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In this study, a new four-unknown quasi-3D shear deformation theory is proposed for studying the vibration responses of functionally graded (FG) beams containing open-edge cracks resting on three-parameter viscoelastic foundations (VEFs). The number of unknowns and governing equations in the current theory has been reduced, making it easier to use. Even less than conventional theories, this theory includes indeterminate integral variables and contains only four unknowns where no shear correction factor is used. The study is conducted with an eye toward a three-parameter foundation that takes into account the effects of the elastic medium's damping coefficient, the Pasternak coefficient, and the Winkler coefficient. The material characteristics of the FG beams are considered to vary in the thickness direction via a power law distribution as a function of the volume fractions of the constituents. The system of differential equations governing the free vibration behavior of FG beams is derived by Hamilton's principle. To satisfy the foundation conditions, the Navier method is used to obtain the analytical solutions of the dynamic response of cracked FG beams resting on viscoelastic foundations. Comparison of the results of the current theory with other results and with data available in the literature demonstrates its accuracy. A detailed parametric study is presented to show the impact of material properties, slenderness ratio, foundation type and foundation damping coefficient, crack depth, and location on the natural frequencies of cracked FG beams resting on VEFs. [ABSTRACT FROM AUTHOR]

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]

In this study, vibration response of functionally graded plate resting on a viscoelastic foundation is studied. An analytical solution is proposed using a high-order shear deformation theory with only four unknowns, which means that the current theory has few unknowns compared to the first and other high shear deformation theories. The present theory uses a displacement field with integer terms instead of derivative terms by also including the shear deformation effect without introducing the shear correction factors. The mathematical model of the foundation used followed the Winkler–Pasternak two-coefficient model, with an additional term added to represent the damping effect. The governing equations were generated using the principle of virtual works. Subsequently, the analytical solution is based on Navier's principle to solve the vibration problem of a simply supported FG plate resting on a viscoelastic foundation. Some numerical results are presented to demonstrate the impact of the material index, the type of elastic foundation and the damping coefficient of the foundation on the dynamic response of FG plates resting on viscoelastic foundations. In the end, it is concluded that the current results with the proposed theory are found to be in good agreement with the other available results and this theory can easily be used to solve the free vibration problems of FGM plates resting on visco-Pasternak medium. [ABSTRACT FROM AUTHOR]

This present paper concerned with the analytic modelling for vibration of the functionally graded (FG) plates resting on non-variable and variable two parameter elastic foundation, based on two-dimensional elasticity using higher shear deformation theory. Our present theory has four unknown, which mean that have less than other higher order and lower theory, and we denote do not require the factor of correction like the first shear deformation theory. The indeterminate integral are introduced in the fields of displacement, it is allowed to reduce the number from five unknown to only four variables. The elastic foundations are assumed a classical model of Winkler-Pasternak with uniform distribution stiffness of the Winkler coefficient (kw), or it is with variables distribution coefficient (kw). The variable's stiffness of elastic foundation is supposed linear, parabolic and trigonometry along the length of functionally plate. The properties of the FG plates vary according to the thickness, following a simple distribution of the power law in terms of volume fractions of the constituents of the material. The equations of motions for natural frequency of the functionally graded plates resting on variables elastic foundation are derived using Hamilton principal. The government equations are resolved, with respect boundary condition for simply supported FG plate, employing Navier series solution. The extensive validation with other works found in the literature and our results are present in this work to demonstrate the efficient and accuracy of this analytic model to predict free vibration of FG plates, with and without the effect of variables elastic foundations. [ABSTRACT FROM AUTHOR]