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

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