In this article, various non‐polynomial higher‐order shear deformation theories are applied for the first time to analyze the free vibration and transient responses of plates with functionally graded material (FGM) supported on an elastic foundation. The shear deformation theories account for the non‐linear variation of the transverse shear strains with various warping functions, namely trigonometric, inverse hyperbolic, and inverse trigonometric ones. These models also inherently satisfy the traction‐free boundary conditions of transverse shear stresses at the top and bottom surfaces of the plates and do not require any shear correction factor. A two‐parameter model, namely Winkler‐Pasternak's elastic foundation model, is utilized to develop the interaction between the FGM plates and the elastic medium. The governing equations of motion are obtained using Hamilton's principle and solved analytically using Navier's solution scheme. Furthermore, the transient responses of the plates are obtained using Newmark's average acceleration method. The applicability of the present theories is established by solving several numerical problems and validating the results with the solutions available in the literature. The effects of various parameters like span‐thickness ratios, aspect ratios, gradation coefficients, mechanical loads, and foundation stiffness on the fundamental frequencies and the transient responses of the plates are thoroughly investigated. The comparison of the results reveals the efficiency of the non‐polynomial functions, and the capability of efficient prediction of the structural responses of the FGM plates at a similar computational cost compared to established models in the literature. Furthermore, the results show that the stiffness of the elastic foundation can tweak the stiffness characteristics of the FGM plate resulting in significant changes in the natural frequencies and more controlled displacement‐time responses.

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This paper proposes new analytical and finite element solutions for studying the effects of elastic foundations on the uncontrolled and controlled static and vibration responses of smart multi-layered laminated composite plates with integrated piezoelectric layers, acting as actuators and sensors. A non-polynomial higher-order plate theory with zigzag kinematics involving a trigonometric function and a local segmented zigzag function is adopted for the first time for modeling the deformation of a smart piezoelectric laminated composite plate supported on an elastic foundation. This model has only five independent primary variables like that of the first-order shear deformation theory, yet it considers the realistic parabolic behavior of the transverse shear stresses across the thickness of the laminated composites plates, and also maintains the continuity conditions of transverse shear stresses at the interfaces of the laminated plates. A two-parameter foundation model, namely Pasternak’s foundation, is used to model the deformation and shear interactions of the elastic foundation. The governing set of equations is derived by implementing Hamilton’s principle and variational calculus. Two different solution methods, namely, a generalized closed-form analytical solution of Navier-type, and a C0 isoparametric finite element (FE) formulation, are developed for solving the governing set of equations. The solutions in the time domain are obtained with Newmark’s average acceleration method. Comprehensive parametric studies are presented to investigate the influence of elastic foundation parameters, piezoelectric layers, loading, and boundary conditions on the static and dynamic responses of the smart composite plates with piezoelectric layers. The effects of the elastic foundations on the vibration control of the smart composite plates are also presented by coupling the piezoelectric actuator and sensor with a feedback controller. Several benchmark results are presented to show the influence of the various material and geometrical parameters on the controlled and uncontrolled responses of the smart plates, and also the significant effect of the elastic foundations on the static and dynamic responses of the smart structures. The results obtained are in very good agreement with the available literature, and it can be concluded that the proposed analytical solution and FE formulation can be efficiently used to model the static and dynamic electro-elastic behavior of smart laminated plates supported on elastic foundations.

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This article is devoted to derive the analytical solution for flexural behavior of general symmetric and anti-symmetric cross-ply laminated composite and sandwich plates subjected to transverse mechanical load using the recently developed trigonometric zigzag theory. The inter-laminar continuity conditions of transverse shear stresses at the layer interfaces of the plate are enforced which is an essential condition for any zigzag model. The governing equations of equilibrium of the boundary value problem derived from the principle of minimum potential energy is reduced to a system of five partial differential equations whose solutions are obtained by Navier’s method. Attempt is made to demonstrate number of numerical problems to compare the results of the zigzag model with the elasticity solutions and with the results of other researchers in one common platform. Though in any solid mechanics problem, the displacement components are the primary unknowns, more attention is paid to the stress determination. Hence, the transverse shear stresses are evaluated using both the constitutive and equilibrium equations.

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The Trigonometric Zigzag theory is utilized in this research for analytically evaluating the forced vibration responses of smart multilayered laminated composite plates with piezoelectric actuators and sensors. This theory, as the name suggests, incorporates a trigonometric function, namely the secant function for describing the nonlinear behavior of transverse shear stresses through the thickness of the smart composite plates. The kinematics for the in-plane displacement components are obtained by superposing a globally varying nonlinear field through the thickness of the plate structure on a piecewise linearly varying zigzag field with slope discontinuities at the layer interfaces. The model also satisfies the inter-laminar continuity conditions of tractions at the interfaces of the multilayered plate. The equations of motion are derived using Hamilton’s principle, and the separation of the variables technique is extended to assume the solutions for the primary variables in space and time and solved analytically using Navier’s solution technique along with Newmark’s time integration scheme. A detailed analytical investigation of the dynamic behavior of the smart laminated plate coupled with piezoelectric materials like PVDF and piezoelectric fiber-reinforced composite (PFRC) is carried out by considering several forms of the time-dependent electromechanical excitations and also covering different geometrical and material features of the smart plate structure. The responses are found to be in close agreement with the elasticity solutions and some new results are also presented to show the dynamic controlling capacity of the piezoelectric layers.