European Journal of Mechanics A/Solids...A thermomechanical shear lag analysis of short fuzzy ﬁber reinforced composite containing wavy carbon nanotubes M.C. Ray*, S.I. Kundalwal

lable at ScienceDirect

European Journal of Mechanics A/Solids 44 (2014) 41e60

Contents lists avai

European Journal of Mechanics A/Solids

journal homepage: www.elsevier .com/locate/ejmsol

A thermomechanical shear lag analysis of short fuzzy fiber reinforcedcomposite containing wavy carbon nanotubes

M.C. Ray*, S.I. KundalwalDepartment of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India

a r t i c l e i n f o

Article history:Received 16 April 2013Accepted 3 October 2013Available online 16 October 2013

Keywords:Shear lag modelFuzzy fiber reinforced compositeNanotube waviness

* Corresponding author. Tel.: þ91 3222282984.E-mail address: [email protected] (M.C.

0997-7538/$ e see front matter � 2013 Elsevier Mashttp://dx.doi.org/10.1016/j.euromechsol.2013.10.001

a b s t r a c t

A novel three-phase shear lag model is derived to study the load transfer characteristics of the shortfuzzy fiber reinforced composite (SFFRC) subjected to the thermomechanical loading. The distinctivefeature of the SFFRC is that the short carbon fiber reinforcements coated with radially aligned carbonnanotubes (CNTs) are uniformly interlaced in the polymer matrix. The main novelty of the shear lagmodel derived in this study is that the interactions between the representative volume elements (RVEs)of the SFFRC, the application of the radial and the thermal loads on the RVE, and the radial as well as theaxial deformations of different orthotropic constituent phases of the SFFRC have been taken into account.Particular emphasis has been placed on investigating the effect of waviness of CNTs on the load transfercharacteristics of the SFFRC when the wavy CNTs are coplanar with either of the two mutually orthogonalplanes. In the absence and the presence of the applied radial and thermal loads on the RVE, the shear laganalysis revealed that if the wavy CNTs are coplanar with the axial plane of the carbon fiber such that theamplitudes of the CNTs are parallel to the length of the carbon fiber then the load transfer characteristicsof the SFFRC are significantly improved over those of the composite with and without the straight CNTs.The limiting value of the effective aspect ratio of the carbon fiber is also found for the efficient loadtransfer to the carbon fiber.

� 2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

The research on the synthesis of molecular carbon structure byan arc-discharge method for evaporation of carbon led to the dis-covery of an extremely thin needle-like graphitic multi-walled CNT(Iijima, 1991). Within a couple of years, Iijima and Ichihashi (1993)discovered the synthesis of single-walled CNT. Researchers prob-ably thought that CNTs may be useful as nanoscale fibers fordeveloping novel nanocomposites, and this conjecture motivatedthem to accurately predict the mechanical and the thermal prop-erties of CNTs (Ruoff and Lorents, 1995; Bandow, 1997; Treacy et al.,1996; Krishnan et al., 1998; Maniwa et al., 2001; Saether et al.,2003; Li and Chou, 2003; Natsuki et al., 2004; Shen and Li, 2004;Kwon et al., 2004; Liu et al., 2005; Chen et al., 2009; Liew et al.,2011). The quest for utilizing such remarkable mechanical andthermal properties of CNTs led to the opening of an emerging areaof research on the development of CNT-reinforced composites(Pipes and Hubert, 2003; Odegard et al., 2003; Liu and Chen, 2003;Griebel and Hamaekers, 2004; LopezManchado et al., 2005; Ashrafi

Ray).

son SAS. All rights reserved.

and Hubert, 2006; Kirtania and Chakraborty, 2009; Meguid et al.,2010; Wu and Chou, 2012). These studies reveal that the excep-tional properties of CNTs can be exploited to develop two-phaseCNT-reinforced polymer matrix composites where CNTs areused as fiber reinforcements. For structural applications, themanufacturing of two-phase unidirectional continuous CNT-reinforced composites in large scale has to encounter some chal-lenging difficulties. Typical among these are the agglomeration ofCNTs, the misalignment and the difficulty in manufacturing longCNTs (Thostenson et al., 2001; Wernik and Meguid, 2010; Meguidet al., 2013). Also, the waviness of CNTs is intrinsic to manymanufacturing processes and plays an important role in the me-chanical behavior of CNT-reinforced composites (Fisher et al., 2002;Berhan et al., 2004; Anumandla and Gibson, 2006; Li and Chou,2009; Shady and Gowayed, 2010; Tsai et al., 2011). It has beenexperimentally observed that CNTs are actually curved cylindricaltubes with a relatively high aspect ratio (Berhan et al., 2004; Tsaiet al., 2011; Shaffer and Windle, 1999; Vigolo et al., 2000; Qianet al., 2000; Zhang et al., 2008; Tyson et al., 2011).

It seems that in comparison to the manufacturing of long CNTsand the dispersion of long CNTs in the polymer matrix, directgrowth of short CNTs on the circumferential surfaces of theadvanced fibers for achieving uniform distribution of CNTs

Delta:1_given name

Delta:1_surname

mailto:[email protected]

http://crossmark.crossref.org/dialog/?doi=10.1016/j.euromechsol.2013.10.001&domain=pdf

www.sciencedirect.com/science/journal/09977538

http://www.elsevier.com/locate/ejmsol

http://dx.doi.org/10.1016/j.euromechsol.2013.10.001



Nomenclature

A) NotationsA Amplitude of the CNT wave (m)½~A1� Matrix of the strain concentration factorsa Radius of the carbon fiber (m)b Radius of the short composite fuzzy fiber (m)[Cnc] Elastic coefficient matrix of the unwound polymer

matrix nanocomposite containing straight CNTs (GPa)½CNC� Elastic coefficient matrix of the unwound polymer

matrix nanocomposite containing wavy CNTs (GPa)[Ci] Elastic coefficient matrix of the ieth phase (GPa)Ciij Elastic coefficients of the ieth phase (GPa)

dn Diameter of the CNT (m)En, Ep Young’s moduli of the CNT and the polymer,

respectively (GPa)E1, E2 Axial and transverse Young’s moduli of the CNT-

reinforced composite, respectively (GPa)[I] Fourth order identity matrixL Half length of the RVE of the SFFRC (m)Lf Half length of the short carbon fiber (m)Ln Length of the straight CNT (m)Lnr Running length of the sinusoidally CNT wave (m)n Number of the sinusoidally CNT wavesqo Radial normal stress applied on the RVE of the SFFRC in

the re direction (GPa)R Radius of the RVE of the SFFRC (m)[Sn] Eshelby tensor for the CNT[T] Transformation matrixui, wi Axial and radial displacements at any point in the ieth

phase along the xe and re directions, respectively (m)(VCNT)maxMaximum CNT volume fraction in the SFFRCVi Volume of the ieth phase (m3)vf Volume fraction of the carbon fiber in the SFFRCvm Volume fraction of the polymer in the polymer matrix

nanocompositevn Volume fraction of the CNT in the polymer matrix

nanocomposite/the CNT-reinforced nanocomposite

B) Greek Symbols{a} Thermal expansion coefficient vector of the SFFRC

(K�1){anc} Thermal expansion coefficient vector of the unwound

polymer matrix nanocomposite containing straightCNTs (K�1)

faNCg Thermal expansion coefficient vector of the unwoundpolymer matrix nanocomposite containing wavy CNTs(K�1)

{ai} Thermal expansion coefficient vector of the ieth phase(K�1)

aiij Thermal expansion coefficients of the ieth phase (K�1)a1, a2 Axial and transverse thermal expansion coefficients of

the CNT-reinforced composite, respectively (K�1)DT Temperature deviation from the reference

temperature (K)3ix , 3iq, 3ir Normal strains along the x, q and r directions,

respectively, in the ieth phase3ixr Transverse shear strain in the ieth phase

q Angle between the radial axis (30eaxis) along whichthe wavy CNT is grown and the 3eaxis in the 2e3plane

ln Wavelength of the CNT wave (m)li11 , li22 Axial and transverse thermal stiffness coefficients,

respectively, of the ieth phase (GN/m2K)nn, np Poisson’s ratios of the CNT and the polymer,

respectivelyn12, n23 Axial and transverse Poisson’s ratios of the CNT-

reinforced composite, respectivelys Axial normal stress applied on the RVE of the SFFRC in

the xedirection (GPa){si} Stress vector of the ieth phase (GPa)fsig Average stress vector of the ieth phase (GPa)six, s

ir Normal stresses along the xe and re directions,

respectively, in the ieth phase (GPa)sixr Transverse shear stress in the ieth phase (GPa)six Average axial stress in the ieth phase (GPa)spfx Average axial stress in the imaginary carbon fiber

made of the polymer material (GPa)spmx Average axial stress in the imaginary short composite

fuzzy fiber made of the polymer material (GPa)s* Nondimensional axial stress along the length of the

carbon fibersi Transverse shear stress at the interface between the

carbon fiber and the PMNC (GPa)so Transverse shear stress at the interface between the

PMNC and the polymer matrix (GPa)s�i Nondimensional transverse shear stress at the

interface between the carbon fiber and the PMNCs�o Nondimensional transverse shear stress at the

interface between the PMNC and the polymer matrixf Angle between the CNT axis at any point and the 3 or

30eaxis which is varying over the linear distancebetween the CNT ends

u Wave frequency of the CNT wave (m�1)

C) Superscriptsc Polymer matrix nanocompositef Carbon fiberm Polymer materialn CNTnc Unwound polymer matrix nanocomposite containing

straight CNTsNC Unwound polymer matrix nanocomposite containing

wavy CNTspf Imaginary carbon fiber made of the polymer materialpm Imaginary short composite fuzzy fiber made of the

polymer materialPMNC Polymer matrix nanocomposite

D) Subscriptsn CNTp Polymer

E) AcronymsCNT Carbon NanotubeRVE Representative Volume ElementSFFRC Short Fuzzy Fiber Reinforced Composite

M.C. Ray, S.I. Kundalwal / European Journal of Mechanics A/Solids 44 (2014) 41e6042

M.C. Ray, S.I. Kundalwal / European Journal of Mechanics A/Solids 44 (2014) 41e60 43

throughout the composite is practically more feasible and advan-tageous (Yamamoto et al., 2009) and provides a means to tailor themultifunctional properties of the existing advanced fiber reinforcedcomposites. The presence of radially aligned CNTs on the circum-ferential surface of the fiber may enhance the multifunctionalproperties of the advanced fiber composite. For example, Downsand Baker (1995) reported the initial research work on the growthof carbon nanofibres on the circumferential surfaces of the com-mercial carbon fibers. They modified the circumferential surfacearea of the carbon fiber by coating with carbon nanofibres andfound the significant enhancement in the adhesion properties be-tween the fiber and the surrounding matrix. Pan et al. (1999)demonstrated the growth of aligned CNTs on the surfaces of film-like iron/silica substrates by means of pyrolysis of acetylene.These CNTs have been grown outwards perpendicularly from thesurfaces of the substrates to form aligned CNTs arrays of uniformexternal 20e40 nm diameter. Bower et al. (2000) have grownradially aligned CNTs on the circumferential surface of the hair-thinoptical fiber glass using high-frequency microwave plasmaenhanced chemical vapor deposition process. Thostenson et al.(2002) synthesized CNTs on the circumferential surfaces of thecarbon fibers using chemical vapor deposition technique and foundthat the presence of CNTs at the fiber/matrix interface improves theinterfacial shear strength of the composite. In their work, the sur-faces of the fibers after each step in the synthesization process ofgrowing CNTs were examined with the scanning electron micro-scopy and found that no pitting was occurred on the fiber surfaces.Ci et al. (2005) have grown coating of aligned CNTs of thicknessw10 mmon the circumferential surfaces of the ceramic fibers with agrowth rate of w0.5 mm/min by using simple chemical vapordeposition process. De Riccardis et al. (2006) synthesized a hybridcomposite in which the carbon fiber is coated with uniform distri-bution of CNTs on its circumferential surface. In their study, pseudo-mechanical tests were performed on different samples (with andwithout CNT coated carbon fibers) and test results revealed goodanchorage between CNTs and carbon fiber. Veedu et al. (2006)fabricated a 3-D composite in which multi-walled CNTs are grownnormal to the surface of the micro-fiber fabric cloth layouts. Themeasured fracture toughness for crack initiation, interlaminar shearsliding fracture toughness, in-plane strength, modulus, toughness,transverse thermal conductivity and damping characteristics of this3-D composite exhibit 348%, 54%, 140%, 5%, 424%, 51% and 514%enhancements, respectively, over those of the base composite. Theirtest results also indicated that the presence of CNTs in the trans-verse (i.e., thickness) direction of this 3-D composite reduces theeffective coefficient of thermal expansion of the composite by 62%as compared to that of the base composite. Electrophoresis tech-nique has been utilized for the selective deposition of multi-walledCNTs and single-walled CNTs on the circumferential surfaces of thecarbon fibers by Bekyarova et al. (2007). For the single-walled CNT/carbon fiber reinforced composites they demonstrated w30%enhancement in the interlaminar shear strength and 2efoldimprovement in the out-of-plane electrical conductivity ascompared to that of the composite without CNTs. A three-phasethermoplastic composite consisting of the main reinforcing glassfiber and the CNT-reinforced nanocomposite matrix has beenfabricated by Vlasveld et al. (2007). They reported that this glassfiber composite with the nanocomposite matrix possess 40%increased flexural strength at elevated temperatureswith respect tothat of the composite without CNTs. Sager et al. (2009) synthesizedhigh density multi-walled CNTs on the T650 PAN-based carbon fi-bers. Four types of fibers were studied: unsized fiber, sized fiber,unsized with aligned multi-walled CNTs and unsized with randommulti-walled CNTs. The interfacial shear strength of CNT coatedcarbon fiber in epoxy was investigated using the single-fiber

composite fragmentation test. Results of fragmentation tests indi-cated that commercially sized fibers have the highest interfacialshear strength while unsized fibers had the lowest. Randomly ori-ented multi-walled CNTs and aligned multi-walled CNTs coatedunsized fibers attained 71% and 11% increase in the interfacial shearstrength over the unsized fibers, respectively.

