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Computational MechanicsSolids, Fluids, Structures, Fluid-Structure Interactions, Biomechanics,Micromechanics, Multiscale Mechanics,Materials, Constitutive Modeling,Nonlinear Mechanics, Aerodynamics ISSN 0178-7675 Comput MechDOI 10.1007/s00466-013-0908-x

Interface characteristics of carbonnanotube reinforced polymer compositesusing an advanced pull-out model

Khondaker Sakil Ahmed & Ang KokKeng

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Comput MechDOI 10.1007/s00466-013-0908-x

ORIGINAL PAPER

Interface characteristics of carbon nanotube reinforced polymercomposites using an advanced pull-out model

Khondaker Sakil Ahmed · Ang Kok Keng

Received: 24 November 2012 / Accepted: 31 July 2013© Springer-Verlag Berlin Heidelberg 2013

Abstract An advanced pull-out model is presented toobtain the interface characteristics of carbon nanotube (CNT)in polymer composite. Since, a part of the CNT/matrix inter-face near the crack tip is considered to be debonded, theremust present adhesive van der Waals (vdW) interaction whichis generally presented in the form of Lennard-Jones potential.A separate analytical model is also proposed to account nor-mal cohesive stress caused by the vdW interaction along thedebonded CNT/polymer interface. Analytical solutions foraxial and interfacial shear stress components are derived inclosed form. The analytical result shows that contribution ofvdW interaction is very significant and also enhances stresstransfer potential of CNT in polymer composite. Paramet-ric studies are also conducted to obtain the influence of keycomposite factors on bonded and debonded interface. Theresult reveals that the parameter dependency of interfacialstress transfer is significantly higher in the perfectly bondedinterface than that of the debonded interface.

Keywords Polymer composites · Stress transfer ·Debonded interface · Analytical pull-out model ·van der Waals interaction

1 Introduction

Carbon nanotubes have stimulated extensive research activ-ities devoted to smart materials including nanocompositesand their applications in the wide range of engineering, mate-rial science, physics and chemistry because of their excep-tional properties [1–9]. It is well known that aspect ratio

K. S. Ahmed (B) · A. K. KengDepartment of Civil & Environmental Engineering, National Universityof Singapore, Singapore 117576, Singaporee-mail: ceeksa@nus.edu.sg

(AR) and surface to volume ratio (SVR) of nanotubes arehigher in magnitudes than those of traditional composites.Carbon nanotubes are also identified to have great potential asreinforcements of high strength and lightweight smart com-posites [6,10,11]. The notable factors that affect the perfor-mance of carbon nanotube (CNT) reinforced composites arethe mechanical properties of nanotubes, their purity, interac-tions with host, dispersion, orientation of CNTs in the matrixand CNT/polymer interface. Similar to conventional fiberreinforced composites, many research studies also suggestedthat the performance of CNT reinforced composites dependscritically on the interfacial properties of the nanocomposite[12–16].

The main contributing factors for interfacial load trans-fer between CNT and polymer are generally chemical bond-ing, mechanical interlocking (friction), electrostatic forceand non-covalent bonding like van der Waals (vdW) inter-actions [17–19]. The presence of chemical bonding leadsthe CNT/polymer interface to be perfectly bonded in whichother factors may be considered to be insignificant. Previousresearch study also revealed that chemisorption to as littleas 5.0 % carbon atoms of the nanotube increases the interfa-cial shear stress by about 1,000 % [20]. However, chemicalbonding at the CNT/matrix interface may be damaged dueto higher local stress intensity on CNT surface [21]. Chem-ical bonding may also be defected due to excessive load-ing, fatigue or improper manufacturing process that lead theinterface to debonded and the stress transferring ability of thistype of debonded interface is controlled by mechanical inter-locking, thermal residual stress, Poisson’s contraction, vdWinteraction and electrostatic energy. Cohesive energy causedby the vdW interaction contributes three orders higher inmagnitude than the electrostatic energy [22]. The influenceof electrostatic interaction is thus considered to be neglectedin the debonded interface.

