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In this study, we have introduced, for the first time, a novel integral quasi-3D higher-order shear deformation theory (HSDT) employing a third-order shape function. This approach is employed to analyze the free vibration characteristics of a cracked porous functionally graded material (FGM) beam supported on a three-parameter elastic foundation (Winkler/Pasternak/Kerr). This new Quasi HSDT introduces a stretching effect that surpasses the capabilities of FSDT and other HDST. The employed shape function satisfies the conditions of shear stress nullity at both the higher and lower facets without the need for correction factors. The study incorporates a mathematical model representing Winkler/Pasternak/Kerr foundation types into the Hamiltonian to derive the equations of motion. The FGM beam studied in this paper is assumed to be composed of materials with a distribution that varies according to a power law along its height. Our results are compared with previous studies and we reinforce our findings with a parametric study assessing the impact of crack attributes on the natural frequencies of the FG plate. This study presents an advanced integral quasi-3D HSDT, applied for the first time, to analyze the behavior of FG beams resting on a three-parameter foundation. [ABSTRACT FROM AUTHOR]

In this paper, wave propagation in functionally graded (FG) porous plates resting on Winkler-Pasternak foundation is studied using a quasi-3D shear deformation theory. The proposed theory has a new displacement field that includes indeterminate integral terms and contains fewer unknown variables taking into account the effect of transverse shear and thickness stretching. The parameters of the elastic foundation are introduced in the present formulation following the mathematical model of Pasternak. In addition, the effect of porosity is studied. The material of the FG plate is inhomogeneous and the material properties are assumed to vary continuously in the thickness direction according to a power law of the volume fraction. The equations governing wave propagation in the plates resting on an elastic foundation are derived using Hamilton's principle. Then, the dispersion relationship between frequency and wave number is solved analytically. A comprehensive numerical result is accomplished to evaluate the effects of the volume fraction index, the porosity, thickness ratio (h/a), and the wave number on the wave propagation in functionally graded porous plates are discussed in detail. [ABSTRACT FROM AUTHOR]