Three-Dimensional Free Vibration Analyses of Preloaded Cracked Plates of Functionally Graded Materials via the MLS-Ritz Method.

In this study, the moving least squares (MLS)-Ritz method, which involves combining the Ritz method with admissible functions established using the MLS approach, was used to predict the vibration frequencies of cracked functionally graded material (FGM) plates under static loading on the basis of the three-dimensional elasticity theory. Sets of crack functions are proposed to enrich a set of polynomial functions for constructing admissible functions that represent displacement and slope discontinuities across a crack and appropriate stress singularity behaviors near a crack front. These crack functions enhance the Ritz method in terms of its ability to identify a crack in a plate. Convergence studies of frequencies and comparisons with published results were conducted to demonstrate the correctness and accuracy of the proposed solutions. The proposed approach was also employed for accurately determining the frequencies of cantilevered and simply supported side-cracked rectangular FGM plates and cantilevered internally cracked skewed rhombic FGM plates under uniaxial normal traction. Moreover, the effects of the volume fractions of the FGM constituents, crack configurations, and traction magnitudes on the vibration frequencies of cracked FGM plates were investigated.

Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates.

This study aimed to develop series analytical solutions based on the Mindlin plate theory for the free vibrations of functionally graded material (FGM) rectangular plates. The material properties of FGM rectangular plates are assumed to vary along their thickness, and the volume fractions of the plate constituents are defined by a simple power-law function. The series solutions consist of the Fourier cosine series and auxiliary functions of polynomials. The series solutions were established by satisfying governing equations and boundary conditions in the expanded space of the Fourier cosine series. The proposed solutions were validated through comprehensive convergence studies on the first six vibration frequencies of square plates under four combinations of boundary conditions and through comparison of the obtained convergent results with those in the literature. The convergence studies indicated that the solutions obtained for different modes could converge from the upper or lower bounds to the exact values or in an oscillatory manner. The present solutions were further employed to determine the first six vibration frequencies of FGM rectangular plates with various aspect ratios, thickness-to-width ratios, distributions of material properties and combinations of boundary conditions.