hygrothermal effects on interfacial stresstransfer characteristics of carbonnanotubes-reinforced

http://jrp.sagepub.com

Composites Journal of Reinforced Plastics and

DOI: 10.1177/0731684406055456 2006; 25; 71 Journal of Reinforced Plastics and Composites

Y. C. Zhang and X. Wang Nanotubes-reinforced Composites System

Hygrothermal Effects on Interfacial Stress Transfer Characteristics of Carbon

http://jrp.sagepub.com/cgi/content/abstract/25/1/71 The online version of this article can be found at:

Published by:

http://www.sagepublications.com

can be found at:Journal of Reinforced Plastics and Composites Additional services and information for

http://jrp.sagepub.com/cgi/alerts Email Alerts:

http://jrp.sagepub.com/subscriptions Subscriptions:

http://www.sagepub.com/journalsReprints.navReprints:

http://www.sagepub.com/journalsPermissions.navPermissions:

http://jrp.sagepub.com/cgi/content/refs/25/1/71SAGE Journals Online and HighWire Press platforms):

(this article cites 20 articles hosted on the Citations

© 2006 SAGE Publications. All rights reserved. Not for commercial use or unauthorized distribution. at Ebsco Electronic Journals Service (EJS) on August 29, 2008 http://jrp.sagepub.comDownloaded from

http://jrp.sagepub.com/cgi/alerts

http://jrp.sagepub.com/subscriptions

http://www.sagepub.com/journalsReprints.nav

http://www.sagepub.com/journalsPermissions.nav

http://jrp.sagepub.com/cgi/content/refs/25/1/71


Hygrothermal Effects on Interfacial StressTransfer Characteristics of Carbon

Nanotubes-reinforced Composites System

Y. C. ZHANG AND X. WANG*

Department of Engineering MechanicsSchool of Naval Architecture, Ocean and Civil Engineering

Shanghai Jiaotong UniversityShanghai 200240, PR China

ABSTRACT: On the basis of the existence of strong bonding between carbon nanotubes (CNTs)and polymer matrix, and considering the axial and radial thermal expansion coefficients of CNTsas nonlinear functions of temperature changes, this article presents an analytical method toinvestigate hygrothermal effects on the interfacial stress transfer characteristics of single-multiwalledCNTs-reinforced composites system under hygrothermal loading by means of thermoelastic theoryand conventional fiber pullout models. According to the known literature, the thermal expansioncoefficient of CNTs is considered as transverse isotropy, and is a nonlinear function of temperaturechanges. The thermal expansion coefficient of polymer matrix is isotropy, and is a linear functionof temperature changes. Numerical examples show that the interfacial shear stress transferbehavior can be described and affected by several parameters such as the temperature changesin CNTs–polymer composite, the moisture concentration changes in polymer matrix, the layernumbers, volume fractions, and chiral vectors of CNTs. From the results obtained it is found thatmismatch of the thermal and moisture expansion coefficients between the CNTs and polymer matrixmay be more important in governing interfacial stress transfer characteristics of CNTs-reinforcedcomposites system.

KEY WORDS: carbon nanotubes–polymer composites, interfacial stress transfer, hygrothermaleffect, thermal expansion coefficient.

INTRODUCTION

SINCE THE DISCOVERY of carbon nanotubes (CNTs) in 1991 [1], much research hasbeen focused on their mechanical properties. In some studies, it was found

that the Young’s modulus, tensile modulus, and fractured strain of CNTs are respec-tively about 1–5TPa, 270–950GPa, and near to 50% [2–4]. These properties suggest thatCNTs may be used as promising reinforcing fibers in high toughness, nanocomposites.

*Author to whom correspondence should be addressed. E-mail: [email protected]

Journal of REINFORCED PLASTICS AND COMPOSITES, Vol. 25, No. 1/2006 71

0731-6844/06/01 0071–18 $10.00/0 DOI: 10.1177/0731684406055456� 2006 SAGE Publications



Recent studies on the mechanical features of CNTs have established that the enhancedstrength, toughness, and mechanical performance of CNTs-reinforced composites systemcritically depends on the interfacial stress characteristics between the reinforcement andthe matrix material [5–7]. If CNTs can give a large interfacial bonding strength with matrixmaterials, a great load transfer ability can be achieved, because a strong bonding allowsshear stress to build up without causing interfacial failure. Thus, the issue on how toqualitatively analyze interfacial stress transfer characteristics between the reinforcementand the matrix material deserve to be discussed and special attention should also be paidas to which parameters may affect the stress transfer characteristics.