The literature reviewed on the hybrid CNT-reinforced advancedcomposites authenticate that the growing of CNTs on the advancedfibers results in the enhancement of the mechanical and the ther-mal properties, and the fiber/matrix interfacial shear strength ofexisting advanced fiber reinforced composites. Thus the currentstatus of progress in research on CNT-reinforced composites bringsto light that the three-phase hybrid CNT-reinforced composite canbe the promising candidate material for achieving structural ben-efits from the exceptionally attractive properties of CNTs. Theadvanced fiber augmented with CNTs is also being called as “fuzzyfiber” (Yamamoto et al., 2009; Garcia et al., 2008; Chatzigeorgiouand Seidel, 2011; Kundalwal and Ray, 2012a) and the resultingcomposite with such fuzzy fiber reinforcements may be called asfuzzy fiber reinforced composite (FFRC). The extensive research onthe growing of CNTs on the circumferential surfaces of theadvanced fibers motivated the authors (Kundalwal and Ray, 2012b)to propose a novel three-phase short fuzzy fiber reinforced com-posite (SFFRC). The distinct constructional feature of such SFFRC isthat the uniformly spaced CNTs are radially grown on the circum-ferential surfaces of the unidirectional short carbon fibers. Recently,Ray et al. (2009) carried out a load transfer analysis of the shortcarbon fiber reinforced composite in which the aligned CNTs areradially grown on the circumferential surfaces of the short carbonfibers. In their study, the short carbon fibers being coated withradially grown CNTswere assumed to touch each other laterally andhence, a two-phase shear lag model was developed. In practice, theCNT coated short carbon fibers may not touch each other laterallyand the corresponding RVE for the shear lag analysis will becomposed of three phases, namely, the carbon fiber, the CNT-reinforced polymer matrix nanocomposite (PMNC) and the poly-mer matrix. In order to establish this novel three-phase SFFRC asthe superior advanced composite for structural applications, loadtransfer analysis must be carried out using this SFFRC. However,studies concerning the load transfer analysis of such hybridadvanced nanocomposite have not yet been reported in the openliterature. Such lack in shear lag studies provided a scope for thepresent research study to develop a novel three-phase shear lagmodel for characterizing the load transfer mechanisms of theSFFRC. Towards this direction, a three-phase shear lag model hasbeen developed in the present study to analyze the load transfercharacteristics of the SFFRC while the RVE of the SFFRC is subjectedto both the axial and the radial loads under temperature field. Theemphasis is also placed on investigating the effect of waviness ofCNTs on the load transfer characteristics of the SFFRC since thewaviness of CNTs is inherent to many manufacturing processes ofthe CNT-reinforced composite and may affect the effective ther-moelastic properties of the CNT-reinforced composite.

The organization of the article is following: Section 2 brieflydescribes the architecture of the SFFRC containing wavy CNTs.Section 3 presents the models of the wavy CNTs being coplanarwith either of the two mutually orthogonal planes in the polymermaterial. In Section 4, development of the Mori-Tanaka model ispresented which provides the effective thermoelastic properties ofthe polymer matrix nanocomposite phase containing wavy CNTs asinput for the shear lag analysis. In Section 5, development of a novelthree-phase shear lag model has been presented. Finally, Sections 6and 7 present numerical results to illustrate the load transfercharacteristics of the SFFRC and conclusions drawn from the pre-sent study, respectively. In Appendix A, the constant coefficients are


presented which are obtained in the course of deriving the shearlag model in Section 5.

2. Architecture of the SFFRC containing wavy CNTs

Fig. 1 (a) represents a lamina of the SFFRC in which the shortfuzzy fibers are uniformly interlaced in the polymer matrix and itsin-plane cross section is illustrated in Fig. 1 (b). The wavy CNTs ofequal length are radially grown on the circumferential surfaces ofthe short carbon fiber reinforcements while they are uniformlyspaced on the circumferential surfaces of the carbon fibers. In thepresent study, the wavy CNTs are modeled as sinusoidal solid CNTfibers (Fisher et al., 2002; Berhan et al., 2004; Anumandla and

Fig. 1. Lamina of the SFFRC and

Gibson, 2006; Tsai et al., 2011). The radially grown wavy CNTseventually reinforce the polymer matrix surrounding the carbonfiber along the direction transverse to the length of the carbon fiber.Thus the combination of the fuzzy fiber with wavy CNTs and thepolymer matrix can be viewed as a circular cylindrical short com-posite fuzzy fiber (SCFF) in which a short carbon fiber is embeddedin the wavy polymer matrix nanocomposite (PMNC) as illustratedin Fig. 2. Also, the short composite fuzzy fibers are assumed to beuniformly spaced over the volume of a lamina of the SFFRC in such away that the three orthogonal principal material coordinate (1e2e3) axes exist in the composite as shown in Fig. 1 (a). It is also to benoted that the local material coordinate (10e20e30) axes as shownin Fig. 2 are considered to estimate the location dependent effective

its in-plane cross section.

Fig. 2. Architecture of the RVE of the SFFRC in which the short composite fuzzy fiber embedded in the polymer material.


thermoelastic properties of the polymer matrix nanocomposite.Subsequently, using appropriate transformations, the locationdependent effective thermoelastic properties of the polymer ma-trix nanocomposite with respect to the local coordinate (10e20e30)system are to be transformed to the principal material coordinate(1e2e3) system prior to derive the three-phase shear lag model.The orientation of the plane of the wavy CNTs is an important issue

Fig. 3. Transverse and longitudinal cross sections of the short composite fuzzy fiber in whilongitudinal plane (i.e., 1e3 or 10e30 plane) of the carbon fiber.

because the planer orientations of thewavy CNTsmay influence theload transfer characteristics of the SFFRC, and hence this issueneeds to be carefully addressed. Two possible planar orientations ofthe wavy CNTs are considered i.e., the wavy CNTs are coplanar witheither the longitudinal plane (i.e., 1e3 or 10e30 plane) or thetransverse plane (i.e., 2e3 or 20e30) of the carbon fiber as shown inFig. 3 (a) and 3 (b), respectively.

ch wavy CNTs are coplanar with either the transverse plane (i.e., 2e3 or 20e30) or the

Fig. 5. RVE of the unwound polymer matrix nanocomposite material containing awavy CNT is coplanar with either the transverse plane (i.e., 2e3 or 20e30) or thelongitudinal plane (i.e., 1e3 or 10e30 plane) of the carbon fiber.


3. Models of the wavy CNTs

Scanning electron microscopy images analyzed by Zhang et al.(2008) and Yamamoto et al. (2009) are illustrated in Fig. 4 (a) and(b), respectively, which clearly demonstrate that the CNTs arehighly curved when dispersed in the polymer matrix. Following theprevious research (Fisher et al., 2002; Berhan et al., 2004;Anumandla and Gibson, 2006; Tsai et al., 2011), the CNT wave isassumed to have the sinusoidal shape. The short composite fuzzyfiber can be viewed to be formed bywrapping the carbon fiber withthe unwound lamina of the polymer matrix nanocomposite. TheRVE of the unwound polymer matrix nanocomposite materialcontaining a wavy CNT has been illustrated in Fig. 5. As shown inFig. 5, the RVE is divided into infinitesimally thin slices of thicknessdy. Averaging the effective thermoelastic properties of these slicesover the length (Ln) of the RVE, the homogenized effective ther-moelastic properties of the unwound polymer matrix nano-composite can be estimated. Each slice can be treated as an off-axisunidirectional lamina and its thermoelastic coefficients can bedetermined by transforming the thermoelastic coefficients. Now,the CNT wave is characterized by

z ¼ AsinðuyÞ or x ¼ AsinðuyÞ; u ¼ np=Ln (1)

according as the CNT wave is coplanar with the 2e3 (20e30) planeor the 1e3 (10e30) plane, respectively. In Eq. (1), A and Ln are theamplitude of the CNT wave and the linear distance between theCNT ends, respectively, and n represents the number of the CNTwaves. The running length of the CNT (Lnr) can be expressed in thefollowing form:

Lnr ¼ZLn0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ A2u2cos2ðuyÞ

qdy (2)

in which, the angle f shown in Fig. 5 is given by

tanf ¼ dz=dy ¼ AucosðuyÞ or tanf ¼ dx=dy ¼ AucosðuyÞ(3)

according as the wavy CNT is coplanar with the 2e3 (20e30) or the1e3 (10e30) plane, respectively. Note that for a particular value of u,the value of f varies with the amplitude of the CNT wave.

Fig. 4. (a) Scanning electron microscopy image of vertically aligned curved CNTs (adapted waligned CNTs grown on the alumina fiber (adapted with permission from Yamamoto et al.,

4. Mori-Tanaka method

This Section presents the micromechanics model based on theMori-Tanaka method to estimate the effective thermoelasticproperties of the polymer matrix nanocomposite containing wavyCNTs which are required as inputs for the shear lag analysis. It maybe noted that the effective thermoelastic properties at any point inthe unwound lamina of the polymer matrix nanocomposite con-taining sinusoidally wavy CNTs where the CNT axis makes an anglef with the 3 (30)eaxis can be approximated by transforming theeffective thermoelastic properties of the unwound lamina of thepolymer matrix nanocomposite containing straight CNTs. Hence, inwhat follows the method of deriving the model for predicting theeffective thermoelastic properties of the unwound lamina of thepolymer matrix nanocomposite containing straight CNTs will bepresented first. Utilizing the elastic properties of the CNT and thepolymer matrix properties, the Mori-Tanaka method (1973) can beemployed to estimate the effective elastic coefficient matrix [Cnc] ofthe unwound polymer matrix nanocomposite containing straightCNTs as follows:

½Cnc� ¼ ½Cm� þ vnð½Cn� � ½Cm�Þ½A1��h

~A1

ihvm½I� þ vn

h~A1

ii�1�(4)

ith permission from Zhang et al., (2008)). (b) Scanning electron microscopy image of(2009)).


in which the matrix of the strain concentration factors isgiven by

h~A1

i¼h½I� þ ½Sn�ð½Cm�Þ�1ð½Cn� � ½Cm�Þ

i�1(5)

CNC11 ¼ Cnc11k4 þ Cnc33l

4 þ 2�Cnc13 þ 2Cnc

55�k2l2; CNC12 ¼ Cnc12k

2 þ Cnc23l2;

CNC13 ¼ �

Cnc11 þ Cnc33 � 4Cnc55�k2l2 þ Cnc13

�k4 þ l4

�;CNC22 ¼ Cnc22;

CNC23 ¼ Cnc12l2 þ Cnc23k

2; CNC33 ¼ Cnc11l4 þ Cnc33k


55�k2l2;


2; CNC55 ¼ �Cnc11 þ Cnc33 � 2Cnc13 � 2Cnc

55�k2l2 þ Cnc55

�k4 þ l4

�;


