Transversely isotropic elastic properties of multi-walled boron nitride nanotubes under a thermal environment

The four loading conditions were used to obtain the five independent elastic constants of MWBNNTs, as shown in figure 2. The stored strain energy (U) in the MWBNNTs during their deformation was calculated using an equation written underneath each diagram in figure 2. An initial strain increment was applied for pulling the MWBNNT along its axis followed by the PE minimization process. The strain rate and time step used were 10⁹ s⁻¹ and 0.5 fs, respectively [38]. The MWBNNTs were deformed in small strain increments (0.01) and we equilibrated their deformed states for a time period of 30 ps in each step to overcome the interatomic fluctuations. The period of 30 ps was found to be enough to equilibrate the deformed MWBNNTs in MD simulations and this agreed well with the existing studies [38, 49, 62]. If variations in temperature and PE of the MWBNNTs are $1{\text{% }}$ after ∼5 ps in the NVT ensemble, then they are considered as equilibrated structures [38, 49, 62]. The loading steps were applied until the fracture occurred in MWBNNT structures and the PEs were recorded across all of the deformation steps (i.e. from initial state to fracture state).

Zoom In Zoom Out Reset image size

2.1.1. Axial Young's modulus (${E_1}$) and major Poisson's ratio (${\nu _{12}}$).

The uniaxial tension loading (figure 2(a)) was applied to determine the axial Young's modulus, major Poisson's ratio and failure behavior of MWBNNTs. In uniaxial tension loading condition, the one end of the tube in the axial (z) direction was restrained, while at the other end, a constant strain rate of 10⁹ s⁻¹ was imposed. The average axial Young's modulus of MWBNNT was determined using the deformation energy density-elastic constant relation in the elastic region. The induced axial stresses in the tube were calculated considering a uniform axial stress distribution along its cross-sectional area (A), by using the following relation:

$$\begin{equation}{{\sigma }_{\text{axial}}} = \frac{1}{{\text{V}}}\frac{{{\text{d}}{{\text{S}}_{\text{e}}}}}{{{\text{d}}{{{\bar \varepsilon }}_{11}}}},\end{equation} \tag{ 1 }$$

where ${{\sigma }_{{\text{axial}}}}$ is the longitudinal stress, ${{\bar \varepsilon }_{11}}$ is the average axial strain and S_e is the stored strain energy in the tube. Subsequently, the strain energy density of MWBNNT can be represented in the form of strain energy per unit volume (U) and is expressed by using the following relations [48, 49]:

$$\begin{equation}{\text{U}} = \frac{1}{2}{{\text{E}}_1}{\bar \varepsilon }_{11}^2,\end{equation} \tag{ 2 }$$

$$\begin{equation}{\text{U}} = \frac{{\Delta {{\text{S}}_{\text{e}}}}}{{{\text{Al}}}},\end{equation} \tag{ 3 }$$

where ${\Delta }{{\text{S}}_{\text{e}}}$ is the change in PE, ${{\bar \varepsilon }_{11}} = {\Delta l}/{\text{l}}$ is the axial strain of the tube, and ${\Delta l}$ is the change in length of the tube after deformation.

The approach of modeling the molecular structure of MWBNNT as an effective 'stick-spiral' system was employed to obtain the major Poisson's ratio (${{\nu }_{12}}$). The complete procedure can be found in [66, 67], and accordingly, the major Poisson's ratio (${{\nu }_{12}}$) can be obtained using the following relation:

$$\begin{equation}{{\nu }_{12}} = - \frac{{{{{\bar \varepsilon }}_{22}}}}{{{{{\bar \varepsilon }}_{11}}}},\end{equation} \tag{ 4 }$$

where ${{\bar \varepsilon }_{{{22}_{\text{l}}}}} = \Delta {\text{C}}/{\text{C}}$ is the average circumferential strain of MWBNNT.

