On the meaning of the chosen set-averaging method within Eshelby-Kröner self-consistent scale transition model: The geometric mean versus the classical arithmetic average

Abstract

Scale-transition models, such as Eshelby-Kröner self-consistent framework, which are often used for predicting the effective behavior of heterogeneous materials or estimating the distribution of local states from the knowledge of the corresponding macroscopic quantities, require the extensive use of set averages. In the present work, the fundamental formalism historically introduced by Kröner is, for the first time, considered from the point of view of both the geometric and the arithmetic set averages methods. It is demonstrated in this paper that the polarization tensors describing the relations existing between the local and the macroscopic mechanical states do have a strong physical meaning when expressed using the geometric average, instead of the classical arithmetic mean. Scale-transition models, such as Eshelby-Kröner self-consistent framework, which are often used for predicting the effective behavior of heterogeneous materials or estimating the distribution of local states from the knowledge of the corresponding macroscopic quantities, require the extensive use of set averages. In the present work, the fundamental formalism historically introduced by Kröner is, for the first time, considered from the point of view of both the geometric and the arithmetic set averages methods. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.