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An Eshelby inclusion of arbitrary shape in a nonlinearly coupled thermoelectric material

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Abstract

In this paper, we present a general method, based on the techniques of analytic continuation and conformal mapping, for the analytic solution of Eshelby’s problem concerned with a two-dimensional inclusion of arbitrary shape in an infinite homogeneous and isotropic nonlinearly coupled thermoelectric plane. The inclusion is subjected to a prescribed uniform electric current-free thermoelectric potential gradient and a uniform energy flux-free temperature gradient. The corresponding boundary value problem is studied in both the physical and image planes. The closed-form general solution is found to be exact provided that the associated mapping function contains only a finite number of terms. Elementary expressions for the internal electric current density and energy flux in the physical plane are obtained. Examples of elliptical, hypotrochoidal and rectangular inclusions are presented to demonstrate the solution method. Interestingly, in this case Eshelby’s uniformity property is found invalid for an elliptical inclusion.

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Acknowledgements

This work is supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (Grant No: RGPIN – 2017 - 03716115112).

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Authors and Affiliations

  1. School of Mechanical and Power Engineering, East China University of Science and Technology, 130 Meilong Road, Shanghai, 200237, China

    Xu Wang

  2. Department of Mechanical Engineering, University of Alberta, 10-203 Donadeo Innovation Centre for Engineering, Edmonton, AB, T6G 1H9, Canada

    Peter Schiavone

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  1. Xu Wang
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  2. Peter Schiavone
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Correspondence to Peter Schiavone.

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Wang, X., Schiavone, P. An Eshelby inclusion of arbitrary shape in a nonlinearly coupled thermoelectric material. Z. Angew. Math. Phys. 74, 48 (2023). https://doi.org/10.1007/s00033-023-01936-8

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  • DOI: https://doi.org/10.1007/s00033-023-01936-8

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