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Thermal Stresses for Functionally Graded Materials (FGMs) Subject to Heat Flux



In this paper, the thermal stress due to heat flux at the far field is derived for an infinitely extended elastic medium which contains a spherical inclusion made of functionally graded materials (FGMs). The 3-D heat conduction equation subject to uniform heat flux at the far field is solved analytically to derive the temperature distribution. Based on the temperature solution, the thermal stress field due to heat flux is obtained by solving a set of two ordinary differential equations using the method of weighted residuals. Unlike the two-phase homogeneous medium, the von Mises stress distribution is continuous at the interface of the FGM-matrix medium.


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Nomura, S. and Krishna, N. 2020, “Analysis of composites reinforced with functionally graded

inclusions,” in Proceedings of the American Society for Composites 2020-Thirty-Fifth Technical

Conference, DEStech Publications.

Filippi, M., Carrera, E. and Zenkour, A.M. 2015. “Static analyses ofFGMbeams by various theories

and finite elements”, Composites Part B: Engineering, vol. 72, 1–9.

Sankar, B.V. 2001. “An elasticity solution for functionally graded beams” J. Composites Science

and Technology, vol. 61, 689–696.

Ravichandran, K.S. 1995. “Thermal residual stresses in a functionally graded material system” J.

Material Science and Engineering: A, vol. 201, 269–276.

Lutz, M. P. and Zimmerman, R. W. 2007. “Thermal stresses and effective thermal expansion

coefficient of a functionally graded sphere,” J. Thermal Stresses, 19:39-54.

Tauchert, T. R. 1968. “Thermal Stresses at Spherical Inclusions in Uniform Heat Flow,” J. Comp.

Matr., 2(4):478–486.

Wolfram, S. 2003. The Mathematica Book, Fifth Edition, Wolfram Media, Inc.

Finlayson, B. 2013. The Method of Weighted Residuals and Variational Principles, Society for

Industrial and Applied Mathematics.

Nomura, S. 2016. Micromechanics with Mathematica. Wiley.

Nomura, S. 2013. “Thermal stress analysis for multiphase composites” Proceedings of the Ameri-

can Society for Composites 2013-Twenty-Eighth Technical Conference, DEStech Publications.


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