Vibrational Study on Multilayer Sandwich Plates: Porous FGM Core, Nanocomposite and Piezoelectric Face Sheets

Subject Areas : Mechanics of Solids

M Pakize ¹ , M Irani Rahaghi ² , Z Khoddami Maraghi ³ , Sh Niknejad ⁴ , A Ghorbanpour Arani ⁵

1 - Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
2 - Department of Solid Mechanics, Faculty of Mechanical Engineeirng, University of Kashan, Kashan, Iran
3 - Faculty of Engineering, Mahallat Institute of Higher Education, Mahallat, Iran
4 - Faculty of Mechanical Engineering, Department of Solid Mechanics, University of Kashan, Kashan, Iran
5 - Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran

Received: 2020-09-13 Accepted : 2021-04-28 Published : 2024-10-12

Keywords: Vibration, Porous, Sandwich plate, Carbon Nanotube, Quasi-3D plate theory,

Abstract :

A quasi-3D sinusoidal shear deformation theory and an analytical solution are presented for free vibration analysis of a 5-layer sandwich plate. The core of the sandwich plate is composed of a functionally graded porous (FGP) material that is distributed in two different types of nonlinear functions. The porous core is surrounded by two randomly oriented straight single walled CNT reinforced layers and two piezoelectric face sheets. Eight various couples of distribution of CNTs are considered for interior layers of sandwich plate. Symmetric distributions include uniform distribution and FG-XX, FG-OO, and FG-VA and asymmetric distributions are FG-XO, FG-UO, FG-UX, and FG-AV. Effective elastic moduli of the nanocomposite layers are calculated by Mori-Tanaka approach; the set of the governing equations are derived using Hamilton’s principle and are solved for the simply supported boundary conditions using Navier’s method. Accuracy of the presented solution is confirmed and effects of different parameters on the natural frequencies of the plate are studied including aspect ratio, porosity parameter, porosity distribution pattern, volume fraction and distribution pattern of CNTs and Winkler and shear coefficients of the foundation.

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Full-Text:



 

Journal of Solid Mechanics Vol. 16, No. 3 (2024) pp. 288-313 DOI: 10.60664/jsm.2024.0091642

Research Paper  Vibration Study on Multilayer Sandwich Plates: Porous FGM Core, Nanocomposite and Piezoelectric Face Sheets

 

M. Pakize 1, M. Irani Rahaghi 2, Z. Khoddami Maraghi 2, S. Niknejad 3,             A. Ghorbanpour Arani 1,31
1 Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran 2 Faculty of Engineering, Mahallat Institute of Higher Education, Mahallat, Iran 3 Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Iran
Received 13 September 2020; Received in revised form 28 April 2021; Accepted 26 October 2022
	ABSTRACT
	A quasi-3D sinusoidal shear deformation theory and an analytical solution are presented for free vibration analysis of a 5-layer sandwich plate. The core of the sandwich plate is composed of a functionally graded porous (FGP) material that is distributed in two different types of nonlinear functions. The porous core is surrounded by two randomly oriented straight single walled CNT reinforced layers and two piezoelectric face sheets. Eight various couples of distribution of CNTs are considered for interior layers of the sandwich plate. Symmetric distributions include uniform distribution and FG-XX, FG-OO, and FG-VA and asymmetric distributions are FG-XO, FG-UO, FG-UX, and FG-AV. Effective elastic moduli of the nanocomposite layers are calculated by Mori-Tanaka approach; the set of the governing equations are derived using Hamilton’s principle and are solved for the simply supported boundary conditions using Navier’s method. Accuracy of the presented solution is confirmed and effects of different parameters on the natural frequencies of the plate are studied including aspect ratio, porosity parameter, porosity distribution pattern, volume fraction and distribution pattern of CNTs and Winkler and shear coefficients of the foundation.
	Keywords: Vibration; Sandwich plate; Quasi-3D plate theory; Porous; Carbon nanotube.