2.1.2. Longitudinal shear modulus (${G_{12}}$).

The torsional loading (figure 2(b)) was applied to obtain the longitudinal shear modulus and torsional failure behavior of MWBNNTs. In the torsional loading, the atoms of all layers at its one end were constrained and at the other end, constant rotation of 0.81° ps⁻¹ was applied for a constant time interval of 442 ps with a time-step of 0.5 fs. The longitudinal shear deformation of MWBNNT during the torsional loading is more complicated than the uniaxial tension test. MWBNNT was restrained in a radial direction to maintain the perfect cylindrical surface to simulate the pure twisting condition. The torsional loading is transferred to the other layers via the interlayer vdW forces. The strain energy density of MWBNNT can be obtained as [38, 48, 49]:

$$\begin{equation}{\text{U}} = \frac{{{{\text{G}}_{12}}{\phi ^2}{\text{J}}}}{{2{\text{l}}}},\end{equation} \tag{ 5 }$$

where φ and J are the torsional angle and second polar moment of area of the tube, respectively. Consequently, the second polar moment of area can be obtained using the following relation [38, 48, 49]:

$$\begin{equation}{\text{J}} = \frac{{\pi }}{{32}}\mathop \sum \limits_{{\text{j}} = 1}^{\text{n}} [{\left( {{\text{D}}_{\text{j}}^{\text{o}}} \right)^4} - {\left( {{\text{D}}_{\text{j}}^{\text{i}}} \right)^4}].\end{equation} \tag{ 6 }$$

2.1.3. Plane-strain bulk modulus (${K_{23}}$).

The 2D plane-strain condition (figure 2(c)) was used to determine the plane-strain bulk modulus and failure behavior of MWBNNTs. To meet the pure plane-strain condition, both ends of the MWBNNT were restrained in the z-direction so that the strain in the longitudinal direction remains zero during the deformation. The plane-strain bulk modulus (${{\text{K}}_{23}}$) of MWBNNT can be determined using the following relation [38, 48, 49]:

$$\begin{equation}{\text{U}} = 2{{\text{K}}_{23}}{{\bar \varepsilon }_{22}}.\end{equation} \tag{ 7 }$$

2.1.4. In-plane shear modulus (${G_{23}}$).

The tension-compression loading (figure 2(d)) can be applied to obtain the in-plane shear modulus and failure behavior of MWBNNTs. In this case, the small strain applied in such a way that the circular cross-section of the tube deformed into an elliptical form; such as:

$$\begin{equation}{\text{r}} = {{\text{R}}_{\text{n}}}\left( {1 + {{{\bar \varepsilon }}_{22}}} \right){\text{cos}}{{\theta }_{\text{i}}} + {{\text{R}}_{\text{n}}}\left( {1 - {{{\bar \varepsilon }}_{22}}} \right){\text{cos}}{{\theta }_{\text{j}}}{\theta } \in \left[ {0,2{\mathrm{\pi }}} \right],\end{equation} \tag{ 8 }$$

where ${{{\bar \varepsilon }}_{22}}$ characterizes the deformation strain from a circle of outer radius R_n to an elliptical shape with the major-axis (${{\text{R}}_{\text{n}}}\left( {1 + {{{\bar \varepsilon }}_{22}}} \right)$ and minor-axis ${{\text{R}}_{\text{n}}}\left( {1 - {{{\bar \varepsilon }}_{22}}} \right)$), and i and j are the normal unit vectors along the x- and y-directions, respectively.

To satisfy the in-plane pure shear condition, both ends of the tube were restrained in the z-direction so that the strain in the longitudinal direction remains zero during the in-plane shear deformation. Then, we followed the same procedure used for obtaining the plane-strain bulk modulus. The in-plane shear modulus was determined using the following relation [38, 48, 49]:

$$\begin{equation}{\text{U}} = 2{{\text{G}}_{23}}{{\bar \varepsilon }_{22}}.\end{equation} \tag{ 9 }$$

Note that other dependent elastic coefficients such as transverse Young's modulus (${{\text{E}}_2}$) and minor Poisson's ratio (${{\nu }_{23}})$ can be determined using the five independent elastic constants [68, 69]; so that:

$$\begin{equation}{{\text{E}}_2} = {{\text{E}}_3} = \frac{{4{{\text{G}}_{23}}{{\text{K}}_{23}}}}{{{{\text{K}}_{23}} + {\psi }{{\text{G}}_{23}}}},\end{equation} \tag{ 10 }$$

$$\begin{equation}{{\nu }_{23}} = \frac{{{{\text{K}}_{23}} - {\psi }{{\text{G}}_{23}}}}{{{{\text{K}}_{23}} + {\psi }{{\text{G}}_{23}}}},\end{equation} \tag{ 11 }$$

$$\begin{equation}{\text{where}}\ \psi = 1 + \frac{{4{{\text{K}}_{23}}{\nu }_{21}^2}}{{{{\text{E}}_1}}}.\end{equation} \tag{ 12 }$$

All the loading conditions applied to MWBNNTs are provided in supplementary information video S1 (available online at stacks.iop.org/NANO/31/395707/mmedia).

