Application of differential cubature method for nonlocal vibration, buckling and bending response of annular nanoplates integrated by piezoelectric layers based on surface-higher order nonlocal-piezoelasticity theory

doi:10.1016/j.cam.2019.112625

Journal of Computational and Applied Mathematics

Volume 369, 1 May 2020, 112625

https://doi.org/10.1016/j.cam.2019.112625 Get rights and content

Highlights

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Vibration, buckling and bending of annular sandwich smart nanoplate are studied.
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The higher order nonlocal theory for size effects is used.
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Surface effects are considered based on Gurtin–Murdoch theory.
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The governing equations are derived based on the layer-wise theory.
•
The differential cubature method is utilized for solution.

Abstract

This paper deals with the vibration, buckling and bending analyses of annular nanoplate integrated with piezoelectric layers at the top and bottom surfaces. The higher order nonlocal theory for size effect and Gurtin–Murdochtheory for surface effects are utilized. The governing equations are derived based on the layer-wise (LW) theory and Hamilton’s principle. The differential cubature method (DCM) as a new numerical procedure is utilized to solve the motion equations for obtaining the frequency, buckling load and deflection. The influences of various parameters such as external voltage, boundary condition, surface stresses, nonlocal parameter, outer to inner radius ratio and core to top layer thickness ratio were shown on the vibration, buckling and bending responses of the nanostructure. The results of vibration, buckling and bending are validated with other published works. The outcomes show that the surface stresses have a significant effect on the increases of the frequency and buckling load and decrease of the deflection.

Introduction

Nanoplate is a two-dimensional structure which has good thermal, mechanical and optical characteristics. These nanostructures can be used in optical devises, nanocomposites, energy store and nano-electro-mechanical systems (NEMS). In addition, in nano size, the surface to volume ratio is large and consequently, the surface effects should be considered. Due to exterior application of nanoplates, the smart layers can be used as sensor and actuator for them. Hence, this study deals with the mechanical analysis of nanoplate with smart layers assuming surface effects.

The study of nanostructures is presented by researches in the recent decades. A novel beam model and modified couple stress theory were utilized by Al-Basyouni et al. [1] to investigate dynamic and bending responses of functionally graded nanobeams. Utilizing visco-nonlocal-piezo-elasticity theory, Kolahchi et al. [2] presented dynamic stability of smart nanoplates based on DC and Bolotin’s methods. Kolahchi [3] investigated vibration, buckling and bending of laminated nanoplates utilizing zigzag, sinusoidal, Mondlin and classical theories. Bouadi et al. [4] applied a new higher order nonlocal theory for buckling analysis of nanoplate utilizing exact solution. The buckling study of orthotropic embedded nanoplates utilizing a refined new plate theory was investigated by Yazid et al. [5] assuming nonlocal effects by Eringen theory. Hamza-Cherif et al. [6] analysed thermos-vibration response of nanobeams in elastic foundation utilizing a numerical solution. Utilizing higher order strain gradient nonlocal theory, Karami et al. [7], [8] studied wave propagation and buckling of nanoshells assuming various boundary condition. Guo et al. [9] studied 3D functionally graded nanoplate under the magnetic load utilizing analytical solution and couple stress theory. Kolahchi et al. [10] investigated postbuckling analysis in quadrilateral defective nanoplates subjected to thermos-magneto-hydro loads utilizing higher order plate theory.

The LW and higher order theories are utilized by researchers for achieving more accurate outcomes. Ferreira et al. [11] investigated vibration and bending of sandwich plates utilizing LW theory and numerical method for solution. A new higher trigonometric order theory was utilized by Bousahla et al. [12] for static behaviour of composite functionally graded plates based on Navier method. Bennoun et al. [13] studied vibration response of sandwich functionally graded plates utilizing refined five-variable plate theory. Vibration and bending of functionally graded plates were presented by Houari et al. [14] utilizing classical, Mindling and Reddy theories. Vibration response of sandwich circular plates was presented by Alipour and Shariyat [15] based on finite element method. Abualnour et al. [16] investigated vibration of functionally graded plates utilizing higher order shear theory assuming stretching influences. Fourn et al. [17] studied wave propagation of functionally graduated plates utilizing a hyperbolic high order theory. Meksi et al. [18] presented vibration, bending and buckling of functionally graded plates applying quasi-parabolic theory. Vibration, buckling and smart control of sandwich laminated conical truncated shells with smart layers were studied by Hajmohammad et al. [19], [20] utilizing LW theory. Moleiro et al. [21] studied static behaviour of laminated plates subjected to thermal load utilizing LW theory. A literature review for mechanical response of laminated and composite structures utilizing LW theory was prepared by Liew et al. [22].