2; aNC11 ¼ anc11k2 þ anc33l

2; aNC22 ¼ anc22 and aNC33 ¼ anc11l2 þ anc33k

2

(8)

where vn and vm represent the volume fractions of the CNT fiberand the monolithic polymer material, respectively, present in theRVE of the polymer matrix nanocomposite while [Sn] representsthe Eshelby tensor and the specific form of the Eshelby tensor forthe cylindrical inclusion given by Qiu and Weng (1990) is utilized.Using the effective elastic coefficient matrix [Cnc], the effectivethermal expansion coefficient vector {anc} for the unwound poly-mermatrix nanocomposite containing straight CNTs can be derivedin the form (Laws, 1973) as follows:

fancg ¼fang þ�½Cnc��1 � ½Cn��1

��½Cn��1 � ½Cm��1

��1

� ðfang � famgÞ(6)

where {an} and {am} are the thermal expansion coefficient vectorsof the CNT fiber and the monolithic polymer material, respectively.The effective elastic coefficients (CNCij ) and the effective thermalexpansion coefficients (aNCij ) at any point in the unwound lamina ofthe polymer matrix nanocomposite where the CNT is inclined at anangle f with the 3 (30)eaxis can be derived in a straightforwardmanner by employing the appropriate transformation law. Thus, ifthe plane of the wavy CNTs is coplanar with the 2e3 (20e30) plane,the effective elastic (CNC

ij ) and thermal expansion (aNCij ) coefficientsat any point in the unwound polymer matrix nanocompositelamina are given by

CNC11 ¼ Cnc11; CNC12 ¼ Cnc12k

2 þ Cnc13l2; CNC13 ¼ Cnc12l

2 þ Cnc13k2;



44�k2l2;

CNC23 ¼ �Cnc22 þ Cnc33 � 4Cnc44

�k2l2 þ Cnc23

�k4 þ l4

�;



44�k2l2;

CNC44 ¼ �Cnc22 þ Cnc33 � 2Cnc23 � 2Cnc

44�k2l2 þ Cnc44

�k4 þ l4

�;


2;CNC66 ¼ Cnc55l2 þ Cnc66k

2;

aNC11 ¼ anc11;aNC22 ¼ anc22k

2 þ anc33l2 and aNC33 ¼ anc22l

2 þ anc33k2

(7)

in which

k ¼ cosf ¼h1þ fnpA=Lncosðnpy=LnÞg2

i�1=2and

l ¼ sinf ¼ npA=Lncosðnpy=LnÞh1þfnpA=Lncosðnpy=LnÞg2

i�1=2

Similarly, if the plane of the wavy CNTs is coplanar with the 1e3(10e30) plane, then the effective elastic (CNC

ij ) and thermal expansion(aNCij ) coefficients at any point of the unwound polymer matrixnanocomposite laminawhere the CNT is inclined at an angle fwiththe 3 (30) eaxis are given by

It is now obvious that the effective thermoelastic properties ofthe unwound polymer matrix nanocomposite lamina containingwavy CNTs vary along the length of the CNT as the value of f varyover the length of the CNT. The average effective elastic coefficientmatrix ½CNC� and the thermal expansion coefficient vector faNCg ofthe lamina of such unwound polymer matrix nanocomposite ma-terial containing wavy CNTs can be obtained by averaging thetransformed elastic coefficients ðCNC

ij Þ and thermal expansion co-efficients ðaNCij Þ over the linear distance between the CNT ends asfollows (Hsiao and Daniel, 1996):

hCNCi ¼ 1

Ln

ZLn0

hCNC

idy and

naNC

o¼ 1

Ln

ZLn0

naNC

ody (9)

It may also be noted that when the carbon fiber is viewed to bewrapped by such unwound polymermatrix nanocomposite lamina,the matrix ½CNC� and the vector faNCg provide the effective prop-erties at a point located in the polymer matrix nanocompositewhere the CNT axis (30eaxis) is oriented at an angle q with the 3eaxis in the 2e3 plane as shown in Fig. 2. Hence, at any point in thepolymer matrix nanocomposite surrounding the carbon fiber, theeffective elastic coefficient matrix ½CPMNC� and the effective thermalexpansion vector faPMNCg of the polymer matrix nanocompositewith respect to the 1e2e3 coordinate system turn out to be loca-tion dependant and can be determined by the followingtransformations:

hCPMNCi ¼ ½T��T

hCNCi½T��1 and

naPMNC

o¼ ½T��T

naNC

o(10)

where, ½T� ¼

26666664

1 0 0 0 0 00 m2 n2 mn 0 00 n2 m2 �mn 0 00 �2mn 2mn m2 � n2 0 00 0 0 0 m �n0 0 0 0 n m

37777775

with

m ¼ cosq and n ¼ sinqFrom Eq. (10) it is obvious that the effective thermoelastic

properties at any point of the polymer matrix nanocomposite sur-rounding the carbon fiber with respect to the principle materialcoordinate axes of the SFFRC vary over an annular cross section ofthe polymer matrix nanocomposite phase of the RVE of the shortcomposite fuzzy fiber. However, without loss of generality, itmay beconsidered that the volume average of these effective thermoelasticproperties over the volume of the polymer matrix nanocomposite


can be treated as the constant effective elastic coefficient matrix[CPMNC] and the constant effective thermal expansion vector {aPMNC}of the polymermatrix nanocomposite containing sinusoidally wavyCNTs surrounding the carbon fiber with respect to the 1e2e3 co-ordinate axes of the SFFRC and are given by

½Cc� ¼ 1

2p�b2 � a2

�L

ZL�L

Z2p0

Zba

hCPMNC

irdrdqdx and

facg ¼ 1

2p�b2 � a2

�L

ZL�L

Z2p0

Zba

naPMNC

ordrdqdx

(11)

5. Development of the three-phase shear lag model

Figs. 2 and 3 illustrate the cylindrical RVEs of the SFFRC based onwhich the three-phase shear lag model is derived. The cylindricalcoordinate (r, q and x) system is considered in such a way that theaxis of the RVE coincides with the xeaxis while CNTs are alignedalong the redirection. The model is derived by dividing the RVEinto three zones. The portion of the RVE in the zone �Lf�x�Lfconsists of three concentric cylindrical phases, namely, the carbonfiber, the polymer matrix nanocomposite and the polymer matrix.

sfr

��r¼a

¼ scr��r¼a;s

fxr

��r¼a

¼ scxr��r¼a ¼ si; scr

��r¼b ¼ smr

��r¼b; s

cxr

��r¼b ¼ smxr

��r¼b ¼ so;

uf��r¼a ¼ ucjr¼a;u

cjr¼b ¼ umjr¼b;wf��r¼a ¼ wcjr¼a and wcjr¼b ¼ wmjr¼b

(16)

The RVE of the SFFRC has the radius R and the length 2L. The radiusand the length of the carbon fiber are denoted by a and 2Lf. Theinner and outer radii of the polymer matrix nanocomposite phaseare a and b, respectively. The portions of the RVE in thezones �L�x��Lf and Lf�x�L are considered to be composed of animaginary fiber, an imaginary polymer matrix nanocomposite andthe polymer matrix phase. The radius of the imaginary fiber is alsodenoted by a while the inner and outer radii of the imaginarypolymermatrix nanocomposite phase are also represented by a andb, respectively. Thus the shear lag model derived for thezone�Lf�x�Lf can be applied to derive the shear lag models for thezones �L�x��Lf and Lf�x�L.

In what follows, the shear lag model for the zone �Lf�x�Lf isfirst derived. As shown in Fig. 3, a tensile stress s is applied to theRVE along the xedirection at x ¼ �L while the RVE is subjected to aradial normal stress qo at r ¼ R. The governing equations for anaxisymmetric RVE problem in terms of cylindrical coordinates (r, qand x) are given by

vsirvr

þ vsixrvx

þ sir � siq

r¼ 0 and

vsixvx

þ 1rv�rsixr

�vr

¼ 0;

i ¼ f ; c andm(12)

while the relevant constitutive relations are

six ¼ Ci11 3

ix þ Ci

12 3iq þ Ci

13 3ir � li11DT;

sir ¼ Ci13 3

ix þ Ci

23 3iq þ Ci

33 3ir � li22DT and

sixr ¼ Ci66 3

ixr; i ¼ f ; c and m

(13)

in which li11 ¼ Ci11a

i11 þ Ci

12ai22 þ Ci

13ai33 and li22 ¼ Ci

12ai11þ

Ci22a

i22 þ Ci

23ai33

In Eqs. (12) and (13), the superscripts f, c and m denote,respectively, the carbon fiber, the polymer matrix nanocompositeand the monolithic polymer matrix. For the i-th constituent phase,six and sir represent the normal stresses in the x and r directions,respectively; 3ix, 3iq and 3ir are the normal strains along the x, q and rdirections, respectively; sixr is the transverse shear stress, 3ixr is thetransverse shear strain, Ci

ij are the elastic constants, li11 and li22 arethe axial and the transverse thermal stiffness coefficients, respec-tively, and aiij are the coefficients of thermal expansion. The strain-displacement relations for an axisymmetric problem relevant tothis RVE are

3ix ¼ vui

vx; 3

iq ¼ wi

r; 3

ir ¼ vwi

vrand 3

ixr ¼ vui

vrþ vwi

vx(14)

in which ui and wi represent the axial and the radial displacementsat any point in the i-th phase along the x and the r directions,respectively. The traction boundary conditions are given by

smr��r¼R ¼ qo and smxr

��r¼R ¼ 0 (15)

and the continuity conditions are

where si is the transverse shear stress at the interface between thecarbon fiber and the polymer matrix nanocomposite while so is thetransverse shear stress at the interface between the polymermatrixnanocomposite and the polymer matrix. It may be noted here thatthese interfacial shear stresses can be attributed to the interactionsbetween the adjacent staggered short carbon fibers (Gao et al.,2003; Ji and Gao, 2004). The consideration of the radial load qoon the RVE also accounts for the lateral extensional interactionsbetween the adjacent short composite fuzzy fibers. The averageaxial stresses in different phases are defined as

sfx ¼ 1pa2

Za0

sfx2prdr; scx ¼ 1

p�b2 � a2

� Zba

scx2prdr and

smx ¼ 1p�R2 � b2

� ZRb

smx 2prdr

(17)

Now, making use of Eqs. (12) and (15)e(17), it can be derivedthat

vsfxvx

¼ �2asi;

vscxvx

¼ 2ab2 � a2

si �2b

b2 � a2so and

vsmxvx

¼ 2bR2 � b2

so

(18)

It is evident from Eq. (18) that the gradients of scx and smx withrespect to the axial coordinate (x) are independent of the radialcoordinate (r). Hence, as the radial dimension of the RVE is verysmall, it is reasonable to assume that (Nairn, 1997)


vscxvx

¼ vscxvx

andvsmxvx

¼ vsmxvx

(19)

Thus using the equilibrium equations given by Eq. (12), thetransverse shear stresses in the polymer matrix nanocompositephase and in the polymer matrix phase can be expressed in termsof the interfacial shear stresses si and so, respectively, as follows:

scxr ¼ arsi þ

12r

�a2 � r2

� vscxvx

(20)

smxr ¼�R2

r� r

bR2 � b2

so (21)

Also, since the RVE is an axisymmetric problem, it is usuallyassumed (Nairn, 1997) that the gradient of the radial displacementswith respect to the xedirection is negligible and so, from theconstitutive relations given by Eq. (13) and the strain-displacementrelations given by Eq. (14) between sixr and 3ixr , one can write

vuc

vr¼ 1

Cc66scxr and

vum

vr¼ 1

Cm66smxr (22)

Solving Eq. (22) and satisfying the continuity condition at r ¼ aand r ¼ b, respectively, the axial displacements of the polymermatrix nanocomposite phase and the polymer matrix phase alongthe xedirection can be derived as follows:

uc ¼ ufa þ A1si þ A2so (23)

um ¼ ufa þ A3si þ A4so and (24)

ufa ¼ uf��r¼a

(25)

in which Ai (i ¼ 1, 2, 3 and 4) are the constants of the displacementfields of the polymer matrix nanocomposite and the polymer ma-trix, and are explicitly shown in Appendix A.