It is well known that the interfacial minimal characteristics scale length of CNTs-reinforced composites system is 1–100 nm. At present, approaches that were frequentlyused to analyze and interpret the interfacial stress transfer characteristics of CNTs-reinforced composites system can be separated into nanometer experimental mechanics,molecular dynamic (MD) simulation, and continuum media mechanics based on differentpoints of views. Wagner et al. [9,10] and Qian et al. [11] studied the effect of interfacialadhesive action on strength and toughness of CNTs-reinforced composites system usingTEM image and scanning electron micrographs. However, it is difficult to qualitativelyanalyze the interfacial stress transfer characteristics of CNTs-reinforced compositessystem directly using the experimental method due to the enhanced strength and verysmall diameter of CNTs. MD stimulation is an effective research technique that simulatesaccurately the physical properties of structures at the atomic-scale level. Results froma pullout simulation show that the interfacial shear strength of CNTs–polymer compositesystem is about 160MPa [7], which is much higher than those of most carbon fiberreinforced polymer composite systems. However, the computational problem here is thatthe time steps involved in the MD simulations are limited by the vibration modes of theatoms to be of the order of femto (10�15) seconds. So even after a million time steps, wecan reach only the nanosecond range, in which period most of the thermal, mechanical,physical, or magnetic events have not even started [12,13]. For basically computing theelement of interfacial stress transfer in CNTs-reinforced composites system, because thescale length of CNTs is 1–100 nm (tube diameter), the minimal characteristics scale lengthof the polymeric matrix is micron scale length. It is impractical to perform thesesimulations on a single processor. Yakobson et al. [14] solved the axial compressionbuckling problem of SWNT using the MD simulation and the continuum methods ofmechanics, respectively. Comparing the results from the two different methods, it wasfound that the all-buckling modes achieved by MD simulations can also be predicted byusing continuum methods of mechanics. Xiao and Zhang [8] investigated the effects oftube length and diameter on the distributions of tensile and interfacial shear stress of asingle-walled CNTs in epoxy matrix. Lau [3] has explored the interfacial bondingcharacteristics of CNTs-reinforced composites system by using the conventional fiberpullout model and continuum media mechanics. Results obtained suggest that the changein trends of allowable pullout force and interfacial shear stress are in good agreement withthe corresponding experimental data [6]. Therefore, continuum methods of mechanicscould be effectively attempted to solve interfacial stress transfer characteristics betweenCNTs and polymeric matrix.

Ruoff and Lorents [15] presented that the thermal expansion coefficient of CNTs had animportant effect on the mechanical properties of CNTs. Lau [16] investigated the failuremechanism of CNTs-reinforced composites pretreated in different temperature environ-ments. These previous publications suggested that the thermal expansion coefficient of

72 Y.C. ZHANG AND X. WANG



CNTs was essentially isotropic [15,16]. By contrast with the discussion, Kwon et al. [17],and Maniwa and Fujiwara [18] reported the axial and radial thermal expansioncoefficients of CNTs are, respectively, nonlinear functions of temperature change byutilizing molecular dynamic simulations and CNTs possess the characteristics ofcontraction at low temperature changes (�400K) and expansion only at highertemperatures (400� 1000K) [17]. So far, few works have been reported to study theinterfacial stress transfer characteristics of CNTs-reinforced composites considering thehygrothermal environmental effects. Hence, it is significant for practical CNTs-reinforcedcomposites’ applications to understand the hygrothermal effects on interfacial stresstransfer characteristics of CNTs-reinforced composites system.

Omitting van der Waals force interaction and considering transverse isotropiccharacteristic of thermal expansion coefficients of CNTs, this article presents ananalytical method to investigate hygrothermal effects on interfacial stress transfercharacteristics of CNTs-reinforced composites system by means of thermoelastic theoryand conventional fiber pullout models. Young’s modulus of multi-CNTs in roomtemperature is obtained by means of local density approximation model and elastic shelltheory. In this study, the thermal expansion coefficient of CNTs is considered as anonlinear function of temperature change, the thermal expansion coefficient of polymermatrix is isotropic and a linear function of temperature changes, and the moistureconcentration change in CNTs is neglected. From the results obtained, it is found thatthe mismatch of the thermal and moisture expansion coefficients between the CNTsand polymer matrix may be more important in governing interfacial stress transfercharacteristics of CNTs-reinforced composite systems. The current work can providehelpful information for describing the stress transfer mechanism of CNTs–polymercomposites under hygrothermal loading, which has not been previously discussedelsewhere.

BASIC FORMULATION AND SOLUTION

A mechanics model for the pullout test of CNTs-reinforced composites undertemperature change (�T ) and moisture concentration change (�C ) environments, isshown in Figure 1(a) and (b). Here, CNTs with external radius RN and length L arelocated at the center of a coaxial cylindrical polymer matrix with an outer radius b.Multiwalled carbon nanotubes (MWNTs) could be imagined as a group of co-axialcircular shells packed together with uniform interval spacing d and the effective wallthickness h of single-walled carbon nanotubes (SWNTs).

The effective cross-sectional area of CNTs can be expressed as [3]:

Aeff ¼ 2�h N�0 þXNc¼1

dðc� 1Þ

( )ð1Þ

where, non-relaxed radius of the CNTs, �0, can be expressed as [19]:

�0 ¼a0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ðm2 þ n2 þ 2mnÞ

p2�

ð2Þ

Carbon Nanotubes-reinforced Composites System 73



(m, n) is the chiral vector of CNTs which is shown in Figure 1(c), a0 (1.42 A) is the C–Cbond distance and d (�3.4 A) is the interval spacing of single CNTs, respectively.The external radius of the MWNTs shown in Figure 1(a) is expressed as:

RN ¼ �0 þ dðN� 1Þ ð3Þ

Since the role of the CNTs in advanced composite structures is to take up all stressesthat are transferred from the matrix through the interfacial shear stress [7,8], a well-knownclassical fiber pullout model was used to study the interfacial stress transfer problemof CNTs-reinforced composites system and a three-dimensional, axial-symmetricthermoelastic model was proposed to analysis the stress transfer behaviors between theCNTs and polymer matrix.

