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1 Introduction

Knowledge of mechanical properties is quite important in the design of various kinds of materials. Due to their excellent physical, mechanical, and electrical properties [1], nanostructures have attracted much attention among the scientists/researchers to develop innovatory applications in the field of nanomechanics. Proper understanding of their mechanical behavior is a key factor in the production of such engineering structures. Among these nanostructures, single walled carbon nanotubes viz. nanobeams attract more attention due to their great potential in engineering applications such as nanowires, nanoprobes, atomic force microscope (AFM), nanotube resonators [2], nanoactuators [3], and nanosensors etc. Various applications concerning CNT reinforced structure like reinforced beam, plate etc. can be found in the literatures [4–11]. Ever since the discovery of the fullerene, the family of carbon nanostructures has been steadily expanded. Singlewalled carbon nanotubes (SWCNTs) are one of the important member in this family. Carbon nanotubes (CNTs) are made up of graphene with long hollow cylindrical shape. By analyzing all the mechanical properties of carbon nanotubes one may say that it is the strongest materials available in nature. Carbon nanotubes possess different properties in axial and radial directions even if graphene sheets have 2D symmetry. Experimentally it has been found that CNTs are very strong in the axial direction and rather soft in radial direction. A standard single-walled carbon nanotubes can withstand a pressure up to 25 GPa without deformation and after that They undergo a transformation to become super hard nanotubes. Maximum pressures measured using current experimental techniques are around 55 GPa. However, these new super hard phase nanotubes have very high mechanical strength that may be collapsed at an even higher, albeit unknown, pressure. Schematic representation of SWCNT is given below.

Figure 1:

Schematic representation of SWCNT

Structural members with variable cross section are frequently used in civil, mechanical, and aeronautical engineering to satisfy architectural requirements. In practical cases such as space structures, this type of vibration analysis plays an important role in design. Many engineers currently design light slender members with variable cross sections to construct ever-stronger and ever-lighter structures. Unfortunately, design engineers are lacking proper knowledge on the design of nonuniform structural elements since most of the design specifications are available for uniform elements. Hence there is a need for vibration analysis of nonuniform structural elements. So far, three approaches have been developed to study mechanical behaviors of structural elements. These are atomistic, semi-continuum and continuum models. Continuum mechanics are categorized into classical continuum mechanics and nonclassical continuum mechanics. In classical continuum model, lattice spacing between individual atoms is not taken into consideration. The significant influence caused by small scale effects such as electric force, chemical bond and van der waals force are neglected when classical continuum model is considered. Both experimental and atomistic simulation results show that at nanoscale, the small length scale effect (such as lattice spacing between individual atoms) may not be neglected. Hence various nonclassical continuum theories like strain gradient theory, couple stress theory, micropolar theory and nonlocal elasticity theory have been developed to incorporate size effect by introducing an intrinsic length scale. Among these theories, nonlocal elasticity theory proposed by Eringen [12] has been widely applied in the vibration of nanobeams. The small scale effect depends on the crystal structure in lattice dynamics and the nature of physics under investigation. Conducting experiments with nanoscale size is quite complicated and expensive. Therefore, development of appropriate mathematical models for vibration analysis of nanobeams is an important issue concerning its applications. In this respect, authors have investigated various studies either by numerically or by analytically. The value of small scale effect which plays a vital role in the nonlocal elasticity theory has been calibrated using molecular dynamics [13, 14]. When the size of beam is of nanoscale dimension, nonlocal impact plays a significant role in the prediction of natural frequencies and vibrating modes especially higher order natural frequencies and vibrating modes [15]. One may find significant role of nonlocal effects in nanoscale devices [16, 17]. Analytical results of Euler–Bernoulli shows that frequency parameters decrease with increase in scaling effect parameter [18]. Reddy [19] investigated analytical solutions for various beam theories like Euler–Bernoulli, Timoshenko, Reddy and Levinson in case of simply supported boundary condition. Aydogdu [20] has given a general expression for the displacement fields of all the wellknown beam theories and analytical solution for simply supported boundary condition has also been presented. Some numerical methods like meshless [21], Differential Quadrature [22, 23] and Rayleigh–Ritz have also been used in various problem related with vibration of nanobeams. Behera and Chakraverty [24] applied differential quadrature method to study vibration analysis of nanobeams based on nonlocal theories and one may get further detail in the book [25]. Similarly, Tornabene et al. surveyed several methods under the heading of strong formulation finite element method (SFEM) which can be found in [26] and also one can find numerical approach for beams with variable thickness based on DQ method in [27].

