Resonance analysis of composite curved microbeams reinforced with graphene nanoplatelets

https://doi.org/10.1016/j.tws.2020.107407Get rights and content

Highlights

  • The resonances of curved microbeams reinforced with GNPs are studied.

  • The modified strain gradient theory is used to build the size-dependent model.

  • The Navier-type solution procedure is utilized to solve the whole problems.

  • Three different types of GPNs distributions are considered.

Abstract

In this paper, the forced resonance vibration analysis of curved micro-size beams made of graphene nanoplatelets (GNPs) reinforced polymer composites is presented. The approximating of the effective material properties is on the basis of Halpin–Tsai model and a modified rule of mixture. The Timoshenko beam theory is applied to describe the displacement field for the microbeam. To incorporate small-size effects, the modified strain gradient theory, possessing three independent length scale coefficients, is employed. Hamilton principle is applied to formulate the size-dependent governing motion equations, which then is solved by Navier solution method. Ultimately, the influences of length scale coefficients, opening angle, weight fraction and the total number of layers in GNPs on composite curved microbeams corresponding to different GNPs distribution are discussed in detail through parametric studies. It is shown that, the resonance position is significantly affected by changing these parameters.

Introduction

There are several factors that are gained by designers to provide an optimum and safe sketch to manufacturers. One of them is the resonance phenomenon. Resonance is a key and almost destructive factor in the operation of engineering or biological parts which may be made of curved beams as basic elements of them. The applications of parts are in rotors, turbine blades, bridges, rockets body, radiobiological processes, and so on. Hence, a lot of attention has been done to these structures whose made of advanced composite materials, laminas and especially composites, and nanocomposites reinforced by carbon-based materials (see in Refs. [[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]]).

Carbon-based materials are the most popular constructive elements in the 21st century. This is because of its excellent mechanical, thermal, electrical, and optical properties. Hence, carbon-based materials have become the focus of interdisciplinary research and have potential application prospects in various fields. Although both carbon nanotubes (CNTs) and graphene nanoplatelets (GNPs) are derivatives of carbonaceous materials, GNPs as a two-dimensional counterpart of them supplied even better-enhancing impacts considering dispersion at a low concentration [[20], [21], [22], [23], [24]].

Composites and nanocomposites and carbon-based materials reinforced of them, etc. Inside its polymeric matrix have been analyzed by many researchers available in the current applicable field [[8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]]. Arefi and his co-workers [8] analyzed the thermo-mechanical buckling behavior of GNPs reinforced composite microplate based on modified strain gradient theory. Wang et al. [9] investigated the vibration of GPNs reinforced composite beams loaded by two moving masses using various beam theories. Mao et al. [10] proposed a size-dependent dynamics of GNPs reinforced microplate with piezoelectric face sheets. Based on nonlocal strain gradient theory, Sahmani et al. [14] discussed the nonlinear vibrations of the micro/nano-plates reinforced with GPNs. Arefi et al. [15] solved the free vibration of a polymeric based composite nanoplates reinforced with GPNs. Yang and his co-authors [25] investigated the post-buckling of multilayer graphene-platelet reinforced composite beams under mechanical loading. Thermal affected stability phenomenon of GNPs reinforced composite plates was reported by Wu et al. [26]. Fazelzadeh et al. [27] studied the vibrational behavior of GNPs reinforced doubly-curved nanocomposite shell in thermal environment. Karami and his co-workers reported a resonance study on GNPs reinforced polymeric nanocomposite shells and plates [11, 12]. Liu et al. [28] performed three-dimensional static and dynamic responses of GPNs reinforced circular nanoplates. Employed a four-variable refined plate theory, Thai et al. [29] systematically studied the buckling, static bending and free vibration behaviors of GPNs reinforced composite plates. Habibi et al. [30] presented the vibrational behavior of electrically cylindrical nanoshells reinforced with GPNs. Although some achievements have been made in the research of GNPs reinforced composites and nanocomposites, there is no study reporting the resonance phenomenon of composite curved microbeam reinforced with GPNs, even at the macroscale.

The conventional continuum theories do not take into account the small-scale effects which have been proved by a lot of experiments. Fleck and his colleagues [31] found that reducing diameter in a thin copper wire from 170 × 10−6 m to 12 × 10−6 m, affected significantly its shear strength. Stölken, and Evans [32] reported by a decrement in the thickness of the ultra-thin beams, its stiffness increases significantly. Therefore, in order to describe this phenomenon of sub-micron structures and due to the time and cost consuming of experimental tests and also the complexity of simulations such as molecular dynamics, non-classical continuum theories such as nonlocal elasticity [33], strain gradient elasticity [34, 35], nonlocal strain gradient elasticity [36, 37], coupled stress/-strain elasticities [38, 39], and surface elasticity [40] have been propounded by the researchists. Among them, Lam et al. [34] proposed an improved modified strain gradient theory (MSGT) which contains three independent size-dependent coefficients. As soon as this theory was put forward, it caused widespread utilization among researchers (see in Refs. [[41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55]]).