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Pull-out tests have been widely used to measure inter-face characteristics of conventional fiber reinforced compos-ites for many years [23–26]. However, it is very difficult toexamine experimental investigation for CNT pull out testfrom polymer matrix due to the challenges involved in grip-ing, manipulation and stress, strain measurement. As pre-liminary study, analytical studies on pull-out test can servethe purpose by considering all practical factors involved atthe different types of interface. Pull-out model on CNT rein-forced composite have been conducted by using moleculardynamic (MD) simulations [16,27]. Though MD simulationis generally accepted to be more accurate, it is however highlytime consuming and costly. Recently, some researchers haveproposed various pull-out models for carbon nanotube rein-forced composites using continuum mechanics approach[28–31]. Some of these models consider either the case offrictionally bonded interface that are valid only for weakCNT/matrix interface or non-bonded interface without defin-ing the source of interface strength. The pull-out models pro-posed by Tan and Kin [28] and Natsuki et al. [32] investi-gated the interfacial shear stress transfer for multi-walledcarbon nanotubes (MWCNT) in composites. Though theyconsidered the inter-wall van der Waals interaction in theMWCNT they ignored the major portion of vdW interac-tion that actually arises between the nanotube and polymer.The study carried out by Daniel Wagner [30], the interfacialshear strength in polymer composites reinforced by SWNThas been estimated using a modified Kelly–Tyson approachwhich however assumes the interfacial shear stresses to beuniform all through the length.

In addition, most of the previous pull-out models aredeveloped based on the stress free end condition. However,CNT embedded in polymer matrix having a perfectly bondedinterface condition should experience equal matrix’s stress atthe embedded end of CNT. A perfectly bonded CNT/matrixinterface using RVE concept was considered by [35]. Theirstudy was however on a shear lag model which assumes theCNT to be fully embedded within the matrix and is aimed atestimating the average stress components of the matrix andCNT. On the other hand, a pull-out test model is designed toestimate the critical pull-out force, the mechanism of stresstransfer from CNT to matrix as well as the factors that influ-ence the composite behavior. No complete research workon CNT pull-out model considering vdW based debondedinterface near the crack tip and perfectly bonded interfacein the remaining embedded length has been reported in theliterature.

This study aims to develop a complete pull-out model inorder to illustrate the actual stress transferring mechanismof CNT reinforced polymer composite. In nanotube pull-outtest, a portion of the interface near the crack tip of the embed-ded fiber is expected to be debonded due to generation largeinterfacial shear stress at the tip. Previous pull-out models

also show that maximum interfacial shear stress generallydevelops near crack tip [28–31,33,34]. This study also con-siders the interface to be debonded near the crack tip and per-fectly bonded interface in the remaining part of the embeddednanotube. In order to compute the cohesive stress caused bythe effect of vdW interaction in the debonded region of inter-face, a separate model has also been proposed using contin-uum analysis. Closed form analytical solutions are derivedthat can be used to determine different stress components ofthe nanocomposite. Contribution of vdW interaction alongthe length of the debonded interface has been investigated.A parametric study has also been conducted to determine theeffect of key composite parameters. In order to serve the pur-pose of applying nanotube as reinforcement in wide range ofpolymer composite, the proposed model is also expected toprovide the solution for any percentage of debonded lengthwhich includes completely debonded interface as well.

2 Proposed analytical pull-out model

A schematic diagram of the proposed pull-out model is shownin Fig. 1. A cracked section showing the propagation of inter-face crack along the length of nanotube in polymer compositeis presented in Fig. 1a. A 3D cylindrical representative vol-ume element (RVE) selected from a cracked section of thecomposite to define the current pull-out model is presentedin Fig. 1b. The debonded CNT/matrix interface is presentedin Fig. 1c to represent the mechanical interlocking and vander Waals interaction. The z and r coordinates are assignedalong the axial and radial directions of the CNT, respectively.The pull-out model comprises of a CNT of radius a partiallyembedded within a cylindrical matrix of radius b. L is thetotal embedded length of CNT with a debond length l fromthe free end. Thus, the remaining portion of embedded CNTof length (L − l) is considered to be perfectly bonded withpolymer. F is the axial normal force applied at the open endof the CNT.

In this study, it is considered that the CNT be replaced byan effective solid fiber having the same length and outer diam-eter. The modulus of the effective fiber E f can be expressedin terms of elastic modulus of the nanotube Et as follows[2,35,36]

E f = Ant

AefEt = 2at − t2

a2 Et (1)

in which t denotes the thickness of the nanotube.The governing equilibrium equations for the pull-out

problem in terms of polar coordinates (r, θ, z) may be writ-ten as

dσrr

dr+ dτr z

dz+ σrr−σθθ

r= 0 (2a)

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Fig. 1 A schematic diagram ofthe pull-out model forimperfectly bonded interface

dσzz

dz+ 1

r

d(rτr z)

dr= 0 (2b)

where, σzz, σrr , σθθ , τr z are the axial, radial, hoop and shearstress components, respectively and εrr , εzz, εθθ , γr z theircorresponding strain components, respectively.