To validate the MD results, the obtained values of elastic moduli of MBNNTs were first compared with the existing results, as summarized in table 1. Note that the elastic moduli of MWBNNTs were obtained using the averaged axial and radial deformations. Here, the tensile strength is identified as the maximum stress prior to the failure and Young's modulus was determined as the slope of the σ– curve under the uniaxial tensile test within the elastic limit. As shown in table 1, the obtained values of elastic moduli are in reasonable agreement with the previous experimental results reported by Chopra and Zettl [16] and Nautiyal et al [43].

Table 1. Comparison of axial Young's modulus and major Poisson's ratio of MWBNNTs.

			Young's Modulus ${{\mathbf{E}}_1}$ (TPa)
	Diameter (Å)		Present			Major Poisson's ratio $\,{{\mathbf{\upsilon }}_{12}}$
MWBNNT Configuration	Internal Diameter	External Diameter	AR ∼ 5	AR ∼ 10	Ref.	>AR ∼ 5	AR ∼ 10
A_DWBNNT	6.9549	13.8336	1.344	1.351	1.36 [16]	0.166	0.175
A_TWBNNT	6.9549	20.7293	1.219	1.225	1.22 [43]	0.159	0.168
A_FWBNNT	6.9549	27.6292	1.102	1.110	1.10 [16]	0.153	0.162
Z_DWBNNT	6.4189	13.5721	1.379	1.386	1.40 [16]	0.169	0.178
Z_TWBNNT	6.4189	20.7404	1.248	1.257	1.22 [43]	0.164	0.171
Z_FWBNNT	6.4189	27.1152	1.144	1.151	1.15 [43]	0.157	0.165

In this study, both armchair and zigzag types of MWBNNTs with almost the same diameters were considered. BNNTs with their geometrical characteristics are shown in figure 1. First, the MD simulations were performed on the tubes under the uniaxial loading (figure 2(a)) until they fractured. The stress–strain (σ–) curve and variation of PE of MWBNNTs under uniaxial loading are shown in figure 3. It can be observed from figures 3(a)–(c) that the σ– curves for tubes with AR 5 and 10 follow the same trend and almost overlap each other up to the proportional limit and thus, the elastic regions predict almost the same value of elastic moduli. From figures 3(a)–(c), no obvious yielding is observed before the failure of MWBNNTs. The trends of results of σ– and energy–strain (U–) plots are found to be similar to those obtained for BNNTs in the existing MD and experimental studies [36, 38, 43]. For the sake of brevity, we considered only MWBNNTs with AR ∼ 10 to study the effect of temperature and chirality on their mechanical properties. Figures 3(d)–(f) compare the variation of PEs of both types of MWBNNTs. As expected, it can be observed that the MWBNNTs with a higher number of layers show higher PE than smaller ones irrespective of the chirality. This is attributed to the higher volume of tubes, which stores higher PE.

