Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

57, 1, pp. 207-219, Warsaw 2019
DOI: 10.15632/jtam-pl.57.1.207

INVESTIGATION OF CRACK RESISTANCE IN EPOXY/BORON NITRIDE
NANOTUBE NANOCOMPOSITES BASED ON MULTI-SCALE METHOD

Hossein Hemmatian

Department of Mechanical Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran

e-mail: hoseinhemmatian@gmail.com

Mohammad Reza Zamani, Jafar Eskandari Jam

Faculty of Mechanical Engineering, Malek-Ashtar University of Technology, Tehran, Iran

Boron nitride nanotubes (BNNTs) possess superior mechanical, thermal and electrical pro-
perties and are also suitable for biocomposites. These properties make them a favorable
reinforcement for nanocomposites. Since experimental studies on nanocomposites are time-
consuming, costly, and require accurate implementation, finite element analysis is used for
nanocomposite modeling. In this work, a representative volume element (RVE) of epo-
xy/BNNTnanocompositesbasedonmulti-scalemodeling is considered.Thebonds ofBNNT
are modeled by 3D beam elements. Also non-linear spring elements are employed to simu-
late the van der Waals bonds between the nanotube and matrix based on the Lennard-
-Jones potential.Young’s and shearmodulus ofBNNTs are in ranges of 1.039-1.041TPaand
0.44-0.52TPa, respectively.Three fracturemodes (opening, shearing, and tearing)havebeen
simulated and stress intensity factors have been determined for a purematrix andnanocom-
posite by J integral.Numerical results indicate that by incorporation ofBNNT in the epoxy
matrix, stress intensity factors of three modes decrease. Also, by increasing the chirality
of BNNT, crack resistance of shearing and tearing modes are enhanced, and stress inten-
sity factor of opening mode reduced. BNNTs bridge the crack surface and prevent crack
propagation.

Keywords: boron nitride nanotube, epoxy, fracturemodes, finite elementmodel, multi-scale
method

1. Introduction

Nanostructures as a new class of materials are prevalently used in the recent years. One of the
most commonly used nanostructure is carbon nanotube (CNT), and one similar structure ne-
wer than CNTs is boron nitride nanotube (BNNT) (Chopra et al., 1995). BNNTs, like CNTs,
have extraordinary mechanical properties (Chopra and Zettl, 1998), high thermal conductivity
(Chang et al., 2005), and good resistance against oxidation at high temperature (Chen et al.,
2004). Despite their similar structures, BNNTs have different properties because BNNTs are
composed of various atoms (Fereidoon et al., 2015).Metal, semiconductor or insulator characte-
ristics of CNTs are highly depending on chirality, diameter, and number of walls, while BNNTs
behave independently as an insulator for low electric fields (Khaleghian and Azarakhshi, 2016;
Molani 2017). BNNTs are also found to be nontoxic to health and environment due to their che-
mical inertness and structural stability. Therefore, BNNT is particularly suitable for biological
applications.
The elastic properties ofBNNTshave been theoretically investigated inmanyworks. Slightly

different results were presented, all of which indicated a very highYoung’smodulus, but slightly
smaller than CNTs. So BNNTs can be widely used as a structural reinforcement of matrix
materials (Zhi et al., 2010).



208 H. Hemmatian et al.

First-principles, tight-binding, density functional and classical molecular mechanics appro-
aches have been performed to characterize properties of BNNTs. Young’s modulus of multi-
-walled boron nitride nanotubes (MWBNNTs)was obtained 1.22 0.24TPausing thermal vibra-
tion amplitude analysis (Chopra and Zettl, 1998). Many researchers employed the tight-binding
method for calculating the axial Young’s moduli of zigzag and armchair BNNTs (Verma et al.,
2007). It is also observed that zigzag nanotubes have a higher Young’s modulus than armchair
ones. In another study, Akdim et al. indicated that Young’s modulus of BNNTs varied in the
range of 0.71∼ 0.83TPa andwas slightly dependent on the tube diameter (Akdim et al., 2003).
Using ab-initio calculations based on the density functional theory (DFT), Young’smodulus

of DWBNNT was calculated and the estimated values for (2,2) and (7,7), (2,2) and (9,9) were
821 and 764GPa, respectively (Fakhrabad and Shahtahmassebi, 2013). Also, Young’s modulus
of SWBNNTswith vacancy and functionalization defects was calculated byGriebel et al. (2009)
using molecular dynamics (MD) simulation. They found that Young’s modulus decreased with
increasing defect concentration.