Thermal load ∆T

MWNT dzMatrix

(b)

(a)

F

RN

z

Matrix

Layer spacing d

Layer thickness, h

MatrixL

Moisture load ∆C

(c)

r0

Nsz (z)

msz (z)

N Nsz (z) + dsz (z)

m msz (z) + dsz (z)

t (z)

Armchair tubule [5, 5]

Zigzag tubule [5, 0]

Chiral tubule [5, 3]

Figure 1. (a) Schematic diagram of the CNTs-reinforced composites system under hydrothermal environment;(b) fiber pullout model; and (c) hexagonal lattice of carbon atoms with different types of chiral vector.




Axial-symmetric equilibrium equations for CNTs or polymer matrix, are expressed as:

@�zj@zþ@�rzj@rþ�rzjr¼ 0 ð4Þ

@�rj@rþ@�rzj@zþ�rj � �

�j

r¼ 0 ð5Þ

Considering hygrothermal effects and assuming thermal-mechanical properties of CNTsexhibit transverse isotropy in its cross-section. Therefore, the circumferential thermalexpansion coefficient of CNTs is equal to the radial thermal expansion coefficientof CNTs, �rNðT Þ ¼ �

�NðT Þ. Thus, the relationships among stress, strain, and displacement

in CNTs or matrix can be expressed as:

"rj ¼@urj@r¼

1

Ej�rj � �jð�

zj þ �

�j Þ

h iþ �rj ð�TjÞ�Tj þ �j�Cj ð6aÞ

"�j ¼urjr¼

1

Ej��j � �jð�

rj þ �

zj Þ

h iþ ��j ð�TjÞ�Tþ �j�Cj ð6bÞ

"zj ¼@uzj@z¼

1

Ej�zj � �jð�

rj þ �

�j Þ

h iþ �zj ð�TjÞ�Tj þ �j�Cj ð6cÞ

rzj �@uzj@r¼

2ð1þ �jÞ

Ej�rzj ð6dÞ

In the above formula, the subscript, j, may be ‘N’, which represents the equationsfor CNTs or may be ‘m’, which represents the equations for polymeric matrix.Superscripts, r, �, and z, denote the components of stresses and strains in the cylindricalcoordinate axes, respectively. Ej, �j, �

zNðT Þ, �

rNðT Þ and ��NðT Þ are Young’s modulus,

moisture expansion coefficients, the axial, radial, and circumferential thermal expansioncoefficients of CNTs, respectively. The thermal load in CNTs is equal to the thermal loadin polymer matrix, where �TN ¼ �Tm ¼ �T. The moisture concentration changes inCNTs can be neglected where �CN ¼ 0, and the moisture concentration changein polymer matrix expressed as �Cm ¼ �C.

Based on experimental investigations by Kwon et al. and Maniwa and Fujiwara [17,18],from Figure 2 the relationships between axial and radial thermal expansion coefficients(�zN½K

�1�, �rN½K�1�) of CNTs and temperature change are expressed as:

�rNð�T Þ ¼ 1:15� 10�9�T� 6:12� 10�11�T2 þ 1:25� 10�13�T3 � 6:43� 10�17�T 4 ð7aÞ

�zNð�T Þ ¼ 5:38� 10�8�Tþ 5:71� 10�11�T2 þ 9:79� 10�15�T3 � 7:97� 10�18�T 4 ð7bÞ

The Young’s modulus of CNTs, and the Young’s modulus and thermal expansioncoefficient of polymeric matrix are expressed as a linear function of temperature changes,which are respectively [20,21]

EN ¼ E 0Nð1� 0:0005�T Þ, Em ¼ E 0

mð1� 0:0003�T Þ ð8Þ

�m ¼ �0mð1þ 0:001�T Þ ð9Þ

where, E 0N, E

0m, and �

0m are, respectively, Young’s modulus of CNTs, and Young’s

modulus and thermal expansion coefficient of polymer matrix in a reference temperature.




On the basis of local density approximation model [19] and classic elastic shell theory,the Young’s modulus of multiwall CNTs in reference temperature, E 0

N can be expressedas [19]:

E 0N ¼

N

N� 1þ RE 0SNTR ð10Þ

0 200 400 600 800 1000

−1.2

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

0.2

∆T [k]

a z [K−1]×10−5N

0 200 400 600 800 1000

−0.35

−0.30

−0.25

−0.20

−0.15

−0.10

−0.05

0.00

0.05

∆T [k]

Na r [K−1]×10−5

(a)

(b)

Figure 2. (a) The axial thermal expansion coefficient �zN½K�1� of CNTs dependent on temperature change and

(b) the radial thermal expansion coefficient �rN½K�1� of CNTs dependent on temperature change.




where, N, E 0SNT are, respectively, layer numbers of CNTs and Young’s modulus

of single-CNTs in a reference temperature. R(¼ h/d ) denotes the thickness to spacingratio of CNTs. Obviously, E 0

N ¼ E 0SNT for N¼ 1.

