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6th International Conference from “Scientific Computing to Computational Engineering”

6th IC-SCCE

Athens, 9-12 July, 2014

© IC-SCCE

TENSILE BEHAVIOR OF BORON-NITRIDE NANORIBBONS VIA A NON-LINEAR

SPRING BASED STRUCTURAL MECHANICS APPROACH

G.I. Giannopoulos1, S.K. Georgantzinos2, D.-P.N. Kontoni 3 and P.A. Kakavas4

1 Department of Mechanical Engineering, 3,4 Department of Civil Engineering,

Technological Educational Institute of Western Greece,

1 M. Alexandrou Str., Koukouli, GR-26334 Patras, Greece

e-mail(1): ggiannopoulos@teipat.gr, e-mail(3): kontoni@teipat.gr , e-mail(4): kakavas@teipat.gr

web page: http://www.teipat.gr , www.teipat.gr/civil , www.teipat.gr/civil/kontoni

2 Mechanical Engineering and Aeronautics Department, University of Patras, GR-26500, Greece

e-mail(2): sgeor@mech.upatras.gr, web page: www.mead.upatras.gr, http://mdl.mech.upatras.gr

Keywords: Boron nitride, Nanoribbon, Finite element analysis, Structural mechanics, Stress-strain behavior,

Mechanical properties.

Abstract. Boron nitride, like graphene, can be formed as one-atom thick sheets or as nanotubes, then cut into

nanoribbons with their atoms arranged in a hexagonal lattice. Two-dimensional hexagonal boron-nitride

nanoribbons have been extensively investigated due to their excellent mechanical properties and high thermal

conductivity. They also resist chemical change and are unaffected by high temperatures, leading researchers to

believe that they could be consummate nanomaterials. The dimensions of boron-nitride nanoribbons as well as

the shape of their edges, which may be armchair or zigzag, may affect the overall behavior of the nanoribbons.

In the present paper, the mechanical behavior of different sized zigzag and armchair boron nitride nanoribbons

is numerically investigated and predicted by using a structural mechanics approach based on the Brenner

potential for boron nitride bonds. According to the proposed method, appropriate spring elements are combined

in nanoscale in order to simulate the interatomic interactions appearing within boron-nitride nanostructure.

The study focuses on the prediction of tensile stress-strain behavior of boron-nitride nanoribbons of different

sizes and edge shapes as well as the estimation of significant corresponding material properties such as Young’s

modulus, Poisson’s ratio, tensile strength stress and tensile failure strain. The numerical results, which are

compared with corresponding data given in the open literature where possible, demonstrate thoroughly the

important influence of size and chirality of a narrow boron nitride monolayer on its mechanical behavior.

1 INTRODUCTION

A Boron-Nitride (BN) monolayer in hexagonal state (h-BN) consists of equal numbers of boron (B) and

nitrogen (N) atoms and it has similar honeycomb structure as graphene monolayer (h-C). The properties of few-

nanometer-thick BN multilayer sheets (BN flakes) have attracted the interest of many researchers over the last

several years [1]. Although individual atomic planes of BN were also isolated [2] and investigated by transmission

electron microscopy (TEM) [3] and atomic force microscopy (AFM) [4], the scientific interest concerning BN

monolayers has been rather limited in comparison with the research activity towards its “sister” material,

graphene. There has been a lot of effort in characterizing the mechanical behavior of BN sheets as well as BN

nanotubes via theoretical studies based on density functional theory (DFT) [5-10], tight-binding approximation [11],

molecular dynamics (MD) [12], continuum mechanics (CM) [13,14], and molecular mechanics (MM) [15,16]. Due to

the complexity of conducting real measurements in nanoscale, fewer are the relevant experimental approaches

[17-20].

