Free Vibration and Buckling Analyses of Nanobeam Embedded in Pasternak Foundation

1. Kumar ^a* , I. Ali ^b

^a Department of Mathematics, Government Girls Degree College, Behat-247121, Uttar Pradesh, India

^b Department of Mathematics, Government P.G. College, Gopeshwar-246401, Uttarakhand, India

1.     Introduction

After the invention of carbon nanotubes (Iijima [1]), small-scale structures and devices have been developed due to advancements in nanotechnology and nanoscience. Nanobeams and carbon nanotubes are used in micro-electromechanical systems (MEMS), nano-electromechanical systems (NEMS), microactuators, transistors and microsensors, etc. Due to the wide applications of nanotechnology and nanoscience in modern science and technology, researchers have carried out extensive research on nanomaterials. Nanobeams are used in cancer treatment. To propose new designs, an analysis of the mechanical behavior of nanobeams becomes necessary. This problem can be solved by atomistic mechanics (Baughman et al. [2]), hybrid atomistic-continuum mechanics (Wang et al. [3]), and continuum mechanics approach. The first two approaches are computationally expensive and time-consuming. Therefore, the continuum mechanics approach has been used by researchers to model nano-systems as rods, beams, plates, and shells. In this approach, crystal material architecture is replaced by a continuous medium having homogeneous properties. This continuous medium can predict the overall response of the nanomaterial. Hence, new constitutive laws are required to study the nanoscale size effects. Up to a certain size, classical theories can be used to study the behavior of structures. Eringen’s nonlocal elasticity theory [4-7] has been proposed to incorporate the nanoscale effect. The work related to free vibration and buckling of a nanobeam has been reported in the literature as follows: Lu et al. [8] derived frequency equations and mode functions of a nonlocal Euler-Bernoulli beam. Xu [9] investigated free transverse vibration of nano-to-micron scale beams using the integral equation approach. Wang et al. [10] solved the free vibration problem of micro/nano beams analytically for different combinations of classical boundary conditions. Challamel and Wang [11] presented a small length scale effect for a nonlocal cantilever beam. Murmu and Pradhan [12] used the differential quadrature method to obtain the natural frequencies of a nonuniform cantilever nanobeam. Roque et al. [13] studied the bending, buckling, and free vibration of Timoshenko nanobeams using the meshless method. Mohammadi and Ghannadpour [14] used Chebyshev polynomials in the Rayleigh-Ritz method to study the free vibration of Timoshenko nanobeams. Li et al. [15] investigated the dynamics and stability of transverse vibration of nonlocal nanobeams under variable axial load. Vibration analysis of Euler-Bernoulli nanobeams has been presented by Eltaher et al. [16] using the finite element method. Behra and Chakraverty [17] used orthogonal polynomials in the Rayleigh-Ritz method to obtain frequency parameters and mode shapes of Euler and Timoshenko nanobeams. Rahmani and Pedram [18] presented a closed-form solution for the vibration behavior of functionally graded Timoshenko nanobeams. In another paper, Behra and Chakraverty [19] applied the differential quadrature method to study the free vibration of nanobeams based on various nonlocal theories. Ebrahimi and Salari [20] studied free flexural vibration of functionally graded nanobeams using the differential transform method. Ebrahimi et al. [21] investigated the vibrational characteristics of functionally graded nanobeams using the differential transform method. Tuna and Kirca [22] studied free vibration and buckling of nonlocal Euler-Bernoulli beams utilizing the Laplace transform method. Hamza-Cherif et al. [23] employed a differential transform method to study the free vibration of a single-walled carbon nanotube resting on an elastic foundation under thermal effect. Aria and Friswell [24] used the finite element method to examine the free vibration and the buckling behavior of functionally graded Timoshenko nanobeams. Nikam and Sayyad [25] presented closed-form solutions for bending, buckling, and free vibration of simply supported nanobeams. Arian et al. [26] studied the effect of the thickness-to-length ratio on the frequency ratio of Timoshenko nanobeams and nanoplates. Ufuk [27] used the Ritz method in free transverse vibration analysis of cantilever nanobeam with intermediate support. Wang et al. [28] presented an analytical buckling analysis of micro and nano roads/tubes based on nonlocal Timoshenko beam theory. Reddy [29] reformulated