From Figure 1(a) it is seen that the embedding length between CNTs and polymericmatrix is L, and the CNTs is undertaking applied pullout force F. Assuming there areno debonding and sliding between the CNTs and polymer matrix, the pullout stress is

�pullout ¼F

Aeffð11Þ

Owing to the existence of crack opening, the axial applied stress on the other end(z ¼ L) equals zero. Thus, the two end stress boundary conditions of this CNTs-reinforcedcomposite model are expressed as:

�zNð0Þ ¼ �a, �zNðLÞ ¼ 0, �zmð0Þ ¼ 0, �zmðLÞ ¼ �a ð12Þ

where, ¼ R2N=ðb

2 � R2NÞ expresses the transverse area ratio between the CNTs and

polymer matrix.The equilibrium equations between the axial stresses of CNTs or polymer matrix and

the interfacial shear stress are expressed as, respectively

d�zmdz¼

2

RN�RN,

d�zNdz¼ �

2

RN�RN ð13a, bÞ

where, axial stresses, �zN, �zm are independent of radial distance in the cross-section, and

satisfy

�pullout ¼ �zN þ

�zm

ð14Þ

Based on a previous work of Gao and Mai [22], the radial and circumferential stresses ofCNTs are expressed as:

�rNðzÞ ¼ ��NðzÞ ¼ qðzÞ ð15Þ

From Equations (4) and (13b), the shear stress of CNTs can be written as:

�rzN ¼r

RN�RNðzÞ ð16Þ

Since the moisture change in CNTs can be neglected, from Equations (6b), (6c), and(13b), the displacement expressions of the CNTs only considering the temperature changeeffect is solved as:

urNr¼

1

ENð1� �NÞqðzÞ � �N�

zN

� �þ �rN�T ð17aÞ

@uzN@z¼

1

EN�zN � 2�NqðzÞ� �

þ �zN�T ð17bÞ




For the polymer matrix, the inner and outer boundary conditions of cylindrical shellmodel are

�rmðRN, zÞ ¼ qðzÞ, �rzm ðRN, zÞ ¼ �RNðzÞ ð18aÞ

�rmðb, zÞ ¼ 0, �rzm ðb, zÞ ¼ 0 ð18bÞ

From Equations (4), (13a), (18a), and (18b), the shear stress in polymer matrix can beexpressed as:

�rzm ðr, zÞ ¼ðb2 � r2Þ

RNr�RNðzÞ ðRN � r � bÞ ð19Þ

Substituting Equation (19) into Equations (6b) and (6c), gives

�rmðr, zÞ þ ��mðr, zÞ ¼ 2B1ðzÞ �

2ð1þ �mÞ

�m

RN

d�rzRN

dzb2 ln r�

r2

2

� �ð20Þ

From Equations (6), (18), and (20), we have

@½r2�rmðr, zÞ�

@rþ

4RN

d�RN

dz½4b2rð2� 1 þ 21 ln rÞ � 42r

3� � 2B1ðzÞ ¼ 0 ð21Þ

where 1 ¼ 2ð1þ �mÞ=�m, 2 ¼ ð1þ 2�mÞ=�m.Integrating Equation (21) for r, yields

�rmðr, zÞ ¼ �

4RN

d�RN

dz½b2ð2� 1 þ 21 ln rÞ � 2r

2� þ B1ðzÞ þB2ðzÞ

r2ð22Þ

where B1(z) and B2(z) are functions of z, and can be determined by the boundaryconditions in Equations (18a), (18b), which are

B1ðzÞ ¼ � qðzÞ �

4RN

d�RNdz

21b2 ln

b

RN

� �þ 2ðR

2N � b2Þ

� ��

þ

4RN

d�RNdz

b2ð2� 1 þ 21 ln b� 2Þ ð23Þ

B2ðzÞ ¼ b2 qðzÞ �

4RN

d�RNdz

21b2 ln

b

RN

� �þ 2ðR

2N � b2Þ

� �� ð24Þ

Utilizing Equations (20), and (22)–(24), �rmðr, zÞ, ��mðr, zÞ can be easily obtained.




From Equations (6a) and (6b), the displacement components in polymer matrix areexpressed as:

urmr¼

Em�m 1�

b2

r2

� �� 1�

b2

r2

� �qðzÞ �

�m�zmðzÞ

�

þ

4RN

d�RN

dz21b

2ð1� �mÞ lnb

r

� �þ ln

b

RN

� �� þ 21b

2ð1þ �mÞb2 lnðb=RNÞ

r2

�

� 2 ðb2 þ r2Þ 1þ

R2N

r2

� �� mðb

2 � r2Þ 1�R2

N

r2

� ��

þ 4b2 � 21ðb2 � r2Þ

�þ �m�Tþ �m�C ð25Þ

@urm@r¼

1

Em�zmðzÞ þ 2�mqðzÞ � 2�m

4RN

d�RN

dz�21b

2 lnb

r

� �þ ln

b

RN

� ��

þ2ðR2N þ b2Þ � 2b2 þ 1ðb

2 � r2Þ

�þ �m�Tþ �m�C ð26Þ

In the fully bonded region, the continuous conditions of axial and radial deformationsat the interface are given by

urmðRN, zÞ ¼ urNðRN, zÞ, uzmðRN, zÞ ¼ uzNðRN, zÞ ð27Þ

A combination of Equations (18a), (18b), and (24)–(27) leads to a differential equation forthe axial stress of CNTs

d2�zNðzÞ

dz2� A1�

zNðzÞ ¼ A2�pullout þ A3�

zT þ A4�

rT þ A5�C ð28Þ

where �zT ¼ ð�zN � �mÞEm�T, �rT ¼ ð�

rN � �mÞEm�T, �C ¼ Em�m�C, and A1, A2, A3, A4,

and A5 are, respectively, expressed in Appendix.Utilizing Equation (16a) and solving the differential equation (28), the axial stress of

