Elsevier

Composite Structures

Volume 180, 15 November 2017, Pages 211-220
Composite Structures

Differential quadrature method for vibration analysis of electro-rheological sandwich plate with CNT reinforced nanocomposite facesheets subjected to electric field

https://doi.org/10.1016/j.compstruct.2017.07.015Get rights and content

Abstract

In this study, free vibration analysis of smart sandwich plate rested on Winkler-Pasternak foundation is investigated. Sandwich plate is made of electro-rheological (ER) fluid core embedded within two nanocomposite layers which are included ZnO matrix and carbon nanotubes (CNTs) fiber. Due to electrical properties of core and nanocomposite facesheets, the external electric fields are applied to them, separately. The material properties of ER core are determined by Don and Yalcintas models. Also, Eshelby-Mori-Tanaka approach is used to obtain the material properties of nanocomposite facesheets. Hamilton’s principle is utilized to derive the governing equations of motion. Numerical characteristics of the differential quadrature method (DQM) are shown through solving selected ER sandwich plate with CCCC and SSSS boundary conditions, core to facesheets thickness ratios, volume fractions of CNTs, external voltage and Winkler-Pasternak foundation coefficients. The results obtained for ER sandwich plate show that DQM has very good accordance with results of finite element method available in literature. Also, it is observed that increasing the volume fraction of CNTs in facesheets leads to increase the stability of ER sandwich plate. These finding can be employed to design building smart structures and machines.

Introduction

Creating sandwich structure with electro-rheological (ER) core when it is subjected to an electric field is very useful to change the vibration response. ER fluids are the colloidal suspensions which are made of particles with electrical properties. These smart materials under the external electric field can be changed from liquid to quasi-solid phase and so, the stiffness of sandwich structure increase. As the most important features of ER fluids can be pointed to quick response, good reversibility and controllable performance which make them proper for use in various devices and structures.

In order to study ER materials behaviors in different conditions are done very efforts. As one of the first works in an experimental investigation, Choi et al. [1] vibration control and active damping applications for smart structures includes ER fluids. Then, Choi et al. [2] investigated forced vibration response of laminate composite beams which are consisting of metal or polystyrene outer strips and ER fluid filler. They observed that applying an electric field of 2 kV/mm increase the frequency of the various resonance modes. Stanway et al. [3] studied applications of electro-rheological fluids in vibration control. Coincide with them, Lee [4] presented finite element formulation (FEM) of a sandwich beam with embedded ER Fluids. He modeled the nonlinear constitutive relation of the ER fluid in quasi-static shear by an exponential function and then obtained a linear relation for shear modules of ER fluids.

With the start of twentieth century, more attentions to these groups of smart materials were attracted. The dynamic stability problems of a sandwich beam with a constrained layer and an ER fluid core were investigated by Yeh et al. [5]. They developed FEM and harmonic balance method to achieve the dynamic stability regions of a sandwich beam. In other work, the vibration analysis of a sandwich plate with a constrained layer and ER fluid core were done by Yeh and Chen [6]. They examined the influences of the natural frequencies, static buckling loads, and loss factors on the stability of these structures. Yeh and Shih [7] presented formulae for the critical load, natural frequencies, and loss factor of a simply supported sandwich beam with ER fluid core. The parametric instability and dynamic response of the beam subjected to periodic axial force were studied in their research. Lu and Meng [8] conducted an experimental and analytical investigation about the dynamic characteristics of a flexible sandwich plate filled with ER fluid. They carried out a laser holographic interference experiment and modal testing to recognize natural frequencies, modal damping and shapes of the sandwich plate subjected to different electric fields applied to the ER fluid domain. Narayana and Ganesan [9] analyzed the free vibration and damping characteristics of skew sandwich plate consisting of composite stiff layers with viscoelastic or ER core using FEM. Also, they investigated the effect of electric field on the frequency and loss factor. The vibration and damping analysis of composite sandwich box column containing viscoelastic or ER core with constrained layer were studied by Ramkumar and Ganesan [10]. They employed the FEM based modal strain energy method for predicting the modal loss factor and frequency of this system. Dynamic behavior analysis of a sandwich beams with ER core and functionally graded (FG) materials facesheets were carried out by Allahverdizadeh et al. [11]. They proposed a procedure to estimate the complex shear modulus of ER core and validated the results by comparative studies in the literature. Tabassian and Rezaeepazhand [12] concentrated on dynamic stability of smart sandwich beams with ER core resting on Winkler elastic foundation subjected to harmonic axial loads. They considered the influences of different parameters include beam geometry, foundation stiffness, static load, applied voltage and properties of ER core on critical dynamic loads and stability regions of the beam. Manoharan et al. [13] considered dynamic characterization of sandwich plate made of laminated composite facesheets and magneto-rheological (MR) fluid core. They illustrated that increasing magnetic field intensity causes to grow natural frequencies of sandwich plate.

