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Mechanics of Materials 41 (2009) 279–292

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Mechanics of Materials

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Curved-fiber pull-out model for nanocomposites. Part 1: Bondedstage formulation

Xinyu Chen a, Irene J. Beyerlein c, L. Catherine Brinson a,b,*

a Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USAb Department of Materials Science and Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USAc Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 January 2008Received in revised form 27 November 2008

0167-6636/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.mechmat.2008.12.004

* Corresponding author. Address: Department of Ming, Northwestern University, 2145 Sheridan RoadUSA. Tel.: +1 847 467 2347; fax: +1 847 510 0540.

E-mail address: [email protected] (L.C

This is the first part of two papers in which an analytical curved-fiber pull-out model fornanocomposites is proposed. In nanotube-reinforced polymer composites, nanotubes aretypically curved and entangled, a reinforcement morphology that will greatly impact thethermomechanical properties of the material. As the first step to explicitly take intoaccount nanotube curvature and study its effect on nanocomposite mechanical properties,we develop a pull-out model in which the fiber has constant curvature. The model includesthe entire pull-out process, namely the bonded, debonding, and sliding stages. In this firstpaper we formulate the bonded stage based on classic shear lag model assumptions anddevelop a 3D finite element model to verify assumptions. The results from a parametricstudy indicate that fibers with more curvature and longer embedded length need higherdebond initiation force. The finite element results and analytical results show agreementboth qualitatively and quantitatively.

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1. Introduction

1.1. Problem statement

Since the emergence of nanocomposites, intensive workin synthesis, characterization and modeling has providedbetter understanding of the material’s mechanical perfor-mance (Ajayan et al., 2003; Andrews et al., 2002; Breuerand Sundararaj, 2004; Buryachenko et al., 2005; Colemanet al., 2006; Fisher and Brinson, 2006; Valavala and Ode-gard, 2005). For instance, it has been consistently observedthat small amounts of nanotubes can increase stiffnessabove that of the base polymer (Chang et al., 2005; Cole-man et al., 2003; Goh et al., 2003; Liu et al., 2004; Qianet al., 2000; Velasco-Santos et al., 2003; Zeng et al.,2004). The effect of nanoparticles on composite toughnesshas also been studied (Andrews and Weisenberger, 2004).

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echanical Engineer-, Evanston, IL 60208,

. Brinson).

Significant improvements in toughness have been ob-served in some spherical nanoparticle systems (Cotterellet al., 2007; Naous et al., 2006; Ragosta et al., 2005; Xuet al., 2008). For example Ash et al. (2002) demonstrated78% increase in ductility of PMMA with addition of 5 wt%of nano-alumina (39 nm diameter) particles. In contrast,the results for nanoplate and nanotube reinforced poly-mers have varied a great deal. Some researchers (Moni-ruzzaman et al., 2006; Yasmin et al., 2006; Zheng et al.,2004) observed substantial losses in ductility and hencetoughness, while others report minor improvements intoughness (Gojny et al., 2004, 2005; Ma et al., 2007). In afew cases, significant improvement on toughness withtube-based reinforcement has been observed (Blondet al., 2006; Chen et al., 2005; Yang et al., 2007). Fiedleret al. (2006) measured a 45% increase in fracture toughnessof CNT/epoxy composites with 0.3% of amino-functional-ised double-walled carbon nanotubes. Dondero and Gorga(2006) reported with 0.25 wt% MWNT polypropylenematrix’s toughness increases 32%.

Given the inherently large strain capability of nano-tubes, it should be possible to consistently design a

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280 X. Chen et al. / Mechanics of Materials 41 (2009) 279–292

nanotube composite with significantly improved fracturetoughness. Limited success to date and the wide rangeof experimentally observed results calls for a betterunderstanding of the underlying deformation mecha-nisms governing nanocomposite fracture. Such under-standing is critical for design of the nanocompositemicrostructure (nanotube–polymer interface, nanotubevolume fraction, etc) for enhanced toughness.

One important toughness mechanism is nanotube pull-out. As in conventional fiber pull-out, there are threestages in nanotube pull-out. In the first stage, called thebonded stage, the nanotube and the matrix are well-bonded. As the pull-out force increases to a certain thresh-old value, the debonding stage begins. During debonding,part of the nanotube moves along the debonded interfaceresisted by a friction force, while the rest of the nanotubestays well bonded to the matrix. When debonding extendsto the entire interface, sliding occurs. In this final stage, theentire nanotube slides through the matrix resisted by fric-tional forces.