The radial displacements in the three constituent phases can beassumed as (Hashin and Rosen, 1964)

wf ¼ Af r; wc ¼ Acr þ Bcr

and wm ¼ Amr þ Bmr

(26)

where Af, Ac, Bc, Am and Bm are unknown constants. Invoking thecontinuity conditions for the radial displacement at the interfacesr ¼ a and r ¼ b, the radial displacement in the polymer matrixnanocomposite phase can be augmented as

wc ¼ a2

b2 � a2

�b2

r� rAf �

b2

b2 � a2

�a2

r� rAm

� 1b2 � a2

�a2

r� rBm (27)

264A11 A12 A13

A21 A22 A23

A31 A32 A33

3758><>:

Af

Am

Bm

9>=>; ¼ sf

x

Cf11

8><>:

Cf12 � Cc13Cc13 � Cm12�Cm12

9>=>;þ

8><>:

�Cc13A1

Cc13A1 � Cm12A3

�Cm12A3

Substituting Eqs. (23), (24), (26) and (27) into Eq. (14) andsubsequently, employing the constitutive relations (13), the ex-pressions for the normal stresses can be written in terms of theunknowns Af, Am and Bm as follows:

sfx ¼ Cf11vufavx

þ 2Cf12Af � lf11DT (28)

sfr ¼ Cf12

Cf11

sfx þ"Cf23 þ Cf

33 �2�Cf12

�2Cf11

#Af � lf22DT (29)

scx ¼ Cc11

Cf11

sfx � 2Cc

12a2

b2 � a2þ 2Cf

12Cc11

Cf11

!Af þ

2Cc12b

2

b2 � a2Am

þ 2Cc12

b2 � a2Bm þ Cc

11A1vsivx

þ Cc11A2

vsovx

� lc11DT (30)

scr ¼Cc13

Cf11

sfx þ"Cc13

a2

b2 � a2

�b2

r2� 1

þ Cc

33a2

b2 � a2

�� b2

r2� 1

� 2Cf12C

c13

Cf11

#Af þ

� Cc

13b2

b2 � a2

�a2

r2� 1þ Cc

33b2

b2 � a2

��a2

r2þ 1�

Am � lc22DT þ� Cc

231

b2 � a2

�a2

r2� 1

þ Cc33

1b2 � a2

�a2

r2þ 1

�Bm þ Cc

13A1vsivx

þ Cc13A2

vsovx

(31)

smx ¼Cm11

Cf11

sfx �2Cf

12Cm11

Cf11

Af þ 2Cm12Am þ Cm

11A3vsivx

þ Cm11A4

vsovx

� lm11DT

(32)

smr ¼ Cm12

Cf11sfx �

2Cf12C

m12

Cf11Af þ

�Cm11 þ Cm12

�Am þ �Cm12 � Cm11

�Bmr2

þ Cm12A3vsivx

þ Cm12A4vsovx

� lm22DT

(33)

Invoking the continuity conditions sfr��r¼a

¼ scr��r¼a and scr

��r¼b ¼

smr��r¼b, and satisfying the boundary condition smr

��r¼R ¼ qo, the

following equations for solving Af, Am and Bm are obtained:

9>=>;

vsivx

þ

8><>:

�Cc13A2�Cc13A2 � Cm12A4

�Cm12A6

9>=>;

vsovx

þ

8><>:

001

9>=>;qo þ

8><>:

lc22 � lf22lm22 � lc22

lm22

9>=>;DT

(34)


Solving Eq. (34), the solutions of the constants of the radialdisplacements Af, Am and Bm can be expressed as:

Af ¼ k11sfx þ k12

vsivx þ k13

vsovx þ k14qo þ k15DT

Am ¼ k21sfx þ k22

vsivx þ k23


Bm ¼ k31sfx þ k32

vsivx þ k33


(35)

The expressions of the coefficients Aij and kij are presented inAppendix A.

Now, making use of Eqs. (30), (32) and (35) in the last twoequations of (17), respectively, the average axial stresses in thepolymer matrix nanocomposite phase and the polymer matrixphase are written as follows:

scx ¼ L1sfx þ L2

vsivx

þ L3vsovx

þ L4qo þ L5DT (36)

smx ¼ L6sfx þ L7

vsivx

þ L8vsovx

þ L9qo þ L10DT (37)

The constants Li (i ¼ 1, 2, 3,., 10) appearing in the above twoequations are presented in Appendix A.

Now, satisfying the equilibrium of force along the axial (x) di-rection at any transverse cross section of the RVE, the followingequation is obtained:

pR2s ¼ p�R2 � b2

�smx þ p

�b2 � a2

�scx þ pa2sfx (38)

Differentiating first and last equations of (18) with respect to x,we have

vsivx

¼ �a2v2sfxvx2

(39)

vsovx

¼ R2 � b2

2bv2smxvx2

(40)

Use of Eqs. (36)e(40), yields

L11sfx þ L12

v2sfxvx2

þ L13v2smxvx2

þ L14qo þ L15DT � R2s ¼ 0 (41)

where

L11 ¼ a2 þ�b2 � a2

�L1 þ

�R2 � b2

�L6;

L12 ¼ �ða=2Þh�

b2 � a2�L2 þ

�R2 � b2

�L7i;

L13 ¼�R2 � b2

2b

h�b2 � a2

�L3 þ

�R2 � b2

�L8i;

L14 ¼�b2 � a2

�L4 þ

�R2 � b2

�L9 and

L15 ¼�b2 � a2

�L5 þ

�R2 � b2

�L10

b ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2

�� L17 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðL17Þ2 � 4L18

q s; a ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2

�� L17 �

ffiffiffiffiffiffiffiffiffiðL17

qs

bpf ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2

�� Lpf17 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Lpf17�2 � 4Lpf18

r sand apf ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2

�� Lp1

s

Deriving the expression for vso=vx from Eq. (36) and substitut-ing the same into Eq. (37) and then using Eq. (39), the followingresult for smx is obtained:

smx ¼�L6 �

L1L8L3

sfx þ

�L8L3

scx þ

�a2

��L2L8L3

� L7

v2sfxvx2

þ�L9 �

L4L8L3

qo þ

�L10 �

L5L8L3

DT

(42)

Differentiating Eqs. (38) and (42) twice with respect to x andusing the resulting equations in Eq. (41), the governing equation forthe average axial stress in the carbon fiber coated with radiallygrown aligned CNTs is obtained as follows:

v4sfxvx4

þ L17v2sfxvx2

þ L18sfx � L19sþ L20qo þ L21DT ¼ 0 (43)

The coefficients Li (i ¼ 17, 18, 19, 20 and 21) appeared in theabove differential equation have been explicitly presented inAppendix A.

Following the above procedure, the governing equation for theaverage axial stress (spfx ) in the imaginary fiber made of the poly-mer material lying in the zones �L�x��Lf and Lf�x�L can bewritten as

v4spfxvx4

þ Lpf17v2spfxvx2

þ Lpf18spfx � Lpf19sþ Lpf20qo þ Lpf21DT ¼ 0 (44)

In the above equation, the expressions for Lpf17, Lpf18, L

pf19, L

pf20

and Lpf21 are similar to those of expressions L17, L18, L19,L20 and L21, respectively. But these are to be derived byconsidering Cf

ij ¼ Ccij ¼ Cmij . Solutions of Eqs. (43) and (44) aregiven by:

sfx ¼ L22sinhðbxÞ þ L23coshðbxÞ þ L24sinhðaxÞ þ L25sinhðaxÞþ ðL19=L18Þs� ðL20=L18Þqo � ðL21=L18ÞDT

(45)

spfx ¼ Lpf22sinh�bpfx

�þ Lpf23cosh

�bpfx

�þ Lpf24sinh

�apfx

�þ Lpf25cosh

�apfx

�þ�Lpf19=L

pf18

�s�

�Lpf20=L

pf18

�qo

��Lpf21=L

pf18

�DT (46)

where

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÞ2 � 4L18

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif7 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Lpf17�2 � 4Lpf18

r (47)


The constants L22, Lpf22, L23, L

pf23, L24, L

pf24, L25 and Lpf25 are to be

evaluated from the following end conditions:

spfx ¼ s at x ¼ �L andvspfxvx

¼ 0 at x ¼ �L (48)

sfx ¼ spfx at x ¼ �Lf andvsfxvx

¼ vspfxvx

at x ¼ �Lf (49)

Utilizing the end conditions given by Eq. (48) in Eq. (46), theconstants Lpf22, Lpf23, Lpf24 and Lpf25 can be explicitly expressed asfollows:

Lpf22 ¼ 0 (50)

Lpf23 ¼ �apf sinh

�apf L

�bpf sinh

�bpf L

�cosh

�apf L

�� apf sinh�apf L

�cosh

�bpf L

�

�(Lpf19Lpf18

s� Lpf20Lpf18

qo �Lpf21Lpf18

DT � s

)

(51)

bm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2

�� L34 þ


q s; am ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2

�� L34 �


q s

bpm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2

�� Lpm34 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Lpm34

�2 � 4Lpf35

q sand apm ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=2

�� Lpm34 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Lpm34

�2 � 4Lpm35

q s (58)

Lpf24 ¼ 0 (52)

Lpf25 ¼ �bpf sinh

�bpf L

�bpf sinh

�bpf L

�cosh

�apf L


�cosh

�bpf L

�

�(Lpf19Lpf18

s� Lpf20Lpf18

qo �Lpf21Lpf18

DT � s

)

(53)

Substituting Eq. (50)e(53) in Eq. (46), the final solution for spfx isobtained as follows:

spfx ¼

"apf sinh

�apf L

�cosh

�bpfx

�� bpf sinh

�bpf L

�cosh

�apfx

�bpf sinh

�bpf L

�cosh

�apf L


�cosh

�bpf L

�#

�(Lpf19Lpf18

s� Lpf20Lpf18

qo �Lpf21Lpf18

DT� s

)þ Lpf19s� Lpf20qo � Lpf21DT

Lpf18(54)

Similarly, utilizing the end conditions given by Eq. (49) in Eq.(45), the constants L22, L23, L24 and L25 are evaluated and the samehave been explicitly shown in Appendix A. Substitution of Eq. (45)into the first equation of (18) yields the expression for the carbonfiber/polymer matrix nanocomposite interfacial shear stress asfollows:

si ¼ �a2½bL22coshðbxÞ þ bL23sinhðbxÞ þ aL24coshðaxÞ

þ aL25sinhðaxÞ� (55)

Following the procedure for deriving the solutions of theaverage axial stresses sfx and spfx in the carbon fiber and theimaginary fiber, respectively, the solutions for the average axialstresses smx and spmx in the polymer matrices surrounding the shortcomposite fuzzy fiber and the imaginary short composite fuzzy fi-ber, respectively, can be derived as follows:

smx ¼ L39sinh�bmx

�þ L40cosh�bmx

�þ L41sinhðamxÞþ L42coshðamxÞ þ ðL36=L35Þs� ðL37=L35Þqo� ðL38=L35ÞDT (56)

spmx ¼ Lpm39 sinh�bpmx

�þ Lpm40 cosh�bpmx

�þ Lpm41 sinhðapmxÞþ Lpm42 coshðapmxÞ þ �Lpm36 =Lpm35

�s� �Lpm37 =Lpm35

�qo

� �Lpm38 =Lpm35�DT

(57)

where

The constants L39, Lpm39 , L40, L

pm40 , L41, L

pm41 , L42 and Lpm42 are to be

evaluated from the boundary conditions given in Eqs. (48) and (49).Also, the coefficients (L34)�(L38) appeared in Eqs. 56e58 are pre-sented in Appendix A. Finally, substitution of Eq. (56) into the lastequation of (18) yields the expression for the short composite fuzzyfiber/polymer matrix interfacial shear stress as follows:

so ¼�R2 � b2

b2 � a2

�bmL39cosh

�bmx

�þ bmL40sinh�bmx

�þ amL41coshðamxÞ þ amL42sinhðamxÞ (59)

6. Results and discussion

Armchair (10, 10) CNT, carbon fiber and polymer matrix areconsidered for evaluating the numerical results. Their materialproperties, available in Ref. (Kwon et al., 2004; Chen et al., 2009;Honjo, 2007; Villeneuve et al., 1993; Peters, 1998) are listed inTable 1. Unless otherwise mentioned, the values of the geometricalparameters of the RVE of the SFFRC are taken as 2a ¼ 10 mm, Lf/a¼20, L/Lf¼1.1 and R/b¼1.1. Since the investigations by the earlierresearchers (Maniwa et al., 2001; Kwon et al., 2004; Chen et al.,2009) revealed the strong temperature dependence of the ther-moelastic coefficients of CNTs, the variation of the thermoelasticcoefficients of the armchair (10, 10) CNT with the temperaturedeviation is considered here. But, the elastic properties of carbonfiber and polymer are reported to be marginally dependant on thetemperature deviation (Mallick, 1997). Hence, the temperaturedependence of the elastic properties of carbon fiber and polymer

Table 1Material properties of the constituent phases of the SFFRC.