Recently worthwhile studies are published in the field of ER fluids. Soleymani et al. [14] studied free vibration analysis of an ER sandwich plate with four simply-supported end conditions. They determined the natural frequencies and loss factor for the electric fields as well as the ratio of different thicknesses based on Navier analytical method. Vibration characteristics of FGER sandwich beams were investigated by Allahverdizadeh et al. [15]. They compared the results of FEM and experimental test on a fabricated ER fluid composite beam and obtained god compliance between them. Hasheminejad and Aghayi Motaaleghi [16] presented supersonic flutter control of a sandwich curved panel of rectangular plan form with an ER fluid core. They used the classical Hamilton's principle and the Galerkin method to derive the equations of motion of this system. Pawlus [17] studied dynamic response of three-layered Annular Plate with ER core subjected to time-dependent forces. They analyzed this problem as analytically and numerically using the orthogonalization method and the finite difference method. Finally, Ivanov et al. [18] investigated the dielectric properties and flow of ER fluids upon a dynamic shear in electric fields.

Studying available resources in the literature reveal that no investigation can be found regarding vibration analysis of sandwich plate with ER core and carbon nanotubes (CNTs) reinforced nanocomposite facesheets. Smart sandwich plate is subjected to electric field and is rested on Winkler-Pasternak foundation. The material properties of ER core are determined by Don and Yalcintas models and for nanocomposite facesheets is used from Eshelby-Mori-Tanaka approach. Based on Hamilton’s principle, the governing equations of motion are derived and numerical solution is conducted by differential quadrature method (DQM) for the first time.

Section snippets

Sandwich plate modeling

As shown in Fig. 1, consider a rectangular sandwich plate with length a, width b and thickness h which is rested on Winkler-Pasternak foundation. Sandwich plate is made of ER fluid core with thickness hc and nanocomposite facesheets with thickness ht and hb, respectively. It is assumed that nanocomposite facesheets reinforced by straight and long CNT fibers. Also, three layers of sandwich plate is subjected to electric field. Cartesian coordinate system (x, y, z) is considered to derive the

Hamilton's principle

The governing equations of motion using the principle of minimum potential energy for the ER sandwich plate with CNT reinforced nanocomposite facesheets resting on Winkler-Pasternak foundation under electric field, is calculated as [27]:δt1t2[U-K-Σ]dt=δt1t2[(Ut-Kt)+(Uc-Kc)+(Ub-Kb)-Σ]dt=0,where U, K and Σ are strain energy, kinetic energy and work done by external work.

The strain energy of ER core can be obtained as [13]:Uc=12Vc(τyzcγyzc+τxzcγxzc)dV,

Also, the strain energy of nanocomposite

Differential quadrature method

In engineering domain, there are a lot of numerical methods to solve the initial and boundary values problems. FEM, Galerkin method, finite difference method (FDM) and DQM are some of the common numerical methods. FEM and FDM for higher-order modes need to a great number of grid points. Thus, these solution methods for all these points require to more CPU time, while the DQM has several advantages that are listed as below:

  • DQM is a powerful method which can be used to solve numerical problems in

Numerical results and discussion

In this paper, the shear storage and shear loss modulus of ER core subjected to the external electric field are obtained by experimentally data in literature which are introduced as Don and Yalcintas models [24], [25]:Don:G=15000E2andG=6900.Yalcintas:G=50000E2andG=2600E+1700.where E is the electric field in kV/mm.

Zinc oxide (ZnO) is considered as the piezoelectric matrix of nanocomposite facesheets with SWCNTs fibers. The material properties of ZnO matrix are defined in Table 1 [34].

Conclusion

This study investigated free vibration analysis of smart sandwich plate with ER core and CNT reinforced nanocomposite facesheets rested on Winkler-Pasternak foundation. Due to electrical properties of core and nanocomposite facesheets, the external electric fields applied to them. Don and Yalcintas models used to determine the material properties of ER core and also, Eshelby-Mori-Tanaka approach for nanocomposite facesheets. Hamilton’s principle utilized to derive the equations of motion and

Acknowledgments

The authors would like to thank the reviewers for their comments and suggestions to improve the clarity of this article.

This work was supported by University of Kashan [grant number 574600/31].

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