Keywords

1 Introduction

A boron nitride nanotube (BN-NT) can be geometrically formed by rolling up a hexagonal boron nitride (BN) layer or a carbon nanotube (CNT) [1] in which alternating B and N atoms entirely substitute for C atoms as shown in Fig. 1.

Fig. 1
figure 1

Schematic illustration of: a (9, 0) BN zigzag tube; b (5, 5) BN armchair tube

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Various techniques have been used to synthesize BN-NTs, including arc-discharge, chemical vapor deposition, laser ablation, ball-milling methods (see, e.g., the review by [2]). BN-NTs exhibit good mechanical properties with high elastic modulus of ~0.5–1 TPa and tensile strength of ~61 GPa [3]. They possess distinguishable chemical and thermal stability with high oxidation resistance up to 900 °C in air [4], wide bandgaps independent of tube structures [5, 6], excellent thermal conductivity [7]. BN-NTs are also an effective violet and ultra-violet light emission material [8, 9]. Potential applications of BN-NTs include nanofillers in polymeric [10] and metallic [11] composites, optoelectronic fields [8], radiation shielding in space vehicles [12]. Potential applications of BN-NTs need a comprehension of the mechanical properties and performance of BN-NTs under various loading conditions. BN-NTs under compression [13,14,15], tension [16, 17], torsion [16, 18,19,20], and bending with two fixed or simple supports [21, 22] have been investigated. So far, theoretical studies of the buckling behavior under bending of BN-NTs seem unexplored. It should be noted that the buckling behavior of CNTs under bending has been investigated by continuum methods, atomistic simulations, and multi-scale approach; see, for example, [23, 24] and references therein.

The present work investigates through molecular dynamics finite element method (MDFEM) the buckling behavior of BN-NTs under bending. The critical bending buckling angle and moment are studied with respect to the length–diameter ratios of BN-NTs.

2 Framework for Analysis

Tersoff potential is used to model the B-N interatomic interactions [25]. The potential energy E of the atomic structure is a function of atomic coordinates as below:

$$E = \sum\limits_{i} {E_{i} = \frac{1}{2}\sum\limits_{i \ne j} {V_{ij} } } ,V_{ij} = f_{C} \left( {r_{ij} } \right)\left[ {f_{R} \left( {r_{ij} } \right) + b_{ij} f_{A} \left( {r_{ij} } \right)} \right];$$
(1a)
$$f_{R} \left( {r_{ij} } \right) = A_{ij} { \exp }\left( { - \lambda_{ij}^{I} r_{ij} } \right),f_{A} \left( {r_{ij} } \right) = - B_{ij} { \exp }\left( { - \lambda_{ij}^{II} r_{ij} } \right);$$
(1b)
$$f_{C} \left( {r_{ij} } \right) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {r_{ij} \le R_{ij} ;} \hfill \\ {\frac{1}{2} + \frac{1}{2}\cos \left( {\pi .\frac{{r_{ij} - R_{ij} }}{{S_{ij} - R_{ij} }}} \right),} \hfill & {R_{ij} \le r_{ij} \le S_{ij} ;} \hfill \\ {0,} \hfill & {r_{ij} \ge S_{ij} ;} \hfill \\ \end{array} } \right.$$
(1c)
$$b_{ij} = \chi_{ij} \left( {1 + \beta_{i}^{{n_{i} }} \zeta_{ij}^{{n_{i} }} } \right)^{{ - {1 \mathord{\left/ {\vphantom {1 {2n_{i} }}} \right. \kern-0pt} {2n_{i} }}}} ,\zeta_{ij} = \sum\limits_{k \ne i.j} {f_{C} \left( {r_{ik} } \right)\omega_{ik} g\left( {\theta_{ijk} } \right)} ,g\left( {\theta_{ijk} } \right) = 1 + {{c_{i}^{2} } \mathord{\left/ {\vphantom {{c_{i}^{2} } {d_{i}^{2} }}} \right. \kern-0pt} {d_{i}^{2} }} - c_{i}^{2} /\left[ {d_{i}^{2} + \left( {h_{i} - \cos \theta_{ijk} } \right)^{2} } \right]$$
(1d)
$$\lambda_{ij}^{I} = {{\left( {\lambda_{i}^{I} + \lambda_{j}^{I} } \right)} \mathord{\left/ {\vphantom {{\left( {\lambda_{i}^{I} + \lambda_{j}^{I} } \right)} 2}} \right. \kern-0pt} 2},\lambda_{ij}^{II} = {{\left( {\lambda_{i}^{II} + \lambda_{j}^{II} } \right)} \mathord{\left/ {\vphantom {{\left( {\lambda_{i}^{II} + \lambda_{j}^{II} } \right)} 2}} \right. \kern-0pt} 2},A_{ij} = \sqrt {A_{i} A_{j} } ,B_{ij} = \sqrt {B_{i} B_{j} } ,R_{ij} = \sqrt {R_{i} R_{j} } ,S_{ij} = \sqrt {S_{i} S_{j} }$$
(1e)

Here, the lower indices i, j, and k label the atoms of the system, where interaction between atoms i and j is modified by a third atom k. \(r_{ij}\) is the distance between atoms i and j; f A and f R are the attractive and repulsive pairwise terms; f C is a cutoff function to ensure the nearest-neighbor interactions; R ij and S ij denote the small cutoff distance and the large one, respectively; b ij is a bond-order parameter, depending on the local coordination of atoms around atom i. Further detail of the Tersoff potential is given in [25]. Force field parameters are taken from the work by Sevik et al. [26] for B-N interactions.

