Simplified approach to the derivation of the relationship between Hill polarization tensors of transformed problems and applications

This work aims at proposing a general framework based on an intrinsic tensor formalism to relate Hill polarization tensors of elastic problems associated by a linear transformation without detailing the resolution of the problems. Such an approach somehow casts a new light on the calculation of some Hill tensors already known in the literature by means of other methods or other presentations. In particular it allows to avoid the calculation of any Green tensor or derivatives and it boils down to very compact expressions which are rather easy to implement. The application of the method to the case of an arbitrary ellipsoid embedded in an elliptically orthotropic matrix recalls that the Hill tensor can be built from an associated isotropic case but also puts in evidence that the elliptically orthotropic case is the only one that can be derived from an isotropic counterpart. Another application of the method shows that the Hill tensor of a spheroid embedded in a coaxial transversely isotropic matrix can be derived from an associated problem involving a sphere in a modified transversely isotropic matrix. This approach provides new simple and compact expressions for the Hill tensor from which the known limits corresponding to flat spheroid or strongly prolate shapes are retrieved. (C) 2020 Elsevier Ltd. All rights reserved.