The Eshelby, Hill, Moment and Concentration Tensors for Ellipsoidal Inhomogeneities in the Newtonian Potential Problem and Linear Elastostatics

Abstract

One of the most cited papers in Applied Mechanics is the work of Eshelby from 1957 who showed that a homogeneous isotropic ellipsoidal inhomogeneity embedded in an unbounded (in all directions) homogeneous isotropic host would feel uniform strains and stresses when uniform strains or tractions are applied in the far-field. Of specific importance is the uniformity of Eshelby's tensor S. Following Eshelby's seminal work, a vast literature has been generated using and developing Eshelby's result and ideas, leading to some beautiful mathematics and extremely useful results in a wide range of application areas. In 1961 Eshelby conjectured that for anisotropic materials only ellipsoidal inhomogeneities would lead to such uniform interior fields. Although much progress has been made since then, the quest to prove this conjecture is still not complete; numerous important problems remain open. Following a different approach to that considered by Eshelby, a closely related tensor P = SD 0 arises, where D 0 is the host medium compliance tensor. The tensor P is associated with Hill and is of course also uniform when ellipsoidal inhomogeneities are embedded in a homogeneous host phase. Two of the most fundamental and useful areas of applications of these tensors are in Newtonian potential problems such as heat conduction, electrostatics, etc. and in the vector problems of elastostatics. Knowledge of the Hill and Eshelby tensors permit a number of interesting aspects to be studied associated with inhomogeneity problems and more generally for inhomogeneous media. Micromechanical methods established mainly over the last half-century have enabled bounds on and predictions of the effective properties of composite media. In many cases such predictions can be explicitly written down in terms of the Hill tensor, or equivalently the Eshelby tensor and can be shown to provide excellent predictions in many cases.Of specific interest is that a number of important limits of the ellipsoidal inhomogeneity can be taken in order to be employed in predictions of the effective properties of, for example, layered media and fibre reinforced composites and also to the cases when voids and cracks are present. In the main, results for the Hill and Eshelby tensors are distributed over a wide range of articles and books, using different notation and terminology and so it is often difficult to extract the necessary information for the tensor that one requires. The case of an anisotropic host phase is also frequently non-trivial due to the requirement of the associated Green's tensor. Here this classical problem is revisited and a large number of results for problems that are felt to be of great utility in a wide range of disciplines are derived or recalled. A scaling argument leads to the derivation of the Eshelby tensor for potential problems where the host phase is at most orthotropic, without the requirement of using the anisotropic Green's function. The Concentration tensor A linking interior fields to those imposed in the far-field is de...

The Newtonian potential inhomogeneity problem: non-uniform eigenstrains in cylinders of non-elliptical cross section

“…The problem is then solved in the scaled domain and mapped back to physical space [7]. Interest here is for general shaped inhomogeneities, and therefore we consider the isotropic problem only.…”

“…Of course, this merely changes the sign of the Green's function, but it is important in terms of lifting information directly from one reference to another. We follow the convention in [7,28]. Equation (2.5) is an integral equation for the potential gradient e and determining this insideD, which we shall denote as e 1 then tells us the field everywhere.…”

“…where A = [I + (κ − 1)S] −1 is known as the concentration tensor [7]. For imposed fields that are of polynomial form a polynomial eigenstrain of the same order can be employed to represent an inhomogeneity [28].…”

“…On the other hand, this being true for a uniform specific far-field strain is known as the strong Eshelby conjecture. Given the similarity of the Newtonian potential and linear elastostatics problems, differing mainly by their tensorial order, they are often discussed simultaneously [7], and hence the use of this term in both contexts. The strong conjecture in the context of the potential problem is true in two dimensions [8,9], but it is not true in dimensions greater than two.…”

“…These can be shown to be uniform for homogeneous ellipsoidal inhomogeneities. These tensors arise specifically in the context of micromechanics and effective medium theory, with the aim of predicting the effective properties of inhomogeneous media [7,11]. Speaking in the context of thermal conductivity as we do in this article, components of the Eshelby tensor written with respect to Cartesian coordinates in the isotropic case take the form:…”