Vol. 13, No. 2, 2018

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Eshelby inclusion of arbitrary shape in isotropic elastic materials with a parabolic boundary

Xu Wang, Liang Chen and Peter Schiavone

Vol. 13 (2018), No. 2, 191–202
Abstract

We employ analytic continuation and conformal mapping techniques to derive analytic solutions for Eshelby’s problem of an elastic inclusion of arbitrary shape in an isotropic elastic plane with parabolic boundary. The region of the physical (z-) plane lying below the parabola is mapped (conformally) onto the lower half of the image (ξ-) plane. The corresponding boundary value problem is then analyzed in the ξ-plane. A second conformal mapping, which maps the exterior of the region occupied by the (simply-connected) inclusion in the ξ-plane onto the exterior of the unit circle, is then used to construct an auxiliary function of ξ which, when used together with analytic continuation, allows us to extend our analysis to an inclusion of arbitrary shape.

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Keywords
Eshelby inclusion, parabolic boundary, conformal mapping, analytic continuation, auxiliary function
Milestones
Received: 2 November 2017
Revised: 29 November 2017
Accepted: 5 December 2017
Published: 27 May 2018
Authors
Xu Wang
School of Mechanical and Power Engineering
East China University of Science and Technology
130 Meilong Road
Shanghai, 200237
China
Liang Chen
School of Mechanical and Power Engineering
East China University of Science and Technology
130 Meilong Road
Shanghai, 200237
China
Peter Schiavone
Department of Mechanical Engineering
University of Alberta
10-203 Donadeo Innovation Center for Engineering
9211-116 Street NW
Edmonton AB T6G 1H9
Canada