Considering the material is isotropic and obeys Hooke’slaw, the constitutive equations may be written as

εzz = 1

E{σzz − ν (σrr + σθθ )} (3a)

εrr = 1

E{σrr − ν (σzz + σθθ )} (3b)

εθθ = 1

E{σθθ − ν (σrr + σzz)} (3c)

γr z = τr z

G(3d)

where E, G and ν are the Young’s modulus, shear modulusand Poisson’s ratio, respectively.

The strain-displacement relationships may be written as

εrr = du

dr(4a)

εzz = dw

dz(4b)

εθθ = u

r(4c)

γr z = du

dz+ dw

dr(4d)

where w, u the axial and radial displacements, respectively.It is to be noted that Eqs. (2), (3) and (4) are valid for boththe effective solid fiber and matrix.

The mechanical equilibrium equation at any section of thereinforced region in the RVE can be written as

πa2σ =a∫

0

σf

zz (2πr) dr +b∫

a

σmzz (2πr) dr (5)

where σ(= F/πa2) denotes the average stress applied in theeffective fiber at z = 0 and the superscripts f and m refersto the effective fiber and matrix, respectively. The averageaxial stresses of CNT and matrix can be expressed as

σfzz = 2

a2

a∫

0

σf

zzrdr (6a)

σmzz = 2

b2 − a2

b∫

a

σmzzrdr (6b)

The boundary conditions of the pull-out model are

σfzz (0) = σ (7a)

σfzz (L) = σm

zz (L) (7b)

σfzz (l) = σ

fl (7c)

σmrr(a) = σ

frr(a) (7d)

σmzz (b) = 0 (7e)

σmrr (b) = 0 (7f)

τf

r z (a) = τmrz (a) = τi (7g)

τmrz (b) = 0 (7h)

τmrz (b) = 0 (7i)

εfz(a,z) = εm

z(a,z) (7j)

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εfθθ = εm

θθ (7k)

umr(a) = u f

r(a) (7l)

wfz (a) = wm

z (a) (7m)

where σfl is the axial stress of CNT at z = l and τi is the

interfacial shear stress of the CNT.Upon integrating Eq. (2b) with respect to r from 0 to a

and applying the boundary conditions given in Eq. (7g) forthe effective fiber, we obtain

dσfzz

dz= −2

aτi (8)

2.1 Interfacial shear stress in CNT/polymer debondedinterface

Since the stress transferring ability in the debonded inter-face is controlled by mechanical interlocking, thermal mis-match, Poisson’s contraction and van der Waals interactions,the interfacial shear stress at this type of interface may bepresented as

τi = μ (q0 − q1 + q2) (9)

where μ is the coefficient of friction at the nanotube matrixinterface to represent the mechanical interlocking at thedebonded CNT/matrix interface. q0, q1 and q2 are the resid-ual stress due to differential thermal contraction, the radialstress due to Poisson’s contraction and the cohesive stresscaused by vdW interaction acting as pressure normal to theCNT surface at the non-bonded interface, respectively.

2.1.1 Thermal residual stress (q0)

This radial (compressive) stress (q0) is caused by the matrixshrinkage due to differential thermal contraction of the con-stituents upon cooling from the processing temperature. Thisresidual stress acts as a uniform pressure over the entireinterface, which can be estimated directly through experi-mental investigation. This uniform radial stress can also bedetermined by using the experimental value of temperaturechange, thermal contraction, volume fraction, Poisson’s ratioand Young’s modulus of the constituents as follows [37]

q0 =Emγm

2

[1 + v f + (vm−v f )γ f E f

E

] (α f − αm

)T⎧⎨

⎩1 −(

1− EmE f

)(1−v f )

2 + γm(vm−v f )2 −

(EmE f

) [v f + (vm−v f )γ f E f

E

]2

⎫⎬⎭

(10)

where T is the change of temperature after thermal cooling;γm, γ f are the volume fractions of matrix and fiber, respec-tively; α f , αm are the thermal coefficients of expansion of

CNT and matrix, respectively; E the axial modulus of elas-ticity of composite which is approximated by

E = γm Em + γ f E f (11)

