#### 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 ($\xi$-) plane. The corresponding boundary value problem is then analyzed in the $\xi$-plane. A second conformal mapping, which maps the exterior of the region occupied by the (simply-connected) inclusion in the $\xi$-plane onto the exterior of the unit circle, is then used to construct an auxiliary function of $\xi$ which, when used together with analytic continuation, allows us to extend our analysis to an inclusion of arbitrary shape.

##### 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