VIBRATION OF SANDWICH BEAMS REINFORCED BY CARBON NANOTUBES UNDER A MOVING LOAD

This work studies the vibration of sandwich beams reinforced with carbon nanotubes (CNTs) under moving point loads. The cores of the beams are homogeneous while their two sides are made of carbon nanotube reinforced composites. The effective properties of two face sheets are estimated through a micromechanical model. A uniform distribution (UD) and four different types of functionally graded (FG) distributions, namely FG-X, FG,  FG-V, FG-O, are considered. Based on a third-order shear deformation theory, a finite element formulation is derived and used to investigate the vibration characteristics of the beams. The effects of carbon nanotube volume fraction, carbon nanotube distribution pattern and moving load velocity on beam vibration behavior are investigated and highlighted. The influence of layer thickness and span-to-height ratio on beam vibration is also examined and discussed.


INTRODUCTION
Carbon nanotubes (CNTs) with high strength, high stiffness, high aspect ratio and low density are excellent reinforcement for composite materials. The analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams has drawn considerable attention from researchers in recent years. Ke et al. [1,2] investigated the nonlinear free vibration and dynamic stability of FG nano beams reinforced by single-walled carbon nanotubes (SWCNTs) using Timoshenko theory. Their results show that an increase in CNT volume fraction leads to higher natural frequencies for both uniformly distributed CNT (UD-CNT) and FG-CNTRC beams. Yas and Heshmati [3] studied free and forced vibrations of an FG nanocomposite beam reinforced by randomly straight SWCNTs under a moving load. Free vibration and buckling analysis of FG-CNTRC Timoshenko beams resting on an elastic foundation are also described by Yas and Samadi [4]. Shen and Xiang [5] presented the nonlinear bending and the thermal postbuckling analyses of CNTRC beams. The obtained results show that the CNT volume fraction has a significant influence on the load-deflection curves of the beam. Based on Timoshenko beam theory, Ansari et al. [6] studied forced vibration of nanocomposite beams reinforced by SWCNTs. The third-order shear deformation theory was adopted by Lin and Xiang [7] in determining vibration frequencies of UD-and FG-CNTRC beams with various boundary conditions. Nejati and Eslampanah [8] employed the two dimensional (2D) elasticity theory to obtain buckling loads and natural frequencies of cantilever FG-CNTRC beams under axial load. Based on the first-order shear deformation beam theory and von Kármán nonlinearity, Wu et al. [9] investigated nonlinear vibration of imperfect shear deformable FG-CNTRC beams. Nonlinear free vibration and post-buckling of FG-CNTRC beams resting on nonlinear foundation were studied by Shafiei and Setoodeh [10]. Recently, Mohseni and Shakouri [11] investigated the free vibration and buckling of FG-CNTRC beams with variable thickness resting on elastic foundations.
Using FG-CNTRCs as facing composition in sandwich constructions to increase strength and stiffness, Wu and Kitipornchai [12] investigated free vibration and elastic buckling of sandwich beams with FG-CNTRC face sheets, giving a detail on the effects of CNT volume fraction, core-to-face sheet thickness ratio, slenderness ratio, and end supports on the free vibration and buckling behavior of sandwich beams. Ebrahimi and Farzamand Nia [13] proposed a higher-order shear deformation beam theory for free vibration analysis of FG-CNTRC sandwich beams in thermal environment. The effects of carbon nanotube volume fractions, slenderness ratio and core-to-face sheet thickness ratio on the vibration of the sandwich beams have been examined.
The influence of material gradation on the vibration of beams carrying a moving load has been investigated in recent years [14,15]. It has been shown that the variation of material properties in spatial directions has a significant influence on both free and forced vibrations of the beams. This topic is further explored in the present work by studying vibration of FG sandwich beams reinforced by CNTs. The core of the sandwich beams is homogeneous while its two face sheets are a FG-CNT reinforced material. The effective properties of the two face sheets are determined by an extended rule of mixture. Five types of CNT distribution, namely UD, FG-X, FG-,  FG-V, FG-O, are considered. A third-order shear deformation finite element formulation is derived and employed to compute natural frequencies and investigate dynamic response of the beams. A parametric study is carried out to highlight the effects of carbon nanotube volume fraction, the type of carbon nanotube distribution, the beam geometry and moving load velocity on dynamic behavior of the sandwich beams. Figure 1 shows a sandwich beam with FG-CNTRC face sheets subjected to a concentrated load 0 F , moving from the left end to the right end of the beam at a constant speed v. In the figure, the Cartesian coordinate system (x,z) is chosen such that the x-axis lies on the beam midplane. The beam consists of three layers, a homogeneous core and two face sheets of CNTRC interfaces between the layers, and the top surface. Five types of distribution of CNTs in the beam cross-section, as shown in Figure 2 and given in Table 1, namely the UD, FG-X, FG-,  FG-V, FG-O, are considered.   ;; ;

CNT m
 are Poisson's ratio of the CNT and matrix, respectively.
The effective elastic and shear moduli of the kth layer are calculated as follows [12]     ( ) ( ) 11 12 12 21 (2) ; in which , cc EG are the elastic and shear moduli of the core material. The effective mass density of the kth layer is defined as where c  is mass density of the core material.