In the present study, the in-plane non-linear stress-strain behavior of several almost square-shaped hexagonal

BN nanoribbons (h-BNNRs) is numerically estimated by using an atomistic structural mechanics (SM)

approach. According to the adopted numerical technique, in order to represent the B-N repulsive and attractive

interatomic interaction a suitable Brenner potential function [21] is utilized. The bond angle bending and bond

angle torsion interatomic interactions are as well included in the formulation by introducing corresponding

appropriate potential functions. The positions of all B and N atoms are represented by in-plane nodes according

to the exact hexagonal molecular structure of the h-BNNR under consideration. Then, atomic nodes are suitably

interconnected via special spring-like elements of appropriate stiffness which are capable of representing

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.

accurately the aforementioned potential energies of each interatomic interaction. In order to convert computed

force distributions to stress responses, the thickness of all considered h-BN monolayers is assumed to be equal

to the interlayer distance in h-BN flakes [22,23]. To the authors’ best knowledge, for the first time, the non-linear

mechanical behavior of h-BNNRs is numerically estimated by using a spring-based structural mechanics

method. By the arisen non-linear tensile stress-strain curves various mechanical properties such as Young's

modulus, Poisson’s ratio, tensile strength and tensile failure strain are being predicted along both armchair and

zigzag directions of the nanomaterials under investigation. Various results are illustrated and compared with

other data published in the literature, where possible.

2 GEOMETRY OF h-BN

The honeycomb lattice of h-BN, which is assumed to have thickness nm333.0

=

t [22,23] in the present

analysis, has a unit cell represented in Figure 1 by the vectors 1

r and 2

r such that:

021 3|||| r== rr (1)

where 0

r is the equilibrium B-N bond distance taken equal to 0.133nm [21]. In this basis any lattice vector r is

represented as:

21 aar mn

+

=

(2)

where n, m are integers. In Cartesian coordinates:

T

01 ]02/32/3[r=r (3)

T

02 ]02/32/3[ −= rr (4)

Figure 1. Geometry of h-BN monolayer

The vectors connecting any atom to its nearest neighbors are:

3/)2( 211 rrδ

−

=

(5)

3/)2( 122 rrδ

−

=

(6)

3/)( 213 rrδ

+

=

(7)

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.

3 FROM POTENTIAL ENERGY TO FORCE FIELD

The total potential energy due to interatomic interactions within a h-BN monolayer, by ignoring the

negligible van der Waals effects, may be written as:

∑∑∑

+

+

=

τθ

UUUU r (8)

where r

U represents the energy due to bond length change,

θ

U the energy due to bond angle bending and

τ

U

the energy due to bond angle torsion. The interatomic potential r

U may be expressed as:

atrrepr UUU

+

=

(9)

where rep

U and atr

U are the repulsive and attractive potential terms, respectively, given by [21]:

)2exp(

12

10rS

S

D

Urep Δ−

−

=

β

(10)

)/2exp(

12

10rS

S

SD

Uatr Δ−

−

−=

β

(11)

where

r

Δ is the bond length change while the rest parameters are constants which are equal to

nm2nN1.01738212

0=D, 1.0769=S and -1

20.43057nm=

β

for the B-N chemical bond. Figure 2a depicts the

variation of interatomic potential energy r

U due to the bond length change. The arisen force r

F between B and

N bonded atoms due to the distance change between them may be evaluated by differentiating Equation (9), the

result of which ( )d(/d rUF rr Δ= ) is illustrated in Figure 2b.

(a) (b)

Figure 2. Variation of (a) potential energy and (b) force due to B-N bond length change

The interatomic potentials

θ

U and

τ

U, respectively, are approximated as:

2

)(

2

1

θ

θθ

Δ= kU (12)

2

)(

2

1

φ

ττ

Δ= kU (13)

where

θ

k and

τ

k are the bond angle bending and bond angle torsion force constants, respectively, while

θ

Δ

and

φ

Δ represent the bond angle bending and bond angle torsion changes, respectively. The force constants

used for the h-BN nanostructure are taken equal to -2

radnmNn6952.0=

θ

k and -2

radnmNn6255.0=

τ

k [15].

Differentiation of Equations (12) and (13), respectively, yields:

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.