Zoom In Zoom Out Reset image size

It can also be observed from figure 3 that the zigzag MWBNNTs are slightly less stiff than armchair ones at a given finite temperature. In the case of armchair MWBNNTs, the failure occurs in singlestage, but zigzag tubes show two-stage failure. In order to explore the deformation mechanics process and fracture behavior, step-by-step snapshots of armchair and zigzag MWBNNTs under uniaxial loadings are shown in figures 4(a) and (b), respectively. It can be observed from figure 4(a) that the failure occurred in single stage in the armchair MWBNNTs, that is, at the strain value of ∼0.49. While the two-stage failure occurred in the zigzag MWBNNTs, that is, first at the strain value of 0.29–0.30 where the initial failure of bonds starts, followed by the final failure, which occurs at the strain value of 0.58 (see figures 3 and 4(b)). To understand this difference, the mechanism of failure initiation of MWBNNTs is well demonstrated in figure 5. A single layer of the armchair and zigzag MWBNNTs, which possesses almost the same diameters, illustrates the mechanics behind the discrepancy between their failure behavior. The failure strain of the zigzag tube is higher than that of the armchair ones at the same temperature, but shows lower strength as a result of strainhardening. In zigzag MWBNNT, bonds (b₁) exist along its axis (see figure 5(b)), and these bonds bear more stress than that of inclined bonds (a₁) in the armchair tubes. The initial bond (bond $b_1$) brakes in the zigzag tube, but such B–N bond braking does not exist in the armchair tube that connects two hexagonal rings, and this fracture process begins after the dislocation phenomenon in the B–N bonds. As shown in figure 5, there are two types of B–N bonds that exist in each tube: (i) bonds a₁ and a₂ in the armchair tubes and (ii) bonds b₁ and b₂ in the zigzag tubes. Under uniaxial deformation, the length of bonds a₁ increased, while the length of bonds a₂ remains unchanged. For the zigzag tubes, the length of bonds b₁ increased and the length of bonds b₂ reduced. This suggests that bonds a₁ and b₁ act as stress-bearing bonds during the uniaxial deformation. There are two stress-bearing bonds (b₁) in one hexagonal ring of the zigzag tube, while four stress-bearing bonds (a₁) in the case of the armchair tube. The uniaxial stress in the armchair tubes is induced by the stretching and rotation of bonds a₁. For zigzag tubes, only the stretching of bond b₁ occurs. Therefore, armchair tubes show higher failure strength due to the existence of more stress-bearing bonds (a₁) that share more externally applied load. The rotation of bond a₁ releases the axial stretch energy, which makes the armchair tube softer; thus, the stiffness of the armchair tubes is lower than that of the zigzag tubes. It may be noted that the failure of tubes starts from the innermost layer and then reaches the outermost layer through the shear stress in the longitudinal direction, as shown in figure 6. These shear stresses are induced by the change in the interlayer vdW forces that exist between the layers of MWBNNTs. The induced stress magnitude is higher in the innermost layer of the tube and it decreases as the number of layers increases and reaches the lower magnitude of stress in the outermost layer.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Subsequently, a set of MD simulations were performed on MWBNNTs under torsional loading to investigate the effect of temperature and number of layers until they fractured. Figure 7 shows the variation of PE with the twisting angle (θ) of the tubes. The variation of PE with twisting angle of all the tubes shows nearly the same trends, but the failure strain for zigzag tubes is found to be less than that of armchair tubes. Snapshots of the failure behavior of armchair and zigzag DWBNNTs under the torsion loading with incremental angles are illustrated in figure 8. The armchair and zigzag DWBNNTs first twist within the elastic limit at angles of 277° and 245°, as demonstrated in figures 8(a) and (b), respectively. In this region, DWBNNT distorts in such a manner that the cross-section twisted in the xy-plane without changing the shape. The other snapshots demonstrate the systematic step-by-step failure of DWBNNTs at different twisting angles. As the twisting increases, the DWBNNT begins to buckle and its cross-section does not remain circular. The initiation of localized failure of B–N bonds in the armchair and zigzag DWBNNTs can be observed in figure 8. The torsional stress induced in each hexagonal ring of armchair and zigzag DWBNNTs is different and is shown by the marked ellipses in figure 8. As the twisting increases, the cross-section of the tubes becomes increasingly flat and the deformation becomes no longer reversible. The torsional stresses are borne by only one B–N bond and six B–N bonds per hexagonal ring of the armchair and zigzag DWBNNTs, respectively, during the initiation of fracture of bonds (see marked ellipses). The BN ring opposes the early failure of armchair DWBNNT and hence, exhibits higher resistance to twisting deformation compared to that of zigzag tube. Failure of the outermost layer starts at the respective angles of about 437° and 408° spirally along the circumferential surfaces of armchair and zigzag DWBNNTs, and bonds are entirely broken at angles of 689° and 665°, respectively, as illustrated in figures 8(a) and (b).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The step-by-step failure of TWBNNTs and FWBNNTs under torsional loading is provided in figures S1 and S2 of the supplementary information. It was observed that the failure strain during the twisting increases with an increase in the number of tube layers. The failure of the outer layers of armchair and zigzag TWBNNTs begins at the respective angles of 443° and 419°, and the bonds are entirely broken at angles of 695° and 679°, respectively. The failure of the outers layers of armchair and zigzag FWBNNTs occurs at the critical angles of 574° and 537°, respectively, spirally across their circumferential surfaces, and B–N bonds are entirely broken at angles of 735° and 717°.

