Free Vibration and Buckling Analyses of Nanobeam Embedded in Pasternak Foundation

Document Type : Research Article

Authors

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

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

Abstract

In this paper, free transverse vibration and buckling analyses of a nanobeam are presented by coupling the Euler-Bernoulli beam (EBT) theory and Eringen’s nonlocal elasticity theory. The nanobeam is embedded in the Pasternak foundation. Hamilton’s energy principle is used to derive governing differential equations. The Lagrange polynomial-based differential quadrature method (PDQM) and a harmonic differential quadrature method (HDQM) are used to convert the governing differential equation and boundary conditions into a set of linear algebraic equations. The first three frequencies and the lowest critical buckling loads for clamped-clamped, clamped-simply supported, and simply supported-simply supported boundary conditions are obtained by implementing the bisection method through a computer program written in C++. The impacts of nonlocal Eringen’s parameter (scaling effect parameter), boundary conditions, axial force, and elastic foundation moduli on frequencies are examined. The effects of nonlocal Eringen’s parameter, boundary conditions, and elastic foundation moduli on critical buckling load are also studied. A convergence study of both versions of DQM is conducted to validate the present analysis. A comparison of frequencies and critical buckling loads with those available in the literature is presented.

Keywords

Main Subjects


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

 

KEYWORDS

 

ABSTRACT

Free vibration;

Buckling;

Nonlocal;

Nanobeam;

Euler-Bernoulli beam theory;

Pasternak foundation;

PDQM;

HDQM.

In this paper, free transverse vibration and buckling analyses of a nanobeam are presented by coupling the Euler-Bernoulli beam (EBT) theory and Eringen’s nonlocal elasticity theory. The nanobeam is embedded in the Pasternak foundation. Hamilton’s energy principle is used to derive governing differential equations. The Lagrange polynomial-based differential quadrature method (PDQM) and a harmonic differential quadrature method (HDQM) are used to convert the governing differential equation and boundary conditions into a set of linear algebraic equations. The first three frequencies and the lowest critical buckling loads for clamped-clamped, clamped-simply supported, and simply supported-simply supported boundary conditions are obtained by implementing the bisection method through a computer program written in C++. The impacts of nonlocal Eringen’s parameter (scaling effect parameter), boundary conditions, axial force, and elastic foundation moduli on frequencies are examined. The effects of nonlocal Eringen’s parameter, boundary conditions, and elastic foundation moduli on critical buckling load are also studied. A convergence study of both versions of DQM is conducted to validate the present analysis. A comparison of frequencies and critical buckling loads with those available in the literature is presented.

 

 

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]):

 

(1)

where  is normal stress.

Normal strain  in terms of deflection is represented as

 

(2)

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

 

(3)

where M is the bending moment and is defined as

 

(4)

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

 

(5)

The potential energy due to the Pasternak foundation is represented as

 

(6)

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

Work done by the compressive load p is expressed as

 

(7)

Using Hamilton’s principle

 

(8)

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

 

 

(9)

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

 

(10)

where  and  are classical stress and nonlocal modulus, respectively.

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

 

(11)

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:

 

(12)

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

 

(13)

where E is Young’s modulus and

 

(14)

is the second moment of the area.

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

 

 

 

(15)

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:

 

 

(16)

where  is the critical buckling load. 

2.1. Boundary Conditions

Boundary conditions at the ends  and  are given as

For C-C beam

 

 

(17a-17d)

For C-S beam

 

 

 

 

 

For S-S beam

 

 

 

(19a-19d)

 

 

 

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:

 

(20)

 

(21)

 

   and   

(22)

 

   and   

(23)

 

         and

 

(24)

2.3. HDQM (Civalek [59])

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

 

(25)

 

(26)

 

   and   

(27)

 

(28)

 

(29)

 

 

(30)

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

 

 

(31)

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

 

 

 

 

(32)

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

 

 

 

(33)

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

 

(34a-34d)

 

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

 

 

 

 

 

(35a-35d)

and

 

 

 

 

 

 

 

(36a-36d)

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

 

 

 

 

10

13

14

15

16

17

Mode

C-C

I

5.0562

5.0562

5.0562

5.0562

5.0562

5.0562

II

7.4587

7.4640

7.4640

7.4640

7.4640

7.4640

III

9.6622

9.5491

9.5481

9.5475

9.5475

9.5475

 

C-S

I

4.4727

4.4728

4.4728

4.4728

4.4728

4.4728

II

6.8302

6.8326

6.8325

6.8325

6.8325

6.8325

III

9.2086

8.9567

8.9573

8.9560

8.9560

8.9560

 

S-S

I

4.0068

4.0068

4.0068

4.0068

4.0068

4.0068

II

6.2130

6.2169

6.2169

6.2169

6.2169

6.2169

III

8.5510

8.3685

8.3675

8.3667

8.3667

8.3667

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

 

 

 

 

10

13

14

15

16

17

Mode

C-C

I

5.0650

5.0564

5.0566

5.0563

5.0562

5.0562

II

7.4664

7.4651

7.4640

7.4641

7.4640

7.4640

III

9.5614

9.5477

9.548

9.5475

9.5475

9.5475

 

C-S

I

4.4859

4.4702

4.4737

4.4722

4.4730

4.4728

II

6.8247

6.8355

6.8316

6.8332

6.8323

6.8325

III

8.9794

8.9547

8.9572

8.9556

8.9563

8.9560

 

S-S

I

3.9863

4.0068

4.0056

4.0068

4.0065

4.0068

II

6.2266

6.2169

6.2173

6.2169

6.2169

6.2169

III

8.3869

8.3667

8.3675

8.3667

8.3669

8.3667

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

 