2.1.2 Radial stress due to differential Poisson’s ratio (q1)

The radial stress due to Poisson’s contraction is generallyformed at the interface due to the fact the fiber has a smallerPoisson’s ratio than matrix. When an axial tensile stress isapplied at the remote end of the RVE, contraction takes placeat the CNT/polymer interface and hence a compressive radialstress generates at the interface acting normal to the fiber. Theradial compressive stress caused by this contraction, q1 maybe written as given by [38]

q1 = αν f σfzz (a, z) − νmσm

zz (a, z)

α(1 − ν f

)+ 1 + νm + 2γ(12)

where

γ = a2

b2 − a2 (13)

α = Em/E f (14)

2.1.3 van der Waals interaction

One of the major challenges of this study is to account the vander Waal interaction at the debonded CNT/polymer interfaceusing continuum based approach. The major difficulty arisesin calculating each atom interaction because RVE model typ-ically involves billions of atoms or even more. To meet thischallenge, this study assumes that the carbon nanotube isplaced in an infinite polymer as proposed by [18] and esti-mates total cohesive energy caused due to vdW interactionat the CNT/matrix interface. In order to account individualatom’s interaction accurately, this study simplifies the com-putation by considering the number of atoms per unit sur-face area of CNT and number of molecules per unit volumeof polymer. The vdW interactions between two non-bondedatoms is usually represented by the Lennard-Jones potentialV (r) as follows

V (r) = 4 ∈(

δ12

d12 − δ6

d6

)(15)

where d is the distance between non-bonded pair of atoms ormolecules; δ the characteristics bond length between CNTand –CH2– units of the polymer;

√2 δ is the equilibrium

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Fig. 2 An analytical model to determine cohesive stress caused byvdW interaction

distance between the atoms and ε the bond energy at theequilibrium distance.

A 2D CNT/polymer model is proposed to determine thecohesive stress due to vdW interaction as shown in Fig. 2. Inthe model, h is the equilibrium distance of the matrix withrespect to CNT in the infinite polymer and Oi the averageinterface gap beyond equilibrium distance. Therefore, thecohesive energy stored in an area d A of single walled nan-otube embedded in a polymer volume dV due to the van derWaals interaction can be written as

ϕd A = ncd A∫

V (d) n pdV (16)

where n p, nc are the number of polymer molecules per unitvolume and number of atoms per unit area of nanotube,respectively. By substituting Eq. (15) into Eq. (16), and thenintegrating over the entire volume, the total cohesive energycan be simplified as follows

ϕ = 2π

3ncn p ∈ δ3

(2δ9

15 (h + Oi )9 − δ3

(h + Oi )3

)(17)

Now, differentiating Eq. (17) with respect to Oi , the normalcohesive stress due to van der Waals interactions may bepresented as follows [18]

q2 = d ϕ

d Oi

= 2πn pnc ∈ δ2

⎧⎪⎨⎪⎩

1(0.4

16 + Oi

δ

)4 − 0.4(0.4

16 + Oi

δ

)10

⎫⎪⎬⎪⎭(18)

Substituting q1 and q2 from Eq. (12) and Eq. (18) respec-tively, into Eq. (9) leads to

τi = μ

(q0 − αν f σ

fzz (a, z) − νm σm

zz (a, z)

α(1 − ν f

)+ 1 + νm + 2γ

+2πn pnc ∈ δ2

⎧⎪⎨⎪⎩

1(0.4

16 + Oi

δ

)4 − 0.4(0.4

16 + Oi

δ

)10

⎫⎪⎬⎪⎭

⎞⎟⎠

(19)

Substituting Eq. (19) into Eq. (8) and then using Eq. (5), thegoverning differential equation may be obtained as

dσfzz

dz= 2μk

aσ

fzz − 2μk

a

[qo

k− γ νm σ

αv f + γ vm

+2πn pnc ∈ δ2

k

⎧⎪⎨⎪⎩

1(0.4

16 + Oi

δ

)4 − 0.4(0.4

16 + Oi

δ

)10

⎫⎪⎬⎪⎭

⎤⎥⎦

(20)

where

k = αv f + γ vm

α(1 − v f

)+ 1 + vm + 2γ(21)

2.1.4 Solution for debonded interface (0 ≤ z ≤ l)

The solution of the differential equation in the debondedregion (0 ≤ z ≤ l) can be obtained by using the bound-ary condition as stated in Eq. (7). Thus the average axialstress of CNT in debonded region may be obtained as,

σfzz = σ −

⎡⎢⎣q0

k+ 2πn pnc ∈ δ2

k

⎧⎪⎨⎪⎩

1(0.4

16 + Oi

δ

)4

− 0.4(0.4

16 + Oi

δ

)10

⎫⎪⎬⎪⎭− αv f σ

αv f + γ vm

⎤⎥⎦ exp

2μkza (22)