Then, MD simulations were performed on MWBNNTs under in-plane bi-axial tension and shear loadings until their fracture. The change in the PE with radial strain during the in-plane biaxial loading on MWBNNTs is shown in figure 9. It can be observed that the armchair tubes store more PE than zigzag ones. Unlike the trend found in figure 3, the failure of MWBNNTs under in-plane loading occurs in two stages: primary and secondary.

Zoom In Zoom Out Reset image size

In order to explore this fracture behavior, snapshots of the deformation behavior of DWBNNTs are shown in figure 10. It can be observed from figure 10 that the initial failure starts from the inner layer of DWBNNTs and this is considered as primary failure, which begins at strain values of 0.41 and 0.30 for armchair and zigzag DWBNNTs, respectively. The secondary failure phase begins with the additional application of load and this results in the high PE storage in the DWBNNTs at low rate of strain increment, and then the complete failure occurs at strain values of 0.49 and 0.41 for armchair and zigzag DWBNNTs, respectively. Figure 11 depicts the failure initiation mechanics of a single layer of armchair and zigzag DWBNNTs having almost similar diameters. During the in-plane biaxial deformation, the stresses such as radial stress (σ_r), circumferential stress (σ_c) and stress due to vdW interactions between interlayers of MWBNNT are generated. It can be observed from figure 11 that the radial stress and stresses due to vdW forces are uniform throughout the tube, and these stresses do not create any differences in the failure mechanism of tubes. It is clearly observed in figures 11(a) and (b) that the circumferential stresses are tensile in nature and depend on the chirality of tube; thus, only the circumferential stress is the leading cause of failure of armchair and zigzag tubes. Under the biaxial deformation of zigzag MWBNNT, the component of circumferential stress transmits to the bonds b₁ only, while in armchair MWBNNT, the component of circumferential stress transmits to both bonds a₁ and a_2. It can be observed from figures 11(b) and (c) that there are four circumferential stress-bearing bonds (b₁) in one hexagonal ring of zigzag tube, while six circumferential stress-bearing bonds (a₁ and a₂₎ in armchair tube. The stress-bearing bond per hexagonal ring is less in the case of zigzag tube; thus, the failure strain of zigzag MWBNNTs is lower than that of the armchair tubes. For the sake of brevity, the step-by-step failure of TWBNNTs and FWBNNTs under biaxial tension are provided in the supplementary figures S3 and S4. It was observed that the failure strain decreases with the increase in number of tube layers.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Furthermore, the in-plane shear load was applied to MWBNNTs (see figure 3(d)). During the deformation, the circular cross-section of MWBNNT deforms into an elliptical form, and the bonds get aligned to major and minor axes, which bear the maximum tension and compression stresses, respectively. Figure 12 shows the change in PE during the deformation of MWBNNTs under in-plane shear. The same pattern was observed in the case of in-plane biaxial tension loading (figure 9). Moreover, in this case the failure of tubes also occurs in two stages. This is again attributed to the fact that the inner tube fails first followed by the outer layer, as shown in figure 13. This figure clearly reveals that all layers of MWBNNT deform into elliptical shape, and the loads get transferred to inner layers via vdW forces. During the in-plane shear loading, circumferential stress is a function of both tensile and compression stress components shared by each bond of the hexagonal ring of DWBNNT, which is well explained in figure 14. It can be observed from figure 14 that the maximum load-bearing bonds are a₂ and b₁ in armchair and zigzag DWBNNT, respectively. A zigzag MWBNNT fails well before that of an armchair tube because the high stress-bearing bonds per hexagonal ring are four in the case of zigzag and two in the case of armchair MWBNNTs. The interlayer shear stress at the major and minor axes are zero due to the pure tension and compression loads applied across these axes, respectively. This can be observed from figure 13, which clearly depicts that the failure occurs at the major and minor axes. For the sake of brevity, the step-by-step failure of TWBNNTs and FWBNNTs under in-plane shear are provided in the supplementary figures S5 and S6. It was observed that the failure strain decreases with the increase in number of tube layers. For the sake of clarity, the deformation process of MWBNNTs under four loading conditions is provided as supplementary information video S2.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