 

 

 

 

 

 

0

0.1

0.2

0.3

0.4

 

 

 

 

 

 

Mode

-3

3

-3

3

-3

3

-3

3

-3

3

1

0

I

4.8170

4.6427

4.7119

4.4725

4.4740

4.0547

4.2233

3.5302

4.0158

2.9427

 

 

II

7.9240

7.7814

7.2571

7.0188

6.2603

5.7842

5.5576

4.7433

5.1047

3.8372

 

 

III

11.0512

10.9396

9.3728

9.1401

7.6302

7.1119

6.6414

5.7228

6.0581

4.6072

 

10

I

5.0692

4.9229

5.0431

4.8528

4.9852

4.6988

4.9273

4.5449

4.8830

4.4251

 

 

II

8.1452

8.0148

7.6084

7.4037

6.8721

6.5259

6.4215

5.9490

6.1701

5.6052

 

 

III

11.2299

11.1237

9.7262

9.5189

8.3129

7.9246

7.6364

7.0905

7.3007

6.6405

 

30

I

5.4780

5.3657

5.5558

5.4172

5.6941

5.5092

5.8005

5.5797

5.8676

5.6238

 

 

II

8.5381

8.4262

8.1924

8.0304

7.7561

7.5225

7.5171

7.2384

7.3947

7.0906

 

 

III

11.5632

11.4663

10.3357

10.1642

9.3206

9.0525

8.9129

8.5873

8.7377

8.3804

 

50

I

5.8067

5.7145

5.9546

5.8436

6.2073

6.0667

6.3970

6.2356

6.5156

6.3415

 

 

II

8.8809

8.7824

8.6724

8.5368

8.4120

8.2313

8.2736

8.0684

8.2043

7.9865

 

 

III

11.8693

11.7800

10.8533

10.7058

10.0788

9.8693

9.7993

9.5586

9.6901

9.4338

10

0

I

4.8370

4.6650

4.7333

4.4975

4.4989

4.0881

4.2529

3.5802

4.0501

3.0273

 

 

II

7.9285

7.7862

7.2630

7.0253

6.2695

5.7958

5.5707

4.7642

5.1215

3.8764

 

 

III

11.0528

10.9413

9.3756

9.1430

7.6353

7.1181

6.6491

5.7348

6.0682

4.6301

 

10

I

5.0864

4.9416

5.0606

4.8724

5.0033

4.7204

4.9461

4.5687

4.9022

4.4509

 

 

II

8.1494

8.0192

7.6135

7.4093

6.8790

6.5340

6.4299

5.9597

6.1797

5.6180

 

 

III

11.2315

11.1253

9.7286

9.5215

8.3168

7.9292

7.6415

7.0968

7.3065

6.6482

 

30

I

5.4917

5.3802

5.5689

5.4313

5.7062

5.5226

5.8120

5.5926

5.8787

5.6365

 

 

II

8.5417

8.4299

8.1965

8.0347

7.7609

7.5278

7.5224

7.2443

7.4002

7.0970

 

 

III

11.5646

11.4678

10.3378

10.1663

9.3234

9.0556

8.9161

8.5908

8.7410

8.3842

 

50

I

5.8182

5.7265

5.9653

5.8548

6.2167

6.0768

6.4056

6.2448

6.5237

6.3503

 

 

II

8.8841

8.7858

8.6758

8.5404

8.4158

8.2354

8.2776

8.0727

8.2084

7.9909

 

 

III

11.8707

11.7814

10.8551

10.7077

10.0810

9.8716

9.8017

9.5612

9.6926

9.4365

1000

0

I

6.2618

6.1852

6.2150

6.1160

6.1165

5.9689

6.0243

5.8288

5.9568

5.7247

 

 

II

8.3843

8.2646

7.8372

7.6506

7.0957

6.7842

6.6478

6.2287

6.4003

5.9049

 

 

III

11.2317

11.1255

9.6624

9.4509

8.1392

7.7229

7.3664

6.7465

6.9595

6.1704

 

10

I

6.3824

6.3110

6.3694

6.2782

6.3409

6.2093

6.3131

6.1448

6.2922

6.0976

 

 

II

8.5725

8.4612

8.1213

7.9545

7.5384

7.2825

7.2080

6.8884

7.0342

6.6758

 

 

III

11.4022

11.3009

9.9869

9.7961

8.7172

8.3848

8.1443

7.7062

7.8719

7.3657

 

30

I

6.6018

6.5387

6.6467

6.5673

6.7290

6.6197

6.7943

6.6607

6.8365

6.6868

 

 

II

8.9138

8.8158

8.6131

8.4744

8.2434

8.0509

8.0465

7.8224

7.9472

7.7063

 

 

III

11.7214

11.6285

10.5549

10.3941

9.6148

9.3719

9.2465

8.9571

9.0901

8.7759

 

50

I

6.7982

6.7414

6.8920

6.8213

7.0595

6.9652

7.1907

7.0787

7.2751

7.1518

 

 

II

9.2178

9.1300

9.0323

8.9127

8.8034

8.6466

8.6831

8.5068

8.6233

8.4372

 

 

III

12.0160

11.9299

11.0436

10.9038

10.3143

10.1194

10.0545

9.8326

9.9536

9.7182

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

 

 

 

 

 

 

 

0

0.1

0.2

0.3

0.4

 

 

 

 

 

 

Mode

-3

3

-3

3

-3

3

-3

3

-3

3

 

1

0

I

4.0655

3.7797

3.9825

3.6454

3.7920

3.3102

3.5875

2.8782

3.4149

2.3753

 

 

 

II

7.1586

6.9764

6.5937

6.3298

5.7313

5.2570

5.1121

4.3304

4.7087

3.5081

 