 

1    INTRODUCTION

D

UE to their excellent multi-functionality offered by low specific weight, the efficient capacity of energy dissipation and enhanced recyclability, porous materials are receiving worldwide interests as advanced engineering materials in mechanical, aerospace and civil engineering. There are a considerable number of papers regarding the mechanical analysis of FGP structures. Chen et al. [1] studied static bending and mechanical buckling analyses of shear deformable FGP beams. They focused on the influences of the porosity parameter and slenderness ratio of the beam on the bending and buckling characteristics of FGP beams. In another study, they studied free and forced vibration analysis of FGP beams [2]. They presented a parametric study to examine the influences of porosity parameter, porosity distribution pattern, slenderness ratio and boundary condition on the natural frequencies and dynamic response of FGP beams. In another work, They investigated nonlinear free vibration analysis of sandwich beams with an FGP core [3]. This time, they presented a parametric study to examine the influences of porosity parameter, slenderness ratio and thickness ratio on the linear and nonlinear natural frequencies. Size-dependent nonlinear vibration analysis of FGP tapered microbeams was investigated by Shafiei et al. [4]. They studied the effects of amplitude, material length scale, rate of thickness and porosity parameter on the nonlinear fundamental frequency of FGP tapered microbeams. Ebrahimi and Jafari [5] focused on the thermo-mechanical vibration characteristic of FGP beams in a thermal environment. They investigated the influences of porosity parameter, porosity distribution pattern, thermal load, boundary conditions and power-low exponent on the natural frequencies of FGP beams. Using a refined four-variable plate theory, Barati and Zenkour [6] studied electro-thermo-mechanical vibrational analysis of FGP piezoelectric plates. They discussed the effects of porosity distribution pattern, applied voltage, thermal load, material gradation, geometrical parameters and boundary conditions on the natural frequencies of FGP plates. Ebrahimi and Barati [7] studied size-dependent and porosity-dependent vibration analysis of magneto-electro-elastic FGP nanobeams resting on a two-parameter elastic foundation. They presented some numerical results to show the effects of porosity parameter, elastic and shear coefficients of foundation, magnetic potential, applied voltage, scale coefficient, material gradation and slenderness ratio on the natural frequencies of FGP nanobeams. Free vibration analysis of CNT-reinforced cylindrical shells made of FGP materials with temperature-dependent properties under thermal load was investigated by Safarpour et al. [8]. They presented a parametric study to show the influences of length to radius ratio, porosity parameter, thermal load and boundary conditions on the natural frequency of FGP cylindrical shells. Using a quasi-3D hyperbolic shear deformation plate theory, Shahsavari et al. [9] studied free vibration behavior of FGP plates resting on elastic foundations. They presented comprehensive parametric results to study the influences of volume fraction index, porosity fraction index and geometrical parameters on the natural frequencies of FGP plates. Daikh and Zenkour [10] proposed a new porosity for bending analysis of various functionally graded sandwich plates. They used a New higher-order deformation theory to derive the field equations of the FG sandwich plate. Numerical results presented to show the effect of the material distribution, the sandwich plate geometry and the porosity on the deflections and stresses of FG sandwich plates. A quasi-3D tangential shear deformation theory was employed by Amir et al. [11] to analyze the free vibration behavior of a three-layered FG porous micro plate integrated by nanocomposite faces in a hygrothermal environment that rested on Pasternak foundation. They used Hamilton’s principle and Navier’s solution to extract and solve motion equations, respectively. Akbari et al. [12] presented a numerical solution for free vibration analysis of sandwich cylindrical panels made of a saturated FGP core and two similar homogenous face sheets. They focused on the influence of various parameters on the natural frequencies such as geometrical parameters of the panel, porosity parameter, porosity distribution pattern, thickness ratio and compressibility of pore fluid.

