Free vibrations of thin functionally graded plates with microstructure

• Vibrations of thin functionally graded plates with microstructure are investigated.
• The microstructure size of the plates is of an order of the plate thickness.
• The tolerance modelling technique is applied and the asymptotic-tolerance model is proposed.
• Vibrations are described by differential equations with slowly-varying coefficients.
• Higher frequencies are obtained by the tolerance and asymptotic-tolerance models.

There are considered functionally graded plates with a microstructure. The size of the microstructure is of an order of the plate thickness. It is assumed that the functionally graded macrostructure of these plates is only along one axis. Because the effect of the microstructure on dynamic behaviour of the plates under consideration plays a crucial role the tolerance modelling method is applied, cf. Kaźmierczak and Jędrysiak, Engng. Struct., 2013, cf. Jędrysiak, Thin-Walled Struct., 2013. This method makes it possible to obtain model equations with smooth functional coefficients, describing the effect of the microstructure size. In this paper it is shown that this effect can be taken into account in two models: the asymptotic-tolerance model and the tolerance model. Applications of these models with a comparison to results by the asymptotic model are presented for various boundary conditions and different distribution functions of material properties in an analysis of free vibration frequencies of functionally plate strips. Moreover, a certain verification for chosen cases is also shown by using the finite element method.