 

 

III

10.2760

10.1435

8.7705

8.5286

7.1604

6.6543

6.2284

5.3495

5.6740

4.2981

 

 

10

I

4.4351

4.2257

4.4020

4.1661

4.3280

4.0326

4.2531

3.8955

4.1945

3.7858

 

 

 

II

7.4343

7.2727

6.9738

6.7535

6.3289

5.9920

5.9283

5.4830

5.7039

5.1771

 

 

 

III

10.4861

10.3616

9.1342

8.9213

7.8214

7.4461

7.1736

6.6556

6.8449

6.2233

 

 

30

I

4.9708

4.8292

4.9924

4.8377

5.0270

4.8478

5.0497

4.8500

5.0621

4.8483

 

 

 

II

7.9073

7.7745

7.5880

7.4194

7.1786

6.9553

6.9549

6.6943

6.8427

6.5603

 

 

 

III

10.8717

10.7604

9.7535

9.5801

8.7899

8.5329

8.3815

8.0736

8.1961

7.8604

 

 

50

I

5.3684

5.2589

5.4239

5.3054

5.5165

5.3833

5.5854

5.4409

5.6287

5.4767

 

 

 

II

8.3066

8.1930

8.0812

7.9428

7.8028

7.6313

7.6607

7.4694

7.5943

7.3922

 

 

 

III

11.2198

11.1188

10.2733

10.1256

9.5151

9.3149

9.2184

8.9913

9.0909

8.8502

 

10

0

I

4.0985

3.8207

4.0176

3.6910

3.8326

3.3706

3.6352

2.9682

3.4701

2.5278

 

 

 

II

7.1648

6.9830

6.6016

6.3387

5.7432

5.2724

5.1288

4.3578

4.7301

3.5591

 

 

 

III

10.2781

10.1457

8.7738

8.5322

7.1665

6.6619

6.2377

5.3641

5.6863

4.3261

 

 

10

I

4.4606

4.2552

4.4281

4.1969

4.3555

4.0665

4.2820

3.9330

4.2247

3.8266

 

 

 

II

7.4398

7.2786

6.9805

6.7608

6.3378

6.0025

5.9391

5.4966

5.7160

5.1932

 

 

 

III

10.4880

10.3636

9.1371

8.9245

7.8261

7.4516

7.1797

6.6632

6.8519

6.2326

 

 

30

I

4.9890

4.8490

5.0104

4.8574

5.0446

4.8674

5.0671

4.8696

5.0794

4.8679

 

 

 

II

7.9118

7.7792

7.5932

7.4249

7.1847

6.9620

6.9615

6.7018

6.8497

6.5682

 

 

 

III

10.8735

10.7622

9.7559

9.5826

8.7932

8.5365

8.3853

8.0779

8.2002

7.8650

 

 

50

I

5.3829

5.2743

5.4380

5.3205

5.5298

5.3977

5.5983

5.4548

5.6413

5.4903

 

 

 

II

8.3105

8.1971

8.0855

7.9472

7.8076

7.6363

7.6657

7.4748

7.5994

7.3977

 

 

 

III

11.2214

11.1205

10.2754

10.1278

9.5177

9.3177

9.2213

8.9943

9.0939

8.8534

 

1000

0

I

5.9722

5.8895

5.9467

5.8555

5.8927

5.7838

5.8418

5.7162

5.8043

5.6663

 

 

 

II

7.7595

7.6179

7.3316

7.1437

6.7516

6.4796

6.4040

6.0623

6.2136

5.8240

 

 

 

III

10.4988

10.3748

9.1193

8.9055

7.7608

7.3758

7.0738

6.5297

6.7169

6.0506

 

 

10

I

6.1014

6.0251

6.0888

6.0049

6.0614

5.9620

6.0347

5.9212

6.0145

5.8911

 

 

 

II

7.9792

7.8496

7.6159

7.4493

7.1431

6.9162

6.8751

6.6047

6.7350

6.4375

 

 

 

III

10.6963

10.5792

9.4456

9.2540

8.2980

7.9888

7.7712

7.3768

7.5178

7.0703

 

 

30

I

6.3339

6.2673

6.3444

6.2712

6.3614

6.2759

6.3727

6.2769

6.3789

6.2761

 

 

 

II

8.3702

8.2588

8.1045

7.9672

7.7752

7.6017

7.6014

7.4053

7.5161

7.3073

 

 

 

III

11.0611

10.9555

10.0122

9.8523

9.1366

8.9092

8.7768

8.5113

8.6163

8.3307

 

 

50

I

6.5402

6.4806

6.5711

6.5057

6.6239

6.5484

6.6641

6.5807

6.6897

6.6010

 

 

 

II

8.7117

8.6136

8.5178

8.4001

8.2825

8.1401

8.1644

8.0077

8.1096

7.9452

 

 

 

III

11.3926

11.2962

10.4963

10.3581

9.7926

 

 

 

9.6096

9.5219

9.3167

9.4065

9.1903

 

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

 

 

 

 

 

 

 

0

0.1

0.2

0.3

0.4

 

 

 

 

 

 

Mode

-3

3

-3

3

-3

3

-3

3

-3

3

1

0

I

3.3637

2.8800

3.3047

2.7837

3.1658

2.5340

3.0110

2.1894

2.8756

1.7397

 

 

II

6.4002

6.1613

5.9303

5.6234

5.1867

4.6973

4.6355

3.8724

4.2700

3.1268

 

 

III

9.5037

9.3445

8.1657

7.9091

6.6878

6.1893

5.8167

4.9733

5.2951

3.9890

 

