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Eshelby force and power for uniform bodies

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Abstract

Inspired by the seminal works of Eshelby (Philos Trans R Soc A 244A:87–112, 1951, J Elast 5:321–335, 1975) on configurational forces and of Noll (Arch Ration Mech Anal 27:1–32, 1967) on material uniformity, we study a thermoelastic continuum undergoing volumetric growth and in a dynamical setting, in which we call the divergence of the Eshelby stress the Eshelby force. In the classical statical case, the Eshelby force coincides with the negative of the configurational force. We obtain a differential identity for the modified Eshelby stress, involving the torsion of the connection induced by the material isomorphism of a uniform body, which includes, as a particular case, that found by Epstein and Maugin (Acta Mech 83:127–133, 1990). In this identity, the divergence of the modified Eshelby stress with respect to this connection of the material isomorphism takes the name of modified Eshelby force. Moreover, we show that Eshelby’s variational approach (1975) can be used to formulate not only the balance of material momentum, but also the balance of energy. In this case, we find that what we call Eshelby power is the temporal analogue of the Eshelby force, and we obtain a differential identity for the modified Eshelby power. This leads to concluding that the driving force for the process of growth–remodelling is the Mandel stress. Eventually, we find that the relation between the differential identities for the modified Eshelby force and modified Eshelby power represents the mechanical power expended in a uniform body to make the inhomogeneities evolve.

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Change history

  • 28 October 2020

    In the original paper, we erroneously used the total potential energy (Helmholtz free energy of the material plus potential energy of the external forces) in the entropy inequality.

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Acknowledgements

Libyan Ministry of Higher Education (MFA), Natural Sciences and Engineering Research Council of Canada through the NSERC Discovery Programme (ME, SF).

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Authors and Affiliations

  1. Graduate Programme in Mechanical Engineering, The University of Calgary, Calgary, Canada

    Mawafag F. Alhasadi

  2. Department of Mechanical and Manufacturing Engineering, The University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4, Canada

    Marcelo Epstein & Salvatore Federico

Authors
  1. Mawafag F. Alhasadi
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  2. Marcelo Epstein
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  3. Salvatore Federico
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Corresponding author

Correspondence to Salvatore Federico.

Additional information

This work is dedicated to the memory of Prof. Gaetano Giaquinta (Catania, Italy, 25 November 1945–13 August 2016). Prof. Giaquinta was a professor of Structure of Matter at the Università di Catania (Italy) and a multiform intellectual. He has had a profound influence on his students, both from the scientific and the humanistic points of view, and has left an unbridgeable hole in their hearts.

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Alhasadi, M.F., Epstein, M. & Federico, S. Eshelby force and power for uniform bodies. Acta Mech 230, 1663–1684 (2019). https://doi.org/10.1007/s00707-018-2353-6

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  • DOI: https://doi.org/10.1007/s00707-018-2353-6

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