Euler-Bernoulli, Timoshenko, Reddy, and Levinson beam theories using the nonlocal theory for bending, buckling, and vibrations of beams. Pradhan and Phadikar [30] solved buckling, bending, and vibration problems of nonhomogeneous nanotubes using the differential quadrature method. Aydogdu [31] proposed a generalized nonlocal beam theory to study the bending, buckling, and free vibration of nanobeams. Thai [32] presented analytical solutions for bending, buckling, and free vibration of nanobeams based on a nonlocal shear deformation beam theory. Eltaher et al. [33] used Galerkin finite element method to study static deflection and buckling response of functionally graded nanobeams for different combinations of boundary conditions. Ebrahimi and Salari [34] developed a differential transform method solution for vibrational and buckling analysis of functionally graded nanobeams considering the physical neutral axis position. Buckling analysis of two-directional functionally graded Euler-Bernoulli nanobeam has been presented by Nejad et al. [35] using the generalized differential quadrature method. Safarabadi et al. [36] studied the effect of surface energy on the free vibration of rotating nanobeam using the differential quadrature method. Mohammadi et al. [37] used the differential quadrature method to present the hygro-mechanical vibration of a rotating viscoelastic nanobeam resting on a visco-Pasternak foundation subjected to nonlinear temperature variation. An asymptotic solution for critical buckling load of Euler-Bernoulli nanobeam using Eringen’s two-phase nonlocal theory has been presented by Zhu et al. [38]. Khaniki and Hosseini-Hashemi [39] presented a buckling analysis of tapered nanobeam using nonlocal strain gradient theory and the generalized differential quadrature method. Finite element analysis of bending, buckling, and free vibration problems of Euler-Bernoulli nanobeam has been presented by Tuna and Kirca [40] using Eringen’s nonlocal integral model. Buckling analysis of nonuniform nonlocal strain gradient beams has been presented by Bakhshi et al. [41] using the generalized differential quadrature method. Stability analysis of simply supported nonlocal Euler-Bernoulli beam with varying cross-sections and resting on the Pasternak foundation has been presented by Soltani and Mohammadi [42]. Xu and Zheng [43] solved the buckling problem of the nonlocal strain gradient Timoshenko beam in closed form. The differential quadrature method and the differential transform method have been used by Jena and Chakraverty [44] for the buckling analysis of nanobeams. Ragb et al. [45] obtained natural frequencies of a piezoelectric nanobeam resting on a nonlinear Pasternak foundation using different versions of the differential quadrature method. Nazmul and Devnath [46] derived closed-form solutions for bending and buckling of functionally graded nanobeam using the Laplace transform. Karmakar and Chakravarty [47] studied the thermal vibration of an Euler nanobeam resting on the Winkler-Pasternak foundation using the differential quadrature method and the Adomian decomposition method. Zewei et al. [48] used the differential quadrature method in free vibration, buckling, and dynamic stability analyses of Timoshenko micro/nanobeams resting on the Pasternak foundation under axial load. Civalek et al. [49] studied the stability of restrained FGM nonlocal beams using the Fourier series. Jalaei et al. [50] presented an analytical transient response of porous viscoelastic functionally graded nanobeam subjected to dynamic load and magnetic field. Beni [51] investigated free vibration and static torsion of an electromechanical flexoelectric micro/nanotube. Beni and Beni [52] studied the dynamic stability of an isotropic viscoelastic/piezoelectric Euler-Bernoulli nanobeam using the Galerkin method. Numanoğlu et al. [53] presented a thermo-mechanical vibration analysis of Timoshenko nanobeam using the nonlocal finite element method. Karmakar and Chakraverty [54] used the Adomian decomposition method and the homotopy perturbation method to study the thermal vibration of nonhomogeneous Euler nanobeam resting on the Winkler foundation.

This paper aims at providing a numerical solution for free transverse vibration and buckling of a nanobeam under axial load and resting on the Pasternak foundation. Eringen’s nonlocal elasticity theory along with the Euler-Bernoulli beam theory is used to develop the mathematical model. The PDQM and the HDQM are used to obtain the first three frequencies and the lowest critical buckling loads for clamped-clamped (C-C), clamped-simply supported (C-S), and simply supported-simply supported (S-S) beams.