Upon differentiating Eq. (22) with respect to z and substi-tuting its derivative into Eq. (8), we obtain interfacial shearstress as follows,

τi = μk

⎡⎢⎣q0

k+ 2πn pnc ∈ δ2

k

⎧⎪⎨⎪⎩

1(0.4

16 + Oi

δ

)4

− 0.4(0.4

16 + Oi

δ

)10

⎫⎪⎬⎪⎭− αv f σ

αv f + γ vm

⎤⎥⎦ exp

2μkza (23)

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2.2 Formulation for the perfectly bonded interface

In view of perfectly bonded interface that the matrix shearstress (τm

rz) has to be compatible with interfacial shear stress(τi ) and the fact that the outer surface of matrix cylinder isstress free, τm

rz at any radial distance r can be derived byintegrating Eq. (2b) to give

τmrz = a

b2 − a2

(b2 − r2)

rτi (24)

As dudz << dw

dr , we may consider dudz + dw

dr ≡ dwdr . Therefore,

Eq. (4d) may be rewritten as

γr z = dw

dr(25)

In view of Eqs. (25) and (3d) for both the fiber and matrixmay then be rewritten as

τf

r z = E f

1 + ν f

dw f

dr(26a)

τmrz = Em

1 + νm

dwm

dr(26b)

By substituting Eq. (26b) into Eq. (24), we obtain

Em

1 + νm

dwm

dr= γ

a

(b2 − r2)

rτi (27)

By integrating Eq. (27) over a to b, we obtain

τi = a

γ

Em(wm

b − wma

)(1 + νm)

(b2ln b

a − a2/2γ) (28)

Finally, by substituting Eq. (28) into Eq. (27) and integratingover a to b, we obtain

wm (r, z) = wma +

(b2ln r

a − (r2 − a2)/2) (

wmb − wm

a

)(b2ln b

a − a2/2γ)

(29)

In view that the axial stress is the predominant stress compo-nent, we may assume that σrr + σθθ << σzz . Equation (3a)may therefore be rewritten as

σf

zz = E fdw f

dz(30a)

σmzz = Em

dwm

dz(30b)

Equation (30b) in view of Eq. (29) becomes

σmzz (r, z) = σm

zz (a, z)

+{b2ln r

a − (r2 − a2)/2} {

σmzz (b, z) − σm

zz (a, z)}

{b2ln b

a − a2/2γ} (31)

Upon substituting Eq. (31) into Eq. (5) and after rearranging,we obtain

σmzz (b, z) =

σ − γ(σ

fzz − σ

)

β+(

1 − 1

β

)σm

zz (a, z) (32)

where

β = b2(1 + γ )ln ba − (3b2 − a2)/4(

b2ln ba − a2/2γ

) (33)

Now, by substituting Eq. (28) into Eq. (8), we obtain

dσfzz

dz= − 2

γ

Em(wm

b − wma

)(1 + νm)

(b2ln b

a − a2/2γ) (34)

By differentiating Eq. (34) with respect to z and making useof σm

zz (b, z) given in Eq. (32), we obtain the following secondorder differential equation

d2σfzz

dz2 = − 2

γ (1 + νm)

σ−σfzz

β− σm

zz (a,z)γβ(

b2ln ba − a2/2γ

) (35)

Since we consider that this part of the CNT/polymer interfaceis perfectly bonded, i.e. ε

fz (a) = εm

z (a), the stress strainrelationship given in Eq. (3a) reduces to

σmzz (a, z) = ασ

fzz (36)

Now, by substituting σmzz (a, z) from Eq. (36) and β from Eq.

(33) into Eq. (35) and after rearranging, we obtain

d2σfzz

dz2 = − 2

a2γ 2 (1 + νm)

γ σ − (α + γ )σfzz( b

a

)4ln b

a − (3b2 − a2)/4a2γ

(37)

which may be simplified and written as

d2σfzz

dz2 − C1σfzz + C1

γ

α + γσ = 0 (38)

where

C1 = 2

a2γ 2 (1 + νm)

[α + γ( b

a

)4ln b

a − (3b2 − a2)/4a2γ

]

(39)

2.2.1 Solution for perfectly bonded region (l < z ≤ L)

This governing differential equation can be solved by usingboundary conditions given in Eq. (7) and hence axial stressof CNT in perfectly bonded region is obtained as