10

I

3.8803

3.5975

3.8423

3.5496

3.7566

3.4393

3.6674

3.3212

3.5952

3.2226

 

 

II

6.7474

6.5458

6.3556

6.1112

5.7831

5.4491

5.4099

4.9893

5.1930

4.7058

 

 

III

9.7524

9.6055

8.5461

8.3242

7.3312

6.9668

6.7148

6.2233

6.3968

5.8125

 

30

I

4.5380

4.3706

4.5145

4.3442

4.4624

4.2856

4.4102

4.2264

4.3693

4.1799

 

 

II

7.3144

7.1582

7.0147

6.8364

6.6092

6.3938

6.3696

6.1270

6.2407

5.9813

 

 

III

10.1995

10.0716

9.1829

9.0058

8.2645

8.0176

7.8563

7.5658

7.6645

7.3497

 

50

I

4.9930

4.8695

4.9753

4.8505

4.9367

4.8087

4.8983

4.7672

4.8686

4.7349

 

 

II

7.7737

7.6445

7.5275

7.3847

7.2069

7.0432

7.0251

6.8477

6.9300

6.7448

 

 

III

10.5946

10.4807

9.7094

9.5605

8.9588

8.7675

8.6452

8.4312

8.5030

8.2775

10

0

I

3.4213

2.9699

3.3653

2.8827

3.2345

2.6622

3.0903

2.3780

2.9659

2.0643

 

 

II

6.4088

6.1709

5.9411

5.6360

5.2028

4.7189

4.6580

3.9106

4.2987

3.1979

 

 

III

9.5063

9.3472

8.1698

7.9137

6.6953

6.1988

5.8281

4.9915

5.3102

4.0240

 

10

I

3.9183

3.6449

3.8814

3.5989

3.7983

3.4934

3.7122

3.3810

3.6427

3.2879

 

 

II

6.7547

6.5538

6.3643

6.1210

5.7947

5.4629

5.4240

5.0073

5.2090

4.7272

 

 

III

9.7549

9.6080

8.5497

8.3280

7.3369

6.9734

6.7222

6.2327

6.4054

5.8240

 

30

I

4.5619

4.3973

4.5387

4.3713

4.4875

4.3139

4.4362

4.2559

4.3961

4.2104

 

 

II

7.3202

7.1643

7.0212

6.8435

6.6170

6.4024

6.3783

6.1367

6.2499

5.9918

 

 

III

10.2017

10.0738

9.1858

9.0088

8.2685

8.0220

7.8609

7.5710

7.6695

7.3554

 

50

I

5.0110

4.8889

4.9935

4.8701

4.9553

4.8288

4.9173

4.7878

4.8880

4.7560

 

 

II

7.7785

7.6495

7.5328

7.3903

7.2129

7.0496

7.0316

6.8547

6.9367

6.7521

 

 

III

10.5965

10.4827

9.7119

9.5630

8.9619

8.7708

8.6487

8.4349

8.5067

8.2814

1000

0

I

5.7941

5.7164

5.7828

5.7047

5.7583

5.6791

5.7342

5.6541

5.7158

5.6349

 

 

II

7.1930

7.0283

6.8764

6.6866

6.4425

6.2086

6.1822

5.9147

6.0406

5.7519

 

 

III

9.7821

9.6366

8.5901

8.3717

7.4005

7.0472

6.8044

6.3352

6.5001

5.9486

 

10

I

5.9169

5.8441

5.9064

5.8332

5.8834

5.8093

5.8609

5.7859

5.8436

5.7680

 

 

II

7.4447

7.2968

7.1617

6.9947

6.7835

6.5853

6.5632

6.3429

6.4458

6.2123

 

 

III

10.0112

9.8757

8.9209

8.7270

7.8963

7.6105

7.4205

7.0704

7.1906

6.8018

 

30

I

6.1420

6.0771

6.1325

6.0673

6.1120

6.0461

6.0919

6.0254

6.0766

6.0095

 

 

II

7.8829

7.7591

7.6474

7.5114

7.3428

7.1885

7.1715

7.0053

7.0822

6.9093

 

 

III

10.4272

10.3076

9.4897

9.3298

8.6753

8.4636

8.3273

8.0863

8.1675

7.9111

 

50

I

6.3447

6.2859

6.3361

6.2771

6.3176

6.2580

6.2994

6.2393

6.2855

6.2250

 

 

II

8.2582

8.1510

8.0550

7.9392

7.7974

7.6694

7.6555

7.5199

7.5824

7.4427

 

 

III

10.7986

10.6912

9.9715

9.8343

9.2876

9.1166

9.0082

8.8202

8.8830

8.6865

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

 

 

 

 

 

0

0.1

0.2

0.3

0.4

 

 

C-C

1

0

39.5544

28.3659

15.3518

10.3878

5.4280

 

30

69.5544

58.3659

45.3518

40.3878

35.4280

 

50

89.5544

78.3659

65.3518

60.3878

55.4280

10

0

40.2376

28.9201

15.7558

10.4530

6.0810

 

30

70.2376

58.9201

45.7558

40.4530

36.0810

 

50

90.2376

78.9201

65.7558

60.4530

56.0810

1000

0

101.1910

61.2950

28.6333

13.6863

7.8113

 

30

131.1910

91.2950

58.6333

43.6863

37.8113

 

50

151.1910

111.2950

78.6333

63.6863

57.8113

 

 

C-S

1

0

20.2733

16.8759

11.2375

7.2283

4.8300

 

30

50.2733

46.8759

41.2375

37.2283

34.8300

 

50

70.2733

66.8759

61.2375

57.2283

54.8300

10

0

21.0149

17.5676

11.8463

7.7780

6.1985

 

