Skip to main content


Weitere Artikel dieser Ausgabe durch Wischen aufrufen

08.03.2016 | Review | Ausgabe 2/2016 Open Access

Journal of Elasticity 2/2016

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

Journal of Elasticity > Ausgabe 2/2016
William J. Parnell


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 \(\mathbf{S}\). 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 \(\mathbf{P}=\mathbf{S}\mathbf{D}^{0}\) arises, where \(\mathbf{D}^{0}\) is the host medium compliance tensor. The tensor \(\mathbf{P}\) 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 \(\boldsymbol{\mathcal{A}}\) linking interior fields to those imposed in the far-field is derived for a wide variety of problems. These results can therefore be used in the various micromechanical schemes.
Directly related to the tensors of Eshelby and Hill is the so-called Moment tensor \(\mathbf{M}\). As well as arising in the literature on micromechanics, this tensor is important in the vast area of research associated with inverse problems and specifically with the problem of identifying an object inside some domain given the application of a specific set of boundary conditions. Due to its fundamental importance and direct link to the Eshelby and Hill tensors, here we state the connection between \(\mathbf{M}, \mathbf{P}\) and \(\mathbf{S}\) in an effort to ensure that the work is of use to as wide a community as possible.
Both tensor and matrix formulations are considered and contrasted throughout. Appendices give various details that illustrate the implementation of both approaches.

Unsere Produktempfehlungen

Springer Professional "Wirtschaft+Technik"


Mit dem Kombi-Abo erhalten Sie vollen Zugriff auf über 1,8 Mio. Dokumente aus mehr als 61.000 Fachbüchern und rund 500 Fachzeitschriften aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Umwelt
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe

Testen Sie jetzt 30 Tage kostenlos.

Premium-Abo der Gesellschaft für Informatik

Sie erhalten uneingeschränkten Vollzugriff auf alle acht Fachgebiete von Springer Professional und damit auf über 45.000 Fachbücher und ca. 300 Fachzeitschriften.

Springer Professional "Technik"


Mit dem Technik-Abo erhalten Sie Zugriff auf über 1 Mio. Dokumente aus mehr als 40.000 Fachbüchern und 300 Fachzeitschriften aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Umwelt
  • Maschinenbau + Werkstoffe

Testen Sie jetzt 30 Tage kostenlos.

Über diesen Artikel

Weitere Artikel der Ausgabe 2/2016

Journal of Elasticity 2/2016 Zur Ausgabe

Premium Partner

BranchenIndex Online

Die B2B-Firmensuche für Industrie und Wirtschaft: Kostenfrei in Firmenprofilen nach Lieferanten, Herstellern, Dienstleistern und Händlern recherchieren.



Grundlagen zu 3D-Druck, Produktionssystemen und Lean Production

Lesen Sie in diesem ausgewählten Buchkapitel alles über den 3D-Druck im Hinblick auf Begriffe, Funktionsweise, Anwendungsbereiche sowie Nutzen und Grenzen additiver Fertigungsverfahren. Eigenschaften eines schlanken Produktionssystems sowie der Aspekt der „Schlankheit“ werden ebenso beleuchtet wie die Prinzipien und Methoden der Lean Production.
Jetzt gratis downloaden!


Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.