θ

θθ

θ

θ

Δ=

Δ

=k

U

M

d

d (14)

φ

φ

τ

φ

φ

Δ=

Δ

=k

U

Md

d (15)

where

θ

M and

φ

M are the arisen bending and torsion moments due to the bond angle bending and bond angle

torsion, correspondingly.

4 FORCE FIELD SIMULATION

In order to represent the entire force field appearing within h-BN nanostructure, two types of two-noded

spring elements of specific local coordinate systems, with six degrees of freedom per node i.e., three translations

and three rotations, are utilized. Firstly, spring elements, called hereafter BN, are utilized for the simulation of

both force r

F variation (Figure 2b) and torsion moment

φ

M variation (Equation 15) due to the bond length

change

r

Δ and bond angle torsion

φ

Δ, respectively. Three of the specific spring elements, denoted by node

pairs oi ′, oj ′′ and on ′′′ , are shown in Figure 3. These elements are described according to a three dimensional

local coordinate system ( zyx ,, ). The x-axis of the specific local coordinate system coincides with the line that

connects the two bonded B and N atoms while

z

-axis is vertical to the h-BN plane. Their displacement stiffness

matrix is defined by:

⎥

⎦

⎤

⎢

⎣

⎡

−

−

=BNBN

BNBN

BN

zyxzyx

zyxzyx

zyx kk

kk

K (16)

where:

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

∞

∞

∞

∞

Δ

=

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

=

00000

00000

00000

00000

00000

00000

)d(

d

00000

00000

00000

00000

00000

00000 2

2

BN

BN

BN

BN

BN

BN

BN

τ

k

r

U

k

k

k

k

k

kr

zrot

yrot

xrot

z

y

x

zyx

k (17)

where infinite symbol is used to define a very high stiffness value capable of leading to a rigid behavior

regarding corresponding nodal translations or rotations.

Figure 3. Spring element representation of interatomic interactions

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.

Secondly, spring elements, called hereafter NBN, are utilized for the simulation of bending moment

θ

M

variation (Equation 14) due to the bond angle torsion

θ

Δ

. Three of the specific spring elements, denoted by

node pairs oo ′′′ , oo ′′′′′ and oo ′′′′ , are depicted in Figure 3. It should be mentioned that in the actual models these

nodes coincide with each other at the atomic positions B and N. Their displacement stiffness matrix is expressed

in the global coordinate system (

z

y

x

,, ) as:

⎥

⎦

⎤

⎢

⎣

⎡

−

−

=NBNNBN

NBNNBN

NBN

xyzxyz

xyzxyz

xyz kk

kk

K (18)

where:

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

∞

∞

∞

∞

∞

=

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

=

θ

k

k

k

k

k

k

k

rotz

roty

rotx

z

y

x

xyz

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

00000

NBN

NBN

NBN

NBN

NBN

NBN

NBN

k (19)

To treat the inherent non-linearity of the problem, arisen by the highly non-linear stiffness term

22BN )d(/d rUk rx Δ= in Equation (17), an incremental-iterative scheme is adopted. According to the utilized

algorithm the load is applied in IN increments. The solution at the current load increment, denoted by

INin ,..,2,1=, is achieved by initially using the solution of the previous converged increment ( 1−in ). The

system of equations is constructed by applying the elemental stiffness equations for every BN and NBN element

and then transformed to the global coordinate system. Next, all equations are assembled according to the

requirements of nodal equilibrium and the following system of equations for increment in is obtained:

ininin FUUK

Δ

=

Δ

)( (20)

where )( in

UK , in

UΔ and in

FΔ are the assembled stiffness matrix, incremental displacement vector and

incremental force vector, respectively, for the increment in . Matrix Equation (20) can be solved via standard

numerical techniques by taking into consideration the imposed boundary conditions. The well-known Newton-

Raphson iterative procedure is applied to obtain a converged solution for the current load increment. The

iterations, during which linear-like equations are solved, are continued until the convergence criterion is

satisfied. In the present work, the convergence is checked via the calculation of the Euclidean norm of the

residual vector. At the end of the iterative process the solution of the current load increment is achieved and thus

the displacement ( ininin UUU Δ+= −1) and force ( ininin FFF

Δ

+

=

−1) distribution results are obtained.