CNTs is expressed as:

�zNðzÞ ¼ !1 sinhð�zÞ þ !2 coshð�zÞ �A2

A1�pullout �

A3

A1�zT �

A4

A1�rT �

A5

A1�C ð29Þ

where

� ¼ffiffiffiffiffiffiA1

pð30aÞ

!1 ¼

(A2

A1�pullout þ

A3

A1�zT þ

A4

A1�rT þ

A5

A1�C

� �

� 1þA2

A1�pullout

� �þA3

A1�zT þ

A4

A1�rT þ

A5

A1�C

� �coshð�LÞ

)

sinhð�LÞ

ð30bÞ




!2 ¼ 1þA2

A1�pullout

� �þA3

A1�zT þ

A4

A1�rT þ

A5

A1�C ð30cÞ

Substituting Equation (29) into Equation (13b), the interfacial shear stress of CNTs andmatrix is given by

�RNðzÞ ¼��RN

2½!1 coshð�zÞ þ !2 sinhð�zÞ� ð31Þ

It is well known that the maximum interfacial shear stress should be located at thestarting point of pullout end where z¼ 0. Therefore, substitution of �RNð0Þ ¼ �max intoEquation (13b), the allowable pullout force, Fmax ¼ Aeff�pullout, which is the maximumapplied force to maintain contact of the CNTs with the polymer matrix, is expressed as(Appendix)

Fmax ¼

Aeff 2�max sinhð�LÞ þA3

A1�zT þ

A4

A1�rT þ

A5

A1�C

� �1� coshð�LÞ½ �RN�

�

1þA2

A1

� �coshð�LÞ �

A2

A1

� �RN�

ð32Þ

NUMERICAL CALCULATIONS AND DISCUSSIONS

The interfacial shear stress in CNTs–polymer matrix composite system is a criticalparameter controlling the efficiency of stress transfer and some of the importantmechanical properties of composite such as effective elastic moduli, tensile strength,and fracture toughness. Hence, numerical calculations are given for a hypotheticalCNT–epoxy composite system to illustrate the interfacial shear stress transfer mechanismunder hygrothermal loading. However, the analysis is equally applicable to other types ofCNTs–composites system. The material properties and geometrical characteristics of theCNT, matrix, and interface are shown as [3,20,21]:

h ¼ 0:75 A, E 0SNT ¼ 4:7TPa ðhE 0

SNT ¼ 360 J=m2Þ, d ¼ 3:4 A, b ¼ 2� 103 nm,

�m ¼ 0:48, �N ¼ 0:34, E 0m ¼ 3:3GPa, a0 ¼ 1:42 A, L ¼ 1:0� 104 nm,

F ¼ 10 nN, �0m ¼ 45� 10�6=8C, �m ¼ 2:68� 10�3=wt%:

Thermal Effect

Figure 3 shows plots of the interfacial shear stress against the embedding length ofSWNT–polymer for different temperature changes, where chiral vector (m, n)¼ (5, 0). It isclear that there is good agreement between present predictions and Lau’s results [3]without considering hygrothermal effects. It is seen that the interfacial maximum shearstress between the CNTs and the matrix (z/L¼ 0) gradually decreases as temperaturechange �T increases. It is also found that the interfacial shear stress in a higherenvironmental temperature change is lower than that in lower environmental temperature




change when embedded length of SWNT–polymer is shorter. The reasons may beattributed to the fact that the volume contraction ratio of CNT is larger than the volumeexpansion ratio of epoxy matrix because CNTs are more sensitive to change oftemperature subject to the same magnitudes of temperature change. Undoubtedly, thechange of volume ratio weakens interactions between CNTs and matrix such as adhesionand interlocking. According to the trend of the curves shown in Figure 3, the effect oftemperature changes on interfacial shear stresses decreases gradually, as the interfacelength increases. The maximum interfacial shear stress is mainly affected by mismatchof thermal expansion coefficient and the elastic modulus between the CNTs and thepolymer matrix.

Figure 4 shows the interfacial shear stress against the embedded length of MWNTpolymer for different thermal environments. Compared with Figure 3, it is seen that thetransfer length of interfacial shear stress under the same thermal environmentssignificantly increases and the magnitude of the maximum interface shear stress decreases,as the number of CNTs in the layers increases. From Figure 4, it is also found that thethermal loading has significant effects on the transfer length of interfacial shear stressand the magnitude of the maximum interface shear stress for MWNT polymer composites.These interfacial maximum shear stress and stress transfer length have serious implicationson the integrity of the CNT-reinforced composites system interface. The interfacialmaximum shear stress at the ends of the interface exposes the possibility of failure.The interfacial shear stress concentration near the ends could initiate CNTs debondingif it exceeds the interfacial shear strength. So thermal effects which result in reductionof the interfacial maximum shear stress in MWNT–polymer composites, avoid stressconcentration phenomena and help structural stability.