The pull-out problem for nanotube-reinforced compos-ites has been studied experimentally, analytically, andnumerically. Individual nanotube pull-out tests have beenperformed using an atomic force microscopy (AFM) stagesto access the interfacial strength. The Wagner group (Bar-ber et al., 2003. 2004, 2006; Cooper et al., 2002; Nurielet al., 2005) successfully traced the pull-out force andnanotube locations to obtain the force–displacement curveand they further were able to calculate average interfacialshear stress and fracture energy for certain nanotube–polymer interfaces. Analytically, continuum mechanicsmodels for conventional fiber/polymer interface, such asthe Kelly and Tyson model, and models based on local den-sity approximation and classical elastic shell theory, havebeen extended to describe nanotube/polymer interfaces(Gao and Li, 2005; Lau, 2003; Wagner, 2002). Xiao and Liao(2004) developed a nanotube pull-out model for the slidingstage by incorporating nanotubes’ nonlinear elastic prop-erty and found the nonlinearity has a great impact on theinterfacial shear stress distribution. Other researchers(Frankland et al., 2002; Frankland and Harik, 2003; Gouet al., 2004; Liao and Li, 2001; Lordi and Yao, 2000; Wonget al., 2003) considered the physical structure of nanotubesand polymer chains at the nanoscale and applied molecu-lar mechanics and molecular dynamics (MD) calculationsto the problem of pull-out, elucidating the stress transfermechanism as a function of the nanotube/polymer inter-face properties. An average interfacial shear stress calcu-lated from MD simulation shows that bonded or non-bonded interactions at the interface can lead to effectivestress transfer from polymer matrix to nanotubes (Frank-land et al., 2002; Gou et al., 2004; Liao and Li, 2001; Wonget al., 2003). Although MD can describe interactions atatomic levels through suitable potential models, it is lim-ited by length and time scales due to the small time stepsrequired. The statistical nature of MD calculations requiresthe MD simulation to run for a sufficiently long time toperform enough sampling for physical properties. Theselimitations make continuum mechanics approaches morefavorable for analyses at length scales in the micronrange.

All the continuum mechanics-based and molecularmechanics-based models above only consider nanotubeswhich are straight and aligned. However, in nanorein-forced polymers, nanotubes are typically curved andentangled in-situ as shown in Fig. 1. The fine, white, hair-like filaments in Fig. 1 are the nanotubes.

The curved fiber morphology will greatly impact ther-momechanical and fracture properties of the compositesystems. While the effects of nanotube curvature on stiff-ness have been addressed (Bradshaw et al., 2003; Fisheret al., 2002, 2003), its influence on ductility and fracturetoughness has yet to be examined at any length scale.

For traditional (larger scale) fiber composites, the effectof reinforcement morphology has been explored in depth.While a weak interface can enhance toughness, it also re-duces strength. A change in the morphology of the fibercoupled with the weak interface can, however, lead to bothhigh toughness and high strength. One example is the socalled bone-shaped-short-fiber composites (Beyerleinet al., 2001; Shuster et al., 1996; Zhu et al., 1999, 2001).Composites reinforced by bone-shaped-short fibers areable to transfer stress effectively through the enlarged fiberends while still providing toughness enhancementsthrough the weak interface. Similarly, we propose thatnanocomposites with appropriately designed interfacesand morphologies may ultimately lead to composites withimproved stiffness, strength and toughness.

For predictive capability and design, it will be importantto account for and understand the effects of nanotubecurvature and entanglement on the critical properties ofnanocomposites, such as toughness and strength.

As a first step, in this two-part series, the curvature ef-fect is added to a shear-lag-based model (Lawrence, 1972)to study nanotube pull-out. Shear lag modeling is a pop-ular and successful scheme to address fiber/matrix inter-face problems in conventional composites. This article isthe first part of the series which presents the formulationfor the bonded stage. It is structured as follows. First abrief review of conventional straight fiber pull-out model-ing is given. Then, a 2D analytical model for singlecurved-fiber pull-out is derived. A 3D finite element sim-ulation model is built to check some of the simplifyingassumptions made in the formulation. With the analyticalmodel, we examine the influence of fiber curvature on theinitial portion of the force–displacement curve when thefiber and matrix are still bonded. Finite element simula-tion results are then compared with those from theanalytical formulation.

1.2. Review: different straight fiber pull-out models

Since straight fibers are prevalent in conventional fiber-reinforced composites, research in modeling single straightfiber pull-out has been extensively carried out. Fig. 2shows a concentric cylinder model commonly used as arepresentative volume element of fiber composite modelsor in single fiber pull-out analyses. Cox (1952) proposedthe original shear lag model based on linear elasticity,which involves three inherent assumptions, namely (1)shear stress is a function of axial displacement; (2) thefiber and matrix stresses and displacements in the axial

Fig. 1. SEM photograph of curved single walled carbon nanotubes in PMMA (functionalized tubes) (Ramanathan et al., 2005). The fine, white, hair-likefilaments are the nanotubes.

Fig. 2. Commonly used concentric cylinder geometry for fiber compositesand single fiber pull-out problems with cylindrical coordinate system.The fiber is the inner cylinder and the matrix is the outer cylinder.

X. Chen et al. / Mechanics of Materials 41 (2009) 279–292 281

direction are independent of radial coordinates; (3) stressin the axial direction is dominant over stress in the othertwo directions. These assumptions are explained in detailin Gao and Li (2005), Nairn (1997). Since then, the shearlag idea has been widely applied to straight fiber pull-outanalysis for different composite systems and has been fur-ther developed with various degrees of approximation(Cox 1952, 1990; Gao and Li, 2005; Gao et al., 1988; Hsueh,1992a; Kerans and Parthasarathy, 1991; Kharrat et al.,2006; Kim et al., 2004; Kim and Mai, 1998; Nairn, 1997;

Nairn and Wagner, 1996; Rosen, 1964; Tsai and Kim,1996; Wu and Davies, 2005; Wu and Yu, 1994). For exam-ple, some researchers assumed zero radial displacementand uniform matrix deformation confined in the cylindergeometry (Hsueh, 1988, 1990; Takaku and Arridge 1973).Gao and Li (2005) developed a shear lag model for carbonnanotube/polymer composites by modeling a cappednanotube as an effective fiber based on molecular structuremechanics. They also modified the ‘free ends’ boundaryconditions in the original Cox model to represent a fullyembedded nanotube. They found that the large aspect ratioof a nanotube can increase interfacial stress transfer, andthus improve the reinforcing effects of nanotubes. Theirpaper is the only effort to apply the shear lag model tonanocomposites to date.