σfzz =

⎡⎣1 −

⎧⎨⎩(

1−α1+γ

)sinh

(√C1 (z − l)

)+(1 − (α+γ )

γ σσ

fl

)sinh

(√C1 (L − z)

)⎫⎬⎭/

sinh(√

C1 (L − l))⎤⎦ γ σ

α + γ(40)

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Interfacial shear stress of CNT in the perfectly bonded regioncan be obtained by using Eq. (8) and Eq. (40) as

τi =⎧⎨⎩(

1−α1+γ

)cosh

(√C1 (z − l)

)−(1 − (α+γ )

γ σσ

fl

)cosh

(√C1 (L − z)

)⎫⎬⎭

× aγ√

C1

2(α + γ )sinh(√

C1 (L − l))σ (41)

3 Analytical results and discussions

The axial and interfacial shear stress of the CNT can be deter-mined by using Eqs. (40) and (41), respectively for the per-fectly bonded interface. On the other hand, the axial andinterfacial shear stress of CNT for the debonded interfacecan be determined by using Eqs. (21) and (22), respectively.The derived analytical solutions are valid to determine stresscomponents for any percentage of debonded length fromthe crack tip. It is observed from the closed form solutionsthat stress transferring of CNT in perfectly bonded region isrelated to the mechanical and geometric properties of the con-stituents as well as applied pull-out stress. In the debondedinterface, both fiber residual stress and cohesive stress dueto vdW force act as pressure normal to embedded CNT. Thecohesive stress by vdW interaction may fluctuate over thelength due to the fact that this cohesive stress is a function ofinterface displacement, (Oi ). This interface displacement isa function of relative radial displacement of CNT and poly-mer matrix that results due to external loading. Since boththe stress components and cohesive stress due to vdW areinterrelated in the debonded interface, it becomes compli-cated to determine stress components accurately. In order tosolve the problem, an iterative approach is used to accountthe relative radial displacement as well as cohesive stress dueto vdW interaction and hence correct stress components areobtained. Since the model is symmetric with respect to itslength, it is to be noted that the radial displacement is samealong the circumference of the tube at a certain value of z.Available experimental data that has been used to obtain ana-lytical results from the derived solution are given in Table 1.The analytical results are compared with existing pull-outmodel for CNT reinforced composite [31,39]. The contribu-tion of vdW interaction is also determined along the lengthof the debonded interface.

3.1 Comparison with previous study

In order to compare the results with existing models, theCNT/matrix interface of the proposed pull-out model is con-sidered to be completely debonded, i.e. the case l = L isconsidered. Figure 3a shows the average axial stress of CNTalong the embedded length of the CNT. It can be seen from

Table 1 Values of the different parameters for the pull-out model

Parameter Value

F 125.6 nN

a 2.0 nm

b 12 nm

L 50 nm

Em 25 Gpa

Et 1, 000 GPa

ν f 0.28

νm 0.34

δ 0.3825 nm

μ 0.48

T 250 ◦C

n p 3.1 × 1028 m−3

nc 3.82 × 1019 m−2

Oi 0.25

the figure that the maximum axial stress occurs at z = 0 andthen gradually decreases towards the end. The figure showsthat the current axial distribution is close to the earlier workdone by Natsuki et al. [31] but a clear deviation from thework done by Xiao and Liao [39]. The current study pre-dicts higher axial stress over the full length of the embeddedfiber than other two studies. This may happen because cur-rent study accounts vdW’s interaction in full order in whicheach atoms or molecules are taken into consideration wherethe other two study account vdW interaction linearly as wellas inter-wall interaction only. The deviation between pull-outmodel proposed by Xiao and Liao [39] and other two maybe explained as their double derivative removes the thermalresidual stress. As a result, their solution becomes indepen-dent of thermal residual stress though they considered ther-mal residual effect in their modeling assumption. However,in friction based model, there should be an effect due to ther-mal residual stress unless thermal coefficients of fiber andmatrix are not equal at same temperature.

Figure 3b shows the comparison of interfacial shear stressdistribution along the normalized length (z/L) of the embed-ded CNT with the previous models. As can be seen from thefigure, the shear stress gradually increases towards the endstarting from an initial value at z = 0. The interfacial shearstress distribution also shows that the prediction of the cur-rent model is close to the previous model proposed by Natsukiet al. [31]. It is interesting to note that the interfacial shearstress distribution of all models coincides nearly at a distance60 % of the embedded fiber.