Figure 5 shows the maximum interfacial shear stress against the thermal loading fordifferent layer numbers of MWNTs. It is found that maximum interfacial shear stresses of

0 50 100 150 200

0

50

100

150

200

nm

(MPa)tRN (z)

2

1

Lau Kin-Tak [3] (∆T=0K, ∆C=0%, N=1)

1: ∆T=30K, ∆C=0%, N=1

2: ∆T=60K, ∆C=0%, N=1

Figure 3. Plots of the interfacial shear stress against the embedding length of SWNT for different thermalloads, where chiral vector (m, n)¼ (5, 0).




SWNT and MWNT decrease with increase of thermal loading. Besides, interactiondirection of the maximum interfacial shear stress for MWNTs–polymer compositeschanges when temperature changes exceed a critical value (critical temperaturechange value for five layer CNTs–polymer composites is 75K and that for ten layer-CNTs–polymer composites is 25K).

(MPa)tRN (z)

0 50 100 150 200 250 300

0

5

10

15

20

25

30

35

nm

3

2

1

1:∆T= 0K, ∆C=0%, N=5

2:∆T=30K, ∆C=0%, N=5

3:∆T=60K, ∆C=0%, N=5

Figure 4. Plots of the interfacial shear stress against the embedding length of MWNT for different thermalloads, where chiral vector (m, n)¼ (5, 0).

0 50 100 150 200 250 300

−50

0

50

100

150

200

∆T [k]3

2

1

1:∆C=0%, N=1

2:∆C=0%, N=5

3:∆C=0%, N=10

(MPa)tRN (0)

Figure 5. Plots of the interfacial maximum shear stress against the thermal load for different layer numbers ofCNTs, where chiral vector (m, n)¼ (5, 0).




Figure 6 shows the distribution of interfacial shear stresses of SWNT with differentchiralities along thermal loading. It can be seen in Figure 6 that the armchair (5, 5) andchiral (5, 3) CNTs–polymer composites appear in the amplitudes of lower interfacial shearstress than that in the interface of the zigzag (5, 0) CNTs–polymer composites due to theirlarger cross-sectional area when the three different types of CNT–polymer composites aresubjected to the same magnitude of temperature change.

The maximum interfacial shear stress and the volume fraction of CNTs compositesagainst different layer numbers of MWNTs for different thermal loads are shown inFigure 7, where chiral vector (m, n)¼ (5, 0). The volume fraction of CNTs increases aslayer numbers of the CNTs and the maximum interfacial shear stress decreases as thevolume fraction of CNTs increases.

The distribution of maximum pullout force, Fmax, for the zigzag (5, 0) CNT–polymerwith different layer numbers, under various thermal loads is shown in Figure 8. It is seenthat the allowable pullout force of CNTs increases with an increasing number of walllayers of CNTs and the magnitudes of temperature change. This may be due to theestablishment of hybridized bonding between CNTs and polymeric matrix because activityof carbon atom increases when environmental temperature rises.

Moisture Effect

Figure 9 shows the interfacial shear stress against the embedded length of MWNT fordifferent moisture concentration changes in polymer matrix. It is clear that the moistureconcentration changes in polymer matrix effect on the change in trend of interfacial shearstress are similar to the thermal load effect.

0 50 100 150 200 250 300

0

50

100

150

200

∆T [k]

3

2

1

1:∆C=0%, N=1, Zigzag (m,n)=(5,0)

2:∆C=0%, N=1, ChiralNT (m,n)=(5,3)

3:∆C=0%, N=1, Armchair (m,n)=(5,5)

(MPa)tRN (0)

Figure 6. Plots of the interfacial maximum shear stress of single-wall carbon nanotube–polymer compositeagainst the thermal load for different chiral vectors.




The relationship between the interfacial maximum shear stress and the change ofmoisture concentration for different chiral vectors of MWNT with 10 layers are shownin Figure 10. It is found that the interfacial maximum shear stress decreases linearly withthe increase of moisture concentration change in polymer matrix because the coefficient ofmoisture expansion is independent of the moisture concentration change.

5 10 15 20 25 300

50

100

150

200

250

0

50

100

150

200

250

VNT% (×10−7)

N

Vol fraction

3

2

1

1:N=1, ∆T=0K, ∆C=0%

2:N=1, ∆T=30K, ∆C=0%

3:N=1, ∆T=60K, ∆C=0%

(MPa)tRN (0)

Figure 7. Plots of the interfacial maximum shear stress and volume fraction of CNTs against the layer numberof CNTs under different thermal loads, where chiral vector (m, n)¼ (5, 0).

5 10 15 200

10

20

30

40

50

60

70

80

21

N

Lau kin-Tak [3], ∆T=0K, ∆C=0%

1:∆T=30K, ∆C=0%

2:∆T=60K, ∆C=0%

F (nN)max

Figure 8. Plots of the allowable pullout force against the layer number of CNTs under different thermal loads,where chiral vector (m, n)¼ (5, 0).