Although there are many works that have been carried out by MSGT, the forced resonance response of GNPs reinforced polymeric composite curved microbeams are steel new, even at the macroscale. In the past decade, due to the wide application of straight and curved beams or columns as structural elements in microelectromechanical systems (MEMS), the investigation on its mechanical behavior has attracted the interest of many researchers (see in Refs. [50, [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71]). Anirudh et al. [59] proposed a comprehensive numerical study on the GNPs reinforced composite curved beams. Arefi et al. [60] presented the nonlinear bending analysis of curved composite nanobeams reinforced with GNPs. Within the framework of MSGT, Ansari et al. [61] as well as Karami et al. [50] studied the linear vibrational behavior of functionally graded material (FGM) curved microbeams. Çalım [62] and Aribas et al. [63] investigated the forced vibrations of curved beams resting on an elastic foundation. Considering a geometrical nonlinearity, Mohamed et al. [64] discussed the free and forced vibration of buckled curved beams supported by a nonlinear elastic foundation. Consider four different beam theories, Polit et al. [65] used Navier solution method to study the bending and buckling of GNPs reinforced curved beams. Sobhy [66] analyzed the buckling and vibrations of GPNs/aluminum curved nanobeams in the magnetic field. Based on the nonlocal strain gradient theory, She et al. [67] proposed a size-dependent model to study the resonance of FGM curved nanobeams. More works can be found in Refs. ([[72], [73], [74]]). It is worth mentioning that, compared to straight beams, the stiffness of curved beams is more suitable for improving the stiffness characteristics of certain components.

Through literature review, we can find that a great deal of attention by researchers have been conducted in analyzing statics and dynamics of GNPs reinforced curved beams in macro/micro/-nano dimension; however, the existing literature lacks the studies of forced resonance vibration characteristics of curved micro-size beams reinforced with GNPs. To solve this problem, in the current work, Timoshenko beam theory is coupled with MSGT to propose a model for forced resonance analysis of GNPs reinforced composite curved microbeams. The effective properties of the microbeam are estimated using Halpin–Tasi model as well as a modified role of mixture. In addition, Navier solution procedure is applied to solve the governing motion equations analytically. In the end, the influences of different parameters on resonance phenomenon are discussed in detail.

Section snippets

Material properties

A composite curved microbeam with length L, mid-surface radius R, thickness h (h << R), and opening angle α is considered in Fig. 1. The matrix of the composite is a polymer, and the GPNs are evenly and even distributed in each layer of polymer matrix material. Halpin-Tsai model is used to estimate the material parameters of the GPNs composite as below [25, 75, 76],Ec(k)=Em8[3(1+ξLηLVGNP(k)1ηLVGNP(k))+5(1+ξWηWVGNP(k)1ηWVGNP(k))]ηL=(EGNP/EM)1(EGNP/EM)+ξLηW=(EGNP/EM)1(EGNP/EM)+ξWin which, EM

Solution procedure

For simply-supported ends, the Navier-type solution procedure is utilized. The following Fourier series are assumed to satisfy the admissible displacement components as below,{u(t,x)w(t,x)ψ(t,x)}=m=1{Umcos(mπxL)Wmsin(mπxL)Ψmcos(mπxL)}sinΩtqdynamic=m=1Qmsin(mπxL)sinΩtwith Ω indicates the excitation frequency, Qm is the load amplitude, Um, Wm and Ψm are the displacement amplitudes. By employing Eq [46]. into Eq. [34], then we have([K]+Ω2[M]){UmWmΨm}={0m=1Qmsin(mπxL)sinΩt0}

Herein, total

Numerical results

The purpose of this paper is to analyze the resonance phenomenon of a curved micro-size beam. The microbeam is made of a polymeric matrix which is reinforced with GNPs. Since that the dimension of the beam is sub-micron, small-size effects should be captured. According to the experimental and theoretical results reported by Lei et al. [35], for a nickel microbeam, the strain gradient coefficients are  = 0 = 1 = 2 = 0.843 × 10−6 m. It should be noticed that there is no available experimental

Conclusions

The complex resonance behavior of composite microbeams reinforced with GPNs was investigated in this paper. Halpin-Tasi model, Timoshenko beam theory, and MSGT were considered in order respectively to estimate the effective material properties and gain the governing motion equations. According to the numerical analysis, the following conclusions were summarized:

  • The resonance position of the composite microbeams can be controlled by changing the radius which is controlled by the opening angle.

Author agreement statement

We the undersigned declare that this manuscript is original, has not been published before and is not currently being considered for publication elsewhere.

We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us.

We understand that the Corresponding Author is the sole contact

Declaration of competing interest

The authors declared that they have no conflicts of interest to this work.

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