Material DT (K) C11 (GPa) C12 (GPa) C13 (GPa) C23 (GPa) C33 (GPa) C66 (GPa) a1 (10�6 K�1) a2 (10�6 K�1)

(10, 10) CNT (Kwon et al., 2004;Chen et al., 2009)

100 1196.2 255.8 255.8 255.8 1196.2 470.2 �28.46 �5.07300 1180 276.2 276.2 276.2 1180 452.3 �62.76 �11.49500 1144.8 332.3 332.3 332.3 1144.8 406.2 �66.98 �12.72

Carbon fiber (Honjo, 2007;Villeneuve et al., 1993)

e 236.4 10.6 10.6 10.7 24.8 25 1.1 6.8

Polymer (Peters, 1998) e 4.09 1.55 1.55 1.55 4.09 1.27 66 66

Table 3Comparison of the coefficients of thermal expansion of the unwound polymer


material is neglected. Volume fraction of CNTs (VCNT) in the SFFRCdepends on the CNT waviness, the carbon fiber diameter and thesurface to surface distance between the adjacent radially alignedCNTs at their roots. The surface to surface distance between the twoadjacent CNTs at their roots is considered as 1.7 nm (Kundalwal andRay, 2011). Recall that the SFFRC lamina can be viewed as beingcomprised of the short composite fuzzy fibers and the polymermatrix. In the SFFRC, the hexagonal packing array of the shortcomposite fuzzy fibers is considered for evaluating the numericalresults while they are not touching each other. Based on the surfaceto surface distance at the roots of two adjacent CNTs as 1.7 nm, theCNT diameter (dn) and the running length of the sinusoidally CNTwave (Lnr), the maximum CNT volume fraction in the SFFRC can bedetermined as (Kundalwal and Ray, 2013):

ðVCNTÞmax ¼ pd2nLnrdðdn þ 1:7Þ2

vf (60)

where dn is in nm and is the diameter of the CNT. The derivation ofEq. (60) can be found in Ref. Kundalwal and Ray (2013). Unlessotherwise mentioned, the values of the carbon fiber volume frac-tion (vf), the wave frequency of the CNT (u) and the magnitude ofthe temperature variation (DT) are considered as 0.4, u¼24p/Ln and300 K, respectively. The diameter of the short composite fuzzy fiber(2b) and the length of the straight CNT become 14.3567 mm and1.5258 mm, respectively, when the value of vf is 0.4. The degree ofthe waviness of the CNT is characterized by the waviness factor, A/Ln. It should be noted that for the straight CNT, the value of thewaviness factor is zero. The maximum amplitude for the armchair(10, 10) CNT wave is considered as A ¼ 50dnwhich yields the upperlimit for the value of waviness factor as A/Ln ¼ 0.0446.

In order to verify the validity of the micromechanics modelbased on the Mori-Tanaka (MT) method derived in Section 4, theengineering constants of the unwound polymer matrix nano-composite containing straight CNTs determined by this methodhave been compared with those of the similar nanocompositecontaining straight CNTs predicted by Liu and Chen (2003). Table 2illustrates these comparisons and it may be observed that the twosets of results are in excellent agreement validating the Mori-Tanaka method derived in this study. The coefficients of thermal

Table 2Comparison of the engineering constants of the unwound polymer matrix nano-composite material with straight CNTs.

En/Epa E1/Ep E2/Ep n12, n13

FEMa MT FEMa MT FEMa MT

5 1.1948 1.1948 1.1737 1.0666 0.3 0.310 1.4384 1.4384 1.3336 1.0912 0.3 0.3

where En and Ep are the Young’s moduli of the CNT and the polymer matrix,respectively; E1 and E2 are the axial Young’s modulus and the transverse Young’smodulus of the unwound polymermatrix nanocomposite, respectively; nn and np arethe Poisson’s ratios of the CNT and the polymer matrix, respectively; n12 and n23 arethe axial Poisson’s ratio and the transverse Poisson’s ratio of the unwound polymermatrix nanocomposite, respectively.

a Liu and Chen (2003): En¼1000 GPa, nn¼0.3, np¼0.3 and CNT volume fraction,vn¼0.04871.

expansion of the unwound polymer matrix nanocomposite con-taining straight CNTs determined by the Mori-Tanaka method havealso been compared with those of the similar nanocomposite pre-dicted by Kirtania and Chakraborty (2009) as illustrated in Table 3.For the effective coefficients of thermal expansion of the nano-composite, the two sets of results are in excellent agreement vali-dating the Mori-Tanaka method utilized in this study. It can beinferred from the comparisons shown in Tables 2 and 3 that theMori-Tanaka method can be reasonably applied to predict thethermoelastic properties of the polymer matrix nanocomposite.

First, the effective thermoelastic coefficients of the polymermatrixnanocomposite are computedbyemploying theMori-Tanakamethod. Estimated values of the effective thermoelastic coefficientsof the polymer matrix nanocomposite are summarized in Table 4.It may be noted that the effective thermoelastic coefficients ofthe polymer matrix nanocomposite have been determined withrespect to the RVE of the polymer matrix nanocomposite. Foranalyzing the load transfer characteristics of the three-phasecomposite, the following nondimensional parameters are used:

s� ¼ sfxs; s�i ¼ si

sand s�o ¼ so

s(61)

The shear lag model derived in Section 5 analyses the loadtransfer mechanisms of the SFFRC incorporating different trans-versely isotropic constituent phases of the SFFRC subjected to thethermomechanical loading which have not been considered in theexisting shear lag studies. However, it may be imperative to justifythe validity of the shear lag model derived in Section 5 consideringdifferent transversely isotropic constituent phases of the SFFRC,and the application of the radial and the thermal loads on the RVE.For this purpose, three-phase finite element shear lag model hasbeen developed using the commercial software ANSYS 11.0 tovalidate the analytical shear lag model derived herein. It should benoted that because of the symmetry, one half of the RVE isconsidered for the finite element simulations. Under the conditionsof an imposed tensile stress (s), radial normal stress (qo) and

matrix nanocomposite material with the straight CNTs.

vn a1(�10�6 K�1) a2(�10�6 K�1)

FEMa MT FEMa MT

0.5 25.203 24.485 69.754 69.9031 15.302 15.072 72.867 72.8793 5.0978 5.182 74.7155 74.5125.45 2.267 2.2978 73.5434 73.1227.9 1.0643 1.117 71.7165 71.1310.3 0.4253 0.48471 69.7879 69.01415.77 �0.3201 �0.25599 65.1728 64.037

a Kirtania and Chakraborty (2009): En¼1000 GPa, nn¼0.2, Ep¼3.89 GPa, np¼0.37,an¼�1.5�10�6K�1 and ap¼58�10�6K�1 where a1 and a2 are the axial coefficient ofthermal expansion and the transverse coefficient of thermal expansion of the un-wound polymer matrix nanocomposite with the straight CNTs, respectively; an andap are the coefficients of thermal expansion of the CNT and the polymer matrix,respectively.

Table 4Effective thermoelastic coefficients of the polymer matrix nanocomposite.

R/b & L/Lf U (nm�1) (vn)max DT(K)

Cc11(GPa)

Cc12(GPa)

Cc13

(GPa)Cc22(GPa)

Cc23(GPa)

Cc33

(GPa)Cc66

(GPa)ac11(10�6 K�1)

ac22(10�6 K�1)

1.1 0 0.1346 300 5 1.8 1.8 58.3 20.3 58.3 2.1 59.3 24.71.1 24p/Ln (1e3 plane) 0.3272 100 216.8 25.4 25.4 27.3 10.2 27.3 26.6 7.64 32.21.1 24p/Ln (1e3 plane) 0.3272 300 211.1 24.8 24.8 26.7 10 26.7 26 �0.659 201.1 24p/Ln (1e3 plane) 0.3272 500 195.7 23.2 23.2 25.2 9.6 25.2 24.2 �1.96 18.51.1 24p/Ln (2e3 plane) 0.3272 300 7 2.4 2.4 120.1 63.4 120.1 26 25.2 7.11.05 24p/Ln (1e3 plane) 0.2562 300 137.6 23.3 23.3 26.8 10 26.8 24 6.64 28.51.15 24p/Ln (1e3 plane) 0.4454 300 335.8 26.2 26.2 27.9 10.6 27.9 28.3 �7.43 4.71.2 24p/Ln (1e3 plane) 0.6741 300 580.9 28.5 28.5 34.6 13.8 34.6 34.2 �13.3 �24.2


thermal load (DT) on the RVE of the SFFRC, the average stresses fsigin the i-th phase of the RVE can be obtained as

nsio

¼ 1Vi

Z nsiodVi; i ¼ f ; c and m (62)

where Vi represents the volume of the i-th phase of the RVE and thefield variable with an overbar represents the average of the fieldvariable. With and without the application of the radial load (qo) onthe RVE, the comparisons of the average axial stress in the carbonfiber (s*) and the interfacial shear stress along the length of thecarbon fiber (s�i ) computed by the analytical shear lag model andthe finite element shear lag model are presented in Figs. 6 and 7,respectively, when the magnitude of the temperature variation,DT ¼ 300 K. It may be observed that the excellent agreement be-tween the results have been obtained verifying the development ofthe analytical shear lag model in the present study. The subsequentresults are presented based on the analytical shear lag model only.

It should be noted that because of the symmetry, distributions ofthe stresses in the zones of the carbon fiber reinforcement and theshort composite fuzzy fiber are plotted for one half of the RVE only.The variations of the axial stress in the carbon fiber (s*), theinterfacial shear stresses along the lengths of the carbon fiber (s*i )and the short composite fuzzy fiber (s�o) are presented in Figs. 8e10,respectively, when the magnitude of qo is zero. It may be observedfrom Fig. 8 that the carbon fiber coated with the wavy CNTs beingcoplanar with the 1e3 plane shares less axial load (s*) than thecarbon fiber coatedwith the straight CNTs (u¼0) or coated with thewavy CNTs being coplanar with the 2e3 plane. The same is true forthe maximum values of s*i as depicted in Fig. 9. This is attributed to

0 20 40 60 80 1001

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

x/Lf

*

Analytical, qo = 0

FEM, qo = 0

Analytical, qo = σ

FEM, qo =

Fig. 6. Analytical shear lag model validation by comparison to FE shear lag model forthe axial stress in the carbon fiber along its length when the wavy CNTs are coplanarwith the 1e3 (10e30) plane (DT ¼ 300 K, Lf/a ¼ 20, D/Db ¼ L/Lf ¼ 1.1).

the fact that the radially grownwavy CNTs being coplanar with the1e3 plane enhance the axial thermoelastic coefficients (Cc11, C

c12,

Cc13, Cc55, C

c66 and ac11) of the polymer matrix nanocomposite matrix

surrounding the carbon fiber. When compared with the resultswithout CNTs, it is observed that almost w35% and w67% re-ductions in the maximum values of s* and s*i occur, respectively, ifthe wavy CNTs are coplanar with the 1e3 plane. On the other hand,the maximum value of s�o for the base composite (VCNT ¼ 0) de-creases compared to that of the SFFRC containing either the straightCNTs or the wavy CNTs as illustrated in Fig. 10.

Next, the application of qo is considered on the RVE of the SFFRCto investigate the load transfer characteristics of the SFFRC. Itshould be noted that the application of qo on the RVE accounts forthe lateral extensional interactions between the adjacent shortcomposite fuzzy fibers. Figs. 11 and 12 illustrate the variations ofthe stresses s* and s�o, respectively, when the magnitude of qo¼s.With the application of the radial load on the RVE, Fig. 11 revealsthat the maximum value of s* for the SFFRC with the wavy CNTsbeing coplanar with the 1e3 plane decreases compared to that ofthe SFFRC containing either the straight CNTs or the wavy CNTsbeing coplanar with the 2e3 plane. Although not presented here,the same is true for the maximum value of s*i whereas themaximum value of s�o for the base composite (VCNT¼0) decreasescompared to that of the SFFRC containing either the straight CNTsor thewavy CNTs as illustrated in Fig. 12. Since the wavy CNTs beingcoplanar with the 1e3 plane significantly improve the load transfercharacteristics of the SFFRC over those of the SFFRC containingeither the wavy CNTs being coplanar with the 2e3 plane or thestraight CNTs, the effects of the temperature deviation (DT), thespacings between the adjacent short composite fuzzy fibers (R/b

0 20 40 60 80 1000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

x/Lf

i*

Analytical, qo = 0

FEM, qo = 0

Analytical, qo =

FEM, qo =

0.1

0.12

0.14

0.16

Fig. 7. Analytical shear lag model validation by comparison to FE shear lag model forthe interfacial shear stress along the carbon fiber length when the wavy CNTs arecoplanar with the 1e3 (10e30) plane (DT ¼ 300 K, Lf/a ¼ 20, D/Db ¼ L/Lf ¼ 1.1).

