Vol. 7, No. 3, 2019

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Eshelby's inclusion theory in light of Noether's theorem

Salvatore Federico, Mawafag F. Alhasadi and Alfio Grillo

Vol. 7 (2019), No. 3, 247–285
Abstract

In a variational setting describing the mechanics of a hyperelastic body with defects or inhomogeneities, we show how the application of Noether’s theorem allows for obtaining the classical results by Eshelby. The framework is based on modern differential geometry. First, we present Eshelby’s original derivation based on the cut-replace-weld thought experiment. Then, we show how Hamilton’s standard variational procedure “with frozen coordinates”, which Eshelby coupled with the evaluation of the gradient of the energy density, is shown to yield the strong form of Eshelby’s problem. Finally, we demonstrate how Noether’s theorem provides the weak form directly, thereby encompassing both procedures that Eshelby followed in his works. We also pursue a declaredly didactic intent, in that we attempt to provide a presentation that is as self-contained as possible, in a modern differential geometrical setting.

We dedicate this work to the memory of our maestro Professor Gaetano Giaquinta (Catania, Italy, 1945–2016), who first taught us Noether's theorem and showed us its unifying beauty.

Keywords
Eshelby stress, energy-momentum tensor, configurational mechanics, inclusion, defect, Noether's theorem, variational principle
Physics and Astronomy Classification Scheme 2010
Primary: 02.40.-k, 11.10.-z, 83.10.Ff, 02.30.Xx, 91.60.Ed
Milestones
Received: 6 February 2019
Revised: 15 June 2019
Accepted: 17 August 2019
Published: 22 December 2019

Communicated by David J. Steigmann
Authors
Salvatore Federico
Department of Mechanical and Manufacturing Engineering
University of Calgary
Calgary, AB Canada
Mawafag F. Alhasadi
Department of Mechanical and Manufacturing Engineering
University of Calgary
Calgary, AB Canada
Alfio Grillo
Deptartment of Mathematical Sciences “G.L. Lagrange”
Politecnico di Torino
Dipartimento di Eccellenza
Torino
Italy