Stability analysis of functionally graded plates based on the three-dimensional theory of elasticity

Abstract

Models available in publications for studying the stability of functionally graded plates are usually tested for some special cases available in publications, and then they are used to study a wide range of issues. This approach is fraught with the danger of making serious mistakes, especially when calculating structures on an elastic foundation. This report will develop two approaches to studying the stability of functionally graded plates in the 3D formulation, which minimize the probability of calculation errors.In the first approach, to construct differential equations of stability, a polynomial approximation of sought-for functions across the structure thickness is used. Its salient feature is assignment of the functions to the outer surfaces of layers, which allows splitting the layers into sublayers with a corresponding increase in the accuracy of calculation results. In the second approach, using the Reissner variational principle, a system of integrodifferential equilibrium equations and the corresponding boundary conditions are obtained without introducing an approximation. For the particular case of hinge-supported plates with a thermal load distributed according to a trigonometric law, the system of integrodifferential equilibrium equations obtained for the first approach and the differential equations of the second approach allow an analytical implementation. In the first approach, the system of differential equations is converted to a system of algebraic equations. The assignment of sought-for functions to the outer surfaces of layers allows one to split the layers into sublayers and thus to reduce the approximation error. Equations of the second approach are transformed into a system of ordinary differential equations for the distribution of required functions across the plate thickness, with an analytical search for the roots of characteristic equations and the corresponding eigenvectors. The same result obtained by the two methods may point to its reliability.In this message, for the particular case of hinged support, the proposed models are implemented analytically. The studies carried out have shown that the proposed applied model in the analysis of the stability of plates made of a functionally graded material with free outer surfaces provided high calculation accuracy without separating the layers into sublayers, even without the transverse compression. We also analysed the possibility of using the proposed approach with a polynomial approximation of the required functions by the thickness of the structure to study stability on an elastic foundation in the form of a half-infinite layer. The half-infinite layer in the calculation by the model (P) was modeled by a layer of finitely thickness. Its thickness was chosen in such a way that the calculation with a fixed and a free lower surface did not differ. The calculation with consideration of each layer within one layer in such task can be considered only as a calculation of the first rude approximation. The separation of the layers into ten sublayers gives the accuracy of the calculation, that is almost equivalent to the approach with the analytical search for the required functions by the thickness of the structure. Further fractination of the base into one hundred and sixty sub-layers gives the coincidence of the calculation results with an error of less than 1%. The presented results can be utilized as a benchmark for further studies of the stability of FGM plates. Further, it is assumed the numerical-analytical implementation of the proposed approaches .

Keywords

• Three-dimensional solution: Stability analysis of functionally graded plates