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Abstract A general solution of the stress concentration in a homogeneous plate with an arbitrary shape hole coated by a functionally graded layer is presented under the remote uniform loads. With using the method of piece-wise homogeneous layers, the functionally graded layer in which the material properties change continuously along the normal direction of the hole is approximately decomposed to N homogeneous layers. When N is chosen to be large enough, the material constants in each layer can be regarded as unchanged. By means of the technique of conformal mapping, N homogeneous layers in -plane are transformed into N concentric circular rings in -plane, and then the complex potentials in each circular ring and the plate in -plane can be given in the form of series with unknown coefficients based on the theory of the complex variable functions. The stress and displacement continuous conditions on the interfaces of each homogeneous layer are used to produce a set of linear equations containing all the unknown coefficients. Through solving these linear equations, the complex potentials can be finally obtained in each layer and the plate. Numerical results of stress distribution around the holes with various shapes including circle, ellipse, triangle, square, rectangle etc. are presented for different varying Young’s modulus. It is shown that the stress concentrations around the elliptical and rectangle holes are more obvious than those of circular and square holes, respectively, and the most obvious one is triangle hole. Moreover, it is most important that the influence of the gradient exponent of Young’s modulus on the stress distributions is noticeable for all shape holes, and the stress concentrations decrease remarkably as the exponent value increases. Therefore, it can be concluded that the existence of the functionally graded layer influence obviously the stress distribution around the holes with various shapes, and the stress concentration can be effectively reduced by choosing proper change ways of the normal elastic properties in the functionally graded layer.
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Received: 29 March 2017
Published: 29 January 2018
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2018, Vol. 39 