In addition to shear lag assumptions, other theoreticalmodels based on linear elasticity have also been developedto study stress transfer in straight-fiber-reinforced com-posites under various assumptions (Hutchinson and Jen-sen, 1990; Marshall, 1992; McCartney, 1989; Mumm andFaber, 1995; Wu et al., 2000). Finite element analyses havealso been conducted for stress distributions along the fiberaxis (Faber et al., 1986; Grande et al., 1988). Someresearchers further modeled stick-slip sliding features ina dynamic fiber pull-out process (Sridhar et al., 2003; Tsaiand Kim, 1996).

The shear lag model has proven to provide good esti-mates for interfacial stress transfer. Due to its mathemati-cal simplicity, it is widely used in straight fiber reinforcedcomposites. Following this background, our pull-out anal-ysis for curved fiber reinforced composites will be builtupon a shear lag model. As in the straight fiber model,which is axisymmetric and essentially 2D, our analyticalmodel for curved-fiber pull-out is also 2D. Although themotivation of this work lies in the observed curvature ofnanotubes embedded in polymer matrix, as shown inFig. 1, our analytical model can also be applied at conven-tional length scales.

1 The detailed derivation of elasticity equations in the current 2Dcurvilinear system can be found in Appendix II.


2. Single curved-fiber pull-out model for bonded stage

In this section, the analytical derivation for singlecurved-fiber pull-out analysis at the bonded stage is firstpresented. Then a 3D finite element model is constructedto test several assumptions made in our analysis.

As mentioned, fiber pull-out includes three stages. Weare interested in connecting the three stages to obtain aforce–displacement curve, which later can be used directlyor indirectly as a bridging law to predict composite tough-ness and fracture behavior. Although there are many analyt-ical models based on shear lag approach, most of them focusonly on one stage, either the bonded stage or the debondingstage. As one of the limited numbers of paper dealing withmore than one stage, Lawrence (1972) modeled the bondedstage based on a shear lag approach and connected it withthe debonding stage and further identified the existence ofprogressive debonding and catastrophic debonding. Hsuehalso has a series of papers applying shear lag to both bondedand debonding stages and the two stages were connected viaa debonding criterion (Hsueh 1988, 1990, 1992a, 1992b). Inthis work, we choose to build upon the Lawrence model withthe following major modifications: (1) As all shear lag mod-els to date, the Lawrence model is for a straight fiber. There-fore, in our analysis it is modified to account for fibercurvature. (2) Lawrence assumed a ‘free end’ at the fiberembedded end, which would not necessarily be accuratefor nanotubes entangled together as in Fig. 1. Thereforeour model denotes a parameter to account for stress due tothis entanglement. (3) In Lawrence’s work, in the debondingstage the debonded part of the fiber was resisted by a con-stant friction stress. In our work, we consider two frictionmodels, a constant friction model and a Coulomb frictionmodel. The last modification is developed in the second partof this series, which focuses on the debonding and slidingstages and the pull-out curve of the entire pull-out process.

2.1. Analytical derivation of single curved-fiber model forbonded stage

Based on the simple shear lag model, Lawrence (1972)analyzed bonded and debonding stages during straight fi-ber pull-out and identified the possibility of progressivedebonding. From his model, we have newly derived theforce–displacement relationship for a straight fiber in thebonded stage subjected to the modification (2) above, andpresent it in Appendix I. It is important to have this solutionin hand as a basis for measuring the curvature effect.

Fig. 3(a) shows the model geometry of a curved fiberembedded in a matrix material. Small strain conditions inthe fiber and matrix are assumed so that this problem fallsinto the scope of linear elasticity and hence the shear lagmodel can be applied. This small-strain assumption shouldbe sufficiently accurate throughout the pull-out analysis,as the fiber and matrix in our model will eventually debondfrom one another and most of the strain will be accommo-dated by the interface. Both fiber and matrix are isotropicand linearly elastic. The Poisson’s effect is neglected to sim-plify the calculation. As a first step to account for fibers withgeneral curvature geometries, our current fiber is assumedto have a constant radius of curvature R. The fiber has a cir-

cular cross-section with radius rf. It is noted that the analysiswould in general allow for noncircular cross-sections, butwe consider circular only here because of our focus on nano-tube reinforcement. Also as shown in Fig. 3(a), here the fiberis assumed to exist normal to the composite surface. Fiberinclination effects will be considered in another paper.Fig. 3 shows the three stages for curved-fiber pull-out. Thispaper derives equations for stage I. As mentioned earlier,stages II and III are derived in the companion paper.

The 2D curvilinear coordinate system used in our cur-rent analysis is shown in Fig. 3(a). s is the direction alongfiber (tangential direction) and r is always perpendicularto s (radial direction). The variation in the hoop directionfor both matrix and fiber is neglected. At the fiber embed-ded end, s = 0. The angle characterizing the fiber geometryis a, and aR, denoted as L later, equals the original fiberembedded length. Our current model is valid for any a va-lue between 0o and 180o. s0 is the fiber length outside thematrix prior to application of the load. Pf is the pull-outforce at the fiber end. In stage I, i.e., the bonded stage,the fiber and matrix are well-bonded. In the current model,the only interaction between fiber and matrix at this stageis through the interfacial shear stress si. Radial compres-sion is not considered here although it is taken into ac-count for the debonding and sliding stages in part II ofthis series, where radial compression is more significant.All equations in the following derivation are in normalizedform to remove any unnecessary dependencies of the formof the solution on parameters. The normalization factor forlength is fiber radius rf and for stress and moduli it is thefiber Young’s modulus Ef. Accordingly, that for force isp(rf)2Ef. The asterisks indicate normalized values.