5 RESULTS AND DISCUSSION

Three models, of almost square shaped h-BNNRs have been developed according to the proposed spring

based SM method (Figure 4). The geometry of the models considered is defined by their length x

l and y

l along

x

and y-axis, respectively.

Figure 4. Structural mechanics models of the three tested h-BNNRs

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.

In order to obtain the stress-strain curve along direction y

x

p,

=

, one edge of the h-BNNR at 0

=

p has

been fixed while a uniform displacement p

lΔ along p-axis has been applied on its opposite edge at p

lq =. The

stress p

σ

and strain p

ε

, with y

x

p,=, has been computed, respectively, from the following formulas:

tl

f

p

Q

np

q

p

∑

=

=1

σ

(21)

p

p

pl

l

Δ

=

ε

(22)

where p

qf is the reaction along p-axis of node q due to the imposed displacement p

l

Δ

along p-axis on edge

at p

lp =. In addition, Q is the total number of nodes that are fixed and belong on the edge at 0=p. Finally, h-

BN thickness is taken equal to the interlayer distance observed within h-BN flakes [22,23], i.e. nm333.0=t.

Figure 5a and b illustrates the )( xx

ε

σ

and )( yy

ε

σ

variations of the h-BNNRs under consideration.

(a) (b)

Figure 5. Tensile stress-strain behavior along (a) x-axis (armchair direction / zigzag edges loaded) and (b) y-axis

(zigzag direction / armchair edges loaded) for three different h-BNNRs sizes

The non-linearity of stress-strain behavior in both in-plane axes becomes evident. However, a better

mechanical response of h-BN monolayer may be observed when its armchair edges are loaded ( y / armchair

direction). In addition, the smaller the BNNR size the higher tensile strength and failure strain. Overall, h-BN

monolayer presents a brittle mechanical behavior with a superior however tensile strength that may reach

75GPa. Table 1 presents all the mechanical properties, i.e Young’s modulus

E

, Poisson’s ratio

ν

, tensile

strength u

σ

and failure strain f

ε

that have been numerically predicted via the proposed formulation in contrast

with other corresponding theoretical and experimental average estimations from the open literature. To enable

the comparisons, a common h-BN thickness, i.e. nm333.0

=

t, has been used. The good agreement between

present analysis and other approaches becomes apparent.

G.I. Giannopoulos, S.K. Georgantzinos, D.-P.N. Kontoni, and P.A. Kakavas.

Table 1. Computed properties in comparison with other corresponding average results available in the literature

6 CONLUSIONS

The mechanical behavior of a nanomaterial is difficult to be simulated due to the presence of complicated

molecular phenomena in nanoscale. Thus, a numerical efficient model is required to treat such problems. In the

present study, a three dimensional, non-linear, spring based SM method has been proposed for the prediction of

the stress-strain response of h-BN monolayer sheets. All interatomic interactions have been included in the

model while special care has been given to the interatomic force arisen due to the B-N bond length change by

adopting a suitable non-linear Brenner potential energy function. The numerical results have shown that h-BN

monolayer:

• Presents in-plane orthotropy.

• Is stronger when its armchair edges are loaded

• Is stronger for smaller sizes.

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Average length (nm)

Mechanical

property

2.59 5.06 9.76

DFT

[7]

DFT

[8]

DFT

[9]

DFT

[10]

MD

[11]

CM

[14]

MM

[15]

Exp.

[20]

x 901.2 792.0 783.4 1004

E (GPa)

y 846.8 814.9 801.7

774.8 838.4 828.2 801.8

1111

997.2 725.2 811

x 0.255 0.296 0.297

ν

y 0.319 0.317 0.315

0.218 0.220 0.210

0.161 0.387

x 61.33 52.03 49.46 63.06

σu (GPa)

y 75.59 72.76 68.36 93.09

78.08

x 0.113 0.107 0.101 0.169

εf

y 0.155 0.153 0.143

0.210

0.210

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