Based on the mentioned literature review, no paper has been found for mechanical analysis of sandwich smart annular nanoplate. In this article, vibration, buckling and bending response of annular nanoplate with smart layers are studied. The small scale effects are considered based on higher order nonlocal theory. The surface stresses of the three layers are assumed utilizing Gurtin–Murdoch theory. Applying LW theory and method of energy, the motion equations are obtained and solved by DCM for calculating the frequency, buckling load and deflection of the nanostructure. The influences of various parameters such as external voltage, boundary condition, surface stresses, nonlocal parameter, outer to inner radius ratio and core to top layer thickness ratio were shown on the vibration, buckling and bending responses of the sandwich smart annular nanoplate.

Section snippets

Surface-higher order nonlocal piezoelasticity theory

As presented in Fig. 1, an annular nanoplate integrated with piezoelectric layers is shown with inner radius of $R_{i}$ , outer radius of $R_{o}$ , thickness of nanoplate $h_{c}$ and thickness of top and bottom face sheets $h_{t}$ and $h_{b}$ , respectively. Utilizing surface higher order nonlocal piezoelasticity theory, we have [2], [10] and [23] $(1 - μ_{1}^{2} \nabla^{2}) (1 - μ_{0}^{2} \nabla^{2}) σ_{i j} = C_{i j m l} ε_{m l} - e_{i j m} E_{m},$ $(1 - μ_{1}^{2} \nabla^{2}) (1 - μ_{0}^{2} \nabla^{2}) D_{i j} = e_{m l i} ε_{m l} + \in_{i m} E_{m},$ $σ_{i j}^{S} = τ_{i j}^{S} + C_{i j m l}^{S} ε_{m l} - e_{i j m}^{S} E_{m},$ $σ_{ℓ z}^{S} = τ_{iz}^{S} w_{, ℓ}, ℓ = r, θ$ $D_{i}^{S} = D_{i}^{0} + e_{m l i}^{S} ε_{m l} + \in_{i m}^{S} E_{m},$ where $σ_{i j}$ and $ε_{i j}$ are stress and strain

DCM

DCM is one of the novel numerical method which can convert the motion equations into the algebraic relations based on the operator of $ℜ$ . This method can be utilized for various boundary condition. Hence, the basic relation of this method is [2] $ℜ f {(x, y)}_{i} \approx \sum_{j = 1}^{N} C_{i j} f (x_{j}, y_{j}),$ where $C_{i j}$ and $N$ present the cubature weighting coefficients and grid point number, respectively. For calculation the cubature weighting coefficients, the following relation can be used $ℜ {\{x^{ν - μ} y^{μ}\}}_{i} = \sum_{j = 1}^{N} C_{i j} f (x_{j}^{ν - μ} y_{j}^{μ}), μ = 0, 1, 2, \dots, ν, ν = 0, 1$

Numerical results and discussion

In this section, bending, buckling and vibration responses of the sandwich annular nanoplate with smart layers are investigated. For this purpose, the Silicon material properties of nanoplate and piezoelectric layers made from Zinc Oxide (ZnO) are presented in Table 1 [24], [25], [26], [27]. In this table, the surface constants of Silicon and ZnO are reported per meter unit. The number of grid points in DCM for convergence of results is 115. The frequency, buckling load and transverse

Conclusion

Vibration, buckling and bending of the sandwich smart annular nanoplate were presented in these articles. The sandwich nanostructure was made from a nanoplate covered by two smart layers under the external applied voltage. The size effects were assumed utilizing higher order nonlocal theory with two parameters. The surface stresses of the layers were considered based on Gurtin–Murdoch theory. The LW theory was utilized for deriving the governing equations based on method of energy. DCM was used

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View full text

Application of differential cubature method for nonlocal vibration, buckling and bending response of annular nanoplates integrated by piezoelectric layers based on surface-higher order nonlocal-piezoelasticity theory

Highlights

Abstract

Introduction

Section snippets

Surface-higher order nonlocal piezoelasticity theory

DCM

Numerical results and discussion

Conclusion

Compos. Struct.

Compos. Struct.

Aerosp. Sci. Technol.

Thin-Walled Struct.

Comput. Math. Appl.

Compos. Struct.

Compos. Struct.

Aerosp. Sci. Technol.

Compos. Struct.

Compos. Struct.

Eur. J. Mech/A Solids

Appl. Math. Model.

A new nonlocal HSDT for analysis of stability of single layer graphene sheet

Adv. Nano Res.

A novel nonlocal refined plate theory for stability response of orthotropic single-layer graphene sheet resting on elastic medium

Smart Struct. Syst.

Vibration analysis of nano beam using differential transform method including thermal effect

J. Nano Res.