Fig. 4 considers the stress equilibrium of a small differ-ential matrix element next to the embedded fiber in the 2Ds–r coordinates. sm

rs is the shear stress at r in the s-direction,si is the shear stress at rf, i.e., interfacial shear stress, in thes-direction. According to equilibrium in the fiber direction,we have for the matrix,

s�i ¼ sm�rs r�; ð1 � r� � rm�Þ; ð1Þ

where sm�rs and s�i are the normalized shear stresses at r*

and at interface, respectively, and the radial position inthe matrix is normalized by rf. rm* is the normalized ‘imag-inary’ matrix radius. rm* is called imaginary because thereare no boundary conditions enforced at the outer bound-aries of the matrix.

From the linear elastic constitutive law, matrix shearstrain is

cmrs ¼

sm�rs

G�m¼ s�i

G�mr�; ð2Þ

where G�m is matrix shear modulus normalized by Ef .The strain–displacement relation in curvilinear coordi-

nates1 gives:

cmrs ¼

R�

R� � r�@um�

r

@s�þ @um�

s

@r�þ um�

s

R� � r�; ð3Þ

where u stands for displacement.

Fig. 5. Stress equilibrium of a differential fiber element in the bondedstage.

Fig. 3. Three stages in pull-out of a single curved fiber. The figure for stage I also illustrates the model geometry. The 2D curved fiber has a constantcurvature R and circular cross-section of radius rf in curvilinear coordinate system s and r. s0 is the free length of the fiber initially not embedded in thematrix.

Fig. 4. Stress equilibrium of a representative matrix segment in thebonded stage. sm

rs is the shear stress at r in the s-direction, si is the shearstress at rf, i.e., interfacial shear stress, in the s-direction.


Consistent implicitly with shear lag assumption (1), thevariation of radial displacement in matrix along s directionis considered negligible, i.e.,

@um�r

@s�� 0 ð4Þ

Substituting Eqs. (3) and (4) into the constitutive law Eq.(2), we obtain the governing equation for matrix.

@um�s

@r�þ um�

s

R� � r�¼ s�i

G�mr�: ð5Þ

Eq. (5) is treated as an ordinary differential equation andits solution is

um�s ðr�Þ ¼ ðR

� � r�Þ gðsÞ � s�iG�mR�

lnr� � R�

r�

� �; ð6Þ

where g(s) is an arbitrary function to be defined by bound-ary conditions.

Considering Eq. (6) at the fiber surface, r* = 1,

um�s ð1Þ ¼ ðR

� � 1Þ gðsÞ � s�iG�mR�

lnð1� R�Þ� �

: ð7Þ

Thus,

gðsÞ ¼ um�s ð1Þ

R� � 1þ s�i

G�mR�lnð1� R�Þ: ð8Þ

Substituting Eq. (8) back to Eq. (6), we obtain the solutionfor the matrix displacement:

um�s ðr�Þ ¼ ðR

� � r�Þ um�s ð1Þ

R� � 1þ s�i

G�mR�lnð1� R�Þr�ðr� � R�Þ

� �: ð9Þ

Rearranging, we obtain the interfacial shear stress ex-pressed as a linear combination of axial matrix displace-ment at r� ¼ rm� and r* = 1in a similar format to thestraight fiber pull-out model as shown in Eq. (7A).

s�i ¼G�mR�

ln ð1�R�Þrm�

ðrm��R�Þ

um�s ðrm�Þ

R� � rm� �um�

s ð1ÞR� � 1

� �: ð10Þ

Considering equilibrium of a fiber segment (see Fig. 5), weobtain,

pðrf�s þ drf�

s Þ cosda2� rf�

s p cosda2þ s�i 2pds� ¼ 0 ð11Þ

where

ds� ¼ R�da ð12Þ

Note that in Fig. 5, and in the above, we have adopted theshear lag assumption that the axial stress in the fiber isuniform across the cross-section (or that the shear modu-lus of the fiber is infinite relative to that for the matrix).This assumption applies to most fiber–polymer matrixsystems.

Fig. 6. (a) Normalized fiber axial stress and (b) normalized interfacialshear stress distribution along normalized fiber axial position in bondedstage for different fiber curvatures. Note that in (a) ats* = 0,rf �

s ¼ r�0 ¼ 1E� 9 from Table 1, which is essentially zero on thescale of these results.

Table 1Parameters for the fiber and matrix used in Fig. 6 (E�m: normalized matrixYoung’s modulus; G�m: normalized matrix shear modulus; L* normalizedfiber embedded length; rm*: normalized imaginary matrix radius; r�0:normalized fiber embedded end stress; P�f : normalized pull-out force).

E�m G�m L* rm* r�0 P�f

1E�2 5E�3 33.3 20 1E�9 2.5E�3


As da ? 0, we obtain the governing equation for the fi-ber in the bonded stage

drf�s

ds�¼ �2s�i : ð13Þ

Interestingly Eq. (13) has the same form for both straightand curved fibers. As we shall see in the sequel, the effectof curvature is actually introduced through s�i .