Due to their extraordinary mechanical, thermal and electrical properties, CNTs can be used as reinforcements of polymer composites. So, since the discovery of CNTs in 1991 by Ijima [13], many researchers have focused on mechanical analysis of FG-CNTRC structures. Ghorbanpour Arani et al. [14] used an analytical approach as well as the finite element method (FEM) and studied buckling analysis of CNT-reinforced laminated composite plates. They focused on the influences of the orientation angle of CNTs, boundary conditions and the aspect ratio on the critical buckling load. Wang and Shen [15] studied nonlinear bending and vibration analyses of sandwich plates with CNT-reinforced composite face sheets resting on an elastic foundation in thermal environments. They presented a parametric study to show the influences of volume fraction of CNTs, core-to-face sheet thickness ratio, temperature elevation and foundation stiffness on the nonlinear bending and vibration characteristics. Using FEM, Zhu et al. [16] presented numerical solutions for bending and free vibration analyses of CNT-reinforced composite plates. They focused on the effects of boundary conditions, the volume fraction of CNTs and the edge-to-thickness ratios on the bending responses, natural frequencies and mode shapes of the plate. Lei et al. [17, 18] used element-free kp-Ritz method and presented numerical solutions for buckling and free vibration analyses of FG CNT-reinforced composite plates. They devoted their results to examine the influences of volume fraction of CNTs, plate width-to-thickness ratio, plate aspect ratio, loading condition and temperature on the critical buckling load and natural frequencies. Bhardwaj et al. [19] studied non-linear dynamic response of CNT-reinforced laminated composite plates. They focused on the effects of volume fraction of CNTs and the aspect ratio of the plate on the non-linear dynamic deflection of laminated composite plates. Based on the three-dimensional theory of elasticity, Alibeigloo [20] investigated bending analysis of FG CNT-reinforced simply supported composite plates embedded in thin piezoelectric layers. He studied the influences of volume fraction and distribution pattern of CNTs, the thickness of piezoelectric layers and length to thickness ratio on the static behavior of the hybrid plate. Using FEM, static bending analysis of FG CNT-reinforced composite plates subjected to non-uniform elevated temperature fields was investigated by Jeyaraj and Rajkumar [21]. They studied the effects of boundary conditions, the distribution pattern of CNTs and type of thermal loading on the static characteristics of the plate. Using a QUAD-8 shear flexible element developed based on higher-order structural theory, Natarajan et al. [22] investigated bending and free vibration analysis of sandwich plates with CNT-reinforced face sheets. The presented a parametric study to examine the effects of volume fraction of CNTs, and core-to-face sheet thickness on the bending and free vibration characteristics of the sandwich plate. Wattanasakulpong and Chaikittiratana [23] studied static bending and free vibration analyses of CNT-reinforced composite plates resting on the Pasternak foundation. They focused on the effects of volume fraction of CNTs, elastic and shear characteristics of the foundation and geometrical parameters of the plate on the static bending and free vibration characteristics of the plate. Free and forced vibration analysis of viscoelastic micro composite beams reinforced by FG CNTs was studied analytically by Mohammadimehr et al. [24]. They presented some parametric results to examine the influences of volume fraction and distribution of CNTs and also damping coefficients on the natural frequencies and dynamic response of microcomposite beams. In another work on the micro-magneto-electro-elastic sandwich panel with a transversely flexible core and functionally graded carbon nanotube–reinforced nanocomposite facesheets, Mohammadimehr et al. [25], studied free vibration analysis of this structure based on high-order sandwich panel theory and modified strain gradient theory. They focused on the influences of the volume fraction, the various distributions of carbon nanotubes, the multi-physical fields, open- and closed-circuit boundary conditions, the material length scale parameters, different face sheet and core thicknesses, and temperature changes on the natural frequency. Their obtained results can be used to prevent the resonance phenomenon.

Ghorbanpour Arani et al. [26, 27] presented numerical solutions for free and forced vibration and supersonic flutter analyses of laminated FG-CNT reinforced cylindrical panels subjected to supersonic yawed flow. They presented parametric results to examine the influences of volume fraction, distribution pattern and orientation of CNTs, yaw angle and geometrical parameters of the panel on the natural frequencies and critical velocity of the panel. Arefi et al. [28] by using the modified couple stress theory as a non-classical continuum method and first-order shear deformation theory, studied the free vibration analysis of elastic three-layered nano-/micro-plate with exponentially graded core and piezomagnetic face sheets. They applied Navier’s solution as an analytical solution to solve seven governing equations of motion.

In the presented paper, an exact solution is presented for free vibration analysis of a 5-layer simply supported sandwich plate consisted of an FGP core, CNT-reinforced interior layers and piezoelectric face sheets resting on Winkler/Pasternak foundation. The porosity of the core is evenly and unevenly repartitioned through the thickness. For its simplicity and accuracy even at a high volume fraction of inclusions, Mori-Tanaka approach is utilized to estimate the effective moduli of the nanocomposite layers. Accuracy of the presented analysis is confirmed and the influences of various parameters on the natural frequencies of the plate are investigated such as aspect ratio, porosity parameter and distribution pattern of pores, volume fraction and distribution pattern of CNTs and also Winkler and shear coefficients of the foundation.

 

2 PROBLEM DESCRIPTION AND THE GOVERNING EQUATIONS

As depicted in Fig. 1, a sandwich rectangular plate of total thickness h and dimensions a×b resting on an elastic foundation is considered. As Fig. 1 shows, the plate is made of an FG-porous core of thickness hc, two FG-CNT reinforced interior layers of thickness hnc and two piezoelectric face sheets of thickness hp. As we’re using the energy method to drive equations, the total potential energy of the structure should be determined. Considering the stress-strain relation of each layer and using quasi-3D sinusoidal theory, strain and kinetic energy of the porous, nanocomposite and piezoelectric layers are calculated separately.