3.2 Contribution of vdW interaction

Before presenting the contribution of the vdW interaction,the cohesive energy produced by vdW force is verified with

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0

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Fig. 3 a Comparison of axial stress distribution of CNT with existingmodels. b Comparison of interfacial shear stress distribution of CNTwith existing models

the molecular mechanics model proposed by Liao and Li[14]. They found that 0.24 eV adhesive energy was producedmainly due to van der Waals force and a minor portion fromelectro static force for 2 nm long and 1.334 nm diameterCNT fully surrounded by 80 molecule polystyrene polymer.Using the same geometric data in Eq. (17) of the current studyshows that the total cohesive energy by vdW interaction is0.25 eV, which is very close to the value obtained by Liaoand Li [14]. The small difference in results may be attributedto the fact that the current model considers the CNT in aninfinite polymer where Lian and Li’s study considers a finitelength of CNT and a limited number of polymer molecules.Note that the vdW interaction between neighbour atoms issix orders larger than that between the non-bonded atomslocated at distant.

Variation of normal cohesive stress along the length of theembedded CNT caused by vdW interaction is presented inFig. 4. In this estimation, the interface is considered to becompletely debonded (i.e. l = L). It can be seen from thefigure that this normal cohesive stress due to vdW interaction

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gradually increases towards the embedded end of the CNT.This may be explained as with the application of load on theopen end of the CNT, both CNT and matrix are displaced inboth axial and radial direction. The relative radial displace-ment of CNT with respect to matrix also changes along thelength of the embedded CNT. Since the relative radial dis-placement that influences the interface displacement is highernear the crack tip, relative smaller normal cohesive stress isobserved at z = 0. The result shows that the minimum andmaximum cohesive stresses are found to be around 146 and165 MPa, respectively, which also ensure that the contribu-tion of vdW interaction is quite significant all through thedebonded interface.

4 Parametric study

Using the analytical solutions of the proposed pull-out model,parametric studies are also conducted for different percentageof debonded interface, CNT/matrix Young’s modulus ratioand CNT/polymer radius ratio to obtain their influences onaxial and interfacial shear stress of CNT.

4.1 Influence of debonded length

Figure 5a, b show the average axial stress of CNT and inter-facial shear stress, respectively along the length for differentpercentage of debonded length. It can be observed from theFig. 5a that the axial stress of CNT decreases towards the endafter starting from a peak value at z = 0. The axial stress dis-tribution also shows that the trends are linear in the debondedregion but a stress saturation zone is found at the middle ofthe embedded length. In addition, the figure shows that withthe increase of debonded length axial stress also increases.This can be illustrated as larger percentage of debond lengthwhich in fact represents longer length of frictionally bonded

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Fig. 5 a Axial stress of CNT along the length for different percentagedebond length. b Interfacial stress along the length of CNT for differentpercentage of debond length

interface ensures less stress transfer than that would happenfor perfectly bonded interface and thus more stress is to becarried by CNT. For an example 0 % debond length meansperfectly bonded interface that insures smallest axial stresswhich indicates maximum stress transfer though interfaceand hence a larger shear stress is also observed in the Fig. 5b.It can be seen from the figure that a jump of shear stressis occurred at the end of debond zone. It is interesting tonote that the shear stress decreases up to 50 % length of theembedded nanotube and then increases towards the end forall three cases.

It is important to note from the Fig. 5a that the stresstransferring of CNT is higher in the case of perfectly bondedinterface than the debonded interface. This can be clarifiedby the comparison of the axial stress distribution for 0 and10 % debonded length. It is clear from the figure that thedifference in axial stress for 0 % (which in fact representsperfectly bonded interface) and 10 % debonded interface isvery significant which ensure that perfectly bonded interfaceis capable to transfer more stress than the debonded inter-face. The result also shows that the difference of axial stress

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Fig. 6 a Axial stress of CNT along the length for different Young’smodulus ratio. b Interfacial shear stress along the length of CNT fordifferent Young’s modulus ratio

between 0 and 10 % debond length are much higher than thatbetween 10 and 20 % debond length. Similar observation isalso found in the interfacial shear stress distribution. How-ever, the deviation is comparatively smaller in the debondregion for 10 and 20 % and very small is in the perfectlybonded zone. Thus, it can be concluded that the stress trans-ferring of CNT significantly reduces due to the presence ofdebonding particularly near the open end of the CNT.