As we know, it is not always possible to find analytical solutions for all set of boundary conditions at the edges. Different researchers used Rayleigh–Ritz method in order to handle all set of classical boundary conditions. Mohammadi and Ghannadpour [28] used Chebyshev polynomials and Behera and Chakraverty [29] used boundary characteristic orthogonal polynomials in the Rayleigh–Ritz method to study vibration of Timoshenko nanobeams. However, it is not always possible to obtain analytical solutions for complicated geometries. Some of the numerical methods such as finite element method [30], Rayleigh–Ritz method and Differential Quadrature (DQ) method have been used by the researchers. For applying these methods, one must have sound knowledge about variational principles. Again, subsequent application of variational principles often requires a proper understanding of principles of mechanics. This has motivated the search for an approximate computational technique. In this context, one may use differential quadrature method which is very efficient, simple and can be applied to both linear and nonlinear problems. Also there is no need of any knowledge about variational principles.

Differential Quadrature (DQ) method has been introduced by Bellman and Casti [31] for the first time. This is an efficient numerical method for the solution of linear and nonlinear partial differential equations. Then Bert et al.[32] applied DQ method in structural problems. Since then, researchers are investigating linear and nonlinear structural problems using DQ method. Different procedure has been used by the authors to implement boundary conditions in the DQ method. Firstly, δ technique was proposed by Bert et al.[33] to implement boundary conditions. In this procedure [34], one boundary condition is implemented at the boundary point and other boundary condition at a distance δ from the boundary point. There are two major drawbacks of this approach. Firstly, since implementation of boundary condition at the δ point is an approximation of the true boundary condition which should be implemented at the boundary, therefore accurate numerical solutions lies on smaller value of δ. Secondly, smaller value of δ causes the solutions to oscillate since weighting coefficient matrices become highly ill conditioned. The above mentioned approach is suitable for clamped end but not suitable for simply supported and simply supported–clamped ends. Seeing demerits of this approach, Bert [35–40] proposed a new approach in applying boundary conditions. In this approach, only one boundary condition is numerically implemented and the other boundary condition is built into the DQ weighting coefficient matrices. Following are some of the advantages of this approach. (i) Boundary conditions are properly satisfied since they are applied at the boundary points. (ii) The effect of δ on the results is eliminated. (iii) Excellent results are obtained with less computational effort.

Though, some of the studies have been done using differential quadrature method, but to the best of the present authors’ knowledge, the article provides first time the frequency parameters of nanobeams with exponentially varying stiffness based on Euler-Bernoulli beam theory. In this article, Euler-Bernoulli beam theory in conjunction with nonlocal elasticity theory have been considered to illustrate the effect of nonuniform parameter, nonlocal parameter, L/h and the boundary condition on the frequency parameter.

2 Review of nonlocal elastic theory

According to nonlocal elasticity theory, the nonlocal stress tensor σ at a point x is expressed as [41]

σ(x)=VK(|xx|,α)τdV(x)(1)( 1 )

where τ is the classical stress tensor, K(|xx|,α) the nonlocal modulus and |xx| the Euclidean distance. The volume integral is taken over the region V occupied by the body. Here α is a material constant which depends on both internal and external characteristic lengths.

According to Hooke’s law

τ(x)=C(x):ε(x)(2)( 2 )

where C is the fourth order elasticity tensor, ε the classical strain tensor and: denotes double dot product.

Eq. (1) is the integral constitutive relation which is quite difficult to solve. Hence equivalent differential form of this equation may be written as [41]

(1α2L22)σ=τ,α=e0aL(3)( 3 )

where ∇2 the Laplace operator, e0 is a material constant which could be determined from experiments or by matching dispersion curves of plane waves with those of atomic lattice dynamics, is an internal characteristic length such as lattice parameter, CC bond length or granular distance while L is an external characteristic length which is usually taken as the length of the nanostructure. The term e0a is called the nonlocal parameter which reveals scale effect in models or it reveals the nanoscale effect on the response of structures.

3 Formulation of non-local Euler-Bernoulli beam theory

The displacements fields are based on

u1=zwx,u2=0,u3=w(x,t)(4)( 4 )

where (u1, u2, u3) are the displacements along x, y and z coordinates respectively, w is the transverse displacement of the point (x, 0) on the mid-plane (z = 0) and t denotes time. Here we have considered free harmonic motion i.e. w(x, t) = w0(x) sin ωt where ω is natural frequency of vibration. The strain-displacement may be given as

εxx=z2w0x2,(5)( 5 )

where εxx is the normal strain.