Fig. 1 Effect of step location on first two natural frequencies (Clamp boundary conditions)

 

2.1 Displacement field

The specified displacement field for this structure is a quasi-3D sinusoidal theory proposed by Zenkour [29,30] in which the stress-free boundary conditions on the top and bottom surfaces of the plate is applied [30]. Using sinusoidal functions first proposed by Levy [31] and assessed by Stein [32]. Simplicity and accuracy of these functions and the fact that they consider both normal and shear deformations caused Zenkour to use and develop them. Based on the quasi-3D sinusoidal shear deformation theory, components of displacement in a rectangular plate can be considered as [29]

(1)

where u1, u2 and u3 are displacement in x, y and z directions respectively and u, v and w show corresponding components of displacement at the middle surface of the plate (z=0). φx and φy stand for rotations around y and x axes, respectively, φz is an unknown function and f(z) is defined as following [33, 34]:

(2)

Based on the strain-displacement relations in x-y-z coordinates, components of the strain can be calculated using Eq. (1) as [35]

(3)

2.2 Porous core

A pore is a void space in a solid structure. A material that contains pores is called porous. Porosity is a fraction of the volume of pores over the total volume. It varies from 0% to 100%. The relation between stress and strain tensors can be specified using Hook’s law. For the porous core following relation can be considered [36]:

(4)

in which γij=2εij indicates to shear strain and

(5)

where Ec, Gc and υc are elasticity modulus, shear modulus and Poisson’s ratio of the porous core respectively. In this paper, two patterns are considered for the distribution of pores which are depicted in Fig. 2. For this cases elasticity modulus, shear modulus and density (ρc) are defined as [37,38]

Even porosity:

(6)

Uneven porosity:

(7)

where E1, G1 and ρ1 are specified in Fig. 2, 0≤e1<1 is porosity parameter which shows the size of pores and

(8)

It is worth mentioning that in porous materials Poisson’s ratio can be considered as a constant value.

2.3 CNT-reinforced layers

As depicted in Fig. 1. The porous core is surrounded by two CNT-reinforced layers. The CNTs are distributed uniformly with randomly orientated directions. According to the rule of mixture, the density of a CNT-reinforced polymer can be calculated as follow:

(9)

in which ρ and V indicate density and volume fraction respectively and subscripts m and r indicate polymer matrix and CNT reinforcement, respectively.

In this paper, five linear types of CNTs distribution patterns are considered. The volume fraction of CNTs for these patterns can be stated as

Upper layer (0.5hc≤z≤0.5hc+hnc):

		(10)
Fig. 2 Distribution patterns of pores.

 

Lower layer (-0.5hc-hnc≤z≤-0.5hc):

(11)

Also, the volume fraction of the polymer matrix can be obtained using the following relation:

(12)

In Eqs. (10) and (11), V*r is the total mass fraction of CNTs which can be calculated based on the mass fraction of CNTs (wr) as follow:

(13)

As the CNTs are orientated randomly, the CNT-reinforced matrix can be considered as an isotropic structure and Eq. (4) can be used for CNT-reinforced layers as well, in which

(14)

where elastic modulus (Enc) and Poisson’s ratio (νnc) of the CNT-reinforced layers can be estimated using the Eshelby–Mori–Tanaka scheme as

(15)

in which Gnc and Knc are shear and bulk moduli of the CNT-reinforced layers respectively and can be calculated based on the Eshelby–Mori–Tanaka scheme as following [39, 40]:

(16)

in which Gm and Km stand for shear and bulk moduli of the isotropic matrix which can be evaluated as

(17)

where Em and νm are elasticity modulus and Poisson’s ratio of the isotropic matrix and αr, βr, δr and ηr are defined as following [41]:

(18)

In Eq. (35), kr, lr, mr, nr and pr are five independent constants known as Hill’s elastic moduli [41].

 

2.4 Piezoelectric layers

For the piezoelectric face sheets, stress-strain relation can be written as follows [42]:

	(19-a)
	(19-b)

where Cij are elastic coefficients of the piezoelectric layers, eij and kij are piezoelectric and dielectric coefficients, D is electric displacement and Ei show electric field which can be calculated using an electric potential function (Ψ=Ψ(x,y,z,t)) as

(20)

In this paper following distribution is considered for electric potential function [37].

(21)

in which ψ0 is the electric potential implied on the top and bottom of the plate and ψ= ψ(x,y,t) is an unspecified function and

(22)

 

2.5 Hamilton’s principle

The set of the governing equations can be derived using Hamilton’s principle as [43]

(23)

in which δ is a variational operator, [t1,t2] is a desired time interval, U is strain energy, T is kinetic energy and Wn.c. is work done by non-conservative forces.

The strain energy can be calculated as

.

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		and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

                                                                                 

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