## Nonlinear Vibration of Embedded Annular Boron Nitride Sheets Using Differential Quadrature Method

Department of Petroleum Engineering, Faculty of Petroleum and Renewable Energy Engineering, UniversitiTeknologi Malaysia, 81310 UTM, Johor, Malaysia### Abstract

One of the most promising materials for nanotechnology is annular Boron Nitride sheets (ABNSs). In this study, however, differential quadrature method (DQM) and nonlocal piezoelasticity theory are used to investigate the nonlinear vibration response of embedded single layered annular Boron Nitride sheets (SLABNSs). The interactions between the SLABNSs and its surrounding elastic medium are simulated by nonlinear Pasternak foundation. A detailed parametric study is conducted to elucidate the influences of the nonlocal parameter, elastic medium, temperature change and maximum amplitude on the nonlinear frequency of the SLABNSs. The results are in good agreement with the previous researches.

### At a glance: Figures

**Keywords:** nonlinear vibration, SLABNS, DQM, nonlocal piezoelasticity theory

*Journal of Mechanical Design and Vibration*, 2014 2 (2),
pp 47-52.

DOI: 10.12691/jmdv-2-2-2

Received May 23, 2014; Revised June 16, 2014; Accepted June 17, 2014

**Copyright**© 2014 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- CheraghiKootiani, Reza. "Nonlinear Vibration of Embedded Annular Boron Nitride Sheets Using Differential Quadrature Method."
*Journal of Mechanical Design and Vibration*2.2 (2014): 47-52.

- CheraghiKootiani, R. (2014). Nonlinear Vibration of Embedded Annular Boron Nitride Sheets Using Differential Quadrature Method.
*Journal of Mechanical Design and Vibration*,*2*(2), 47-52.

- CheraghiKootiani, Reza. "Nonlinear Vibration of Embedded Annular Boron Nitride Sheets Using Differential Quadrature Method."
*Journal of Mechanical Design and Vibration*2, no. 2 (2014): 47-52.

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### 1. Introduction

In recent years, a significant amount of research has been focused on nanostructured materials because of their interesting mechanical and physical properties. Carbon nanotubes (CNTs) have shown great potential for technological applications. It is well established that CNTs possess exceptional mechanical properties such as high strength to weight and stiffness to weight ratios, and enormous electrical conductivity ^{[1]}.

However, their applications are limited in high strength and uniform electronic structures. This may be explained by the fact that the electrical properties of CNTs are unstable and range from metallic to semiconducting depending upon the radius and chirality of the tubes ^{[2]}. Boron nitride (BN) nanotubes, unlike CNTs, are all semiconductors with a constant wide band gap of 5.5 eV, when their diameter is greater than 9.5A. Also, small BN nanotubes show interesting size-dependent electronic and magnetic properties ^{[3]}.

In the 1960s, Boron nitride sheets and relative structures in hexagonal state (h-BN) have been studied experimentally. It has been found that BN exists in several forms: fibers, nanomesh, nanotubes and nanosheets. Different types of BN include armchair, zigzag and chairal. BN nanostructures exhibit electrically insulated properties, strong chemical and thermal stability but, at the same time, excellent thermal conductivities and mass sensing capabilities ^{[4]}.

These properties of BNs, makes them promising candidate materials in a large variety of nanosized electronic and photonic devices. Furthermore, BNs can be used for reinforcement in composite structures due to their high thermal conductivity, high resistance to oxidation at elevated temperatures, and outstanding mechanical properties ^{[4]}.

Conducting experiments with nanoscale size specimens is found to be difficult and expensive. Therefore, development of appropriate mathematical models for nanostructures is an important issue concerning application of Nano-structures. Vibration of nanostructures is of great importance in nanotechnology. Understanding vibration behavior of nanostructures is the key step for many NEMS devices like oscillators, clocks and sensor devices.

Behfar and Naghdabadi ^{[5]} studied the nanoscalevibrational analysis of a multi-layered graphene sheets (MLGS) embedded in an elastic medium. Liew ^{[6]} proposed a continuum-based plate model to investigate the vibration behavior of MLGSs that are embedded in an elastic matrix. Pradhan ^{[7]} presented analytical solutions for vibration of the nanoplates such as GSs. They employed nonlocal theories to bring out the effect of the nonlocal parameter on natural frequencies of the nanoplates. Shen ^{[8]} presented nonlinear vibration behavior for a simply supported, rectangular, single layer grapheme sheet (SLGS) in thermal environments. Ansari ^{[9]} developed a nonlocal plate model which accounts for the small scale effects to study the vibrational characteristics of MLGSs with different boundary conditions embedded in an elastic medium. On the basis of the constitutive equations of nonlocal elasticity, the Mindlin type equations of motion coupled together through the van der Waals interaction are derived.