Using principle of virtual displacement, we may obtain the governing equation as

d2Mdx2=m0ω2w0(6)( 6 )

where M is the bending moment and is given by M=AzσxxdA. Here σxx is the normal stress, A the cross sectional area and m0 is mass moment of inertia which is defined by m0=AρdA=ρA, where ρ denotes the density of beams. Nonlocal constitutive relation may be expressed as

Mμd2Mdx2=EId2w0dx2(7)( 7 )

where μ = (e0a)2 is the nonlocal parameter, I is the second moment of area and E is Young’s modulus. Here e0 and a denote material constant and internal characteristic length respectively. Substituting Eq. (6) in Eq. (7), M in nonlocal form may be obtained as

M=EId2w0dx2+(e0a)2(m0ω2w0)(8)( 8 )

Using Eq. (8) in Eq. (6), governing equation in terms of displacement is rewritten as

μρAω2EId2w0dx2+d4w0dx4=ρAω2w0EI(9)( 9 )

Let us introduce following non-dimensional terms

X=xLW=w0Lλ2=ρAω2L4EI=frequencyparameter.α=e0aL=scalingeffectparameter.

Next, we assume an exponential variation of the flexural stiffness (EI)since the flexural stiffness of the SWCNT may not be constant for a geometrically non-uniform beam model. Based on the proposed exponential variation, the flexural stiffness is defined as

EI=(EI0)eηX

where I0 is the mass moment of area at the left end and η is a positive constant.

Using the above non-dimensional terms in Eq. (9), we obtain the nondimensionalized form of the governing differential equation as

eηXd4WdX4=λ2(WμL2d2WdX2)(10)( 10 )

Next we introduce an overview of the differential quadrature method.

4 Differential Quadrature Method

The derivatives of displacement function W(X) at a given discrete point i are approximated as [24]

Wi=j=1NAijWj(11)( 11 )Wi=j=1NBijWjWi=j=1NCijWjWiIV=j=1NDijWj

where i = 1, 2, ..., N and N is the number of discrete grid points.

Here Aij, Bij, Cij and Dij are the weighting coefficients of the first, second, third and fourth derivatives respectively.

4.1 Determination of weighting coefficients

Computation of weighting coefficient matrix A = (Aij) is the key step in the DQ method. In the present investigation, we have used Quan and Chang’s [30] approach to compute weighting coefficients Aij. As per this approach, matrix A = (Aij) may be computed by the following procedure.

For ij

Aij=1XjXiΠkikjk=1NXiXkXjXk(12)( 12 )i=1,2,,Nj=1,2,,N

for i = j

Aii=kik=1N1XiXki=1,2,,N.(13)( 13 )

Once weighting coefficients of first order derivatives are computed, weighting coefficients of higher order derivatives may be obtained by simply matrix multiplication as follows.

B=Bij=k=1NAikAkj(14)( 14 )C=Cij=k=1NAikBkj(15)( 15 )D=Dij=k=1NAikCkj=k=1NBikBkj.(16)( 16 )

4.2 Selection of mesh point distribution

We assume that the domain 0 ≤ X ≤ 1 is divided into (N-1) intervals with coordinates of the grid points given as X1, X2,..., XN. These Xis have been computed by using Chebyshev-Gauss-Lobatto grid points. That is

Xi=12[1cos(i1N1Π)].

One may note that, here nonuniform grid points have been used.

4.3 Application of boundary conditions

Above matrices A, B, C and D are converted into modified weighting coefficient matrices A¯, B¯, C¯ and D¯ as per the boundary conditions which will be shown below. Four boundary conditions such as SS, CS, CC and CF are considered in the present analysis. The letters S, C, and F denote simply supported, clamped and free edge conditions respectively.

Let us now denote

A¯1=[0A1,2A1,N0A2,2A2,N0AN,2AN,N]A¯2=[A1,1A1,2A1,N10A2,1A2,2A2,N10AN,1AN,2AN1,N10].

In view of the above, we now illustrate the procedure for finding modified weighting coefficient matrices as per the considered boundary conditions and are discussed below:

4.4 Simply supported

To show how the boundary conditions are implemented in the coefficient matrix, we have now explained below. Eq. (11) may be rewritten in matrix form as

A=[A11A12A1,N1A1,NA21A22A2,N1A2,NAN1AN2AN,N1AN,N]{W1W2WN}(17)( 17 )={W1W2WN}.

Firstly, to apply boundary condition W1 = WN = 0, Eq. (17) becomes

[0A12A1,N100A22A2,N100AN2AN,N10]{W1W2WN}={W1W2WN}(18)( 18 )

Or

{W}=[A¯]{W}.

