The pointwise Eshelby force on the interface between a transformed inclusion and its surrounding matrix

Eshelby showed that the pointwise force F on and normal to the interface between a transformed inclusion and its surrounding matrix is the jump in the normal component of the elastic energy-momentum tensor across the interface. Gavazza later showed, using an entirely different approach, that this thermodynamic driving force F has a much simpler form involving only the average of the stress tensors at adjacent points on opposite sides of the interface and the “transformation strain” tensor. The equivalence of and connection between the two formulae was apparently first shown by Eshelby in a personal letter to Gavazza (attached as an appendix to this paper), although the brevity of the letter makes following Eshelby’s proof a little difficult. Here we expand Eshelby’s hitherto unpublished proof of the equivalence of the two expressions in what we believe is a clearer fashion.