4.2 Influence of CNT/matrix Young’s modulus ratio

Figure 6a, b show the average axial stress and interfacialshear stress along the length of CNT, respectively for dif-ferent matrix/nanotube Young’s modulus ratio. Both of thestress distributions show that the trends are linear up to initial20 % of the debond length but a stress saturation zone is foundat the remaining part of the embedded length. Axial stress dis-tribution shows that as the Young’s modulus ratio increases,axial stress of CNT decreases. This happens because higherEm/E f indicates stronger matrix as well as stronger interfacewhich can transfer more stress to matrix and hence smalleraxial stress results in CNT. Figure 6b shows that interfacial

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Fig. 7 a Axial stress of CNT along the length for different radius ratio.b Interfacial stress along the length of CNT for different radius ratio

shear stress increases in a very mild slope in the debondedregion and a downward jump of shear stress is observed at thepoint of intersection of bonded and debonded interface beforereaching a minimum peak nearly at 60 % of the embeddedlength and then sharply increases towards the end.

It is interesting to note that interfacial shear stressincreases as the Young’s modulus ratio increase in debondedinterface but opposite trends are found in the perfectly bondedinterface. This can be explained as with the increase ofEm/E f , the thermal residual stress which is function ofmatrix Young’s modulus also increases and hence interfa-cial shear stress in the debonded region also increases.

4.3 Influence of radius ratio, b/a (volume fraction)

Figure 7a, b show the average axial stress and interfacialshear stress along the length of CNT, respectively for dif-ferent radius ratio which in fact represents the volume frac-tion of CNT in polymer composites. It can be seen from theFig. 7a that the axial stress decreases linearly in the debondedregion before gradually decreases towards the end. It can beobserved from the Fig. 7b that the shear stress distributionis nearly constant in the debonded interface before resultingan upward jump of stress at the intersection of perfectly and

debonded interface. In the perfectly bonded region, shearstress sharply decreases before reaching a minimum peakand then increases sharply towards the embedded end of theCNT. It is also observed that all the shear stress distribu-tions coincide nearly at the 75 % length of embedded nan-otube. In addition, the figure shows that as the radius ratioincrease shear stress decreases at z = l but opposite behav-ior is observed at the embedded end i.e. large shear stress isfound at z = L . However there is a certain limit of inter-facial shear stress after which crack propagation starts to beenhanced. From this result, it is understood that if the volumefraction is increased, the interface crack will start to propa-gate from the embedded end rather than from the vicinity ofthe debonded zone.

It has been clearly observed from both of the figures thatstress distributions in the debond region are nearly indepen-dent of radius ratio. However, the influence of the volumefraction of the CNT is found to be significant in the perfectlybonded interface. In this region, with the increase of radiusratio axial stress of CNT also increases. This can be explainedby the fact that higher radius ratio indicates smaller volumefraction of CNT that ultimately results comparatively morestress transfer of CNT in the perfectly bonded interface.

5 Conclusion

An analytical pull-out model has been proposed to evaluatethe stress transfer of CNT in polymer matrix by using con-tinuum mechanics approach. Closed form analytical solu-tions are also obtained for the axial and interfacial shearstresses along the length of CNT. The accuracy of the pro-posed analytical model is verified by comparing with theavailable results from existing models in the case of com-pletely debonded interface. Current analytical result shows agood agreement with the recent pull-out model proposed by[31]. The analytical results also shows that vdW interactionof CNT with polymer chains contributes significantly at thedebond region and hence enhances stress carrying potentialof CNT. This study suggests that vdW interaction should betaken into account in investigating debonded interface whichare generally common in CNT reinforce polymer compos-ites. To demonstrate the applications of the newly developedmodel, parametric studies of sample cases are also conducted.The parametric study reveals that stress transfer of CNT issignificantly higher in the perfectly bonded interface thanthat of debonded interface. The results also show that thelength of debonded interface, modulus ratio and relative sizeof RVE (i.e. volume fraction) are controlling factors for nan-otube reinforced composites in the perfectly bonded region.In contrast, the analytical results show that interfacial shearstress transfer is nearly independent on the volume frac-

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tion of nanotube in the debonded interface of the polymercomposite.

This present work is a preliminary attempt to evaluate theinterface characteristics of CNT reinforced composites forovercoming the great difficulty in conducting their experi-mental investigation on a single CNT pull-out test. This studycan also be used as an alternate of molecular mechanics orMDs simulation which are not only time consuming but alsocostly as well as the incapability to execute billions of atoms.Therefore, the proposed pull-out model can play a signif-icant role in designing CNT as reinforcement in polymercomposite.

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