0 0.2 0.4 0.6 0.8 10

0.04

0.08

0.12

0.16

x/Lf

o*

VCNT = 0

= 0, (VCNT)max

= 24 /Ln, 1-3 plane, (VCNT)max


Fig. 10. Variation of the transverse shear stress at the interface between the polymerand the short composite fuzzy fiber along its length (DT ¼ 300 K, qo ¼ 0, Lf/a ¼ 20, D/Db ¼ L/Lf ¼ 1.1).

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

x/Lf

*

VCNT = 0

= 0, (VCNT)max



Fig. 8. Variation of the axial stress in the carbon fiber along its length (DT ¼ 300 K,qo ¼ 0, Lf/a ¼ 20, D/Db ¼ L/Lf ¼ 1.1).


and L/Lf) and the carbon fiber aspect ratio (Lf/a) on the load transfercharacteristics of the SFFRC are studied in case of the SFFRC con-taining wavy CNTs which are coplanar with the 1e3 plane.

The variations of s* and s*i are presented in Figs. 13 and 14,respectively, for different values of the applied radial and thermalloads on the RVE. It may be observed from Figs. 13 and 14 that themaximum values of s* and s*i are significantly decreased with thedecrease in the magnitude of the temperature change and this ef-fect becomes more pronounced if the radial load is present. This isattributed to the fact that the effective thermoelastic properties ofthe polymer matrix nanocomposite matrix surrounding the carbonfiber are improved with the decrease in the magnitude of thetemperature variation (see Table 4). Although not shown here, it isalso found that the variations of DT and qo do notmuch improve theinterfacial shear stress transfer along the length of the short com-posite fuzzy fiber (s�o).

So far, the load transfer characteristics of the SFFRC have beenstudied by considering the values of the geometrical parameters R/b and L/Lf as 1.1. Here, the geometrical parameters R/b and L/Lfrepresent the spacings between the adjacent short composite fuzzy

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

x/Lf

i*

VCNT = 0

= 0, (VCNT)max



Fig. 9. Variation of the transverse shear stress at the interface between the polymermatrix nanocomposite and the carbon fiber along its length (DT ¼ 300 K, qo ¼ 0, Lf/a ¼ 20, D/Db ¼ L/Lf ¼ 1.1).

fibers along their radial and axial directions, respectively, over thevolume of the SFFRC lamina. Practically, the gaps between theadjacent short composite fuzzy fibers interlaced in the polymermatrix can vary over the volume of the SFFRC lamina. The variationof such gaps between the adjacent short composite fuzzy fibers fora particular value of vf would be an important study. For this thefour discrete values of R/b and L/Lf are considered as 1.05, 1.1, 1.15and 1.2 and the effective elastic coefficients of the polymer matrixnanocomposite corresponding to these geometrical parametershave been summarized in Table 4 considering the wavy CNTs to becoplanar with the 1e3 plane and DT ¼ 300 K. The variations of thestresses s*, s*i and s�o are presented in Figs.15e17 for different valuesof R/b and L/Lf, respectively. The maximum values of s* and s*i aredecreased with the increase in the values of R/b and L/Lf as illus-trated in Figs. 15 and 16, respectively, and the reverse is true for themaximum value of s�o as shown in Fig 17. Although not presentedhere, the maximum values of s* and s*i are significantly decreasedwith the application of the radial load on the RVE of the SFFRC andthis effect becomes more pronounced if the value of DT < 300 K.This is attributed to the fact that the effective thermoelastic

0 0.2 0.4 0.6 0.8 11

1.2

1.4

1.6

1.8

2

2.2

x/Lf

*

VCNT = 0

= 0, (VCNT)max



Fig. 11. Variation of the axial stress in the carbon fiber along its length (DT ¼ 300 K,qo ¼ s, Lf/a ¼ 20, D/Db ¼ L/Lf ¼ 1.1).

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

x/Lf

o*

VCNT = 0

= 0, (VCNT)max



Fig. 12. Variation of the transverse shear stress at the interface between the polymerand the short composite fuzzy fiber along its length (DT ¼ 300 K, qo ¼ s, Lf/a ¼ 20, D/Db ¼ L/Lf ¼ 1.1).

0 0.2 0.4 0.6 0.8 10

0.04

0.08

0.12

0.16

0.2

x/Lf

i*

0.12

0.16

0.2

qo = 0, T = 100 K

qo = 0, T = 500 K

qo = , T = 100 K

qo = , ΔT = 500 K

Fig. 14. Variation of the transverse shear stress at the interface between the polymermatrix nanocomposite and the carbon fiber along its length for different temperaturevariations when the wavy CNTs are coplanar with the 1e3 (10e30) plane (Lf/a ¼ 20, D/Db ¼ L/Lf ¼ 1.1).


coefficients of the polymer matrix nanocomposite are significantlyimproved with the increase in the values of R/b and L/Lf whicheventually enhance the load carrying capacity of the polymer ma-trix nanocomposite in the radial direction.

Figs. 6, 8, 11, 13 and 15 revealed that the average axial stress inthe carbon fiber (s*) remains uniform over the 90% length of thecarbon fiber from its center while it decreases sharply near the endof the carbon fiber. On the other hand, the interfacial shear stressesalong the lengths of the carbon fiber (s*i ) and the short compositefuzzy fiber (s�o) reaches their maximum values near the ends of thecarbon fiber and the short composite fuzzy fiber, respectively, andbecome zero at x¼�Lf as illustrated in Figs. 7, 9,10,12,14,16 and 17.The transfer length for the stresses may change with the variationof the carbon fiber aspect ratio. Thus the variation of the carbonfiber aspect ratio for a particular value of vf would be an importantissue. Next, the effect of the variation of the carbon fiber aspectratio on the load transfer characteristics of the SFFRC is investigatedconsidering the four discrete values of Lf/a as 5, 10, 30 and 50keeping the values of R/b and L/Lf as 1.1. It may be observed from

0 0.2 0.4 0.6 0.8 11

1.1

1.2

1.3

1.4

x/Lf

*

qo = 0, T = 100 K

qo = 0, T = 500 K

qo = , T = 100 K

qo = , T = 500 K

Fig. 13. Variation of the axial stress in the carbon fiber along its length for differenttemperature variations when the wavy CNTs are coplanar with the 1e3 (10e30) plane(Lf/a ¼ 20, D/Db ¼ L/Lf ¼ 1.1).

Fig. 18 that in the absence of the radial load (qo), the maximumvalue of s* becomes independent of the aspect ratio of the carbonfiber while the length of the carbon fiber being uniformly stressedincreases as the aspect ratio of the carbon fiber increases. Althoughnot shown here, the maximum value s* is also found to be inde-pendent of the aspect ratio of the carbon fiber in the presence of theapplied radial load on the RVE of the SFFRC (qo>0). On the otherhand, the maximum values of s*i are decreased with the increase inthe values of Lf/a as illustrated in Fig. 19 when the value of qo¼0.Although not shown here, the same is true for the maximumvaluesof s�o. In the presence of the applied radial load on the RVE, themaximum values of s*, s*i and s�o are significantly decreased withthe increase in the values of Lf/a and this effect becomes morepronounced if the value of DT < 300 K. It is important to note thatthe maximum length of the carbon fiber required for the effectivestress transfer becomes independent of the value of Lf/a if its valueis greater than 30 irrespective of the values of qo and DT signifyingthe fact that the lower limiting value of the carbon fiber aspect ratiocan be considered as 30.

0 0.2 0.4 0.6 0.8 10.75

1

1.25

1.5

x/Lf

*

D/Db = L/Lf = 1.05, A/Ln = 0.0340

D/Db = L/Lf = 1.10, A/Ln = 0.0446

D/Db = L/Lf = 1.15, A/Ln = 0.0615

D/Db = L/Lf = 1.20, A/Ln = 0.0841

Fig. 15. Variation of the axial stress in the carbon fiber along its length when the wavyCNTs are coplanar with the 1e3 (10e30) plane (DT ¼ 300 K, qo ¼ 0, Lf/a ¼ 20).

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

x/Lf

i*

D/Db = L/Lf = 1.05, A/Ln = 0.0340

D/Db = L/Lf = 1.10, A/Ln = 0.0446

D/Db = L/Lf = 1.15, A/Ln = 0.0615

D/Db = L/Lf = 1.20, A/Ln = 0.0841

Fig. 16. Variation of the transverse shear stress at the interface between the polymermatrix nanocomposite and the carbon fiber along its length when the wavy CNTs arecoplanar with the 1e3 (10e30) plane (DT ¼ 300 K, qo ¼0 , Lf/a ¼ 20).

0 0.2 0.4 0.6 0.8 11

1.05

1.1

1.15

1.2

1.25

1.3

1.35

x/Lf

*

Lf/a = 5

Lf/a = 10

Lf/a = 30

Lf/a = 50

Fig. 18. Variation of the axial stress in the carbon fiber along its length for differentaspect ratios of the carbon fiber when the wavy CNTs are coplanar with the 1e3 (10e30)plane (DT ¼ 300 K, qo ¼ 0, D/Db ¼ L/Lf ¼ 1.1).


7. Conclusions

In the present study, a novel three-phase shear lag model hasbeen developed to analyze the load transfer characteristics of theshort fuzzy fiber reinforced composite (SFFRC) in the presence ofthe temperature field. The distinct constructional feature of theSFFRC is that the short fuzzy fibers are dispersed in the polymermatrix over the volume of the SFFRC lamina. Such short fuzzy fiberreinforcements are coated with radially aligned CNTs on theircircumferential surfaces. The followingmain conclusions are drawnfrom the investigations carried out in this study:

1. Developed shear lagmodel in the present study can describe theload transfer mechanisms of any hybrid advanced nano-composite incorporating different orthotropic constituent pha-ses of the composite subjected to the thermomechanical loadingwhich have not been previously considered elsewhere in theexisting shear lag studies. Conjointly, the present shear lagmodelallowsone to incorporate the application of the radial load on theRVE which accounts for the lateral extensional interactions

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x/Lf

o*

D/Db = L/Lf = 1.05, A/Ln = 0.0340

D/Db = L/Lf = 1.10, A/Ln = 0.0446

D/Db = L/Lf = 1.15, A/Ln = 0.0615

D/Db = L/Lf = 1.20, A/Ln = 0.0841

Fig. 17. Variation of the transverse shear stress at the interface between the polymerand the short composite fuzzy fiber along its length when the wavy CNTs are coplanarwith the 1e3 (10e30) plane (DT ¼ 300 K, qo ¼ 0, Lf/a ¼ 20).

between the adjacent RVEs. Such lateral extensional interactionsbetween the adjacent RVEs are intrinsic to many manufacturingprocesses of the short fiber composites and play an importantrole in the mechanical behavior of the short fiber composites.

2. If the amplitudes of the sinusoidallywavy CNTs are parallel to theaxes of the carbon fibers (i.e., 1e3 plane), the load transfercharacteristics of the SFFRC are significantly improved comparedto that of the SFFRC containing either the straight CNTs or thewavy CNTs being coplanar with the 2e3 plane irrespective of themagnitudes of the applied radial and thermal loads on the RVE.

3. The load transfer characteristics of the SFFRC containing wavyCNTs to be coplanarwith the1e3planeare significantly improvedfor the higher values of the applied radial load on the RVE and thelower values of the magnitude of the temperature variation.

4. With and without the application of the radial and the thermalloads on the RVE, the load transfer characteristics of the SFFRCare significantly improved with the increase in the spacingsbetween the adjacent short composite fuzzy fibers uniformlyinterlaced in the polymer matrix.

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

x/Lf

i*

Lf/a = 5

Lf/a = 10

Lf/a = 30

Lf/a = 50

Fig. 19. Variation of the transverse shear stress at the interface between the polymermatrix nanocomposite and the carbon fiber along its length when the wavy CNTs arecoplanar with the 1e3 (10e30) plane (DT ¼ 300 K, qo ¼ 0, D/Db ¼ L/Lf ¼ 1.1).


5. The minimum effective value of the carbon fiber aspect ratiorequired for the effective load transfer is found as 30. Beyond thevalue of the carbon fiber aspect ratio as 30, the values of themaximum stresses (s*, s*i and s�o) become independent of thecarbon fiber aspect ratio irrespective of the magnitudes of theapplied radial and thermal loads on the RVE.

