Acta Mech DOI 10.1007/s00707-016-1734-y ORIGINAL PAPER Mawafag F. Alhasadi · Salvatore Federico Relation between Eshelby stress and Eshelby fourth-order tensor within an ellipsoidal inclusion Received: 23 May 2016 / Revised: 8 September 2016 © Springer-Verlag Wien 2016 Abstract The pioneering work by John D. Eshelby in the 1950s and the 1960s on the theory of materials with defects has opened the doors to what today we call configurational mechanics or, in his honour, Eshelbian mechanics. Two of the main results that Eshelby obtained in this field are the use of the elastic energy- momentum tensor to calculate the net force on a defect and the study of materials with inclusions from the geometrical point of view. In Continuum Mechanics, the energy-momentum tensor is now commonly referred to as the Eshelby stress and is the physical quantity that captures the presence of singularities, such as point defects, inclusions, dislocations. In the study of materials with inclusions, Eshelby established a method for the calculation of the strain and stress fields, which entails a fourth-order tensor that relates the strain in the inclusion to the virtual strain (transformation strain or eigenstrain) defining the geometrical misfit between inclusion and matrix. Surprisingly, perhaps, the scientific communities in these two streams of research seem to have had little or no interaction, i.e. virtually all those researchers that have worked in terms of the Eshelby stress have never used the Eshelby fourth-order tensor, and vice versa. To the best of our knowledge, there exists no explicit mathematical relation between the two objects. Therefore, the objective of this paper is to study the relationship between the Eshelby stress and the Eshelby fourth-order tensor within an ellipsoidal inclusion, in the infinitesimal theory of elasticity. Of the three cases that shall be analysed, the first two are commonly referred to as “homogeneous inclusion” and “inhomogeneous inclusion” in the literature, while we refer to the latter as to “general inclusion”, since it describes both the other two as particular cases. 1 Introduction The work by Eshelby [1–4] in the mechanics of materials with defects has turned out to be of fundamental importance since at least the 1970s. We are particularly interested in two developments of Eshelby’s work: the “material” mechanics that arises from the study of what Eshelby called the energy-momentum tensor, and the problem of the ellipsoidal inclusion. Eshelby [1] studied the theory of the configurational force on a singularity and showed that this force can be expressed in terms of the surface integral of the tractions caused by the energy-momentum tensor as shown in Eq. (21). This configurational force is the “driving force” for the repositioning of a defect or, in In memory of Prof Gerard A. Maugin (1944-12-02, 2016-09-23), a lighthouse in Continuum Mechanics. M. F. Alhasadi Graduate Programme in Mechanical Engineering, The University of Calgary, 2500 University Drive NW, Calgary, AB T2N 1N4, Canada S. Federico (B ) Department of Mechanical and Manufacturing Engineering, The University of Calgary, 2500 University Drive NW, Calgary, AB T2N 1N4, Canada E-mail: salvatore.federico@ucalgary.ca