Plots of the interfacial maximum shear stress against the moisture concentration changefor different layer numbers of MWNTs are shown in Figure 11. The effect of moistureconcentration change in polymer matrix on the interfacial maximum shear stress in CNTs–polymer composite is linear, and is equal for different MWNTs–polymer compositesbecause the moisture concentration changes appear only in polymer matrix.

0 5 10 15 20

0

2

4

6

8

10

∆C%

3

2

1

1:∆T=0%, N=10, Zigzag (m,n)=(5,0)2:∆T=0%, N=10, ChiralNT (m,n)=(5,3)3:∆T=0%, N=10, Armchair (m,n)=(5,5)

(MPa)tRN (0)

Figure 10. Plots of the interfacial maximum shear stress against the moisture concentration change fordifferent chiral vectors, where layer numbers of CNTs, N¼10.

0 100 200 300 400 500

0

2

4

6

8

10

nm

3

2

1

1:N=10, ∆T=0K, ∆C=0%2:N=10, ∆T=0K, ∆C=3%3:N=10, ∆T=0K, ∆C=6%

(MPa)tRN (0)

Figure 9. Plots of the interfacial shear stress against the embedding length of MWNT under different moistureconcentration change, where chiral vector (m, n)¼ (5, 0).




CONCLUSIONS

In this article, interfacial stress transfer characteristics of CNTs–polymer compositesunder hygrothermal environments have been presented and discussed. According toresults obtained, mismatch in thermal and moisture expansion coefficients and Young’smoduli between the CNTs and polymer matrix may be more important in governinginterfacial characteristics of CNTs–polymer composites subjected to hygrothermalloading. It is noted that the effects of thermal load and moisture concentration changeon interfacial stress transfer characteristics of CNTs–polymer composites are similar.Example calculations and discussions show that temperature changes or moistureconcentration changes will cause some significant phenomena as follows:

(1) Increasing of temperature change or moisture concentration change results in thedecrease of the interfacial maximum shear stress at the ends of the interface, whichavoids effectively stress concentration phenomena at the interface of CNTs–polymercomposites, and helps the structural stability of CNTs–polymer composites.

(2) Thermal effect has a larger effect on interfacial shear stress characteristics ofMWNTs–polymer composites. It is interesting that interaction direction of themaximum interfacial shear stress for MWNTs–polymer composites changes when theamplitude of temperature changes exceeds a critical value.

(3) It is seen that the amplitude of the maximum shear stress in the armchair (5, 5) CNTs–polymer composite is lower than those of maximum shear stresses in the chiral (5, 3)and zigzag (5, 0) CNTs–polymer composites under the same hygrothermal loading.From a practical applications point of view, the armchair (5, 5) CNTs is moreappropriate to be used for the reinforced fiber of CNTs–polymer composites.

0 5 10 15 20

−2

0

2

4

6

8

10

∆C%

3

2

1

1:∆T=0K, N=10

2:∆T=0K, N=15

3:∆T=0K, N=20

(MPa)tRN (0)

Figure 11. Plots of the interfacial maximum shear stress against the moisture concentration change fordifferent layer numbers of CNTs, where chiral vector (m, n)¼ (5, 0).




In the present study, some results are similar to the experimental data [6] relative topullout and fragmentation of CNT–polymer composite system without consideringhygrothermal effects. This work can provide helpful information to describe the stresstransfer mechanism of CNTs–polymer composites under hygrothermal loading, which hasnot been previously discussed elsewhere.

NOMENCLATURE

�0¼ non-relaxed radius of CNTs (m)

m, n¼ chiral vectors of CNTs

�N¼ thermal expansion coefficient of CNTs (/�K)

�m¼ thermal expansion coefficient of polymer matrix (/�K)

�m¼moisture expansion coefficient of polymer matrix (/wt%)

F¼ pullout force (N)

RN¼ external radius of CNT (m)

b¼ external radius of matrix (m)

L¼ interface length between CNTs and polymer matrix (m)

d¼ interval spacing of MWNT (m)

h¼ effective wall thickness of CNTs (m)

N¼ layer numbers of MWNT

VNT¼ volume fraction of CNTs

¼ transverse area ratio between the CNTs and matrix

ESNT¼ elastic modulus of SWNT (N/m2)

EN¼ elastic modulus of MWNT (N/m2)

�N ¼Poisson’s ratio of CNTs

�m ¼Poisson’s ratio of matrix

Fmax¼ allowable pullout force (N)

�T, �C¼ temperature changes (�K) and moisture concentration changes (%)

ACKNOWLEDGMENT

The authors thank the National Science Foundation of China (10272075) for thefinancial support for this project.

