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Eshelby's seminal work on the ellipsoidal inclusion problem leads to theconjecture that the ellipsoid is the only inclusion possessing the uniformityproperty that a uniform eigenstrain is transformed into a uniform elasticstrain. For the three-dimensional isotropic medium, the weak version of theEshelby conjecture has been substantiated. The previous work of Ammari et al.substantiates the strong version of the Eshelby conjecture for the cases whenthe three eigenvalues of the eigenstress are distinct or all the same, whereasthe case where two of the eigenvalues of the eigenstress are identical and theother one is distinct remains a difficult problem. In this work, we study thelatter case. To this end, firstly, we present and prove a necessary conditionfor a convex inclusion being capable of transforming a single uniformeigenstress into a uniform elastic stress field. Since the necessary conditionis not enough to determine the shape of the inclusion, secondly, we introduce aconstraint that is concerned with the material parameters, and prove that thereexist combinations of the elastic tensors and uniform eigenstresses such thatonly an ellipsoid can have the Eshelby uniformity property for thesecombinations simultaneously. Finally, we provide a more specificallyconstrained proof of the conjecture by proving that for the uniform strainfields constrained to that induced by an ellipsoid from a set of specifieduniform eigenstresses, the strong version of the Eshelby conjecture is true fora set of isotropic elastic tensors which are associated with the specifieduniform eigenstresses. This work makes some progress towards the completesolution of the intriguing and longstanding Eshelby conjecture forthree-dimensional isotropic media.

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