A finite element formulation using four-unknown incorporating nonlocal theory for bending and free vibration analysis of functionally graded nanoplates resting on elastic medium foundations

Tran, Van Ke; Pham, Quoc-Hoa; Nguyen-Thoi, T.

doi:10.1007/s00366-020-01107-7

Cited by 53 publications

(11 citation statements)

References 48 publications

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“…It worth remarking that Vaccaro et al [33] has recently adopted the essence of Eringen nonlocal theory [29] to develop a rational nonlocal elastic foundation model and named it the displacement-driven nonlocal foundation model. Different structural-mechanics models have cooperated with this nonlocal theory to include the material small-scale effect [34][35][36][37][38][39][40][41][42]. However, several researchers have noticed peculiar responses obtained with these small-scale structural-mechanics models and considered them as a "paradox" [43][44][45][46].…”

Section: Introductionmentioning

confidence: 99%

Flexibility-based stress-driven nonlocal frame element: formulation and applications

Limkatanyu

Sae-Long

Sedighi

et al. 2022

Engineering with Computers

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Section: Introductionmentioning

confidence: 99%

Flexibility-based stress-driven nonlocal frame element: formulation and applications

Limkatanyu

Sae-Long

Sedighi

et al. 2022

Engineering with Computers

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“…Based the above-mentioned works, we found that the majority of researchers used an analytical solution to investigate the behavior of homogeneous and FGM nanostructures. Such analytical solutions, however, are very limited when the geometry, gradation of materials properties, boundary conditions, and loading conditions become even more complicated [57][58][59][60][61]. To this end, numerical methods such as finite element method (FEM), isogeometric analysis (IGA), and meshless methods (MMs) represent valid alternatives to analyze the complex behavior of size-dependent continuum FGM nanostructures.…”

Section: Introductionmentioning

confidence: 99%

“…Similarly, Phung-Van et al [66] performed a static and vibration analysis of porous power-law FG nanoplates by combination of IGA and HSDT. Tran et al [58] proposed a new nonlocal four-noded quadrilateral finite element based on a HSDT to investigate the bending and free vibration responses of FG nanoplates resting on elastic media.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal vibration of functionally graded nanoplates using a layerwise theory

Belarbi

Houari

et al. 2022

Mathematics and Mechanics of Solids

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This work studies the size-dependent free vibration response of functionally graded (FG) nanoplates using a layerwise theory. The proposed model supposes not only a higher-order displacement field for the core but also the first-order displacement field for the two face sheets, thereby maintaining an interlaminar displacement continuity among layers. Unlike the conventional layerwise models, the number of variables is kept fixed and does not increase for an increased number of layers. This is a very important feature compared to conventional layerwise models and facilitates significantly the engineering analyses. The material properties of the FG nanoplate are graded continuously through the thickness direction in accordance with a power-law function. The Eringen’s nonlocal elasticity theory is here adopted to relax the continuum axiom required in classical continuum mechanics and hence hopeful to capture the small size effects of naturally discrete nanoplates. The equations of motion of the problem are obtained via a classical Hamilton’s principle. The present layerwise model is implemented with a computationally efficient C0- continuous isoparametric serendipity elements and applied to solve a large-scale discrete numerical problem. The robustness and reliability of the developed finite element model are demonstrated by a comparative evaluation of results against predictions from literature. The comparative studies show that the proposed finite element model is: (a) free of shear locking, (b) accurate with a fast rate convergence for both thin and thick FG nanoplates, and (c) excellent in terms of numerical stability. Moreover, a detailed parametric analysis checks for the sensitivity of the vibration response of FG nanoplates to the aspect ratio, length-to-thickness ratio, nonlocal parameter, boundary conditions, power-law index, and modes shapes. Referential results are also reported, for the first time, for natural frequencies of FGM nanoplates which will serve as benchmarks for further computational investigations.

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“…For this paper, the displacement component defines the middle plane of a viscoelastic FGP nanoplate. According to the document [52], the displacement field (U x , U y , U z ) is expressed as follows:…”

Section: Four-unknown a New Inverse Hyperbolic Shear Deformation Theorymentioning

confidence: 99%

“…For this paper, the displacement component defines the middle plane of a viscoelastic FGP nanoplate. According to the document [52], the displacement field (

U_{x}, U_{y}, U_{z}

) is expressed as follows:

\{\begin{matrix} U_{x} (x, y, z, t) = u_{0} (x, y, t) - z w_{, x}^{b} - f (z) w_{, x}^{s} \\ U_{y} (x, y, z, t) = v_{0} (x, y, t) - z w_{, y}^{b} - f (z) w_{, y}^{b} \\ U_{z} (x, y, z, t) = w^{b} + w^{s} \end{matrix}

where

U_{x}, U_{y},

and

U_{z}

are respectively the displacement components in the x‐ , y‐ , and z‐ directions;

u_{0}

and

v_{0}

are the in‐plane displacement components of the middle surface, respectively; and

w^{b}

and

w^{s}

are, respectively, the bending and shear components. A new inverse hyperbolic shear distribution function

f (z)

is employed.…”

Section: Theoretical Formulationmentioning

confidence: 99%

A novel finite element model in nonlocal transient analysis of viscoelastic functionally graded porous nanoplates resting on viscoelastic medium

Giang,

Thuy,

Van

2023

Math Methods in App Sciences

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The transient analysis of viscoelastic functionally graded porous (FGP) nanoplate resting on the viscoelastic medium using the finite element method (FEM) is studied in this work. The nanoplate is made from a material that varies in thickness and has mechanical qualities that change depending on the porosity. The whole of the mechanical system rests on a Pasternak medium, which is modeled in accordance with a Kelvin–Voigt viscoelastic model. The general plate motion balance equations are developed by the application of the higher‐order shear plate theory (HSDT) with a new inverse hyperbolic function, the nonlocal hypothesis, and Hamilton's principle. In order to develop a four‐node quadrilateral element, Lagrangian and Hermitian interpolation functions are used. These functions are employed to define the main variables that correspond to the in‐plane displacements and the transverse displacements, respectively. The direct integration method developed by Newmark is used to calculate the dynamic responses of the plate. The outcomes that were anticipated by the essay have been confirmed by reputable sources. The dynamic response characteristics of a viscoelastic FGP nanoplates that is placed on a viscoelastic medium are explored, and a number of factors that influence these features are discovered and explained.

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A finite element formulation using four-unknown incorporating nonlocal theory for bending and free vibration analysis of functionally graded nanoplates resting on elastic medium foundations

Cited by 53 publications

References 48 publications

Flexibility-based stress-driven nonlocal frame element: formulation and applications

Flexibility-based stress-driven nonlocal frame element: formulation and applications

Nonlocal vibration of functionally graded nanoplates using a layerwise theory

A novel finite element model in nonlocal transient analysis of viscoelastic functionally graded porous nanoplates resting on viscoelastic medium

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