30

51.0149

47.5676

41.8463

37.7780

36.1985

 

50

71.0149

67.5676

61.8463

57.7780

56.1985

1000

0

74.4955

55.8308

25.0525

12.7116

7.3664

 

30

104.4950

85.8308

55.0525

42.7116

37.3664

 

50

124.4950

105.8310

75.0525

62.7116

57.3664

 

 

S-S

1

0

9.9709

9.0843

7.1774

5.3282

3.9280

 

30

39.9709

39.0843

37.1774

35.3282

33.9280

 

50

59.9709

59.0843

57.1774

55.3282

53.9280

10

0

10.8828

9.9963

8.0893

6.2400

4.8399

 

30

40.8828

39.9962

38.0893

36.2400

34.8399

 

50

60.8828

59.9962

58.0893

56.2400

54.8399

1000

0

64.8087

53.6346

25.8330

12.3344

6.9380

 

30

94.8087

83.6346

55.8330

42.3344

36.9380

 

50

114.8087

103.6346

75.8330

62.3344

56.9380

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

Boundary condition

Reference

 

 

 

Mode

I

Mode

II

Mode

III

C-C

Demir [61]

0

0

0

4.73004

7.8532

11.0856

 

Ebrahimi and Salari [34]

 

 

 

4.7299

7.8525

10.9934

 

Present

 

 

 

4.7300

7.8532

10.9956

 

Demir [61]

0

0

0.05

4.69433

7.64178

10.4625

 

Present

 

 

 

4.6943

7.6418

10.4042

 

Demir [61]

0

0

0.2

4.27661

6.03520

7.28636

 

Present

 

 

 

4.2766

6.0352

7.3840

 

Demir [61]

10000

0

0.2

10.08260

10.31634

10.64047

 

Present

 

 

 

10.0826

10.3163

10.6723

 

Rahbar-Ranji and Shahbaztabar [62]

2.5

2.5*

0

5.3200

8.3815

11.4280

 

Present

 

 

 

5.3224

8.3821

11.4282

 

Rahbar-Ranji and Shahbaztabar [62]

10000

2.5*

0

10.1943

11.0546

12.8252

 

Present

 

 

 

10.1943

11.0546

12.8251

C-S

Wang, Zhang, and He [10]

0

0

0

3.9266

7.0686

10.2102

 

Ebrahimi and Salari [34]

 

 

 

3.9265

7.0679

10.2081

 

Present

 

 

 

3.9266

7.0686

10.2102

 

Wang, Zhang, and He [10]

0

0

0.3

3.2828

4.7668

5.8371

 

Present

 

 

 

3.28284

4.7668

5.8371

 

Wang, Zhang, and He [10]

0

0

0.5

2.7899

3.8325

4.6105

 

Present

 

 

 

2.7899

3.8325

4.6105

S-S

Demir [61]

0

0

0

3.14159

6.28319

9.42394

 

Present

 

 

 

3.1416

6.2832

9.4248

 

Demir [61]

0

0

0.05

3.12251

6.13706

8.96310

 

Present

 

 

 

3.1225

6.1371

8.9639

 

Demir [61]

0

0

0.2

2.89083

4.95805

6.45140

 

Present

 

 

 

2.8908

4.9581

6.4520

 

Demir [61]

10000

0

0.2

10.01741

10.14776

10.40748

 

Present

 

 

 

10.0174

10.1478

10.4076

 

Rahbar-Ranji and Shahbaztabar [62]

10000

2.5*

0

10.0842

10.5806

11.9042

 

Present

 

 

 

10.0842

10.5806

11.9042

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

Boundary condition

Reference

 

 

C-C

Ghannadpour, Mohammadi, and Fazilati [63]

0

39.4784

 

Pradhan and Phadikar [30]

 

39.4784

 

Nejad, Hadi and Rastgoo [35]

 

39.4784

 

Zhu, Wang and Dai [38]

 

39.47842

 

Present

 

39.4784

 

Ghannadpour, Mohammadi, and Fazilati [63]

0.2

15.3068

 

Nejad, Hadi and Rastgoo [35]

 

15.3068

 

Present

 

15.3069

C-S

Ghannadpour, Mohammadi, and Fazilati [63]

0

20.1907

 

Pradhan and Phadikar [30]

 

20.1907

 

Nejad, Hadi and Rastgoo [35]

 

20.1907

 

Zhu, Wang and Dai [38]

 

20.19073

 

Present

 

20.1907

 

Ghannadpour, Mohammadi, and Fazilati [63]

0.2

11.1697

 

Nejad, Hadi and Rastgoo [35]

 

11.1697

 

Present

 

11.1697

S-S

Sari, Al-Kouz and Atieh [64]

0

9.8696

 

Reddy [29]

 

9.8696

 

Ebrahimi and Salari [34]

 

9.8696044

 

Eltaher, Emam and Mahmoud [33]

 

9.86973

 

Wang and Cai [3]

 

9.8696

 

Nejad, Hadi and Rastgoo [35]

 

9.8696

 

Zhu, Wang and Dai [38]

 

9.86960

 

Present

 

9.8696

 

Sari, Al-Kouz and Atieh [64]

0.1

8.9830

 

Reddy [29]

 

8.9830

 

Eltaher, Emam and Mahmoud [33]

 

8.98312

 

Present

 

8.9830

 

Sari, Al-Kouz and Atieh [64]

 

8.2426

 

Reddy [29]

 

8.2426

 

Eltaher, Emam and Mahmoud [33]

 

8.24267

 

Present

 

8.2426

 

Sari, Al-Kouz and Atieh [64]

0.2

7.0762

 

Reddy [29]

 

7.0761

 

Eltaher, Emam and Mahmoud [33]

 

7.07614

 

Nejad, Hadi and Rastgoo [35]

 

7.076

 

Present

 

7.0761

 

Sari, Al-Kouz and Atieh [64]

 

6.6085

 

Reddy [29]

 

6.6085

 

Present

 

6.6084

 

 

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.