For the second derivative, one has

{W}=[A]{W}.

Using above relation for {W′}

{W}=[A][A¯]{W}=[B¯]{W}withB¯=[A][A¯].

Now, since W1=WN=0, third order derivative of the displacement function may be obtained as follows.

{W}=[A¯]{W}=[A¯][B¯]{W}=[C¯]{W}.

Here, we have used the above relation for W″ and [C¯]=[A¯][B¯]

Similarly, for fourth order derivative,

{WIV}=[A]{W}=[A][C¯]{W}=[B¯][B¯]{W}=[D¯]{W}

where [D¯]=[B¯][B¯]or[D¯]=[A][C¯].

Clamped-simply supported: Proceeding in the similar fashion as that of SS, we have

{W}=[A¯]{W}{W}=[A¯1]{W}=[A¯1][A¯]{W}=[B¯]{W}

with [B¯]=[A¯1][A¯].

{W}=[A¯2]{W}=[A¯2][B¯]{W}=[C¯]{W}

with [C¯]=[A¯2][B¯].

{WIV}=[A¯]{W}=[A¯][C¯]{W}=[D¯]{W}

with [D¯]=[A][C¯].

Clamped-Clamped: In this case, one may find

{W}=[A¯]{W}{W}=[A¯]{W}=[A¯][A¯]{W}=[B¯]{W}

with [B¯]=[A¯][A¯].

{W}=[A]{W}=[A][B¯]{W}=[C¯]{W}

with [C¯]=[A¯][B¯].

{WIV}=[A]{W}=[A][C¯]{W}=[D¯]{W}

with [D¯]=[A][C¯] .

Clamped-free: Finally, for CF, one have

{W}=[A¯1]{W}{W}=[A¯1]{W}=[A¯1][A¯1]{W}=[B¯]{W}

with [B¯]=[A¯1][A¯1].

{W}=[A¯2]{W}=[A¯2][B¯]{W}=[C¯]{W} 

with [C¯]=[A¯2][B¯]

{WIV}=[A¯2]{W}=[A¯2][C¯]{W}=[D¯]{W} 

with [D¯]=[A¯2][C¯]

It is noted that while substituting value of the derivatives in the governing differential equations, one has to use [A¯], [B¯], [C¯], [D¯] as per the boundary condition.

Substituting the value of Eq. (11) into Eq. (10), one may obtain generalized eigenvalue problem as

[S]{W}=λ2[T]{W}(19)( 19 )

where S is the stiffness matrix and T is the mass matrix.

Figure 2:

Variation of frequency parameters with number of terms for SS condition.

5 Numerical results and discussions

In this section, frequency parameters λ have been obtained by solving Eq. (19) using MATLAB program developed by the authors. The DQ method has been employed and the boundary conditions are implemented in the coefficient matrix. It may be noted that the following parameters are taken for the computation:

E = 30 × 106 Pa, L = 10 nm, Poisson’s ratio (ν) = 0.3, KS=56, non-uniform parameter (η) = 0.2 and unless mentioned Lh=10.

5.1 Validation

In order to validate the present method, we compare our results of frequency parameter (λ) with those available in the literature [19, 20, 30]. For this purpose, we consider uniform nanobeam, viz. η = 0. For this purpose, the same parameters are used as in Refs [19, 20, 30]. Comparison of the fundamental frequency parameter for SS nanobeam based on Euler-Bernoulli beam theory has been shown in Table 1 for different nonlocal parameters (μ). In this case aspect ratio (L/h) is taken as 10. Similarly, in Table 2, fundamental frequency parameter (λ) for SS nanobeam is compared with Aydogu [20] and Eltaher [30] with aspect ratio (L/h) as 20. From these tables, one may observe close agreement of results with those available in the Refs [19, 20, 30].

Table 1:

Comparisons of first fundamental frequency parameter (λ) for SS Nano beam.

5.2 Convergence

A convergence study is carried out for finding minimum number of grid points to obtain the convergence of the results. To show how the solution is affected by the grid points, variations of the frequency parameters (λ) with the number of grid points (N) are shown in Figures 2–5. In these figures, non-local parameter is taken as 1, the values of non-uniform parameter are taken as 0.2, 0.4, 0.6 and 0.8 with L = 10 nm. The convergence is shown for SS, CS, CC and CF conditions. From this figure, one may observe that the convergence is achieved as we increase the number of grid points. It may also be noted that ten grid points are sufficient to compute the desired results.

