DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix

Abstract

A simple mechanical model for buckling behavior of boron nitride nanotube (BNNT) surrounded by an elastic matrix is presented. A nonlocal-continuum model is proposed for BNNT using the Euler–Bernoulli beam theory on an elastic matrix. The elastic matrix surrounded of the BNNT is modeled via linear spring model using the Winkler and Pasternak elastic foundation models. The equation is obtained by variational approach for buckling and has been solved by two different approaches. Separation of variables and method of discrete singular convolution are used for computations. The influences of some geometric parameters of BNNT on buckling behavior are investigated in detail. The effect of mode numbers and nonlocal parameter on buckling behavior of BNNT has also been investigated. Finally, some parametric results are presented for BNNT buckling. It is noticed that the present DSC approach can predict accurately the buckling loads for nano-scaled structures.

Introduction

Micro- and nano-sized systems have been widely used modern industries such as biomedical devices, nano electro mechanical applications, biochemical purpose, mechanical actuators and nano sensors. Nano and micro scaled structures are generally modeled continuous mechanical systems such as beams, bars, tubes and plates. These systems have been used in micro-electro mechanical systems (MEMS) for high frequency and high sensitive purposes due to their ultra-mechanical, thermal and electrical properties. Nanowires, nanobridges, nanotubes, micro and nanosensors, atomic force microscope, and micro-/nano- electromechanical devices are widely used in different engineering and industrial applications [1], [2], [3], [4], [5], [6]. For example, atomic force microscope and micro bridges are generally used to detect the frequency, stress and forces at the nano-level applications. Thus, understanding the mechanical behaviors such as vibration and stability of these devices is an important task for design.

In general, continuum model is chosen for modeling by researchers and designer for such systems due to its computational efficiency and simplicity than the atomic modeling or simulation. By this time, different types of size dependent continuum theory are developed and used for modeling. Micropolar elasticity, Cosserat elasticity, couple stress theory, strain gradient elasticity, surface energy theory, stress gradient theories are generally used such a nano and microsystem modeling. Nonlocal elasticity theory developed by Eringen [7] is also generally used in the past ten years. Unlike the Cauchy elasticity theory, the nonlocal elasticity theory assumes that the stress at a point of any continuum body is a function of strains at all points in the given continuum media. Due its generality and simplicity, many researchers have been applied this theory for modeling of nano/micro sized mechanical or biological systems based on the rod/bar (axial vibration), beam or plate theories [8], [9], [10], [11], [12], [13], [14], [15]. Scale effects play an important role for modeling of nano scaled systems, such as: atomic force microscope, nano bridges, carbon nanotubes, nanowires and micro-electro mechanical systems (MEMS), nano actuators and sensors. Among the many nano/micro systems, carbon nanotubes (CNTs) and boron nitride [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] nanotubes (BNNTs) and micro bridges [31], [32] have attracted considerable attention nowadays.

BNNT have found some applications in different area. Thus, some literature studies on micro or nano-scaled elastic tubes have been mentioned as follows. Ajori and Ansari [24] have investigated the torsional buckling of BNNT via molecular dynamics. Arani et al. [25] have presented some results for BNNT under the electro-thermo-mechanical effects. Effects of structural defects on compressive buckling load are given by Shokuhfar and Ebrahimi-Nejad [26]. Torsional vibration of BNNT has also investigated by Ansari and Ajori [27]. Nonlinear vibration of reinforced composite has been solved for CNT by Guo and Zhang [73]. Tornabene et al. [74] investigated the effect of agglomeration on the natural frequencies of FG carbon nanotubes. Free vibration of CNT on elastic foundation is presented by Rahmanian et al. [75]. An efficient shell model is proposed for vibration problem of single and double walled CNT by Brischetto et al. [76]. Effect of slip condition on CNT conveying fluid is investigated via nonlocal elasticity [77].

It is common place observation that by some experimental and theoretical studies, length scale parameters or size effects play a significant role in mechanical behavior of such nano- and micro-scaled systems. As mentioned before, classical elasticity theory does not take into consideration the size effect of microstructure during the formulation. In order to introduce the size effect to the governing equations, material length scale parameters must be taken into account. Atomistic and molecular dynamic simulation models or hybrid atomistic-continuum models are computationally expensive. Furthermore, controlled experimental studies are very difficult task for nano-scale devices in many conditions. So, size-dependent continuum model is very used by researchers for modeling of micro and nano-scaled systems.

In this manuscript, buckling of BNNT on elastic matrix is presented. The motivation for performing the present study is to give a simple and efficient method for determination of buckling loads of BNNT for different geometric and foundation parameters. The governing equation of buckling based on the nonlocal elasticity theory is derived using the variational formulation. The obtained governing equation is solved by the method of discrete singular convolution (DSC). Some parametric results have been presented.