While the density functional theory (DFT) calculations and molecular dynamics (MD) simulations are time-consuming, molecular dynamic finite element methods (MDFEMs), sometimes known as atomic-scale finite element methods or atomistic finite element methods, have been developed to analyze nanostructured materials in a computationally efficient way; see, for example, [27, 28]. To achieve the atomic positions of the BN-NT under specific boundary conditions, molecular dynamic finite element method (MDFEM) is here adopted. In MDFEM, atoms and atomic displacements are considered as nodes and translational degrees of freedom (nodal displacements), respectively. Both first and second derivatives of system energy are used in the energy minimization computation, hence it is faster than the standard conjugate gradient method which uses only the first-order derivative of system energy as discussed in [27]. The stiffness matrices of these elements are established based upon interatomic potentials. Similar to conventional finite element method, global stiffness matrix is assembled from element stiffness matrices. Hence, relations between atomic displacement and force can be derived by solving a system of equations. Further detailed numerical procedure of MDFEM and our specific development for Tersoff potentials are available in our previous work [29] and references therein. Initial positions of atoms are generated by using the B-N bond length of 1.444 Å taken from previous MD simulations [30] at optimized structure at 0 K with the same force field. (5, 5) BN armchair and (9, 0) BN zigzag tubes are considered. Difference in diameters of these two tubes is less than 4%. The diameter is about 0.717 and 0.689 nm for (9, 0) BN armchair and (5, 0) BN zigzag tubes, respectively.

3 Results and Discussion

Figure 2 shows the variations of the bending moment versus the bending angle of (9, 0) and (5, 0) BN tubes with the length–diameter ratio L/D = 30. The bending angle θ is here defined as the angle between two planes containing the two ends of the tube under bending. It can be seen from Fig. 2 that the bending moment increases monotonously with an increase of the bending angle up to a critical value, and then the bending moment drops suddenly, demonstrating a brittle fracture. The critical bending angle of the (9, 0) BN zigzag tube is approximately 66.6o, 136.6o, 210.6o, and 278.0o for L/D = 10, 20, 30, and 39, respectively. The critical bending angle of the (5, 5) BN armchair tube is about approximately 121.7o, 142.9o, 176.7o, and 181.0o for L/D = 10, 20, 30, and 40, respectively.

Fig. 2
figure 2

Variations of the bending moment versus the bending angle of the (9, 0) and (5, 0) BN tubes with L = 30D

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Figure 3 shows the effects of tube’s length on the variations of the critical bending moments, critical bending angle, and critical bending curvature of these two tubes. The critical bending moment and the critical bending curvature of (9, 0) BN tubes increase with an increase of the tube’s length in the range L/D = 10–40, whereas the critical bending moment and the critical bending curvature decrease when increasing the length of the (5, 5) BN tube. The critical bending angles of these two tubes increase with increasing the tube’s length.

Fig. 3
figure 3

Variations versus the tube length of: a the critical bending angle; b critical bending curvature; and c critical bending moment

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The critical bending moment, the critical bending curvature, and critical bending angle of (5, 5) tubes are higher than those of (9, 0) tubes when L/D = 10, hence short (5, 5) tubes resist better than short (9, 0) tubes under bending. Whereas long (9, 0) tubes undergo bending better than long (5, 5) tubes; the critical bending moment, the critical bending curvature, and critical bending angle of (9, 0) tubes are higher than those of (5, 5) tubes at L/D = 30 and 40 as indicated in Fig. 3. Figure 4 shows the post-buckling shapes of (9, 0) BN zigzag tubes with L/D = 10, 20, and 30. Snapshots under progressive bending are depicted in Figs. 5 and 6 for (9, 0) BN zigzag tube with L/D = 39 and (5, 5) BN armchair tube with L/D = 20, respectively.

Fig. 4
figure 4

Post-buckling shapes of (9, 0) BN zigzag tubes: a L/D = 10, θ = 91.67°; b L/D = 20, θ = 171.89°; and c L/D = 30, θ = 213.37°

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Fig. 5
figure 5

Snapshots of a (9, 0) BN zigzag tube with L/D = 39 under a bending angle of: a 279.37o; and b 279.49o

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Fig. 6
figure 6

Snapshots of a (5, 5) BN armchair tube with L/D = 20 under a bending angle of: a 143.35o; b 143.58o; c 143.70o; and d 143.81°

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4 Conclusions

We present the simulation results of the buckling behavior of BN nanotubes under bending with the use of MDFEM. We have found that the tube length affects significantly the bending behavior of the tube. All tubes exhibit brittle fracture under bending. The buckling takes place in the middle of the compressive side of the tube. More investigation should be done to analyze in details the buckling behavior of the BN tubes.