Young’s modulus of BNNTswas reported to be 1.1-1.3TPa from an experimental test (Bet-
tinger et al., 2002). In another experimental effort, Young’smodulus ofMWBNNTwas obtained
895GPa (Wei et al., 2010). Suryavanshi et al. (2004) applied the electric-field-induced resonance
method and specified Young’s modulus as 0.8TPa.

Polymer nanocomposite combining polymers and nano-filler components have attracted re-
search attention from the academic and industrial communities due to their diverse functio-
nal applications, good processing and relatively low cost (Mohammadimehr and Mahmudian-
-Najafabadi, 2013). It is reported that nano-fillers such as particles and platelets can change
the crack propagation direction and consequently stop this (Rozenberg andTenne, 2008). Crack
deflection as a result of nano-sized reinforcements in a matrix has been reported to have a si-
gnificant role in toughening (Sun et al., 2009). Nano-fillers can stop crack propagation along the
original direction and also result in branching if agglomeration is minimized (Rozenberg and
Tenne, 2008).

Lee et al. (2013) investigated the boron nitride nanoflake (BNNF) modification on epoxy
resin. It was noted that strength of epoxy resin increased while Young’s modulus did not si-
gnificantly change. The highest strength increase was obtained at 0.3wt.% BN content while
the highest toughness increase was achieved with 0.5wt.% BN content. In another work, Ulus
et al. (2014) produced and investigated mechanical properties of boron nitride nanoplatelets
(BNNP)-multiwall carbon nanotubes/epoxy hybrid nanocomposites. Young’s modulus and ten-
sile strength values were obtained via tensile tests. It is seen that tensile strength of epoxy resin
increased from 60MPa to 75MPa (25% increases) at 0.5wt.% BNNP content.

Applications of boron nitride nanotubes/epoxy nanocomposites to adhesive joints and com-
posite laminates were reported by Jakubinek et al. (2016). Nanocomposites containing up to
7wt.% BNNTs were fabricated by planetary mixing. The effects of BNNT loading on viscosity,
tensile properties and fracture toughness were determined. The elastic modulus of nanocompo-
site increased progressively with the BNNT loading up to 5wt.%. While ultimate strain only
decreasedwithBNNTaddition, the fracture toughness also reached amaximumaround 5wt.%.

GhorbanpourArani et al. (2012a) analyzed the electro-thermo-elastic stress of a piezoelectric
polymeric thick-walled cylinder reinforcedbyBNNTs.Theyalso investigated the electro-thermo-
-mechanical axial buckling behavior of a piezoelectric polymeric cylindrical shell reinforcedwith
a double-walled boron-nitride nanotube using the principle of minimum total potential energy
approach in conjunction with the Rayleigh-Ritz method (Ghorbanpour Arani et al., 2012b).
Bending and free vibration of a nonlocal functionally graded nanocomposite Timoshenko beam
model reinforced by SWBNNTwere reported based on amodified coupled stress theory (Davar
andSadri, 2016). Also, the effects ofBNNTs on the elasticmodulus of beta tricalciumphosphate
and hydroxyapatite were analyzed using a RVE model. The predicted elastic moduli of the



Investigation of crack resistance in epoxy/boron nitride nanotube nanocomposites... 209

β-TCP-BNNTsandHA-BNNTs composites showed24.1%and26.3%enhancement, respectively
(Davar andSadri, 2017).Theyalso investigated the effect ofBNNTson the stress-intensity factor
(KI) of a semi-elliptical surface crack in a wide range of matrices using a finite element model.
The results showed that a highermismatch difference between the elasticmodulus of thematrix
and BNNTs resulted in further reduction inKI value (Mortazavi et al., 2013).
Experimental studies on nanocomposites are time consuming, costly, and require accurate

implementation. Therefore, analytical, computational and theoretical approaches are attractive
methods of predictingmechanical properties of composites. Researchers usually employ a small
partof thewhole composite,which is called theunit cell orRVEtoavoid expensiveandenormous
computational calculations (Gojny et al., 2005).

