Uniformity of the Green operator and Eshelby tensor for hyperboloidal domains in infinite media

Abstract

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.