圆体具有均匀的定位张量，所有试图找到共享相同属性的其他有限域形状的尝试失败了，并且提供了有价值的证据证明不存在任何一个。由于这种“均匀性”也适用于无限圆柱体和层作为扁长球体和扁球体的极限，我们研究了双曲面域的情况，其中无限圆柱体和小片也是极限。作为二次曲面家族的成员，当嵌入无限介质时，双曲面预计也具有均匀的 Eshelby 张量和 Green 算子，并且具有无界性和非凸曲率的特定特征。使用作者对各种包含形状以及有限和无限模式应用的 Radon 变换方法，因为形状函数的均匀性（域指示函数的逆 Radon 变换）意味着相关 Green 算子和 Eshelby 的均匀性张量，我们检查轴对称双曲面的形状函数。我们确定两片类型的那些是均匀的，而单片类型的那些不是，带有不均匀性的附加颈部相关术语。接下来在考虑具有类似弹性（包括电介质）特性的各向同性嵌入介质时检查 Green 算子。关于算子（非）均匀性的结果对应于关于形状函数的结果。已建立的算子均匀性特征意味着所有 Eshelby 导出的椭球特性的有效性。然而，确定格林算子解需要克服无限双曲平面截面的问题（算子有限性），同时还要注意正定性。对选项进行了比较，从中获得的关于这两个问题的令人满意的解决方案在数学和机械基础上提出了质疑理论和实践问题。在进一步研究进行中的同时，也指出了一些应用轨迹。对选项进行了比较，从中获得的关于这两个问题的令人满意的解决方案在数学和机械基础上提出了质疑理论和实践问题。在进一步研究进行中的同时，也指出了一些应用轨迹。对选项进行了比较，从中获得的关于这两个问题的令人满意的解决方案在数学和机械基础上提出了质疑理论和实践问题。在进一步研究进行中的同时，也指出了一些应用轨迹。