# Bending, buckling and free vibration of nonlocal FG-carbon nanotube-reinforced composite nanobeams: exact solutions

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## Abstract

This paper investigates the bending, buckling and free vibration behaviors of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanobeams by considering small-scale effect. The governing equations of motion of a Timoshenko beam under a general loading are derived utilizing the nonlocal elasticity theory. The equations governing bending and stretching behavior of CNTRC nanobeams are uncoupled to a fifth-order ordinary differential equation with respect to the rotation of cross-section for the static cases of bending and buckling. This uncoupling makes it possible to develop exact solutions for transverse deflection and buckling load of CNTRC nanobeams. Using differential operator method, the decoupled sixth-order differential equations in terms of the kinematic variables are obtained for vibration analysis. By setting the coefficients matrix in the corresponding system of homogenous algebraic equations to zero, an algebraic frequency equation is derived. Finally, based on the presented closed-form solutions, parametric studies are carried out to assess the effects of CNT distribution, nonlocal parameter and type of boundary conditions on the deflection, buckling and natural frequency of CNTRC nanobeams. Findings show that nonlocal effect on the mechanical behavior of nanobeams is strongly dependent on boundary conditions and loadings. It is seen that cantilever nanobeams become harder by taking into account nonlocal effect, contrary to clamped and simply supported nanobeams. In addition, the influence of CNT distribution on the mechanical behavior of cantilever beams is more significant than that of simply supported and clamped beams.

## Keywords

FG-carbon nanotube-reinforced composite beams Nonlocal elasticity theory Buckling analysis Free vibration Small-scale effects## 1 Introduction

### 1.1 Small-scale effect

The small-scale effect, which has been justified by various empirical observations of mechanical behavior in small-scale structures [1, 2, 3], plays a crucial role in optimal design of micro- and nanoelectromechanical systems (MEMS and NEMS) e.g. atomic force microscopes, chemical sensing device, actuators, and pumps. Owing to lack of the length scale parameters in its constitutive equations, the classical continuum theory cannot appropriately estimate the design parameters of micro- and nanostructures such as natural frequencies, maximum deflections and buckling loads. Consequently, several nonclassical continuum theories have been proposed to account for the small-scale effect on the mechanical behavior of small-scale structures and eliminate the differences between results determined by theoretical and experimental methods [3, 4, 5]. Concerning the submicron structures, the dispersion phenomenon which is the consequence of long-range intermolecular forces, was detected in propagations of waves with short wavelengths in elastic bodies [5, 6, 7]. To capture long-range effects, Eringen [6] considered that the nonlocal strain of points under translational motion is the same with that of the classical theory, but the stress at a point is relevant to the strain in a region near that point. In recent years, researchers’ attention has been devoted to survey static [8, 9, 10, 11, 12], vibration [13, 14], buckling and postbuckling [15, 16, 17], dynamic [18, 19, 20, 21] and thermomechanical [22, 23, 24, 25, 26] behavior of micro-and nanostructures according to the nonclassical continuum theories such as the nonlocal, modified couples stress and modified strain gradient theories.

### 1.2 Carbon nanotubes reinforced composites

The initial idea of carbon nanotubes (CNTs), as a type of novel materials with excellent thermal, electrical and mechanical properties especially low density, extraordinary strength and corrosion resistance was proposed by Iijima [27]. By tradition, composites reinforced by carbon, glass, aramid or basalt fibers have a wide range of applications in structural systems in civil, mechanical, marine, aerospace engineering and many other modern industries. Recently, CNTs which can provide good interfacial bonds, have been utilized instead of traditional fibers for the reinforcement of matrix phases in composites. One of major applications of CNTs is to design nanosensors owing to their exceptional mechanical properties which leads to a reachable ultrahigh frequency range up to the terahertz order and a possible ultrahigh sensitivity. In order to analyze the mechanical behavior of CNT reinforced composites and simulate their effective material properties, various experiments and theoretical techniques have been presented namely, molecular dynamics (MD) simulation [28, 29], representative volume element (RVE) [30], rule of mixture [31, 32, 33], and experimentations [34, 35, 36]. In recent decade, many demands have been raised for production of multilayer MEMS and NEMS with variable properties which are used in thermal environment [37]. Consequently, manufacturing technologies were extended to make functionally graded (FG) layers in micron and submicron dimensions with the desired electrical and mechanical properties at their bottom and top sides. Therefore, many researchers have focused on the thermal [38, 39, 40] and mechanical [41, 42, 43, 44] behavior of FG micro- and nanostructures. With the rapid advancement of manufacturing technology, CNTs are used as the favorite reinforcements for polymer nanolayers utilized in different engineering applications.

