TSDT vs CPT and FSDT for Free Vibration Analysis of Functionally Graded Incompressible Plates

Abstract

This article studies vibrational behavior of incompressible functionally graded plates through using classical, first-order shear, and third-order shear deformation plate theories. The plate material properties are assumed to vary continuously in the direction of thickness according to a power law function. The equations of motion are derived together with the continuity equations and are then solved analytically for rectangular plates with simple supports. The natural frequencies for the homogeneous state of the plate are compared with the results of other theories presented in the previous works. This comparison shows that in transverse vibration analysis, classical and first-order shear theories for incompressible plates are not as accurate as for compressible ones. It is shown that this issue is due to the fact that according to the classical and the shear theory of the first order, unlike the higher order theories, the hydrostatic pressure that appears in the incompressible materials cannot carry bending (transverse) loads. Therefore, the equivalent bending stiffness of the plate and as a result its bending frequencies decrease. Consequently, to analyze the vibration behavior of plates made of functionally graded incompressible materials, whether the plate is either thin or thick, higher order theories should be used. Finally, third-order shear deformation theory has been taken into consideration to study the vibration behavior of rectangular plates made of functionally graded incompressible materials, and detailed parametric studies have been carried out based on it.