Appendix A

The constants Ai (i¼ 1, 2, 3 and 4) appeared in Eqs. (23) and (24)are expressed as follows:

A1 ¼ aCc66ðb2�a2Þ

b2ln r

a � 12

�r2 � a2

��;

A2 ¼ � bCc66�b2 � a2

� a2ln ra� 12

�r2 � a2

��

A3 ¼ aCc66�b2 � a2

� b2ln ba� 12

�b2 � a2

��and

A4 ¼ bCm66�R2 � b2

� R2ln rb� 12

�r2 � b2

�� bCc66�b2 � a2

� a2ln ba� 12

�b2 � a2

��(A1)

The constants Aij and Kij appeared in Eqs. (34) and (35) areexpressed as follows:

K11 ¼ ð1=DÞhðA22A33 � A23A32Þ

�Cf12 � Cc13

�þ ðA13A32 � A12A33Þ

�C

K21 ¼ �ð1=DÞhðA21A33 � A23A31Þ

�Cf12 � Cc13

�þ ðA13A31 � A11A33Þ

�K31 ¼ ð1=DÞ

hðA21A32 � A22A31Þ

�Cf12 � Cc13

�þ ðA12A31 � A11A32Þ

�C

K12 ¼ ð1=DÞ�� ðA22A33 � A23A32ÞA1Cc13 þ ðA13A32 � A12A33Þ

�Cc13A

K22 ¼ �ð1=DÞ�� ðA21A33 � A23A31ÞA1Cc13 þ ðA13A31 � A11A33Þ

�Cc13


�Cc13A


�Cc13A

K23 ¼ �ð1=DÞ�� ðA21A33 � A23A31ÞA2Cc13 þ ðA13A31 � A11A33Þ

�Cc13


�Cc13A

K14 ¼ ð1=DÞ½ðA12A23 � A13A22Þ�; K24 ¼ �ð1=DÞ½ðA11A23 � A13A21ÞK34 ¼ ð1=DÞ½ðA11A22 � A12A21Þ�;

K15 ¼ ð1=DÞhðA22A33 � A23A32Þ

�lc22 � lf22

�þ ðA13A32 � A12A33Þ

�lm2

K25 ¼ �ð1=DÞhðA21A33 � A23A31Þ

�lc22 � lf22

�þ ðA13A31 � A11A33Þ

�K35 ¼ ð1=DÞ

hðA21A32 � A22A31Þ

�lc22 � lf22

�þ ðA12A31 � A11A32Þ

�lm2

A11 ¼ Cc23 � Cf23 � Cf33 � Cc33b2þa2b2�a2 �

2Cf12C

c13

Cf11

þ 2ðCf12Þ2

Cf11

; A12 ¼ 2Cc33b

2

b2�a2 ;

A13 ¼ 2Cc33

b2�a2; A21 ¼ 2Cc33a

2

b2�a2 þ2Cc

13Cf12

Cf11

� 2Cm12C

f12

Cf11

A22 ¼ Cm11 þ Cm12 � Cc23 � Cc33b2þa2b2 �a2; A23 ¼ 1

b2

�Cm12 � Cm11 � Cc23 � Cc33

bb

A31 ¼ �2Cf12C

m12

Cf11

; A32 ¼ Cm11 þ Cm12 and A33 ¼ 1R2

�Cm12 � Cm11

�

in which

D ¼ A11ðA22A33 � A23A32Þ þ A12ðA31A23 � A21A33Þþ A13ðA21A32 � A22A31Þ

The constants Li (i ¼ 1, 2, 3,., 10) appeared in Eqs. (36) and (37)are expressed as follows:

c13 � Cm12

�� ðA12A23 � A13A22ÞCm12i;

Cc13 � Cm12�� ðA11A23 � A13A21ÞCm12

i;

c13 � Cm12

�� ðA11A22 � A12A21ÞCm12i;

1 � Cm12A3�� ðA12A23 � A13A22ÞA3C

m12 ;

A1 � Cm12A3�� ðA11A23 � A13A21ÞA3C

m12 ;

1 � Cm12A3�� ðA11A22 � A12A21ÞA3C

m12 ;

2 � Cm12A4�� ðA12A23 � A13A22ÞA4C

m12 ;

A2 � Cm12A4�� ðA11A23 � A13A21ÞA4C

m12 ;

2 � Cm12A4�� ðA11A22 � A12A21ÞA4C

m12 ;

�;

2 � lc22�þ ðA12A23 � A13A22Þlm22

i;

lm22 � lc22�þ ðA11A23 � A13A21Þlm22

iand

2 � lc22�þ ðA11A22 � A12A21Þlm22

i

(A3)

2þa22�a2

(A2)

L1 ¼ Cc11

Cf11

þ B1k11 þ B2k21 þ B3k31; L2 ¼ B1k12 þ B2k22 þ B3k32 þ B4;

L3 ¼ B1k13 þ B2k23 þ B3k33 þ B5; L4 ¼ B1k14 þ B2k24 þ B3k34;

L5 ¼ B1k15 þ B2k25 þ B3k35 � lc11; L6 ¼ Cm11

Cf11� 2Cm

11Cf12

Cf11

k11 þ 2Cm12k21;

L7 ¼ �2Cm11C

f12

Cf11

k12 þ 2Cm12k22 þ Cm11A3; L8 ¼ �2Cm

11Cf12

Cf11

k13 þ 2Cm12k23 þ Cm11B6

L9 ¼ �2Cm11C

f12

Cf11

k14 þ 2Cm12k24 and L10 ¼ �2Cm

11Cf12

Cf11

k15 þ 2Cm12k25 � lm11

(A4)


in which

B1 ¼ �

2Cc12a

2

b2�a2 þ2Cc

11Cf12

Cf11

!; B2 ¼ 2Cc

12b2

b2�a2 ; B3 ¼ 2Cc12

b2�a2;

B4 ¼ aCc11

Cc66ðb2�a2Þ2

b4ln b

a þ a2b2 � 14

�3b4 þ a4

��;

B5 ¼ � bCc11

Cc66ðb2�a2Þ2

a2b2ln b

a � 14

�b4 � a4

��and

B6 ¼ bCm66ðR2�b2Þ2

R4ln R

b þ R2b2 � 14

�3R4 þ b4

�� bCc66ðb2�a2Þ

a2ln b

a � 14

�b2 � a2

��

The coefficients Li (i ¼ 17, 18, 19, 20 and 21) of the differentialequation given by Eq. (43) are expressed as follows:

L16 ¼ �a2

L2L8L3

� L7

�; L17 ¼ 1

L16

L1L8L3

þ�

a2b2�a2

L8L3� L6 �

�R2�b2

b2�a2

L8L12L3L13

� L12L13

�;

L18 ¼ � 1L16

�R2�b2

b2�a2

L8L3þ L11

L13

�; L19 ¼ � 1

L16

�R2�b2

b2�a2

R2L8L3L13

þ R2

L13

�and

L20 ¼ � 1L16

�R2�b2

b2�a2

L8L14L3L13

þ L14L13

�and L21 ¼ � 1

L16

�R2�b2

b2�a2

L8L15L3L13

þ L15L13

�(A5)

The constants L22, L23, L24 and L25 evaluated by using the endconditions given by Eq. (49) and Eq. (45) are explicitly expressed asfollows:

N23 ¼ bpf sinh�bpf L

�nasinh

�aLf�cosh

�apf Lf

�� apf sinh

�apf Lf

�cos

�asinh�aLf�cosh

�bpf Lf

�osþ asinh

�aLf�n

apf sinh�apf L

�co

��L19L18

s� L20L18

qo �L21L18

DT�þhabpf sinh

�bpf L

�sinh

�aLf�n

cos

�nsinh

�bpf L

�sinh

�apf Lf

�� sinh

�apf L

�sinh

�bpf Lf

�oþ aapf s

�(Lpf19Lpf18

s� Lpf20Lpf18

qo �Lpf21Lpf18

DT

)

D1 ¼nbpf sinh

�bpf L

�cosh

�apf L

�� apf sinh

�apf L

�cosh

�bpf L

�o�na

N25 ¼ bpf sinh�bpf L

�nbsinh

�bLf�cosh

�apf Lf

�� apf cosh

�bLf�sinh

�� bsinh

�bLf�cosh

�bpf Lf

�osþ bsinh

�bLf�n

apf sinh�apf L

�co

� L21L18

DT�þhbbpf sinh

�bpf L

�sinh

�bLf�n

cosh�apf L

�� cosh

�

� sinh�apf L

�sinh

�bpf Lf

�oþ apfbsinh

�bLf�sinh

�apf L

�ncosh

L22 ¼ 0; L23 ¼ N23D1

; L24 ¼ 0 and L25 ¼ N25D1

(A8)

in which

h�aLf�o

sþ apf sinh�apf L

�nbpf sinh

�bpf Lf

�cosh

�aLf�

sh�bpfL

�� bpf sinh

�bpf L

�cosh

�apf L

�oh�apf L

�� cosh

�apf Lf

�oþ apfbpf cosh

�aLf�

inh�aLf�sinh

�apf L

�ncosh

�bpf L

�� cosh

�bpf Lf

�oi

sinh�aLf�cosh

�bLf�� bsinh

�bLf�cosh

�aLf�o

apf Lf�o

sþ apf sinh�apf L

�nbpf cosh

�bLf�sinh

�bpf Lf

�sh�bpf L

�� bpf sinh

�bpf L

�cosh

�apf L

�o��L19L18

s� L20L18

qo

apf Lf�o

þ apfbpf cosh�bLf�n

sinh�bpf L

�sinh

�apf Lf

��bpf Lf

�� cosh

�bpf L

�oi�(Lpf19Lpf18

s� Lpf20Lpf18

qo �Lpf21Lpf18

DT

)


The coefficients Li (i¼ 34, 35, 36, 37 and 38) appeared in the Eqs.(56)�(58) are expressed as follows:

L34 ¼ 1L33

a2L28L26

þ �b2 � a2��L2

L7þ L28L31

L26

þ �R2 � b2

��;

L35 ¼ 1L33

a2L27L26

þ �b2 � a2��L27L31

L26

�;

L36 ¼ 1L33

� a2R2

L26��b2 � a2

�R2�L31L26

�;

L37 ¼ 1L33

a2L29L26

þ�b2 � a2

��L29L31L26

�and

L38 ¼ 1L33

a2L30L26

þ�b2 � a2

��L30L31L26

�(A10)

in which

L26 ¼ a2L7L11L6

þ L12; L27 ¼ �L11L6; L28 ¼

�R2�b2

2b

L8L11L6

� L13;

L29 ¼ L9L11L6

� L14; L30 ¼ L10L11L6

� L15; L31 ¼ L1 � L2L6L7

L32 ¼�R2�b22b

L3 � L2L8

L7

�and L33 ¼ �

b2 � a2�L32

References

Anumandla, V., Gibson, R.F., 2006. A comprehensive closed form micromechanicsmodel for estimating the elastic modulus of nanotube-reinforced composites.Compos.: Part A Appl. Sci. Manuf. 37 (12), 2178e2185.

Ashrafi, B., Hubert, P., 2006. Modeling the elastic properties of carbon nanotubearray/polymer composites. Compos. Sci. Technol. 66, 387e396.

Bandow, S., 1997. Radial thermal expansion of purified multiwall carbon nanotubesmeasured by X-ray diffraction. Jpn. J. Appl. Phys. Part 2 36 (10B), 1403e1405.

Bekyarova, E., Thostenson, E.T., Yu, A., Kim, H., Gao, J., Tang, J., Hahn, H.T., Chou, T.W.,Itkis, M.E., Haddon, R.C., 2007. Multiscale carbon nanotube-carbon fiber rein-forcement for advanced epoxy composites. Langmuir 23 (7), 3970e3974.

Berhan, L., Yi, Y.B., Sastry, A.M., 2004. Effect of nanorope waviness on the effectivemoduli of nanotube sheets. J. Appl. Phys. 95 (9), 5027e5034.

Bower, C., Zhu, W., Jin, S., Zhou, O., 2000. Plasma-induced alignment of carbonnanotubes. Appl. Phys. Lett. 77, 830e832.

Chatzigeorgiou, G., Seidel, G.D., 2011. Lagoudas DC. Effective mechanical propertiesof “fuzzy fiber composites. Compos. Part B 43, 2577e2593.

Chen, X., Wang, X., Liu, B.Y., 2009. Effect of temperature on elastic properties ofsingle-walled carbon nanotubes. J. Reinf. Plast. Compos. 28 (5), 551e569.

Ci, L.J., Zhao, Z.G., Bai, J.B., 2005. Direct growth of carbon nanotubes on the surfaceof ceramic fibers. Carbon 43, 855e894.