 

Fig. 2. (a)First mode (b) second mode (c) third mode of C-C nanobeam for

 

Fig. 3. (a)First mode (b) second mode (c) third mode of C-S nanobeam for

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

 

Fig. 4. (a)First mode (b) second mode (c) third mode of S-S nanobeam for
 (∆);
 (×)

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.

 

Fig. 5.  First three modes of  C-C  nanobeam for
First mode _____ ; second mode - . - . - ; third mode - . . - . .

 

Fig. 6.  First three modes of  C-S  nanobeam for
First mode _____ ; second mode - . - . - ; third mode - . . - . .

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.

 

Fig. 7.  Critical buckling load  (a) C-C   (b) C-S   (c) S-S
nanobeam for  _____ ;  ---- ;

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.

 

Fig. 8. Critical buckling load  (a) C-C   (b) C-S   (c) S-S
nanobeam for

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.

 

Fig. 9. Critical buckling load  (a) C-C   (b) C-S   (c) S-S
nanobeam for

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

 

Cross-section area of the beam

 

Clamped edge

 

Weighting coefficients of mth order

 

Flexural rigidity

 

Young’s modulus of the plate material

 

Second moment of area

 

Nonlocal modulus

 

Foundation parameters

 

Winkler and shear foundation stiffnesses

 

Length of the beam

 

Bending moment

 

Number of grid points

p

Compressive load

 

Critical buckling load

 

Simply supported edge

 

Time

 

Kinetic energy of the beam

 

Strain energy of the beam

 

Transverse deflection

 

Non-dimensional transverse deflection

 

Potential energy due to Pasternak foundation

 

Work done by the compressive load

 

Nonlocal parameter

 

Circular frequency

 

Density of beam material

 

Normal stress

 

Normal strain

 

Frequency parameter

 

The Laplace operator

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.

References

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[35] Nejad, M.Z., Hadi, A., Rastgoo, A., 2016. Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams based on nonlocal elasticity theory. International Journal of Engineering Science, 103, pp.1-10.

[36] Safarabadi, M., Mohammadi, M., Farajpour, A., Goodarzi, M., 2015. Effect of surface energy on the vibration analysis of rotating nanobeam. Journal of Solid Mechanics, 7 (3), pp. 299-311.

[37] Mohammadi, M., Safarabadi, M., Rastgoo, A., Farajpour, A., 2016. Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment. Acta Mechanica, 227, pp. 2207-2232.

[38] Zhu, X., Wang, Y. and Dai, H.H., 2017. Buckling analysis of Euler–Bernoulli beams using Eringen’s two-phase nonlocal model. International Journal of Engineering Science, 116, pp.130-140.

[39] Khaniki, H.B., Hosseini-Hashemi, S., 2017. Buckling analysis of tapered nanobeams using nonlocal strain gradient theory and a generalized differential quadrature method. Materials Research Express, 4(6), pp.065003.

[40] Tuna, M., Kirca, M., 2017. Bending, buckling and free vibration analysis of Euler–Bernoulli nanobeams using Eringen’s nonlocal integral model via finite element method. Composite Structures, 179, pp.269-284.

[41] Bakhshi, K.H., Hosseini-Hashemi, S., Nezamabadi, A., 2018. Buckling analysis of nonuniform nonlocal strain gradient beams using generalized differential quadrature method. Alexandria Engineering Journal, 57, pp.1361-1368.

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[46] Nazmul, I.M., Devnath, I., 2021. Closed-form expressions for bending and buckling of functionally graded nanobeams by the Laplace transform. International Journal of Computational Materials Science and Engineering, 10(2), pp.2150012.

[47] Karmakar, S., Chakraverty, S., 2021. Differential quadrature and Adomian decomposition methods for solving thermal vibration of Euler nanobeam resting on Winkler-Pasternak foundation. Journal of Mechanics of Materials and Structures, 16 (4), pp.555-572.

[48] Zewei, L., Baichuan, L.Bo, C., Xiang, Z., Yinghui, L., 2022. Free vibration, buckling and dynamical stability of Timoshenko micro/nano-beam supported on Winkler-Pasternak foundation under a follower axial load. International Journal of Structural Stability and Dynamics, 22(09), pp.2250113

[49] Civalek, Ӧ., Uzun, B., Yaylı, M. Ӧ., 2022. An effective analytical method for buckling solutions of a restrained FGM nonlocal beam. Computational & Applied Mathematics, 41(2), pp.1-20.         

[50] Jalaei, M.H., Thai, H-T., Civalek, Ӧ., 2022. On viscoelastic transient response of magnetically imperfect functionally graded nanobeams. International Journal of Engineering Science, 172, pp.103629.

[51] Beni, Y.T., 2022. Size dependent torsional electro-mechanical analysis of flexoelectric micro/nanotubes. European Journal of Mechanics - A/Solids, 95, pp.104648.

[52] Beni, Z.T., Beni, Y.T., 2022. Dynamic stability analysis of size-dependent viscoelastic/piezoelectric nano-beam. International Journal of Structural Stability and Dynamics, 22(5), pp.2250050.

[53] Numanoğlu, H.M., Ersoy, H.,  Akgöz, B.,  Civalek, Ӧ., 2022. A new eigenvalue problem solver for thermo-mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method. Mathematical Methods in the Applied Sciences, 45(5), pp. 2592-2614.