### 1.3 A literature review on CNTRC beams

Some studies accomplished on the bending, buckling and vibration behavior of CNTRC beams are reviewed here. By employing the Euler–Bernoulli beam theory and von Kármán geometric nonlinearity, Rafiee et al. [45] analyzed large-amplitude free vibrations of FG-CNTRC beams with surface-bonded piezoelectric layers in thermal environment and subjected to an input voltage. To solve the governing equations of the piezoelectric CNTRC beams, they applied the Galerkin method in conjunction with the multiple time scales method. Ke et al. [46] surveyed dynamic stability behavior of Timoshenko FG-CNTRC beams under axial loading. In their work, the material properties of FG-CNTRCs have been assumed to be determined corresponding to the rule of mixture. In order to solve three governing equations for assessment of the dynamic stability characteristics, they utilized the differential quadrature (DQ) method. In the work of Ansari et al. [47], by taking into account the von Kármán geometric nonlinearity, forced vibration behavior of Timoshenko CNTRC beams has been studied. They discretized the nonlinear governing equations and associated boundary conditions via the generalized differential quadrature (GDQ) method and then employed a Galerkin-based numerical technique to reduce the set of nonlinear partial differential equations into a time-varying set of Duffing-type ordinary ones. By implementing finite element method (FEM), dynamic analysis of Timoshenko and Euler–Bernoulli FG nanocomposite beams reinforced by randomly oriented carbon nanotubes under a moving load has been conducted by Yas and Heshmati [48]. They modelled the material properties via the Eshelby–Mori–Tanaka approach on the basis of an equivalent fiber. By defining the temperature-dependent material properties of fibers and polymeric matrix through a refined rule of mixture, Jam and Kiani [49] presented responses of Timoshenko FG-CNTRC beams subjected to the action of an impacting mass. To derive time history deflection, they used the conventional polynomial Ritz method together with the Runge–Kutta method. By applying the p-Ritz method to extract natural frequencies, Lin and Xiang [50] explored the linear free vibration of FG-CNTRC beams based on the first order and third order shear deformation theories. Shen and Xiang [51] carried out an investigation on the large amplitude vibration, nonlinear bending and thermal postbuckling behavior of CNTRC beams resting on an elastic foundation in thermal environments. They derived the motion equations of CNTRC beams by means of a higher order shear deformation beam theory and solved them by utilizing a two-step perturbation procedure. By accounting for the von Kármán geometric nonlinearity effect, Rafiee et al. [52] assessed thermal post-buckling behavior of Euler–Bernoulli CNTRC beams with surface-bonded piezoelectric layers. Based on the first-order shear deformation beam theory with von-Kármán geometric nonlinearity, Wu et al. [53] conducted an analysis on the thermal post-buckling behavior of FG-CNTRC beams subjected to in-plane temperature change incorporating the effect of imperfection sensitivity. In their study, generic imperfection function has been used to describe different possible imperfections such as sine type, global and localized imperfections. They solved the governing differential equations with the aid of DQ method in conjunction with modified Newton–Raphson technique. Yas and Samadi [54] carried out the free vibration and buckling analysis of Timoshenko CNTRC beams resting on an elastic foundation. In this work, to figure out natural frequencies and buckling loads, the governing equations have been solved through the GDQ method for beams with different boundary conditions.

### 1.4 Present study

From the abovementioned literature review, it is revealed that researches reported on the mechanical analysis of CNTRCs have been conducted on the basis of classical continuum theory, which is not able to account for small-scale effect. In addition, to solve governing equations presented in previous investigations, semi-analytical methods such as Ritz and Galerkin methods or numerical methods have been implemented. The main contribution of current study is to develop size-dependent exact solutions for bending, buckling and free vibration of FG-carbon nanotube-reinforced composite nanobeams in the framework of nonlocal elasticity theory. Material properties of FG-CNTRCs are assumed to be graded in the thickness direction and computed via extended rule of mixture. Based on the nonlocal elasticity theory and Timoshenko beam model, dynamic equilibrium equations of CNTRC nanobeams with arbitrary boundary conditions are derived. For bending and buckling analysis, the bending and stretching coupled equations of CNTRC nanobeams are reduced to a decoupled fifth-order ordinary differential equation with respect to rotation of cross-section which can be solved exactly. For free vibration analysis, the differential operator method is applied to obtain the decoupled sixth-order differential equations governing the in-plane displacement, transverse deflection and the rotation of cross-section. A system of homogenous algebraic equations is obtained for arbitrary boundary conditions and the frequency equation is derived by setting coefficients’ matrix to zero. Detailed numerical results are provided to discuss the effects of CNT distribution, nonlocal parameter, aspect ratio and boundary conditions on the deflection, buckling and natural frequency of FG-CNTRC nanobeams.