To examine the effect of temperature on the elastic and failure behavior of MWBNNTs, we again used four loading conditions considering six different temperature steps: 0, 300, 500, 700, 900 and 1100 K. The temperature was increased from 0 to 1100 K by keeping the constant strain rate. The failure behavior of armchair and zigzag MWBNNTs subjected to uniaxial, twisting and in-plane biaxial loadings is shown in figures 15, 16 and 17, respectively. It can be observed from figure 15 that the trend of stress–strain behavior of DWBNNT, TWBNNT and FWBNNT is almost similar. The failure behavior of MWBNNTs subjected to in-plane biaxial and shear loadings for varied temperature was found to be the same as that demonstrated in figures 9 and 12, and for the sake of brevity, the plots are not shown here.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

It may be observed from figures 15–17 that the temperature profoundly changes the mechanical behavior, especially the failure behavior of armchair and zigzag MWBNNTs. However, no apparent plastic deformation was observed, and thus brittle fracture persists in all cases considered herein. During the deformation under thermal loading, most of the energy is converted into kinetic energy and therefore, this results in the decrease in the stored strain energy, which eventually decreases the failure strain as the temperature increases. At higher temperature, the early formation of cracks occurs. This is due to the creation of topological defects after a critical strain value [70]. The formation of crack depends on the strain and thermal energies of MWBNNTs during the loading. The thermal energy of MWBNNTs increases with the temperature due to the thermal vibration of atoms; thus, the strain energy required for the failure reduces. As a result, the failure strength and strain of MWBNNTs are significantly reduced at a higher temperature irrespective of the type of loading condition.

Figure 15 shows the stress–strain curves of armchair and zigzag MWBNNTs under tension at different temperatures. The stress values at failure strain are considered as tensile strengths of MWBNNTs where the stress–strain curves dramatically drop. The failure strengths of armchair and zigzag DWBNNTs are 147 and 137 GPa, respectively, at 0 K, and these values decrease significantly as the temperature and number of tube layers increase. For instance, the failure strength of armchair and zigzag FWBNNTs is 121.1 and 118.2 GPa, respectively. As the temperature increases, the failure strength of armchair and zigzag DWBNNTs decreases significantly to 93.2 and 85.8 GPa, respectively, at 1100 K. However, armchair MWBNNTs are stronger than zigzag tubes, even at higher temperature. It can be observed from figures 16 and 17 that the temperature significantly reduces the PE of MWBNNTs. The effect of temperature is not similar in both armchair and zigzag MWBNNTs under twisting. The PE of zigzag MWBNNTs under twisting decreases substantially compared to armchair tubes. This finding shows that the armchair MWBNNTs have higher twisting resistance than that of zigzag tubes at a higher temperature. The same is true in the case of MWBNNTs under in-plane biaxial loading, as shown in figure 17.

The expressions previously derived herein were used to determine the values of E₁, ν₁₂, G₁₂, K₂₃ and G₂₃ of MWBNNTs at varied temperatures, as shown in figure 18. The elastic coefficients of MWBNNTs decrease as their number of layers increases. This is due to the influence of included angles of B–N bonds of adjacent BN tube layers via vdW interactions and the failure initiation starts from the innermost tube layer. It can also be observed from figures 15–17 that the failure strength of MWBNNT decreases as its number of layers increases due to the occurrence of slippage between the two adjacent tube layers at finite temperature. The interlayer distance between the two adjacent layers slightly increases with the number of layers [12] and this results in the increase in inter-layer slippage, which eventually decreases the failure strength of MWBNNT. This is again attributed to the influence of included angles of B–N bonds of adjacent BN tube layers via vdW interactions [48]. It can be observed from figure 18(c) that the value of G₁₂ decreases profoundly compared to all other elastic coefficients as the number of tube layers increases. For instance, the values of G₁₂ of zigzag TWBNNT and FWBNNT are reduced by 28% and 37%, respectively, while the respective reductions in the case of armchair TWBNNT and FWBNNT are found to be 20% and 30% compared to zigzag and armchair DWBNNTs. All other constants of armchair and zigzag MWBNNTS are reduced by 4%–8 % and 8%–17%, respectively. As shown in figure 18, the elastic properties decrease with increase in temperature. As the temperature rises, MWBNNTs become softer and less stiff. Moreover, as the temperature increases, the kinetic energy of atoms in the tube increases and they start moving faster, which eventually reduces its strength. The effect of temperature on the elastic properties of armchair and zigzag MWBNNTs is almost similar. Even though the loading was uniformly applied over the whole MWBNNT, the atomic stresses were not uniformly induced, and this is attributed to the thermal fluctuations of atoms at higher temperature (shown in supplementary video S3). The total kinetic energy of MWBNNTs increases with the temperature. Due to the action of forces generated by thermal loading, the B and N atoms move from their original equilibrium positions, resulting in a soft and less rigid-like structure. At higher temperature, this temperature-stimulated softening is attributed to the higher thermal vibrations of B and N atoms. Such thermal vibration of B and N atoms makes MWBNNTs less stiff and due to the thermal agitation, the B–N bonds are more likely to reach the critical bond length and therefore, they break. The temperature-dependent elastic properties of MWBNNTs shown in figure 18 can be described by a simple linear equation as follows:

$$\begin{equation}{M^T} = \,{M^0} - \alpha \times T.\end{equation} \tag{ 13 }$$

Zoom In Zoom Out Reset image size

where M^T is the elastic modulus of MWBNNT at a given finite temperature, α is a material and chirality-dependent constant and M⁰ is the elastic modulus of MWBNNT at 0 K.

The obtained constants are summarized in table 2. Compared with the failure strength and strain, the elastic properties are less sensitive to temperature. The elastic properties of armchair MWBNNTs are more sensitive to temperature than zigzag tubes. It can be observed from figure 18 that the effect of chirality and number of layers on the values of G₁₂ of MWBNNTs at different temperatures is more pronounced than all other elastic coefficients. The reason behind this can be found in figure 7, which shows that the PE of MWBNNTs increases with the number of tube layers under twisting, but their strength decreases due to the vdW interactions between the layers. The reduction rate of shear modulus is 1.955 × 10⁻⁴ to 1.43 × 10⁻⁴ TPaK⁻¹ for armchair and zigzag MWBNNTs, respectively, as the temperature rises from 0 to 1100 K. A similar effect of temperature on the values of K₂₃ of MWBNNTs is observed, as shown in figure 18(e). Table 2 reveals that the temperature significantly affects the values of K₂₃ of MWBNNTs and the reduction rate of K₂₃ is found to be in the range of 0.23 × 10⁻⁴ to 0.18 × 10⁻⁴ TPaK⁻¹.

Table 2. Temperature-dependent elastic properties of MWBNNTs.

MWBNNT	$E_1^0$ (TPa)	α (TPaK−1)	$\nu _{12}^0$ (TPa)	α (TPaK−1)	$G_{12}^0$ (TPa)	α (TPaK−1)	$K_{23}^0$ (TPa)	α (TPaK−1)	$G_{23}^0$ (TPa)	α (TPaK−1)
Armchair MWBNNTs
DWBNNT	1.345	3.13 × 10−4	0.167	0.21 × 10−4	0.641	1.95 × 10−4	0.168	0.23 × 10−4	0.09	0.23 × 10−4
TWBNNT	1.232	3.23 × 10−4	0.161	0.18 × 10−4	0.519	1.70 × 10−4	0.151	0.22 × 10−4	0.079	0.20 × 10−4
FWBNNT	1.094	2.86 × 10−4	0.154	0.17 × 10−4	0.452	1.49 × 10−4	0.141	0.18 × 10−4	0.075	0.20 × 10−4
Zigzag MWBNNTs
DWBNNT	1.392	2.57 × 10−4	0.17	0.21 × 10−4	0.588	1.76 × 10−4	0.174	0.22 × 10−4	0.094	0.22 × 10−4
TWBNNT	1.259	2.42 × 10−4	0.165	0.21 × 10−4	0.473	1.59 × 10−4	0.157	0.20 × 10−4	0.083	0.21 × 10−4
FWBNNT	1.162	2.57 × 10−4	0.158	0.18 × 10−4	0.416	1.43 × 10−4	0.148	0.20 × 10−4	0.078	0.21 × 10−4