The impacts of nonlocal parameters, axial force, and elastic foundation moduli on frequencies are studied. The effects of nonlocal parameters, boundary conditions, and elastic foundation moduli on critical buckling load are also studied.

2.     Formulation of the Problem

Consider an isotropic uniform nanobeam of length  cross-section area  , density and transverse deflection  as shown in Fig. 1. Here, t denotes the time. The beam is resting on the Pasternak foundation and is also subjected to uniform compressive load p.

The fundamental assumptions of the Euler-Bernoulli beam theory (Abbas et al. [55]) are:

• The cross-section of the beam remains plane to the deformed axis of the beam.
• Deformed angles of rotation of the neutral axis are small.

This theory does not account for the effect of transverse shear strain and overpredicts natural frequencies.

Fig. 1. Geometry of a Euler-Bernoulli nanobeam
resting on Pasternak foundation

The strain energy of the beam is expressed as (Wang et al. [56]):

where  is normal stress.

Normal strain  in terms of deflection is represented as

Using (2), strain energy given by (1) becomes

where M is the bending moment and is defined as

The kinetic energy of the beam is expressed as (Wang et al. [56])

The potential energy due to the Pasternak foundation is represented as

where  and  are the Winkler and the Pasternak (shear) foundation stiffnesses, respectively.

Work done by the compressive load p is expressed as

Using Hamilton’s principle

The equation of motion of the beam is obtained as follows:

Based on Eringen’s nonlocal elasticity theory, the nonlocal stress tensor at a point  is given as (Murmu and Adhikari [57])

where  and  are classical stress and nonlocal modulus, respectively.

The differential form of the above integral constitutive relation (Murmu and Adhikari [57]) is

where  is a material constant,  is the Laplace operator and a is the internal characteristic length.

For Euler-Bernoulli nanobeam, the relationship between local and nonlocal stresses (Eringen [5]) given by Eq. (11) can be rewritten as follows:

Multiplying the equation (12) by z dA and integrating over the area A, we obtain

where E is Young’s modulus and

is the second moment of the area.

By using Eqs. (9) and (13), the equation of motion is obtained as follows:

where
 is the nonlocal parameter and  is the frequency parameter.

The buckling problem is obtained by putting  in Eq. (15) and is given as follows:

where  is the critical buckling load. 

2.1. Boundary Conditions

Boundary conditions at the ends  and  are given as

For C-C beam

For C-S beam

For S-S beam

Using the PDQM and the HDQM, free vibration and buckling problems are reduced to eigenvalue problems which are solved for frequencies and critical buckling loads, respectively.

2.2. PDQM (Bert and Malik [58])

The displacement  and its derivatives are approximated by a weighted linear sum of functional values at grid points  as follows:

2.3. HDQM (Civalek [59])

In this method, the weighting coefficients of the first, second, third, and fourth order derivatives are obtained as follows:

The Gauss-Chebyshev-Lobatto grid points (Chang [60]) in the range [0, 1] are given below as:

We obtain a set of   equations in  unknowns  by discretizing the equation of motion (15) at grid points
 as follows:

Similarly, the buckling problem (16) can be discretized as follows:

By discretizing boundary conditions
(17a-17d) for the C-C beam, we obtain a set of four equations in  unknowns  

Equations (32) and (34a-34d) form a set of  algebraic linear equations in  unknowns. This set represents a generalized eigenvalue problem that is solved to obtain frequencies for the C-C beam.

The boundary conditions for C-S and S-S beams are discretized as

Similarly, critical buckling loads for C-C, C-S and S-S beams can be obtained by using equation (33) and discretizing boundary conditions represented by equations
(17a-17d), (18a-18d), and (19a-19d), respectively.

3.     Results and Discussion

The first three frequencies and the lowest critical buckling loads are calculated by solving the corresponding eigenvalue problem and the buckling problem, respectively. The Convergence of results for both versions of DQM with an increasing number of grid points is shown in Tables 1 and 2 for
. The value of N has been fixed as 17 as we get results correct to four decimal places. Results are shown in the Tables (3-6).