APPENDIX

RN ¼ �0 þ dðN� 1Þ ðA-1Þ

A1 ¼�ð1� 2k�NÞ þ ð1� 2k�mÞ

U2 � 2kU1, A2 ¼

�ð1� 2k�mÞ

U2 � 2kU1ðA-2Þ

A3 ¼1

U2 � 2kU1, A4 ¼

�2k

U2 � 2kU1, A5 ¼

2k� 1

U2 � 2kU1ðA-3Þ




� ¼ �ð1� �NÞ þ 1þ 2 þ �m, k ¼��N þ �m

�ðA-4Þ

� ¼Em

EN, 1 ¼

2ð1þ �mÞ

�m, 2 ¼

ð1þ 2�mÞ

�mðA-5Þ

U1 ¼

841b

2 lnb

RN

� �ð1þ Þ � 22ðb

2 þ R2NÞ þ 4b2 � 21ðb

2 � R2NÞ

� ðA-6Þ

U2 ¼�m4

21b2 ln

b

R

� �ð1þ Þ � 2ðb

2 þ R2NÞ þ 2b2 � 1ðb

2 � R2NÞ

� ðA-7Þ

REFERENCES

1. Iijima, S. (1991). Helical Microtubules of Graphitic Carbon, Nature (London), 354: 56–58.

2. Lu, J. P. (1997). Elastic Properties of Single and Multi-layered Nanotubes, Journal of Physics and ChemicalSolids, 58(11): 1649–1652.

3. Lau, K. T. (2003). Interfacial Bonding Characteristics of Nanotubes/Polymer Composites, Chemical PhysicsLetters, 370: 399–405.

4. Yu, M. F., Lourle, O., Dyer, M. J., Moloni, K., Kelly, T. F. and Ruoff, R. S. (2000). Strength and BreakingMechanics of Multi-walled Carbon Nanotubes under Tensile Load, Science, 287: 637–640.

5. Bower, C., Rosen, R. and Jin, L. (1999). Deformation for Carbon Nanotubes in Nanotubes-polymerComposites, Applied Physics Letters, 74(22): 3317–3329.

6. Cooper, C. A. (2000). Detachment of Nanotubes from a Polymer Matrix, Applied Physics Letters, 281(20):3873–3875.

7. Liao, K. and Seam, L. I. (2001). Interfacial Characteristics of a Carbon Nanotubes-polystyrene CompositeSystem, Applied Physics Letters, 79(25): 4225–4227.

8. Xiao, K. Q. and Zhang, L. C. (2004). The Stress Transfer Efficiency of a Single-walled Carbon Nanotubes inEpoxy Matrix, Journal of Materials Science, 39(14): 4481–4486.

9. Wagner, H. D., Lourie, O., Feldman, Y. and Tenne, R. (1998). Stress-induced Fragmentation of MultiwallCarbon Nanotubes in a Polymer Matrix, Applied Physics Letters, 72(2): 188–190.

10. Lourie, O., Cox, D. M. and Wagner, H. D. (1998). Buckling and Collapse of Embedded Carbon Nanotubes,Physical Review Letters, 81(8): 1638–1641.

11. Qian, D. and Dicdkey, E. C. (2000). Load Transfer and Deformation Mechanisms in CarbonNanotube-polystyrene Composites, Applied Physics Letters, 76(20): 2868–2870.

12. Wang, X. and Yang, H. K. (2005). Axially Critical Load of Multiwall Carbon Nanotubes Under ThermalEnvironment, Journal of Thermal Stresses, 28(2): 185–196.

13. Lau, Kin-Tak, Micrcea, Chipara, Ling, H. Y. and Hui, D. (2004). On the Effective Elastic Moduli of CarbonNanotubes for Nanocomposite Structures, Composites Part B: Engineering, 35(2): 95–101.

14. Yakobson, B. I., Brabec, C. J. and Bernholc, J. (1996). Nanomechanics of Carbon Tubes: Instability BeyondLinear Response, Phys. Rev. Lett., 76: 2511.

15. Ruoff, R. S. and Lorents, D. C. (1995). Mechanical and Thermal Properties of Carbon Nanotubes,Carbon, 33(7): 925–930.

16. Lau, K. T. and Shi, S. Q. (2002). Failure Mechanisms of Carbon Nanotube/Epoxy Composites Pretreatedin Different Temperature Environments, Carbon, 40(15): 2965–2968.

17. Kwon, Young-Kyun, Berber, Savas and Tomanek, David (2004). Thermal Contraction of Carbon Fullerenesand Nanotubes, Physical Review Letters, 92(1): 015901–015904.

18. Maniwa, Yutada and Fujiwara, Ryuji (2001). Thermal Expansion of Single-walled Carbon Nanotubes(SWNT) Bundles: X-ray Diffraction Studies, Physical Review B, 64: 241402-1–241402-3.

19. Tu, Z. C. and Yang, Z. C. (2002). Single Walled and Multi-walled Carbon Nanotubes Viewed as ElasticTubes with the Effective Young’s Moduli Dependent on Layer Number, Phys. Rev., B65: 233407.

20. Mallick, P. K. (1997). Composites Engineering Handbook [M], Marcel Dekker, USA.

21. Shen, H. S. (2001). Hygrothermal Effects on the Postbucking of Shear Deformable Laminated Plates,International Journal of Mechanical Science, 43(5): 1259–1281.

22. Gao, Y.-C. and Mai, Y.-W. (1988). Fracture of Fibre-reinforced Materials, ZAMP, 39: 550–572.




hygrothermal effects on interfacial stresstransfer characteristics of carbonnanotubes-reinforced

Documents