A Hyperbolic Shear Deformation Theory for Natural Frequencies Study of Functionally Graded Plates on Elastic Supports

Abstract

This study presents a hyperbolic shear deformation theory for free vibration of functionally graded plates on elastic foundations. The field of displacements is chosen based on the assumptions that axial and transverse displacements consist of components due to bending and shear. The components of the axial shear displacements give rise to the parabolic variation in the shear strain through the thickness, such that the shear stresses vanish on the plate boundaries. Therefore, the shear correction factor is not necessary. The material properties of the functionally graded plate are assumed to vary through the thickness according to the power law of the volume fraction of the constituents. The elastic foundation is modeled as a Pasternak foundation. The equations of motion are derived using Hamilton’s principle. The analytical solutions were established from Navier’s approach, and the results obtained are found to be in good agreement with the solutions of three-dimensional elasticity and with the solutions of the various plate theories. The effects of the power law index, the thickness ratio, and the foundation parameters on the natural frequency of the plates were also evaluated.