The experimental studies on nanocomposites are time-consuming, costly, and require accu-
rate implementation, so the analytical, computational and theoretical approaches are attractive
methods for nanocomposite simulation and predicting mechanical properties. In this work, a
representative volume element (RVE) of epoxy/BNNT nanocomposites based on multi-scale
modeling is considered.

Mechanical behaviors of BNNTs is studied using a three-dimensional finite element (FE)
model, named as the space frame model. Ansari et al. (2015) used DFT calculations to obtain
exact force constants of BNNT which are employed in determining the element properties.
Fundamental to these approaches, BNNTs are considered as geometric space frame structures
and can be analyzed by classical structural mechanics. In this paper, three-dimensional RVEs
with different chirality of BNNT are simulated and analyzed in three fracture modes. In all
fracture modes, the stress intensity factor of nanocomposites is determined and compared with
the purematrix one.

2. Multi-scale modeling

In fracturemechanics basedon the crack surfacesdisplacement, three crackmodes of the opening
mode (tensilemode), shearingmode (slidingmode), tearingmode (out-of-plane) are considered.
Stress intensity factors of RVEs are determined using the J integral technique withANSYSPa-
rametricDesign Language (APDL) based on finite element analysis. Similar loads andboundary
conditions are applied to both the neat matrix and nanocomposite in the three fracturemodes,
and then their stress intensity factors are determined and compared together.

In the bottom-up analysis method, one firstly obtains the effective material constants using
a low-scale such as the nano, meso (Tserpes et al., 2008) or micro RVE model (Gibson et al.,
2007), then applies it in the high-scale FE simulation in which the material is assumed to be
equivalently homogeneous according to the theory of continuummechanics. In the top-down or
global-local method, one firstly finds the properties in the local region of a macro-scale sample
under practical loads, then applies the resultant loads or displacements to the boundary of a
smaller region as a new input.

2.1. Nanotube

BNNTs atoms are bonded together with covalent bonds forming a hexagonal lattice. The
displacement of individual atoms under an external force is constrained by the bonds.Therefore,
the total deformation of the nanotube is a result of the interactions between the bonds. By
considering the bonds as connecting load-carrying elements, and the atoms as joints of the
connecting elements, BNNTsmay be simulated as space-frame structures (Ansari et al., 2015).
The 3D FE model is developed using the ANSYS commercial FE code. The 3D elastic

BEAM4 element is used for modeling the bonds. The properties of these elements are obtained
bya linkage between the potential energy of bonds (froma chemical point of view) and the strain



210 H. Hemmatian et al.

energy ofmechanical elements (fromamechanical point of view).To represent the covalent bond
between boron and nitrogen atoms, a circular beamof length l, diameter d, Young’smodulusE,
and shear modulus G is considered (Ansari et al., 2015). The required properties of the beam
element are given in Table 1.

Table 1.The properties of beam elements for real BNNT (Ansari et al., 2015)

Diameter d 1.648Å

Cross-sectional area A 2.132Å2

Boron-nitrogen bond 1.45Å

Polar inertia momentum Ixx 0.7250Å
4

Inertia momentum Izz = Iyy = I 0.3625Å
4

Young’s modulusE 4.2155 ·10−8N/Å2
Shear modulusG 4.9437 ·10−9N/Å2

Aroutine code has been created using theANSYSmacro language, for automatic generation
of FEmodels.The thickness ofBNNT is considered as 0.34nmand also, the center of theBNNT
wall is placed at the midsection of the tube thickness. The FE meshes, loading and boundary
conditions of (10,10)BNNTwith length of 80Å are shown inFig. 1. Young’s and shearmodulus
of BNNT are found by tension and torsion loading. They are in ranges of 1.039-1.041TPa and
0.44-0.52TPa, respectively. The experimental and theoretical elastic moduli of BNNT are given
in Table 2. The current results are in good agreement with the simulation and experimental
values.