Combining Eq. (13) with Eq. (10) results in

drf�s

ds�¼ �2

G�mR�


ðrm��R�Þ

um�s ðrm�Þ

R� � rm� �um�

s ð1ÞR� � 1

� �: ð14Þ

To obtain the governing equation for rfs , first a second

derivative is taken,

d2rf�s

ds�2¼ �2

G�mR�


ðrm��R�Þ

1R� � rm�

@um�s ðrm�Þ@s�

� 1R� � 1

dum�s ð1Þds�

� �;

ð15Þ

where um�s ð1Þ ¼ uf�

s .Next recall the following strain–displacement relation-

ships in the fiber and in the matrix

ems ðrm�Þ ¼ R�

R� � rm�@um�

s ðrm�Þ@s�

� 1R� � rm� um�

r ðrm�Þ ð16Þ

efs ¼

R�

R� � 1duf�

s

ds�� 1

R� � 1uf�

r ð17Þ

Inserting into Eq. (15) we obtain

d2rf �s

ds�2¼�2

G�mR�


ðrm��R�Þ

ems ðrm�Þ

R�þ um�

r ðrm�ÞR�ðR� � rm�Þ�

efs

R�� uf�

r

R�ðR� �1Þ

" #:

ð18Þ

As in the straight fiber pull-out model, ems ðrm�Þ is regarded

as a virtual matrix strain as if no fiber exists, i.e.,

ems ðrm�Þ � e1s ðrm�Þ ¼ rf�

sE�mrm�2 (Cox 1952; Lawrence 1972; Nairn

1997). We further assume that um�r ðrm�Þ and uf�

r are small.Note that these assumptions are not typical shear lagassumptions as these two terms do not appear in thestraight fiber case. Finally with these assumptions Eq.(18) gives the following governing equation for the axial fi-ber stress:

d2rf�s

ds�2¼ T�rf�

s ; ð19Þ

whereT� ¼ 2G�mln ð1�R�Þrm�

ðrm��R�Þ

1� 1E�mrm�2

� �: ð20Þ

Note that T* contains the curvature effect.To solve Eq. (19) we apply the following two boundary

conditions:

(1) rf�s ðs� ¼ 0Þ ¼ r�0, where r0 denotes the stress at the

fiber embedded end due to its entanglement withother nanotubes.

(2) rf�s ðs� ¼ L�Þ ¼ r�pull, i.e., stress at the pulled end

required to balance the applied stress rpull, whichequals the pull-out force Pf divided by fiber cross-section area.

Therefore, we have for the fiber stress as a function of s*

rf�s ¼

P�fsinhð

ffiffiffiffiffiT�p

L�Þ� r�0 cothð


L�Þ !

sinhðffiffiffiffiffiT�p

s�Þ

þ r�0 coshðffiffiffiffiffiT�p

s�Þ ð21Þ

and likewise for the interfacial shear stress

s�i ¼ �12

P�fsinhð


L�Þ� r�0 cothð


L�Þ !"

�ffiffiffiffiffiT�p

coshðffiffiffiffiffiT�p

s�Þ þ r�0ffiffiffiffiffiT�p


s�Þi: ð22Þ

The distribution of rf�s and s�i along the fiber s* calculated

from Eqs. (21), (22) at a given pull-out force P�f is shownin Fig. 6(a) and (b), respectively, and the correspondingparameters are listed in Table 1. As can be seen, both stres-ses are largest at the pulled end and smallest at the fiberembedded end, which implies debonding will start fromthe pulled end. The interfacial shear stress at the pulledend reduces as the curvature increases as seen from the

Table 2Properties of fiber and matrix used in the 3D FE model (R, radius of fibercurvature; rf, fiber radius; L, fiber embedded length; Ef, fiber Young’smodulus; Em, matrix Young’s modulus).

R (m) rf (m) L (m) Ef (Pa) Em (Pa)

3.03E�2 1.5E�3 4.99E�2 1E12 1E9

Fig. 7. 3D symmetric FE model in the bonded stage with appliedboundary conditions. Front face is the symmetric plane. The red outlineshows the nanotube-matrix interface on the symmetric plane. The orangecolored outline and points represent following boundary conditions: thematrix left surface is fully fixed, the matrix top and bottom surfaces canonly move in the 1-direction, and fiber nodes at the pulled end are given auniform displacement in the 1-direction. The blue colored points repre-sent the nodes on the symmetric plane are symmetrically constrained.(For interpretation of the references to color in this figure legend, thereader is referred to the web version of this paper.)


three curves with different values of R*. This curvature ef-fect is examined in more detail in Section 3.1.

From the pull-out Pf–d curves, where d is the fiber dis-placement, we can begin to see how curvature would affectcomposite toughness. The Pf–d curve can serve as a bridg-ing law in modeling crack propagation.

The fiber displacement d*(I) (where superscript (I) de-notes stage (I) is composed of two parts: the elongation ofthe embedded fiber and that of the original extruded part.

d�ðIÞ ¼Z L�

0ef

sds� þP�f s�0¼1ffiffiffiffiffiT�p

P�f cothð


L�Þ�P�f


L�Þ

� r�0sinhð


L�Þþr�0 coth


L�� !

þP�f s�0

ð23Þ

This displacement for the bonded stage will be used in thecompanion paper, where the full pull-out curve forbonded, debonding, and sliding stages will be determinedand impact on toughness examined.