## 2 Theoretical formulation

### 2.1 Problem definition

*y*-axis. Meanwhile, the variable

*t*is time and

*z*is the distance from mid-plane.

#### 2.1.1 Carbon nanotube-reinforced composites

*m*stand for CNTs and the matrix, respectively. \(\rho_{cnt}\), \(\rho_{m}\), \(V_{cnt}\) and \(V_{m}\) are the densities and volume fractions of CNTs and the matrix. It is to be noticed that the volume fractions are related as \(V_{cnt} + V_{m} = 1\) with each other. Also, \(\eta_{i}\)’s (for \(i = 1,2,3)\) are the CNT efficiency parameters considering the size-dependent effects.

#### 2.1.2 Nonlocal elasticity theory

### 2.2 Motion equations

### 2.3 Exact solutions

The optimal design of beams has an important role for development of MEMS and NEMS technology. The mechanical parameters including maximum deflection and stress, buckling load and frequency responses are selected as design variables in different applications. In this section, exact solutions for bending, buckling and vibration behavior of CNTRC beams with arbitrary boundary conditions are developed which can be used in the conceptual design.

#### 2.3.1 Bending and buckling behavior

Upon substitution of Eq. (24) into Eq. (21), one can get

##### 2.3.1.1 Bending problem

##### 2.3.1.2 Buckling problem

#### 2.3.2 Free vibration behavior

## 3 Numerical results and discussion

CNT efficiency parameters for different values of CNT volume fraction

\(V_{cnt}^{ *}\) | \(\eta_{1}\) | \(\eta_{2}\) | \(\eta_{3}\) |
---|---|---|---|

0.12 | 0.137 | 1.022 | 0.715 |

0.17 | 0.142 | 1.626 | 1.138 |

0.28 | 0.141 | 1.585 | 1.109 |

### 3.1 Bending behavior

The maximum normalized deflections of SS, CC and CF-supported FGX-CNTRC beams under uniform and sinusoidal transverse loads for different values of \({\text{V}}_{\text{cnt}}^{ *}\) based on the nonlocal theory (\(\mu = 2\,{\text{nm}}^{2} )\)

\(V_{cnt}^{ *}\) | Boundary conditions | Maximum normalized deflection, \(\bar{w} \times 10\) | |
---|---|---|---|

Uniform load | Sinusoidal load | ||

0.12 | SS | 0.515866 | 0.463104 |

CC | 0.340608 | 0.332083 | |

CF | 2.781312 | 1.692826 | |

0.17 | SS | 0.323635 | 0.288384 |

CC | 0.203674 | 0.198682 | |

CF | 1.786445 | 1.083878 | |

0.28 | SS | 0.242150 | 0.219264 |

CC | 0.168422 | 0.164122 | |

CF | 1.271270 | 0.777062 |

### 3.2 Buckling behavior

The normalized buckling loads of SS and CC-supported CNTRC beams for different values of \({\text{V}}_{\text{cnt}}^{ *}\)

\(V_{cnt}^{ *}\) | Boundary conditions | \(\mu \left( {{\text{nm}}^{2} } \right)\) | The normalized buckling load, \(\bar{P}\) | ||
---|---|---|---|---|---|

