# An alternative numerical scheme for calculating the thermal stresses around an inclusion of arbitrary shape in an elastic plane under uniform remote in-plane heat flux

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## Abstract

Based on the complex variable method, the decoupled thermoelastic problem of an infinite matrix containing an arbitrarily shaped inclusion subjected to plane deformations and uniform remote heat flux is studied in this paper. The shape of the inclusion is defined by a polynomial conformal mapping. Faber series and Fourier expansion techniques are used to solve the corresponding boundary value problems. Several numerical examples are presented to study the influence of the hardness and the heat conductivity of the inclusion on the concentration of the interfacial Von Mises stress and tangential stress in the matrix for a uniform remote uniaxial heat flux. It is shown that for given thermal expansion coefficients and given heat conductivities of the inclusion–matrix system, the Von Mises stress and the tangential stress in the matrix around the inclusion both increase significantly with increasing hardness of the inclusion whether the inclusion is softer or harder than the matrix. On the other hand, it is found that for given thermal expansion coefficients of the inclusion–matrix system, the Von Mises stress in the matrix around the inclusion decreases significantly with increasing heat conductivity of the inclusion, while the tangential stress concentration first decreases and then increases with increasing heat conductivity of the inclusion whether the inclusion is softer or harder than the matrix.

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## Notes

### Acknowledgements

The authors thank the support from the National Natural Science Foundation of China (Grant Number 11702147).

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