At s� ¼ L�, Eq. (22) yields

s�imax¼�12


P�f cothðffiffiffiffiffiT�p

L�Þ� r�0sinhð


L�Þ

!; ð24Þ

The negative sign indicates that the shear stress acts in theopposite direction from what is illustrated in Fig. 5, whichis physically reasonable.

The peak interfacial shear stress s�i increases with pull-out force. As it grows to a certain value denoted as s�s , deb-onding begins. The formulations for debonding and slidingstages are presented in the companion paper.

To further elucidate the curvature effect we compareour results to that for a straight fiber presented below. Adetailed derivation can be found in Appendix I. In this solu-tion, x denotes the fiber axial direction.

rf�x ¼

P�

sinhðffiffiffiffiffiffiQ �

pL�Þ� r�0 cothð

ffiffiffiffiffiffiQ �

pL�Þ

!sinhð


px�Þ

þ r�0 coshðffiffiffiffiffiffiQ �

px�Þ ð25Þ

s�i ¼ �12

P�




pL�Þ

" !

�ffiffiffiffiffiffiQ �

pcoshð


px�Þ þ r�0


psinhð


px�Þi

ð26Þ

d�ðIÞ ¼ 1ffiffiffiffiffiffiQ�

p P�ð cothðffiffiffiffiffiffiQ �

pL�Þ � P�


pL�Þ

� r�0sinhð


pL�Þþ r�0 cothð

ffiffiffiffiffiffiQ�

pL�Þ!þ P�l�0 ð27Þ

s�i max ¼ �12


pP� cothð


pL�Þ � r�0

sinhðffiffiffiffiffiffiQ�

pL�Þ

!ð28Þ

where

Q � ¼ 2G�mln rm� 1� 1

rm�2E�m

� �ð29Þ

These expressions are of the same form as the correspond-ing expressions for a curved fiber with Q* replacing T*.

Recalling our parameter T* in Eq. (20),

T� ¼ 2G�mln ð1�R�Þrm�

ðrm��R�Þ

1� 1E�mrm�2

� �ð30Þ

Note that when R* goes to infinity, T* becomes Q*, andconsequently Eqs. (21)–(24) converge to the results for astraight fiber, Eqs. (25)–(28).

2.2. 3D Finite element model

In our single curved-fiber pull-out analysis for thebonded stage, several assumptions have been made,including the basic shear lag assumption and negligiblematrix radial displacement. To check the validity of theseassumptions, a symmetric 3D finite element model is con-structed and analyzed. The commercial software, I-DEAS,was used to construct the finite element model and thecommercial finite element package, ABAQUS, was used toperform the finite element simulation.

The fiber and matrix are constructed as one body andmeshed with 10-node quadratic tetrahedron elements inI-DEAS. The pull-out simulation at the bonded stage is thenperformed in ABAQUS. The properties of the fiber and ma-


trix used in analysis are listed in Table 2. These are repre-sentative of a nanotube–polymer matrix composite sys-tem. Applied boundary conditions are as illustrated inFig. 7: the matrix left surface is fully fixed, the matrix topand bottom surfaces can only move in the 1-direction,nodes on symmetric plane are properly constrained, and fi-ber nodes at the pulled end are given a uniform displace-ment in the 1-direction.

Fig. 8. Effects on normalized Pf–d curves by changing following param-eters from their initial values: R* = 70, L* = 33, r�0 ¼ 1e� 9. (a) Radius ofcurvature R*, (b) fiber length L*, (c) fiber embedded end stress r�0.

3. Results and discussion

This section presents numerical results for curved-fiberpull-out, first for analytical model and then for the finiteelement model.

3.1. Parametric study

In this section, a parametric study is performed toexamine the effects of different factors on the curved-fiberpull-out behavior in the bonded stage. All parameters stud-ied are in normalized form and therefore the normalizationfactors, such as the fiber radius rf and fiber Young’s modu-lus Ef, do not need to be considered. The following param-eters are chosen to represent a typical polymernanocomposite: E�m ¼ 1E� 2;G�m ¼ 5E� 3; s�s ¼ 3:5e� 5;rm� ¼ 20: The parameters of interest are fiber radius of cur-vature R*, fiber length L*, and fiber axial stress from entan-gled fibers at the embedded end r�0: The base values ofthese parameters are taken as 70, 33, 1e�9, respectively,and then varied individually within a physically reasonablerange for the simulations. Note that L* is taken as a smallervalue than the typical nanotube length, which can be fromseveral hundreds up to several thousands, because ouranalysis focuses on one curved segment of a nanotube.

When only the bonded stage is considered, the Pf–dcurve ends when the critical interfacial shear stress isreached, and debonding starts. Fig. 8(a) and (b) show theeffects of changing R* and L*, respectively, on the pull-outstress to initiate debonding. The displacements shownare purely from fiber elongation because while bonded,no relative displacement between fiber and matrix is al-lowed. Increasing L*, and to a lesser extent the fiber curva-ture, both increase the debond initiation force with astraight fiber (R* = infinity) requires the smallest pull-outforce to initiate debonding. Due to the same value appliedfor the debonding parameter s�s in both straight and curvedfiber cases, we can infer that given the same pull-out forcethe interfacial shear stress in the curved fibers is smallerthan that in the straight fiber. This result implies that theinterfacial shear stresses build up slower in curved fibersthan those in straight fibers. This would be a nice qualityfor composites with curved fibers since it could lead to en-hanced toughness. In Fig. 8(c) the fiber embedded endstress r�0 is changed from 1e�9 to 1e�4. Increasing thisstress leads to a higher debond initiation force because ofthe larger end stress to overcome. The plot also shows anonzero pull-out distance under zero pull-out force, whichis more obvious in the large r�0 case. This offset is due tofiber elongation from the residual stress r�0. Ideally theparameter r�0 should start from zero when no pull-out

loads are applied and increase with pull-out rather thanthe constant value assumed here. However, the effect ofr�0 within the ranges examined is relatively small. In fact,from Fig. 8, both R* and r�0 have little effect in the bondedstage, and L* has the most significant impact on the pull-out curve. Longer curved segments lead to higher debondinitiation force, which implies potential toughnessimprovement of the nanocomposites as desired.

