New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations

Zaoui, Fatima Zohra; Ouinas, Djamel; Tounsi, Abdelouahed

doi:10.1016/j.compositesb.2018.09.051

Cited by 155 publications

(61 citation statements)

References 99 publications

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“…In this section, we provide a brief overview of the mathematical fundamentals and governing equations of the problem with a special focus on the nonlocal structural response of shallow and deep isotropic doubly-curved nanoshells. The displacement field is modeled based on the combination of exponential, sinusoidal, and cosine strain functions due to their accuracy, as verified in Reference [47]. Figure 1 shows the geometrical scheme of the doubly-curved nanoshell analyzed in this scenario, where h is the thickness of the nanoshell, 1 R and 2 R refer to its curvature radii, a and b stand for its width and length, respectively.…”

Section: Governing Equations Of Doubly-curved Nanoshellsmentioning

confidence: 99%

“…This is in view of the lower accuracy of polynomial functions [47]. The selected shape function follows the shear deformation distribution throughout the thickness, and satisfies the shear stress-free surface conditions without considering any shear correction factor.…”

Section: Governing Equations Of Doubly-curved Nanoshellsmentioning

confidence: 99%

“…This is explored for isotropic doubly-curved nanoshells, where we propose a novel nonlocal shear deformation theory, based on a combination of exponential and trigonometric functions. These functions are selected for their higher accuracy compared to the polynomial functions [47]. The small-scale effect of the nanostructure is, thus, modeled, according to a differential law consequent, but not equivalent to the strain-driven nonlocal integral theory of elasticity supplemented with Helmholtz's averaging kernel, whereby the strain-displacement relations for symmetric nanoshells are based on the Reddy's doubly-curved shells theory.…”

Section: Introductionmentioning

confidence: 99%

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Nonlocal Elasticity Response of Doubly-Curved Nanoshells

Dindarloo

Dimitri

et al. 2020

Symmetry

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In this paper, we focus on the bending behavior of isotropic doubly-curved nanoshells based on a high-order shear deformation theory, whose shape functions are selected as an accurate combination of exponential and trigonometric functions instead of the classical polynomial functions. The small-scale effect of the nanostructure is modeled according to the differential law consequent, but is not equivalent to the strain-driven nonlocal integral theory of elasticity equipped with Helmholtz’s averaging kernel. The governing equations of the problem are obtained from the Hamilton’s principle, whereas the Navier’s series are proposed for a closed form solution of the structural problem involving simply-supported nanostructures. The work provides a unified framework for the bending study of both thin and thick symmetric doubly-curved shallow and deep nanoshells, while investigating spherical and cylindrical panels subjected to a point or a sinusoidal loading condition. The effect of several parameters, such as the nonlocal parameter, as well as the mechanical and geometrical properties, is investigated on the bending deflection of isotropic doubly-curved shallow and deep nanoshells. The numerical results from our investigation could be considered as valid benchmarks in the literature for possible further analyses of doubly-curved applications in nanotechnology.

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Section: Governing Equations Of Doubly-curved Nanoshellsmentioning

confidence: 99%

Section: Governing Equations Of Doubly-curved Nanoshellsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Nonlocal Elasticity Response of Doubly-Curved Nanoshells

Dindarloo

Dimitri

et al. 2020

Symmetry

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“…e stiffness matrix for the element based on the RFSDT is simply obtained by setting the terms corresponding to C s in equations (35) and (36) to zeros. e strain energy U F e given by equation (21) can now be written in the form…”

Section: Element Formulationsmentioning

confidence: 99%

“…Bessaim et al [34] proposed a five-unknown theory by adding hyperbolic functions to the transverse displacements to account for the thickness variation of this displacement. Zaoui et al [35] established the two-dimensional and quasi-3D theories for modeling the free vibration of FGM plates on Pasternak foundation. e theories employed a shear strain shape function which enables to take both the transverse shear and thickness stretching effects into account.…”

Section: Introductionmentioning

confidence: 99%

Free Vibration of FGSW Plates Partially Supported by Pasternak Foundation Based on Refined Shear Deformation Theories

Pham

Nguyen³

2020

Mathematical Problems in Engineering

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A refined third-order shear deformation theory (RTSDT), in which the transverse displacement is split into bending and shear parts, is employed to formulate a four-node quadrilateral finite element for free vibration analysis of functionally graded sandwich (FGSW) plates partially supported by a Pasternak foundation. An element based on the refined first-order shear deformation theory (RFSDT) which requires a shear correction factor is also derived for comparison purpose. The plates consist of a fully ceramic core and two functionally graded skin layers with material properties varying in the thickness direction by a power gradation law. The Mori–Tanaka scheme is employed to evaluate the effective moduli. The elements are derived using Lagrangian and Hermitian polynomials to interpolate the in-plane and transverse displacements, respectively. The numerical result reveals that the frequencies obtained by the RTSDT element are slightly higher than the ones using the RFSDT element. It is also shown that the foundation supporting area plays an important role on the vibration of the plates, and the effect of the material distribution on the frequencies is dependent on this parameter. A parametric study is carried out to highlight the effects of the material inhomogeneity, the foundation stiffness parameters, and the foundation supporting area on the frequencies and vibration modes. The influence of the layer thickness and aspect ratios on the frequencies is also examined and highlighted.

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Assessment of non‐polynomial shear deformation theories for the free vibration and transient analysis of plates with functionally‐graded materials supported on an elastic foundation

et al. 2023

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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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New 2D and quasi-3D shear deformation theories for free vibration of functionally graded plates on elastic foundations

Cited by 155 publications

References 99 publications

Nonlocal Elasticity Response of Doubly-Curved Nanoshells

Nonlocal Elasticity Response of Doubly-Curved Nanoshells

Free Vibration of FGSW Plates Partially Supported by Pasternak Foundation Based on Refined Shear Deformation Theories

Assessment of non‐polynomial shear deformation theories for the free vibration and transient analysis of plates with functionally‐graded materials supported on an elastic foundation

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