Figure 4:

Variation of frequency parameters with number of terms for CC condition.

5.3 Effect of nonlocal parameter

Effect of nonlocal parameter on the first four frequency parameters (λ) of nanobeam is analyzed. The values of nonlocal parameter are taken as 0, 1, 2, 3, 4, 5 nm2. In this analysis, boundary conditions such as SS, CS, CC and CF are taken into consideration. Both tabular and graphical results are presented in this context by taking non-uniform parameter as 0.2 and L = 10 nm. Table 3 shows first four frequency parameters of SS, CS, CC and CF nanobeams for different nonlocal parameters. From this table, it is seen that frequency parameters decrease with increase in nonlocal parameter except first fundamental frequency parameter of CF nanobeams. It is also observed that frequency parameters increase with increase in mode number. One of the interesting observation is that CC nanobeams are having highest frequency parameters

Figure 6:

Variation of frequency parameter with nonlocal parameter.

than other set of boundary conditions at the edges. Figure 6 show variation of frequency parameters with nonlocal parameters for different boundary conditions such as SS, CS, CC and CF.

5.4 Effect of nonuniform parameter

In this subsection, the effect of non-uniformity η on the frequency parameter is depicted. It may be noted that ηtakes positive values only and different values of η gives different cross section of the nanobeams. Figure 7 illustrates the variation of frequency parameter with nonuniform parameter η. In this graph, boundary conditions such as SS, CS, CC and CF are taken into consideration with nonlocal pa-

Table 2:

Comparisons of first fundamental frequency parameter (λ) for SS Nano beam.

Table 3:

First four frequency parameters with different aspect ratio as well as with different nonlocal parameter.

rameter = 1 nm2, L = 10 nm. The values of non-uniform parameter η are taken as 0.2, 0.4, 0.6, 0.8 and 1. One may observe from the graph that the frequency parameter de-

Table 4:

First four frequency parameters with different aspect ratio.

creases with increase in the non-uniform parameter and this decrease is more significant in case of higher modes.

5.5 Effect of length-to-height ratio

Effect of length-to-height ratio (L/h) on the first four frequency parameters has been investigated. First four frequency parameters are given in Table 4 for different L/h (10, 20, 30, 40, 50). In this table, we have considered all the boundary conditions such as SS, CS, CC and CF with μ = 1 nm2 and η = 0.2. Graphical results are illustrated in Figure 8, where variation of first four frequency parameter with has been shown. In this figure, L/h ranges from 10 to 50. It is noticed that frequency parameter increases

Figure 8:

Variation of frequency parameter with aspect ratio or length to height ratio (L/h).

with increase in excepts fundamental frequency of CF condition.

5.6 Effect of boundary condition

For designing engineering structures, one must have proper knowledge about boundary conditions. It may help engineers to have an idea without carrying out detail investigation. Therefore, analysis of boundary conditions is quite important. In this subsection, we have considered the effect of boundary condition on the frequency parameter. Figure 9 depicts variation of fundamental frequency parameter with nonlocal parameter for different boundary conditions. This graph is plotted with and η = 0.2. It is observed from the figure that CC nanobeams are having highest frequency parameter and CF nanobeams are having lowest frequency parameter. It is also seen that frequency

Figure 9:

Variation of frequency parameter with boundary condition

parameters decrease with increase in nonlocal parameter for SS, CS, and CC boundary conditions but frequency parameters increase with nonlocal parameter in case of first mode of CF nanobeams.

6 Conclusions

Vibration analysis of nonuniform nanobeams with exponentially varying stiffness based on Euler-Bernoulli beam theory has been carried out. Differential quadrature method has been employed and the boundary conditions have been substituted in the coefficient matrix. The numerical as well as graphical results are presented to show the effects of the nonuniform parameter, the nonlocal parameter, L/h and the boundary condition on the frequency parameters. One may observe from the graph that the frequency parameter decreases with increase in the non-uniform parameter and this decrease is more significant in case of higher modes. It is found that clamped(CC) nanobeam possesses highest frequency parameters and cantilever(CF) nanobeam possesses lowest among all other types of boundary conditions at the edges. It is also found that the effect of nonlocal parameter is more in higher modes. One of the interesting observation is that fundamental frequency parameter of cantilever(CF) nanobeam does not decrease with increase in nonlocal parameter.

Acknowledgement

The authors are thankful to Defence Research & Development Organization(DRDO), Ministry of Defence, New Delhi, India (Sanction Code: DG/TM/ERIPR/GIA/17-18/0129/020) for the support and funding to carry out the present research work.

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