Pradhan ^{[10]} studied the small scale effect on the vibration analysis of orthotropic SLGSs embedded in elastic medium using nonlocal elasticity theory and DQM. Transverse nonlinear vibration of orthotropic double layered grapheme sheets (DLGSs) embedded in a nonlinear elastic medium under thermal gradient was studied by Ghorbanpour Arani et al. ^{[11]}, using nonlocal elasticity orthotropic plate theory and DQM. Mohammadiet al. ^{[12]} studied the free vibration behavior of circular and annular GS using the nonlocal elasticity theory based on Bessel functions and new version of DQM. All of aforementioned researches are related to non-smart material structures. Salehi-Khojin and Jalili ^{[13]} derived the equation of motion for BNNT-based composites subjected to combined electro-thermo-mechanical loadings using three-dimensional equilibrium equations. Based on nonlocal piezoelasticity theory, transverse vibration and instability of embedded double-walled boron nitride nanotubes (DWBNNTs) conveying viscose fluid was studied by Khodami Maraghi et al. ^{[14]} using DQM. They showed that the electric field effect on the frequency is approximately constant, while it decreases with increasing temperature change. Electro-thermo-mechanical nonlocal wave propagation analysis of embedded DWBNNTs conveying fluid was presented by Ghorbanpour et al ^{[15]} via strain gradient theory. Also, Ghorbanpour et al. ^{[16]} applied DQM and nonlocal piezoelasticity theory for electro-thermal vibration of fluid-conveying DWBNNTs embedded in an elastic medium using cylindrical shell theory. However, to date, no report has been found in the literature on vibration analysis of annular boron nitride sheets. Motivated by these considerations, in order to improve optimum design of smart nanostructures, we aim to investigate the nonlinear nonlocal vibration analysis of SLABNSs embedded in a nonlinear elastic medium using DQM. The influences of nonlocal parameter, aspect ratio, elastic medium coefficients on the frequency of the coupled system are taken into account.

### 2. Basic equations

A schematic of SLABNS resting on a Pasternak foundation is shown in Figure 1. The geometrical parameters of the SLABNS are inner radius a, outer radius b and thickness h. The elastic medium containing spring constants of Winkler-type (K_{1W}) and shear constants of Pasternak-type (K_{G}).

**Figure**

**1.**

**A schematic of SLABNS resting on a Pasternak foundation**

**2.1. Nonlocal Piezoelasticity Theory**

Based on the theory of nonlocal piezoelasticity, the stress tensor and the electric displacement at a reference point depend not only on the strain components and electric-field components at same position but also on all other points of the body. The nonlocal constitutive behavior for the piezoelectric material can be given as follows ^{[17, 18]}:

(1) |

(2) |

Where and are, respectively, the nonlocal stress tensor and local stress tensor, and are the components of the nonlocal and local electric displacement. is the nonlocal modulus. is the Euclidean distance, and is defined that l is the external characteristic length, denotes a constant appropriate to each material, and a is an internal characteristic length of the material. Consequently, is a constant parameter which is obtained with molecular dynamics, experimental results, experimental studies and molecular structure mechanics. The constitutive equation of the nonlocal elasticity can be written as ^{[18]}:

(3) |

Where the parameter denotes the small scale effect on the response of structures in Nano size, and is the Laplacian operator in the above equation. Similarly, Eq. (2) can be written to as:

(4) |

### 3. Motion Equations

Based on the classical plate theory (CPT) which satisfies Kirchhoff assumption, displacement field is represented as ^{[19, 20]}:

(5) |

(6) |

(7) |

Where () denote the total displacements of a point along radial (r), circumferential (θ) and axial (z) directions and () are the displacements of the same points on the mid-plane. Using Eqs. (5)-(7), the nonlinear mechanical strains can be written in term of displacements as:

(8) |

(9) |

(10) |

Constitutive equations for SLABNSs can be expressed as follows ^{[15]}:

(11) |

(12) |

Where [σ], [ε], [D] and [E] are classical stress, strain, electric displacement and electricfield tensors, respectively. Also, [C], [h] and [∈] denote elastic stiffness, piezoelectric and dielectric coefficients matrix, respectively. Considering radial polarization for SLABNS and using Eqs. (5)-(7), the above equations can be rewritten as:

(13) |

(14) |

Where E_{r} in term of electric potential () is given as follow:

(15) |

The total electrostatic and kinetic energies of the SLABNS can be expressed as:

(16) |

(17) |

Where is density of SLABNS. The external work due to surrounding elastic medium is given as:

(18) |

The motion equations of embedded SLABNS can be derived by Hamilton’s principles as follows:

(19) |

Substitute Eqs. (16)-(18) Into Eq. (19), the following motion equations can be derived:

(20) |

(21) |

(22) |

(23) |

Where the force resultants (, , ) and the moment resultants (, , ) can be defined as:

(24) |

(25) |

Furthermore, (I_{0}, I_{2}) are the mass moments of inertia which can be represented as:

(26) |

(27) |

Using the nonlocal piezoelasticity theory and Eq. (13), the nonlocal stress resultant can be rewritten as:

(28) |

(29) |

Where:

(30) |

(31) |

Finally, substituting Eqs. (28) & (29) into Eqs. (20)- (23), yields:

(32) |

(33) |

(34) |

(35) |

Where;

(36) |

(37) |

In the DQM, the derivatives of a function are approximated with weighted sums of the function values at a group of grid points. Apply DQM to Eqs. (20)- (23), yield the following matrix equation;

(38) |

### 4. Results and Discussion

In order to show the effects of nonlocal parameter, surrounding elastic medium and temperature change, the frequency reduction percent and nonlinear frequency ratio are defined as follows;

(39) |

(40) |

Figure 2. Illustrates the FRP versus the nonlocal parameter for four cases including:

Case1: linear vibration analysis of SLABNS.

Case 2: Nonlinear vibration analysis of SLABNS.

Case3: linear vibration analysis of SLAGS.

Case 4: Nonlinear vibration analysis of SLAGS.

As can be seen, the FRP increases with increasing. It means that with increasing, the frequency of the coupled system becomes lower. This is due to the fact that the increase of nonlocal parameter decreases the interaction force between SLABNS atoms, and that leads to a softer structure. It is also concluded that the FRP (or frequency) of SLABNS is lower (or higher) than SLAGS. Hence, application of SLABNS in nanostructures based vibration analysis is better than SLAGS. Furthermore, the FRP of SLABNS and SLAGS in nonlinear vibration response is lower than linear one. It is due the fact that in nonlinear analyzing the accuracy of the obtained results is higher than linear one. The effect of elastic medium on the FRP versus the nonlocal parameter of SLABNS, respectively is shown in Figure 3. Three different cases of elastic medium are considered. Case 1, Case 2 and Case 3 depict the (i) without elastic medium (ii) with Winkler medium (iii) with Pasternak medium, respectively. As can be seen, the FRP (or frequency) increases (or decreases) with increasing nonlocal parameter. It can be observed that the FRP for cases 1 and 3 is maximum and minimum, respectively. In the other words, the frequency of the system for the case of SLABNS embedded in elastic medium is higher than other cases. It is because considering elastic medium increases the stiffness of the system. It is also obvious that the FRP (or frequency) of the case 3 is lower (or higher) than case 2. It is due to the fact that in Winkler medium, a proportional interaction between pressure and deflection of SLABNS is assumed, which is carried out in the form of discrete and independent vertical springs. Whereas, Pasternak medium considers not only the normal stresses but also the transverse shear deformation and continuity among the spring elements. Figure 4 demonstrates the influence of thermal gradient on the FRP with respect to the nonlocal parameter. It could be said however, that FRP increases slightly as thermal gradient is increased. This is perhaps because increasing thermal gradient decreases the structure stiffness. In addition, the effect of thermal gradient on the

FRP becomes more prominent at higher nonlocal parameter.

**Figure**

**2.**

**FRP versus the nonlocal parameter for linear and nonlinear vibration of SLABNS and SLAGS**

**Figure**

**3.**

**The effect of elastic medium on the FRP versus the nonlocal parameter**

**Figure**

**4.**The effect of thermal gradient on the FRP versus the nonlocal parameter

### 5. Conclusion

Based on nonlocal piezoelasticity theory, nonlinear vibration response of SLABNSs surrounded by Pasternak medium was investigated. Electro-thermo-mechanical coupled motion equations were solved using DQM in order to obtain the nonlinear frequency of the system. The effects of the nonlocal parameter, elastic medium, temperature change and maximum amplitude on the nonlinear frequency of the SLABNSs were taking into account. Numerical results show that the frequency of SLABNS was higher than SLAGS. It was also concluded that the FRP of SLABNS and SLAGS in nonlinear vibration response was lower than linear one. Furthermore, with increasing nonlocal parameter, the frequency of the coupled system becomes lower.

### Acknowledgement

I would like to appreciate UniversitiTeknologi Malaysia for their continual support during the course of this paper.

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