[54] Karmakar, S., Chakraverty, S., 2022. Thermal vibration of nonhomogeneous Euler nanobeam resting on Winkler foundation. Engineering Analysis with Boundary Elements, 140,  pp.581-591.

[55] Abbas, W., Bakr, O.K., Nassar, M.M., Abdeen, M.A.M., Shabrawy, M., 2021. Analysis of tapered Timoshenko and Euler-Bernoulli beams on an elastic foundation with moving loads. Journal of Mathematics, https://doi.org/10.1155/2021/6616707.

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[1]   Iijima, S., 1991. Helical microtubules of graphitic carbon. Nature, 354, pp.56-58.
[2]   Baughman, R.H., Zakhidov, A.A., De Heer, W.A., 2002. Carbon nanotubes--the route toward applications. Science, 297, pp.787-792.
[3]   Wang, X., Cai, H., 2006. Effects of initial stress on non-coaxial resonance of multi-wall carbon nanotubes. Acta Materialia, 54(8), pp.2067-2074.
[4]   Eringen, A.C., 1972. Linear theory of non-local elasticity and dispersion of plane waves. International Journal of Engineering Science, 10(5), pp.425-435.
[5]   Eringen, A.C., 1972. Nonlocal polar elastic continua. International Journal of Engineering Science, 10 (1), pp.l-16.
[6]   Eringen, A.C., Edelen, D.G.B., 1972. On nonlocal elasticity. International Journal of Engineering Science, 10(3), pp.233-248.
[7]   Eringen, A.C., 1983. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), pp.4703-4710.
[8]   Lu, P., Lee H., Lu C., Zhang P., 2006. Dynamic properties of flexural beams using a nonlocal elasticity model. Journal of Applied Physics, 99(7), pp.073510(1-9).
[9]   Xu, M., 2006. Free transverse vibrations of nano-to-micron scale beams. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 462(2074), pp.2977-2995.
[10] Wang, C., Zhang, Y., He, X., 2007. Vibration of nonlocal Timoshenko beams. Nanotechnology, 18(10), pp.105401(1-9).
[11] Challamel, N., Wang, C. M., 2008. The small length scale effect for a non-local cantilever beam: A paradox solved. Nanotechnology, 19(34), pp.345703(1-7).
[12] Murmu, T., Pradhan, S., 2009. Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory. Physica E, 41(8), pp.1451-1456.
[13] Roque, C., Ferreira, A., Reddy, J., 2011. Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method. International Journal of Engineering, 49(9), pp.976-984.
[14] Mohammadi, B., Ghannadpour, S., 2011. Energy approach vibration analysis of nonlocal Timoshenko beam theory. Procedia Engineering, 10, pp.1766-1771.
[15] Li, C., Lim, C.W., Yu, J.L., 2011. Dynamics and stability of transverse vibrations of nonlocal nanobeams with a variable axial load. Smart Materials and Structures, 20(1), pp.015023(1-7).
[16] Eltaher, M.A., Alshorbagy, A.E., Mahmoud, F.F., 2013. Vibration analysis of Euler-Bernoulli nanobeams by using finite element method. Applied Mathematical Modelling, 37, pp.4787-4797.
[17] Behera, L., Chakraverty, S., 2014. Free vibration of Euler and Timoshenko nanobeams using boundary characteristic orthogonal polynomials. Applied Nanoscience, 4(3), pp.347-358.
[18] Rahmani, O., Pedram, O., 2014. Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory. International Journal of Engineering Science, 77, pp.55-70.
[19] Behera, L., Chakraverty, S., 2015. Application of Differential Quadrature method in free vibration analysis of nanobeams based on various nonlocal theories. Computers & Mathematics with Applications, 69(12), pp.1444-1462.
[20] Ebrahimi, F., Salari, E., 2015. Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi analytical differential transform method. Composites Part B Engineering, 79, pp.156-169.
[21] Ebrahimi, F., Ghadiri, M., Salari, E., Hoseini, S.A.H., Shaghaghi, G.R., 2015. Application of the differential transformation method for nonlocal vibration analysis of functionally graded nanobeams. Journal of Mechanical Science and Technology, 29(3), pp.1207-1215.
[22] Tuna, M., Kirca, M., 2016. Exact solution of Eringen’s nonlocal integral model for vibration and buckling of Euler-Bernoulli beam. International Journal of Engineering Science, 107, pp.54-67.
[23] Hamza-Cherif, R., Meradjah, M., Zidour, M., Tounsi, A., Belmahi, S., Bensattalah, T., 2018. Vibration analysis of nano beam using differential transform method including thermal effect. Journal of Nano Research, 54, pp.1-14.
[24] Aria, A.I., Friswell, M.I., 2019. A nonlocal finite element model for buckling and vibration of functionally graded nanobeams. Composites Part B, 166, pp.233-246.
[25] Nikam, R.D., Sayyad, A.S., 2020. A unified nonlocal formulation for bending, buckling and free vibration analysis of nanobeams. Mechanics of Advanced Materials and Structures, 27(10), pp.807-815.
[26] Arian, B., Hamidreza, S., Nima, K., 2020. Effect of the thickness to length ratio on the frequency ratio of nanobeams and nanoplates. Journal of Theoretical and Applied Mechanics, 58(1), pp.87-96.
[27] Ufuk, G., 2022. Transverse free vibration of nanobeams with intermediate support using nonlocal strain gradient theory. Journal of Structural Engineering & Applied Mechanics, 5(2), pp.50-61.
[28] Wang, C.M., Zhang, Y.Y., Ramesh, S.S., Kitipornchai, S., 2006. Buckling analysis of micro and nano-rods/tubes based on nonlocal Timoshenko beam theory. Journal of Physics D. Applied Physics, 39, pp.3904-3909.