Fig. 1. FEmeshes of BNNT (10,10) with loading and boundary conditions

2.2. Inter-phase between nanotube and polymer

Thebondingbetween the embeddedBNNTand its surroundingpolymer takes place through
vdW and electrostatic interactions in the absence of chemical functionalization. Since vdW
contributes more significantly by three higher orders of magnitude than electrostatic energy,
the electrostatic interactions can be neglected in comparison with vdW interactions (Gou et
al., 2004). So, only vdW interactions are considered between the BNNT and the matrix. The
vdW forces are most often modeled using the famous Lennard-Jones equation (Battezzatti et
al., 1975)

FV dW =4
ε

r

[

−12
(σ

r

)12
+6
(σ

r

)6]

(2.1)



Investigation of crack resistance in epoxy/boron nitride nanotube nanocomposites... 211

Table 2.Elastic moduli fromBNNT of simulation and experimental works

Elastic moduli [TPa] Reference

Young’s modulus

1.22±0.24 Chopra and Zettl (1998)
1.022-1.112 Fereidoon et al. (2015)
0.862-0.94 Verma et al. (2007)
0.71-0.83 Akdim et al. (2003)
0.895 Fakhrabad and Shahtahmassebi (2013)
0.7-1.2 Gerieble et al. (2009)
0.7-1.2 Bettinger et al. (2002)
1.1-1.3 Wei et al. (2010)
0.764-0.821 Suryavanshi et al. (2004)
1 Chowdhury et al. (2010)

1.039-1.041 Current work

Shear modulus
0.42 Chowdhury et al., 2010
0.44-0.52 Current work

where r is the separation distance between the pair of atoms, ε is the bond energy at the
equilibrium distance, and σ is the van derWaals separation distance. The equilibrium distance
between atoms is

6
√
2σ. By introducing x as the distance from the equilibrium distance, the

Lennard-Jones force is represented in Eq. (2.2)2

x= r− 6
√
2σ

F(X) =−24
ε

σ

[

2
( σ

x+
6
√
2σ

)13
−
( σ

x+
6
√
2σ

)7]

σ=
σn+σm
2

ε=
√
εnεm

(2.2)

where m and n sub-indexes denote the matrix and nanotube, respectively. The Lennard-Jones
potential parameters (ε and σ) of the materials are given in Table 3. BNNT is a synthase
from boron and nitrogen atoms, therefore the Lennard-Jones potential parameters of BNNT
are approximately considered the average of boron and nitrogen parameter values. Also, these
values are represented in Table 3. The Lennard-Jones potential parameters for van der Waals
interaction between theBNNTand epoxy are determined as σ=3.897Å and ε=0.00297nNnm
by replacingL-J parameters ofBNNTand epoxy.Also, the equilibriumdistance betweenmatrix
and nanotube is 0.4374nm.

Table 3. Lennard-Jones potential parameters of the materials

Materials σ [Å] Reference ε Reference

Nitrogen 3.365
Chen et al. (2015)

6.281meV
Chen et al. (2015)

Zhang andWang (2016) Zhang andWang (2016)

Boron 3.453
Chen et al. (2015)

4.16meV
Chen et al. (2015)

Zhang andWang (2016) Zhang andWang (2016)

BNNT 3.409 –
5.2205meV-

–
0.00083642nNnm

Epoxy 4.383 Yang et al. (2014)
1.519kcal/mol-

Gou et al. (2004)
-0.1055nNnm

The vdW interactions between the BNNT and the inner surface nodes of the surrounding
resin are modeled using a 3D non-linear spring element based on the corresponding data of the
force-displacement curve (Hemmatian et al., 2012). COMBIN39 element is used for this purpose



212 H. Hemmatian et al.

and the parameters are adjusted to obtain a translational spring. A macro is written to create
elements between the BNNT and the inner surface of the surrounding resin nodes that their
distance is lower than 0.7nm.

2.3. Matrix

SOLID45 elements are utilized for modeling of the matrix. This element is used for the
3D modeling of solid structures. SOLID45 is defined by eight nodes having three degrees of
freedom at each node: translations in the nodal x, y and z directions. Young’s modulus of the
epoxy matrix is considered as 2.9GPa (Fereidoon et al., 2013). Nanocomposites consisting of
%5 volume fraction of BNNT with length of 80Å are simulated. The FE meshes of RVE used
for crack analysis are shown in Fig. 2. The elements of the crack tip are refined to increase the
accuracy of analysis as represented in Fig. 2.