r�0 is treated as a material parameter in the current for-mulation because it describes the axial stress from boththe bonded matrix and the surrounding entangled nano-tubes. As mentioned above, this value should change dur-ing pull-out rather than a constant value. For a straightfiber, Hsueh et al. (1997) has obtained an analytical solu-tion for the embedded end axial stress as a function ofapplied load, matrix and fiber radius, Poisson’s ratio,Young’s modulus, fiber length, and the distance from fiberembedded end to composite surface. Hsueh’s ‘‘imaginary


fiber” technique could be applied to the curved-fiber mod-el but is not considered here.

3.2. Check of analytical assumptions through FE simulation

The 3D finite element results for the deformation andstress field for bonded stage for the pull-out distance of100 lm are, respectively, shown in Fig. 9(a) and (b). Thestress field in the fiber is not uniform along the hoop direc-tion. The lower surface of the fiber displays a much higherstress than of the upper surface. After sampling the displace-ment at several points near the fiber/matrix interface, it isfound that the radial displacement um

r is not negligible com-

Fig. 9. (a) Deformed fiber and matrix (b) Von Mises stress distribution infiber at the bonded stage when pull-out distance is 100 lm.

pared with the axial displacement ums and the ratio um

r =ums

varies from 0.26 to 7.74. The nodes nearer to the pulledend tend to have a smaller radial displacement over axialdisplacement ratio than other nodes. For the straight fibercase, it is found that the ratio remains small and ranges from1e�5 to 1e�2. The curved fiber case illustrated here is an ex-treme case with a large 90 o curve, and for smaller curvaturesthe small um

r assumption becomes more valid. Thereforerelaxing the assumption for the matrix radial displacementwill improve the analytical model.

In Eq. (4), it is assumed that the variation of matrix ra-dial displacement along the fiber axial direction @um

r =@s ismuch smaller than the variation of matrix axial displace-ment along the radial direction @um

s =@r: To check this, sev-eral sets of data points are extracted near the interface. Theratio of @um

r =@s to @ums =@r varies from 0.0331 to 0.478.

Therefore we can say this shear lag assumption is accept-able, but again the solution could be improved by relaxingthis assumption as well. In spite of these coarse approxi-mations, Fig. 9 shows that our analytical model does accu-rately capture the stress distribution along fiber axisqualitatively. The fiber stress decreases along s-directionfrom the pulled end to the embedded end. Quantitativelyspeaking, with same composite dimension and materialproperties and under same loading condition (pull-out dis-placement is 100 lm), the fiber axial stresses at pull-outend calculated from our analysis (9.9E8 Pa) and from FEsimulation (8.5E8 Pa) are quite close.

4. Conclusions

In this paper, fiber curvature has been added into a shearlag model to analyze the bonded stage in single curved-fiberpull-out. A parametric study of the analytical model showsthat fiber curvature and fiber embedded length have strongeffects on the force–displacement curve. Fibers with morecurvature and longer embedded lengths can help toughenthe composites. 3D finite element results show that asidefrom a stress variation around hoop direction, the currentanalytical model captures the interfacial shear stress distri-bution qualitatively. For the same pull-out distance, the fi-ber stress field obtained from finite element is quite closeto that from our analytical model. However, the finite ele-ment results suggest that in cases of large fiber curvature,the matrix radial displacement should not be ignored com-pared with its axial displacement. Therefore, further workon the analytical model for the bonded stage is warranted,in particular with regard to two issues:

(1) Unlike the straight fiber, curved-fiber pull-out is notan axisymmetric problem. As seen from Fig. 9 (b),the stress field at fiber/matrix interface varies alongthe hoop direction. A 3D analytical model is requiredto take into account the variation in the hoopdirection.

(2) Once the radial compressive stresses are consideredin a newly developed 3D model, matrix deformationcan be analyzed more accurately and provide alter-natives to neglecting the radial displacement.

Fig. 2A. Stress equilibrium of a differential fiber element.


In spite of these possible improvements, the resultsfrom our current analytical model are reasonable and canbe extended to include the debonding and sliding stagesto obtain information on effect of curvature on the fullpull-out scenario applicable to nanocomposites. In our sec-ond paper of this series, debonding and sliding stages areanalyzed and the results are combined with the result forbonded stage in this paper to generate the entire pull-outcurve. The effect of curvature on the pull-out curve is thenstudied.

Acknowledgements

This work is supported by the National Science Founda-tion under Grant No. 0404291. I.J.B. acknowledge supportby a Los Alamos Laboratory Directed Research and Devel-opment Project (No. 20030216) and an Office of Basic En-ergy Sciences Project FWP 06SCPE401.

