DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix
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.
Section snippets
Nonlocal elasticity theory
Classical elasticity theory is not capable to capture the size effect for nano/micro sized modeling. In the classical or macro scaled elasticity the stress state of any body at a point x is related to strain state at the same point x in the classical elasticity. During the classical continuum theory, the constitutive equations of classical (macroscopic) elasticity are an algebraic relationship between stress and strain components. However, this theory is not conflict the atomic theory of
Formulation for buckling problem of BNNTs
A typical BNNT and its continuum model are depicted in Fig. 1, Fig. 2, respectively. BNNT is one of the one-dimensional nanostructures. BNNTs have many novel and potential applications due to their unique physical properties. For the buckling formulation of BNNT, Euler–Bernoulli beam theory is used. The displacement components are as follows:in which the and are the axial and transverse displacements on the neutral axis. The strain-displacement
Discrete singular convolutions (DSC)
The method of discrete singular convolution (DSC) is introduced by Wei [33] in 1999 as efficient and potential numerical approaches based on the some regularized kernels such as Hilbert and delta type transforms. In a mathematical point of view, singular convolutions (SC) are a special class of mathematical transformations [34] which appear in many science and engineering problems, such as the Hilbert, Abel and Radon transforms. Also, convolution theory is used in the some applications of
Results and discussion
In this section, results on buckling of BNNT on two-parameter elastic foundations are presented for different parameters. Firstly, a comparative study for buckling analysis of beams on one parameter elastic foundation has been presented. In order to obtain a reasonable convergence for the buckling, the number of required grid points in related directions of beams in the DSC solution should be determined. To validate the analysis, obtained buckling values for beams on elastic foundation are
Conclusions
Buckling analysis of BNNTs on elastic matrix is investigated. Euler–Bernoulli beam model is used as continuum model for boron nitrite nanotube. The elastic matrix surrounded of the BNNT modeled by Winkler-Pasternak elastic foundation model. The effect of some geometric parameters on buckling loads is investigated. The effect of foundation parameters and nonlocal parameters on buckling is also studied. From the results, it is possible to say that, the effect of nonlocal parameter is to reduce
Acknowledgement
The financial support of the Scientific Research Projects Unit of Akdeniz University is gratefully acknowledged.
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