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Free vibrations of porous functionally graded material (FGM) plates with complex shapes are analyzed by using the R-functions method. The thickness of the plate is variable in the direction of one of the axes. Two types of porosity distributions through the thickness are considered: uniform (even) and non-uniform (uneven). The elastic foundation is defined by two parameters (Winkler and Pasternak). To obtain the mathematical model of the problem, the first-order shear deformation theory of the plate (FSDT) is used. The effective material properties in the thickness direction are modeled by means of a power law. Variational Ritz’s method joined with the R-functions theory is used for obtaining a semi-analytical solution of the problem. The approach is applied to a number of case studies and validated by means of comparative analyses carried out on rectangular plates with a traditional finite element approach. The proof of the efficiency of the approach and its capability to handle actual engineering problems is fulfilled for FGM plates having complex shapes and various boundary conditions. The effect of different parameters, such as porosity distribution, volume fraction index, elastic foundation, FGM types, and boundary conditions, on the vibrations is studied.

Free vibrations of shallow sandwich shells resting on elastic foundations are investigated. It is assumed that the shell consists of three layers of defined thickness. The core is made of ceramics or metal, while the upper and lower layers are made of functionally graded material (FGM). The volume fractions of metal and ceramics are described by the power law. Hence, estimation methods for higher accuracy remain a challenge. The elastic foundation is described by two-parameter Pasternak’s model. Both higher-order shear deformation shell theory (HSDT), which includes interactions with elastic foundation, and the R-functions theory combined with the variational Ritz method are used to study shells with arbitrary planforms. The numerical study is carried out in the framework of the refined third-order theory. The proposed method and developed numerical techniques have been validated on test problems for FG shell with rectangular planform. Furthermore, new results for shallow shells with a cut-out of the complex form are obtained. The effects of the gradient index, boundary conditions, thickness of core and face sheet layers, as well as elastic foundations on fundamental frequencies, are investigated. In addition, we demonstrate how the divergence of the results varies with the gradient index. The considered studied cases show that the natural frequencies depend on the foundation parameters, the thickness of the layers, boundary conditions, and the thickness arrangement.

In this paper, the dynamic behavior of 3D-printed plates with different shapes and boundary conditions is investigated. The natural frequencies and mode shapes were determined using three different methods: the experimental analysis, the finite element method, using Nastran, and the R-functions method. The experimental and theoretical results are compared. The specimens tested included four cases. The test procedure is deeply described, and the material properties of the plates are given. The fixed-fixed configuration shows a better agreement both in the rectangular plate and in the plate with rectangular cuts, and the R-functions method gives better convergence with respect to the experimental and finite element analysis. The simply supported arrangement indicates some uncertainty in the boundary realization of the specimen.

Nonlinear free vibration of functionally graded shallow shells with complex planform is investigated using the R-functions method and variational Ritz method. The proposed method is developed in the framework of the first-order shear deformation shallow shell theory. Effect of transverse shear strains and rotary inertia is taken into account. The properties of functionally graded materials are assumed to be varying continuously through the thickness according to a power law distribution. The Rayleigh–Ritz procedure is applied to obtain the frequency equation. Admissible functions are constructed by the R-functions theory. To implement the proposed approach, the corresponding software has been developed. Comprehensive numerical results for three types of shallow shells with positive, zero and negative curvature with complex planform are presented in tabular and graphical forms. The convergence of the natural frequencies with increasing number of admissible functions has been checked out. Effect of volume fraction exponent, geometry of a shape and boundary conditions on the natural and nonlinear frequencies is brought out. For simply supported rectangular FG shallow shells, the results obtained are compared with those available in the literature. Comparison demonstrates a good accuracy of the approach proposed.

In present work, an effective method to research geometrically nonlinear free vibrations of elements of thin-walled constructions that can be modeled as laminated shallow shells with complex planform is applied. The proposed method is numerical–analytical. It is based on joint use of the R-functions theory, variational methods, Bubnov–Galerkin procedure and Runge–Kutta method. The mathematical formulation of the problem is performed in a framework of the refined first-order shallow shells theory. To implement the developed method, appropriate software was developed. New problems of linear and nonlinear vibrations of laminated shallow shells with clamped cutouts are solved. To confirm reliability of the obtained results, their comparison with the ones known in the literature is provided. Effect of boundary conditions is studied.