Fig. 2. The FE meshes of nanocomposite RVE

Stress intensity factors of the neat matrix and nanocomposite are compared for similar
loading and boundary conditions. The loading and boundary conditions of three fracturemodes
including the opening, shearing and tearing of the nanocomposite with 5Vo.% (5,5) BNNT are
shown in Fig. 3. In order to apply the conditions of the opening mode, the middle nodes of
the RVE are fully built-in (zero displacement and rotation conditions), while the nodes of two
ends are subjected to tensile forces. In shearing and tearing modes, the nodes of the back of
the RVE are fully built-in (zero displacement and rotation conditions), while the middle nodes
are constrained in theZ direction. Shearing and tearing forces are applied to the front nodes of
these RVE’s.

Fig. 3. Loadings and boundary conditions of RVE for three modes



Investigation of crack resistance in epoxy/boron nitride nanotube nanocomposites... 213

Fig. 4. Displacement [Å] contours of the opening mode: neat matrix and nanocomposite



214 H. Hemmatian et al.

Fig. 5. Displacement [Å] contours of the shearingmode: neat matrix and nanocomposite



Investigation of crack resistance in epoxy/boron nitride nanotube nanocomposites... 215

Fig. 6. Displacement [Å] contours of the tearingmode: neat matrix and nanocomposite



216 H. Hemmatian et al.

3. Results and discussion

In this study, fracture analysis of an epoxy/BNNT nanocomposite reinforcedwith four chirality
(5,5), (10,10), (15,15) and (20,20) are implemented. It is observed that when nanotubes are
vertical to the crack path, theminimum stress intensity factor and themaximumeffect on crack
resistance are achieved. In this condition, while the crack is in the middle of RVE (bridging
condition), this effect is stronger. Nanotubes with constant length and different chirality have
been used for the bridging condition.

Displacement contours of the neat matrix and nanocomposite in the opening, shearing and
tearing modes are shown in Figs. 4, 5 and 6, respectively. The dimension of displacement is
Angstrom.

The maximum displacement of RVE is decreased by adding BNNT, and this phenomenon
in the opening mode is evident. Diagrams of normalized stress intensity factors (ratio of the
nanocomposite stress intensity factor to that of the neat matrix) of the fracture modes are
plotted against chirality in Fig. 7.

Fig. 7. Normalized stress intensity factors against chirality

The results indicate that with the addition of BNNT to epoxy, the stress intensity factors
of three modes decrease. Also, by increasing the chirality and consequently, BNNT diameter,
the crack resistance of shearing and tearing modes enhances, and the stress intensity factor of
the opening mode reduces. Boron nitride nanotube bridges the crack path and resists against
crack propagation. On the other hand, the bridging arises in all three modes, and the stress
intensity factor decreases. This phenomenon is reported in experimental research about CNT
too (Mirjalili andHubert, 2010). Therefore, adding theBNNT to thematrix improves the crack
resistance, which is considerable in the opening and tearing modes.

In modeling and simulation, the chirality, length, diameter, and weight percentage of the
nanotube are important. But in the experimental method, the weight percentage and range
of length and diameter is considered. Hence, appropriate verification between the results of
experimental and simulation is difficult. Also, based on literature review, the experimental and
simulation results donot dealwith fracture behavior of BNNT/epoxy in the threemodes.Multi-
scale simulations from nano to macro, or reversely, improve the specification of toughening
mechanisms.



Investigation of crack resistance in epoxy/boron nitride nanotube nanocomposites... 217

4. Conclusions

A three-dimensional FEMof BNNTs has been proposed.Nodes are placed at locations of boron
and nitrogen atoms, and bonds are modeled using three-dimensional elastic beam elements by
considering a linkage between molecular and continuum mechanics. The simulation performed
underminimal computational time by requiringminimal computational power. The determined
elasticmoduli of BNNTs are in a good agreementwith the real parameters. A three-dimensional
study of three fracture modes in epoxy/BNNT has been performed based on a multi-scale
method. Van der Waals bonds between the resin and nanotube are simulated by non-linear
spring elements based on the Lennard-Jones potential.
Stress intensity factors of three fracturemodes have been computed by J integral. The effect

of BNNTon the stress intensity factor of nanocomposites and crack propagation is investigated.
The results indicated thatBNNTshavea significant effect onpreventing crackpropagation.Also,
by adding a nanotube, stress intensity factors decreased and, consequently, the crack resistance
increased. The results indicated that the crack resistance improved by increasing the chirality
(radius). Finally, by addingBNNT to thematrix, the improvement ofmatrix fracture properties
is evident.

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Manuscript received April 14, 2018; accepted for print September 10, 2018