[29] Reddy, J.N., 2007. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45, pp.288-307.
[30] Pradhan, S.C., Phadikar, J.K., 2009. Bending, buckling and vibration analyses of nonhomogeneous nanotubes using GDQ and nonlocal elasticity theory. Structural Engineering and Mechanics, 33(2), pp.193-213.
[31] Aydogdu, M., 2009. A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration. Physica E, 41, pp.1651-1655.
[32] Thai, H.T., 2012. A non-local beam theory for bending, buckling, and vibration of nano beams. International Journal of Engineering Science, 52, pp.56-64.
[33] Eltaher, M. A., Emam S. A., Mahmoud F. F., 2013. Static and stability analysis of nonlocal functionally graded nanobeams. Composite Structures, 96, pp.82-88.
[34] Ebrahimi, F., Salari, E., 2015. A semi-analytical method for vibrational and buckling analysis of functionally graded nanobeams considering the physical neutral axis position. Computer Modeling in Engineering & Sciences, 105, pp.151-181.
[35] Nejad, M.Z., Hadi, A., Rastgoo, A., 2016. Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams based on nonlocal elasticity theory. International Journal of Engineering Science, 103, pp.1-10.
[36] Safarabadi, M., Mohammadi, M., Farajpour, A., Goodarzi, M., 2015. Effect of surface energy on the vibration analysis of rotating nanobeam. Journal of Solid Mechanics, 7 (3), pp. 299-311.
[37] Mohammadi, M., Safarabadi, M., Rastgoo, A., Farajpour, A., 2016. Hygro-mechanical vibration analysis of a rotating viscoelastic nanobeam embedded in a visco-Pasternak elastic medium and in a nonlinear thermal environment. Acta Mechanica, 227, pp. 2207-2232.
[38] Zhu, X., Wang, Y. and Dai, H.H., 2017. Buckling analysis of Euler–Bernoulli beams using Eringen’s two-phase nonlocal model. International Journal of Engineering Science, 116, pp.130-140.
[39] Khaniki, H.B., Hosseini-Hashemi, S., 2017. Buckling analysis of tapered nanobeams using nonlocal strain gradient theory and a generalized differential quadrature method. Materials Research Express, 4(6), pp.065003.
[40] Tuna, M., Kirca, M., 2017. Bending, buckling and free vibration analysis of Euler–Bernoulli nanobeams using Eringen’s nonlocal integral model via finite element method. Composite Structures, 179, pp.269-284.
[41] Bakhshi, K.H., Hosseini-Hashemi, S., Nezamabadi, A., 2018. Buckling analysis of nonuniform nonlocal strain gradient beams using generalized differential quadrature method. Alexandria Engineering Journal, 57, pp.1361-1368.
[42] Soltani, M., Mohammadi, M., 2018. Stability analysis of non-local Euler-Bernoulli beam with exponentially varying cross-section resting on Winkler-Pasternak foundation. Numerical Methods in Civil Engineering, 2(3), pp.1-11.
[43] Xu, X., Zheng, M., 2019. Analytical solutions for buckling of size-dependent Timoshenko beams. Applied Mathematics and Mechanics, 40, pp.953-976.
[44] Jena, S.K., Chakraverty, S., 2019. Differential quadrature and differential transformation methods in buckling analysis of nanobeams. Curved Layer Structures, 6, pp.68-76.
[45] Ragb, O., Mohamed, M., Matbuly, M.S., 2019. Free vibration of a piezoelectric nanobeam resting on nonlinear Winkler-Pasternak foundation by quadrature methods. Heliyon, 5(6), e01856.
[46] Nazmul, I.M., Devnath, I., 2021. Closed-form expressions for bending and buckling of functionally graded nanobeams by the Laplace transform. International Journal of Computational Materials Science and Engineering, 10(2), pp.2150012.
[47] Karmakar, S., Chakraverty, S., 2021. Differential quadrature and Adomian decomposition methods for solving thermal vibration of Euler nanobeam resting on Winkler-Pasternak foundation. Journal of Mechanics of Materials and Structures, 16 (4), pp.555-572.
[48] Zewei, L., Baichuan, L.Bo, C., Xiang, Z., Yinghui, L., 2022. Free vibration, buckling and dynamical stability of Timoshenko micro/nano-beam supported on Winkler-Pasternak foundation under a follower axial load. International Journal of Structural Stability and Dynamics, 22(09), pp.2250113
[49] Civalek, Ӧ., Uzun, B., Yaylı, M. Ӧ., 2022. An effective analytical method for buckling solutions of a restrained FGM nonlocal beam. Computational & Applied Mathematics, 41(2), pp.1-20.         
[50] Jalaei, M.H., Thai, H-T., Civalek, Ӧ., 2022. On viscoelastic transient response of magnetically imperfect functionally graded nanobeams. International Journal of Engineering Science, 172, pp.103629.
[51] Beni, Y.T., 2022. Size dependent torsional electro-mechanical analysis of flexoelectric micro/nanotubes. European Journal of Mechanics - A/Solids, 95, pp.104648.
[52] Beni, Z.T., Beni, Y.T., 2022. Dynamic stability analysis of size-dependent viscoelastic/piezoelectric nano-beam. International Journal of Structural Stability and Dynamics, 22(5), pp.2250050.
[53] Numanoğlu, H.M., Ersoy, H.,  Akgöz, B.,  Civalek, Ӧ., 2022. A new eigenvalue problem solver for thermo-mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method. Mathematical Methods in the Applied Sciences, 45(5), pp. 2592-2614.
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