De Riccardis, M.F., Carbone, D., Makris, T.D., Giorgi, R., Lisi, N., Salernitano, E.,2006. Anchorage of carbon nanotubes grown on carbon fibers. Carbon 44 (4),671e674.

Downs, W.B., Baker, R.T.K., 1995. Modification of the surface properties of carbonfibers via the catalytic growth of carbon nanofibers. J. Mater. Res. 10 (3),625e633.

Fisher, F.T., Bradshaw, R.D., Brinson, L.C., 2002. Effects of nanotube waviness onthe modulus of nanotube-reinforced polymers. Appl. Phys. Lett. 80 (24),4647e4649.

Gao, H., Ji, B., Jäger, I.L., Arzt, E., Fratzl, P., 2003. Materials become insensitive toflaws at nanoscale: lessons from nature. Proc. Natl. Acad. Sci. U. S. A. 100 (10),5597e5600.

Garcia, E.J., Wardle, B.L., Hart, A.J., Yamamoto, N., 2008. Fabrication and multi-functional properties of a hybrid laminate with aligned carbon nanotubesgrown in situ. Compos. Sci. Technol. 68 (9), 2034e2041.

Griebel, M., Hamaekers, J., 2004. Molecular dynamics simulations of the elasticmoduli of polymer-carbon nanotube composites. Computer Methods Appl.Mech. Eng. 193, 1773e1788.

Hashin, Z., Rosen, B.W., 1964. The elastic moduli of fiber-reinforced materials. ASMEJ. Appl. Mech. 31, 223e232.

Honjo, K., 2007. Thermal stresses and effective properties calculated for fibercomposites using actual cylindrically-anisotropic properties of interfacial car-bon coating. Carbon 45 (4), 865e872.

Hsiao, H.M., Daniel, I.M., 1996. Elastic properties of composites with fiber waviness.Compos.: Part A Appl. Sci. Manuf. 27 (10), 931e941.

Iijima, S., 1991. Helical microtubules of graphitic carbon. Nature 354, 56e58.Iijima, S., Ichihashi, T., 1993. Single-shell carbon nanotubes of 1-nm diameter. Na-

ture 363, 603e605.

Ji, B., Gao, H., 2004. Mechanical properties of nanostructure of biological materials.J. Mech. Phys. Sol. 53, 1963e1990.

Kirtania, S., Chakraborty, D., 2009. Evaluation of thermoelastic properties of carbonnanotube-based composites using finite element method. In: Proceedings ofthe International Conference on Mechanical Engineering Dhaka, BangladeshAM-13.

Krishnan, A., Dujardin, E., Ebbesen, E.W., Yianilos, P.N., Treacy, M.M.J., 1998. Young’smodulus of single-walled nanotubes. Phys. Rev. B 58 (20), 14013e14019.

Kundalwal, S.I., Ray, M.C., 2011. Micromechanical analysis of fuzzy fiber reinforcedcomposites. Int. J. Mech. Mater. Des. 7 (2), 149e166.

Kundalwal, S.I., Ray, M.C., 2012a. Effective properties of a novel continuous fuzzy-fiber reinforced composite using the method of cells and the finite elementmethod. Eur. J. Mech. A/Solids 36, 191e203.

Kundalwal, S.I., Ray, M.C., 2012b. Effective properties of a novel composite rein-forced with short carbon fibers and radially aligned carbon nanotubes. Mech.Mater. 53, 47e60.

Kundalwal, S.I., Ray, M.C., 2013. Effect of carbon nanotube waviness on the elasticproperties of the fuzzy fiber reinforced composites. ASME J. Appl. Mech. 80,021010.

Kwon, Y., Berber, S., Tomanek, D., 2004. Thermal contraction of carbon fullerenesand nanotubes. Phys. Rev. Lett. 92 (1), 015901.

Laws, N., 1973. On the thermostatics of composite materials. J. Mech. Phys. Sol. 21(1), 9e17.

Li, C., Chou, T.W., 2003. A Structural mechanics approach for the analysis of carbonnanotubes. Int. J. Sol. Struct. 40 (10), 2487e2499.

Li, C., Chou, T.W., 2009. Failure of carbon nanotube/polymer composites and theeffect of nanotube waviness. Compos.: Part A Appl. Sci. Manuf. 40 (10),1580e1586.

Liew, K.M., Yan, J.W., Sun, Y.Z., He, L.H., 2011. Investigation of temperature effect onthe mechanical properties of single-walled carbon nanotubes. Compos. Struct.93, 2208e2212.

Liu, J.Z., Zheng, Q.S., Wang, L.F., Jiang, Q., 2005. Mechanical properties of single-walled carbon nanotube bundles as bulk materials. J. Mech. Phys. Sol. 53 (1),123e142.

Liu, Y.J., Chen, X.L., 2003. Evaluations of the effective material properties of carbonnanotube-based composites using a nanoscale representative volume element.Mech. Mater. 35, 69e81.

Lopez Manchado, M.A., Valentini, L., Biagiotti, J., Kenny, J.M., 2005. Thermaland mechanical properties of single-walled carbon nanotubes-polypropylene composites prepared by melt processing. Carbon 43 (7),1499e1505.

Mallick, P.K., 1997. Composites Engineering Handbook. Marcel Dekker, USA.Maniwa, Y., Fujiwara, R., Kira, H., Tou, H., Kataura, H., Suzuki, S., Achiba, Y.,

Nishibori, E., Takata, M., Sakata, M., Fujiwara, A., Suematsu, H., 2001. Thermalexpansion of single-walled carbon nanotube (SWCNT) bundles: X-ray diffrac-tion studies. Phys. Rev. B 64 (24), 241402.

Meguid, S.A., Wernik, J.M., Al Jahwari, F., 2013. Toughening mechanisms in multi-phase nanocomposites. Int. J. Mech. Mater. Des. 9, 115e125.

Meguid, S.A., Wernik, J.M., Cheng, Z.Q., 2010. Atomistic-based continuum repre-sentation of the effective properties of nano-reinforced epoxies. Int. J. Sol.Struct. 47, 1723e1736.

Mori, T., Tanaka, K., 1973. Average stress in matrix and average elastic energy ofmaterials with misfitting inclusions. Acta Metall. 21 (5), 571e574.

Nairn, J.A., 1997. On the use of shear-lag methods for analysis of stress transfer inunidirectional composites. Mech. Mater. 26 (2), 63e80.

Natsuki, T., Tantrakarn, K., Endo, M., 2004. Prediction of elastic properties for single-walled carbon nanotubes. Carbon 42 (1), 39e45.

Odegard, G.M., Gates, T.S., Wise, K.E., Park, C., Siochi, E.J., 2003. Constitutivemodeling of nanotube-reinforced polymer composites. Compos. Sci. Technol. 63(11), 1671e1687.

Pan, Z.W., Xie, S.S., Chang, B.H., Sun, L.F., Zhou, W.Y., Wang, G., 1999. Direct growthof aligned open carbon nanotubes by chemical vapor deposition. Chem. Phys.Lett. 299 (1), 97e102.

Peters, S.T., 1998. Handbook of Composites. Chapman and Hall, London.Pipes, R.B., Hubert, P., 2003. Helical carbon nanotube arrays: thermal expansion.

Compos. Sci. Technol. 63 (11), 1571e1579.Qian, D., Dickey, E.C., Andrews, R., Rantell, T., 2000. Load transfer and deformation

mechanisms in carbon nanotube-polystyrene composites. Appl. Phys. Lett. 76(20), 2868e2870.

Qiu, Y.P., Weng, G.J., 1990. On the application of Mori-Tanaka’s theory involvingtransversely isotropic spheroidal inclusions. Int. J. Eng. Sci. 28 (11), 1121e1137.

Ray, M.C., Guzman de Villoria, R., Wardle, B.L., 2009. Load transfer analysis in shortcarbon fibers with radially-aligned carbon nanotubes embedded in a polymermatrix. J. Adv. Mater. 41 (4), 82e94.

Ruoff, R.S., Lorents, D.C., 1995. Mechanical and thermal properties of carbonnanotubes. Carbon 33 (7), 925e930.

Saether, E., Frankland, S.J.V., Pipes, R.B., 2003. Transverse mechanical properties ofsingle-walled carbon nanotube crystals. Part I: determination of elastic moduli.Compos. Sci. Technol. 63 (11), 1543e1550.

Sager, R.J., Klein, P.J., Lagoudas, D.C., Zhang, Q., Liu, J., Dai, L., Baur, J.W., 2009. Effectof carbon nanotubes on the interfacial shear strength of T650 carbon fiber in anepoxy matrix. Compos. Sci. Technol. 69, 898e904.

Shady, E., Gowayed, Y., 2010. Effect of nanotube geometry on the elastic propertiesof nanocomposites. Compos. Sci. Technol. 70 (10), 1476e1481.

http://refhub.elsevier.com/S0997-7538(13)00114-9/sref1




















































































































































































Shaffer, M.S.P., Windle, A.H., 1999. Fabrication and characterization of carbonnanotube/poly(vinyl alcohol) composites. Adv. Mater. 11 (11), 937e941.

Shen, L., Li, J., 2004. Transversely isotropic elastic properties of single-walled carbonnanotubes. Phys. Rev. B 69, 045414.

Thostenson, E.T., Li, W.Z., Wang, D.Z., Ren, Z.F., Chou, T.W., 2002. Carbon nano-tube/carbon fiber hybrid multiscale composites. Appl. Phys. Lett. 91 (9),6034e6037.

Thostenson, E.T., Reng, Z., Chou, T., 2001. Advances in the science and technologyof carbon nanotubes and their composites. Compos. Sci. Technol. 61 (13),1899e1912.

Treacy, M.M.J., Ebbesen, T.W., Gibson, J.M., 1996. Exceptionally high Young’smodulus observed for individual carbon nanotubes. Nature 381, 678e680.

Tsai, C.H., Zhang, C., Jack, D.A., Liang, R., Wang, B., 2011. The effect of inclusionwaviness and waviness distribution on elastic properties of fiber-reinforcedcomposites. Compos.: Part B Eng. 42 (1), 62e70.

Tyson, B.M., Abu Al-Rub, R.K., Yazdanbakhsh, A., Grasley, Z., 2011. Carbon nanotubesand carbon nanofibers for enhancing the mechanical properties of nano-composite cementitious materials. ASCE J. Mater. Civil Eng. 23, 1028e1035.

Veedu, V.P., Cao, A., Li, X., Ma, K., Soldano, C., Kar, S., Ajayan, P.M., Ghasemi-Nejhad, M.N., 2006. Multifunctional composites using reinforced laminae withcarbon-nanotube forests. Nat. Mater. 5, 457e462.

Vigolo, B., Penicaud, A., Coulon, C., Sauder, C., Pailler, R., Journet, C., Bernier, P.,Poulin, P., 2000. Macroscopic fibers and ribbons of oriented carbon nanotubes.Science 290 (5495), 1331e1334.

Villeneuve, J.F., Naslain, R., Fourmeaus, R., Sevely, J., 1993. Longitudinal/radialthermal expansion and poisson ratio of some ceramic fibers as measured bytransmission electron microscopy. Compos. Sci. Technol. 49 (1), 89e103.

Vlasveld, D.P.N., Daud, W., Bersee, H.E.N., Picken, S.J., 2007. Continuous fibre com-posites with a nanocomposite matrix: improvement of flexural and compres-sive strength at elevated temperatures. Compos. Part A 38 (3), 730e738.

Wernik, J.M., Meguid, S.A., 2010. Recent developments in multifunctional nano-composites using carbon nanotubes. ASME Appl. Mech. Rev. 63, 050801.

Wu, A.S., Chou, T.W., 2012. Carbon nanotube fibers for advanced composites. Mater.Today 15, 302e310.

Yamamoto, N., Hart, A.J., Garcia, E.J., Wicks, S.S., Duong, H.M., Slocum, A.H.,Wardle, B.L., 2009. High-yield growth and morphology control of aligned car-bon nanotubes on ceramic fibers for multifunctional enhancement of structuralcomposites. Carbon 47 (3), 551e560.

Zhang, Q., Huang, J.Q., Zhao, M.Q., Qian, W.Z., Wang, Y., Wei, F., 2008. Radial growthof vertically aligned carbon nanotube arrays from ethylene on ceramic spheres.Carbon 46 (8), 1152e1158.























































Documents

European Journal of Mechanics A/Solids...A thermomechanical shear lag analysis of short fuzzy ﬁber reinforced composite containing wavy carbon nanotubes M.C. Ray*, S.I. Kundalwal