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Materials (Basel). 2012 Jan 30;5(2):210-226. doi: 10.3390/ma5020210.

Stresses and Displacements in Functionally Graded Materials of Semi-Infinite Extent Induced by Rectangular Loadings.

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Shandong Key Laboratory of Civil Engineering Disaster Prevention & Mitigation, Shandong University of Science and Technology, Shandong, Qingdao 266590, China.
Department of Civil Engineering, The University of Hong Kong, Hong Kong, China.


This paper presents the stress and displacement fields in a functionally graded material (FGM) caused by a load. The FGM is a graded material of Si₃N₄-based ceramics and is assumed to be of semi-infinite extent. The load is a distributed loading over a rectangular area that is parallel to the external surface of the FGM and either on its external surface or within its interior space. The point-load analytical solutions or so-called Yue's solutions are used for the numerical integration over the distributed loaded area. The loaded area is discretized into 200 small equal-sized rectangular elements. The numerical integration is carried out with the regular Gaussian quadrature. Weak and strong singular integrations encountered when the field points are located on the loaded plane, are resolved with the classical methods in boundary element analysis. The numerical integration results have high accuracy.


FGM; Kelvin solution; Yue’s solution; elasticity; functionally graded materials; multilayered solids; numerical integration; stress analysis

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