UD | FGX | FGO | |||

0.12 | SS | 0 | 17.2929 | 20.3867 | 12.1347 |

2 | 15.8712 | 19.0337 | 10.8132 | ||

4 | 14.6655 | 17.8492 | 9.7513 | ||

CC | 0 | 26.2252 | 27.9262 | 22.6468 | |

2 | 22.1666 | 24.3697 | 18.3146 | ||

4 | 19.3101 | 21.9048 | 14.9908 | ||

0.17 | SS | 0 | 27.2229 | 32.7365 | 18.6032 |

2 | 24.8201 | 30.3636 | 16.4684 | ||

4 | 22.8070 | 28.3115 | 14.7732 | ||

CC | 0 | 43.0812 | 46.5554 | 36.6469 | |

2 | 35.9902 | 40.1306 | 29.1923 | ||

4 | 30.9776 | 35.6381 | 23.6794 | ||

0.28 | SS | 0 | 36.0119 | 43.0624 | 26.5113 |

2 | 33.3770 | 40.5028 | 23.7818 | ||

4 | 31.1014 | 38.2304 | 21.5618 | ||

CC | 0 | 51.4341 | 56.6634 | 47.0009 | |

2 | 44.2744 | 50.1305 | 38.3292 | ||

4 | 39.2666 | 45.6468 | 31.8491 |

### 3.3 Free vibration behavior

The normalized natural frequencies of SS and CC-supported CNTRC beams for different values of \({\text{V}}_{\text{cnt}}^{ *}\)

\(V_{cnt}^{ *}\) | Boundary conditions | \(\mu \left( {{\text{nm}}^{2} } \right)\) | Normalized natural frequency, \(\bar{\omega }\) | ||
---|---|---|---|---|---|

UD | FGX | FGO | |||

0.12 | SS | 0 | 3.9765 | 4.3188 | 3.3288 |

2 | 3.6340 | 3.9468 | 3.0421 | ||

4 | 3.3670 | 3.6569 | 2.8186 | ||

CC | 0 | 5.0116 | 5.1578 | 4.7252 | |

2 | 4.5787 | 4.6968 | 4.3030 | ||

4 | 4.2350 | 4.3547 | 3.9853 | ||

0.17 | SS | 0 | 4.9034 | 5.3787 | 4.0507 |

2 | 4.4811 | 4.9154 | 3.7018 | ||

4 | 4.1519 | 4.5544 | 3.4299 | ||

CC | 0 | 6.3347 | 6.5492 | 5.9378 | |

2 | 5.7719 | 5.9693 | 5.4058 | ||

4 | 5.3433 | 5.5251 | 4.9947 | ||

0.28 | SS | 0 | 5.4426 | 5.9528 | 4.6670 |

2 | 4.9738 | 5.4400 | 4.2650 | ||

4 | 4.6084 | 5.0404 | 3.9517 | ||

CC | 0 | 6.6306 | 6.9484 | 6.4450 | |

2 | 6.0621 | 6.3254 | 5.8664 | ||

4 | 5.6087 | 5.827 | 5.4301 |

*b*/

*h*= 1 and \(V_{cnt}^{*} = 0.12)\) with SS and CC supports based on the nonlocal theory \(\left( {\mu = 2\,{\text{nm}}^{2} } \right)\). For both boundary conditions and various distributions of CNTs, it is seen that the ascending curves of normalized natural frequency converge to the values predicted by Euler–Bernoulli beam model. Similar to buckling behavior of CNTRC nanobeams, the maximum and minimum values of normalized natural frequency belong to FGX and FGO CNTRC beams, respectively.

*b*/

*h*= 1 and \(V_{cnt}^{*} = 0.12)\) with SS and CC supports. It is observed that the difference of nonlocal and classical dimensionless natural frequencies diminishes with increase of \(L/h\) for both boundary conditions. From Figs. 5 and 7, it is concluded that the effect of supports on the values of normalized buckling loads is more significant in comparison with normalized natural frequencies.

## 4 Conclusion

Among the considered CNT distributions, FGX distribution results in the stiffest behavior for nanobeam, while FGO distribution leads to the most flexible behavior.

For the cantilever beams, the influence of CNT distribution on bending CNTRC beams is more significant than simply supported and clamped beams.

Due to nonlocal effect, softening and hardening phenomenon take place for the simply supported and cantilever beams, respectively.

For bending analysis of CC-supported nanobeams under uniform transverse loading, the normalized deflections predicted by the classical and nonlocal elasticity theories are exactly the same.

Softening takes place as the nonlocal effect is considered in the buckling and free vibration analysis of nanobeams with SS and CC supports.

As \(L/h\) increases, the nonlocal and classical natural frequencies for SS and CC beams approach to each other.

## Notes

### Acknowledgements

The authors gratefully acknowledge supports from Iran National Science Foundation (INSF).

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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