MATHEMATICAL FORMULATION
The Shi's third-order shear deformation theory [16] is adopted herewith to formulate the governing equations for the beam. This theory is derived from an elasticity formulation, rather than displacement hypothesis, which gives better results than the first-order and other simple higher order shear deformation theories. The displacements of a point in the beam in the x and z directions, () ,, u x z t and   , , , w x z t respectively, are given by , w x t are, respectively, the displacements in the x and z directions of a point on the x-axis;  is the cross-sectional rotation, and t is the time variable. In Eq. (5) and hereafter, the subscript comma is used to indicate the derivative with respect to the variable that follows.
By using the transverse shear rotation 0  , defined as 0 0,x w   [17], the axial and transverse displacements in Eq. (5) can be rewritten as follows Eq. (6) gives the axial strain xx  and shear strains xz  in the forms The normal and shear stresses are given by linear elastic constitutive law as (7) and (8)  1 () where A = bh is the cross-section area; 11 where the over dot denotes the derivative with respect to time variable; 11 The potential of the moving load (V) is simply given by where   .  is the Dirac delta function, and x is the abscissa, measured from the left end of the beam.

FG-CNTRC SANDWICH BEAM ELEMENT
Consider a two-node beam element with length l. The element vector of nodal displacements (d) contains eight components as ;;  (17) where ne is the total number of elements, and k is the element stiffness matrix.   (19) where m is the element mass matrix.  are evaluated at the current position of the force 0 F . The system of Eq. (21) can be solved by the Newmark method. The average acceleration method which ensures the numerically unconditional stability is adopted herein.

NUMERICAL RESULTS AND DISCUSSION
A FG-CNTRC sandwich beam with simply supported ends is considered in the numerical investigation in this section. Poly-methyl methacrylate (PMMA) with material properties as  acting at the mid-span.

Formulation verification
Before investigating the vibration characteristics of the beam, the accuracy of the finite element model is firstly confirmed.  Tables 2-4 compare the fundamental frequency parameter of the FG-CNTRC sandwich beam with the results of Wu and Kitipornchai [12], where the differential quadrature method has been employed. The frequency parameter is received for sandwich beam with two types of CNT distribution named as FG-V and UD. Very good agreement between the frequency parameter of the present work with that reported in [12] is obtained from Tables 2-4, regardless of the total CNT volume fraction * CNT V , ratio / cf hhand aspect ratio L/h, noting that a Timoshenko beam theory is used to formulate governing equations in [12]. Table 5 lists fundamental frequency parameter of FG-CNTRC sandwich beam for five different types of CNT distribution. As seen from the Table, the frequency parameter  increases with increasing the total CNTs volume fraction * , CNT V especially  more significantly increases for a smaller / cf hh ratio.  Of the five types of the CNT distribution, the FG-V sandwich beam has the highest frequency, while the FG- sandwich beam gives the smallest result. It is easy to see that the results obtained for beam with UD, FG-X, FG-O distributions are very close together. Moreover, the decrease of the ratio / cf hhleads to the increase in the frequency parameter, especially at smaller values of the aspect ratio L/h. This is because the sandwich beam gets higher stiffness corresponding to a smaller ratio /. cf hh Besides, the influence of the aspect ratio L/h on the frequency parameter  is also seen in the Table 5, where the frequency parameter increases with decreasing aspect ratio L/h.  The difference of three types of CNT distribution is clearly seen from the figure, especially for the smaller values of / cf hh (Figure 3a, c). The mid-span deflection of the sandwich beams corresponding to the distribution FG-V of CNT is the smallest, while that of the sandwich beam with FG- type distribution is the highest. For both values of the / cf hh ratio, the mid-span deflection increases with increasing moving load velocity. In addition, the sandwich beam is subjected to more vibration cycles when it is under load with lower moving velocities. This can be explained by the lower ratio of the moving load speed to the critical speed as in case of the isotropic beams [19]. As can be observed from Figure 3, the mid-span deflection of the FG-V beam insignificantly increases when increasing the / cf hh ratio, regardless of the moving velocity.