Tables (3-5) present the first three frequencies while Table 6 shows critical buckling loads for C-C, C-S, and S-S nanobeams for different combinations of parameters. A comparison of frequencies in special cases is presented in Table (7) while a comparison of critical buckling loads in special cases is shown in Table (8). It is evident from the comparison tables that the results are in good agreement with those available in the literature.

Table 1. Convergence of frequency parameter of Euler-Bernoulli nanobeam using PDQM

Table 2. Convergence of frequency parameter  of Euler-Bernoulli nanobeam using HDQM

Table 3. First three values of frequency parameter  of C-C Euler-Bernoulli nanobeam

Table 4. First three values of frequency parameter  of C-S Euler-Bernoulli nanobeam

Table 5. First three values of frequency parameter  of S-S Euler-Bernoulli nanobeam

Table 6. Critical buckling load parameter  of Euler-Bernoulli nanobeam

Table 7. Comparison of frequency parameter  of Euler-Bernoulli nanobeam

Table 8. Comparison of the lowest non-dimensional critical buckling load
parameter  for Euler-Bernoulli nanobeam without foundation

The first three frequencies and critical buckling loads are also presented through graphs via Figs. (2-9). The effect of nonlocal parameter
along with the shear foundation parameter on frequency  is shown in Figs. (2-3) for
 for C-C and C-S beams, respectively.

Figure 4 shows the variation of frequency with respect to nonlocal parameters for different combinations of Winkler and shear foundation parameters.

The effect of the nonlocal parameter along with the Winkler foundation parameter on frequency is shown in Figs. (5-6) for C-C and C-S beams for fixed values of load parameter and shear foundation parameter.

Figure 7 shows the effect of the nonlocal parameter along with the shear foundation parameter on critical buckling loads for two different values of the Winkler foundation parameter for C-C, C-S, and S-S beams.

Figure 8 depicts a three-dimensional variation of critical buckling load for different values of the Winkler foundation parameter and nonlocal parameter keeping the shear foundation parameter constant for C-C, C-S, and S-S beams.

Figure 9 presents a three-dimensional variation of critical buckling load for different values of shear foundation parameter and nonlocal parameter keeping the Winkler foundation parameter constant for C-C, C-S, and S-S beams.

4.     Conclusions

Free transverse vibration and buckling of an Euler-Bernoulli nanobeam resting on the Pasternak foundation have been studied on the basis of Eringen’s nonlocal elasticity theory. The PDQM and the HDQM are used to obtain the first three values of the frequency parameter and the lowest critical buckling load. A computer program in C++ is developed to calculate the results. In this analysis, C-C, C-S, and S-S boundary conditions have been considered. The study shows that nonlocal parameters, boundary conditions, axial force parameters, and elastic foundation moduli have considerable impacts on the results. So, it is concluded from the present study that

• Frequencies decrease with an increase in nonlocal parameters except for frequency in the fundamental mode for C-C and C-S boundary conditions. The important observation is that for certain values of the Pasternak foundation parameter, the frequency in fundamental mode first increases and then decreases for C-C and C-S boundary conditions. The variation of frequency with nonlocal parameter is not monotonic in the first mode. In the second and third modes, frequency continuously decreases with an increase in the value of .
• The maximum frequency is observed for C-C boundary conditions followed by C-S and S-S boundary conditions, respectively.
• The Winkler and the Pasternak foundation parameters have significant effects on the free vibration and stability behaviour of nanobeams. The critical buckling load increases with an increase in foundation moduli and decreases with an increase in the nonlocal parameter .
• The greatest buckling load is observed for C-C boundary conditions followed by C-S and S-S boundary conditions, respectively.
• The results obtained by the PDQM and the HDQM are identical.

Nomenclature

Algorithm of C++

Step 1: Generation of grid points

Step 2: Generation of weighting coefficients

Step 3: Discretization of governing equation at grid points

Step 4: Implementation of boundary conditions

Step 5: Implementation of bisection method to obtain frequencies and critical buckling loads.

Conflicts of Interest

The authors declare that there is no conflict of interest between the authors.

Acknowledgment

The authors are thankful to the learned reviewers for their critical comments to improve the quality of the paper.

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