Appendix I. Single straight fiber pull-out in the bondedstage

Here a single straight fiber pull-out model based on theshear lag by Lawrence (1972) is reviewed and the corre-sponding pull-out force-displacement relation is derived.As shown in Fig. 2, straight fiber pull-out is consideredan axisymmetric problem, in which all the stress, strainand displacement components depend only on radial andaxial coordinates. A similar model geometry is also shownin Fig. 1A. Fiber and matrix are co-cylinders with diametersof df and dm, respectively. Initially, the fiber has a length ofl0 extruding out of matrix and has an embedded length of L.

All equations in the following derivation are in normal-ized form to remove any unnecessary dependencies of theform of the solution on some parameters. The normaliza-tion factor for length is fiber radius rf. That for stress andmoduli is the fiber Young’s modulus Ef and accordingly,that for force is p(rf)2Ef. The asterisks indicate normalizedvalues.

From Fig. 2A, we have

drf�x

dx�¼ �2s�i : ð1AÞ

Fig. 1A. Model geometry of single straight fiber pull-out in the bondedstage. Fiber and matrix both have a circular cross-section of diameter df

and dm, respectively, in axisymmetric coordinate system x and r. L is thefiber embedded length. l0 is the free length of the fiber initially notembedded in the matrix. P is the applied pull-out force at the fiber pulledend.

Matrix equilibrium shown in Fig. 3A generates

s�i ¼ sm�rx r�: ð2AÞ

Strain–displacement gives

cmrx ¼

dum�x

dr�þ dum�

r

dx�� dum�

x

dr�¼ sm�

rx

G�m¼ s�i

r�G�m; ð3AÞ

based on basic shear lag assumption, implied by assump-tion (1):

dum�x

dr�� dum�

r

dx�: ð4AÞ

Integrating Eq. (3A) on both sides,Z um�x ðrm�Þ

uf�x

dum�x ¼

Z rm�

1

s�iG�m

dr�

r�: ð5AÞ

We have

um�x ðrm�Þ � uf�

x ¼s�iG�m

ln rm�: ð6AÞ

Reorganizing, we get

s�i ¼ G�mum�

x ðrm�Þ � uf�x

ln rm� : ð7AÞ

Combined with the equilibrium equation for the fiber Eq.(1A), we have

drf�x

dx�¼ �2G�m

um�x ðrm�Þ � uf�

x

ln rm� : ð8AÞ

In order to get an ODE of rfx, the displacement is related to

stress through strain–displacement relation and an elastic,isotropic constitutive law.

d2rf�x

dx�2¼ 2G�m

ln rm� ðefx � em

x ðrm�ÞÞ ¼ 2G�mln rm� rf�

x �rm�

x ðrm�ÞE�m

� �;

ð9AÞ

Fig. 3A. Stress equilibrium of a representative matrix segment in thebonded stage.

Fig. 4A. 2D curvilinear system s and r and x–y Cartesian coordinatesystem.


where rmx is regarded as a virtual matrix stress generated

under pull-out stress as if there is no fiber, i.e.,

rm�x ðrm�Þ ¼ P�

rm�2 ¼rf�

x

rm�2 : ð10AÞ

Substituting Eq. (10A) into (9A),

d2rf�x

dx�2¼ 2G�m

ln rm� 1� 1rm�2E�m

� �rf�

x � Q �ðrm�Þrf�x : ð11AÞ

The general solution for Eq. (11A) is

rf�x ¼ A sinhð

ffiffiffiffiffiffiQ�

px�Þ þ B coshð


px�Þ: ð12AÞ

Using the following boundary conditions:

1Þ rf�x ð0Þ ¼ r�0 ) B ¼ r�0; ð13AÞ

where r�0 denotes the stress from neighboring nanotubesat the embedded end. Note that the classic shear lag as-sumes a free-end rf�

x ð0Þ ¼ 0 and thus has a simpler formsolution than we obtain here.

2Þ rf�x ðL

�Þ ¼ r�pull ) A ¼r�pull � r�0 coshð


pL�Þ


pL�Þ

; ð14AÞ

where r�pull is the applied stress at the pulled end and itequals the normalized pull-out force P*.

Substituting Eqs. (13A), and (14A) into Eq. (12A), wehave the fiber axial stress

rf�x ¼

P�




pL�Þ

!sinhð


px�Þ

þ r�0 coshðffiffiffiffiffiffiQ �

px�Þ;

ð15AÞ

and the interfacial shear stress

s�i ¼ �12

P�




pL�Þ

!"

�ffiffiffiffiffiffiQ�

pcoshð


px�Þ þ r�0


psinhð


px�Þ#: ð16AÞ

When r�0 is set to zero, Eq. (16A) has the same form as gi-ven by Lawrence (1972). In our formulation we have ananalytical form for the arbitrary constants in his equation.Note that s�i max is reached at x* = L*.

s�i max ¼ �12


pP� cothð


pL�Þ � r�0


pL�Þ

!:

ð17AÞ

When s�i max ¼ s�s , debonding begins, where s�s is the criticalshear stress for fiber-matrix separation.

Fiber displacement is the sum of elastic elongation ofthe embedded and the extruded part.

d�ðIÞ ¼Z L�

0ef

xdx� þP�l�0¼1ffiffiffiffiffiffiQ�

p

P�cothðffiffiffiffiffiffiQ �

pL�Þ

� P�


pL�Þ� r�0


pL�Þþr�0 cothð


pL�Þ!þP�l�0

ð18AÞ

Appendix II. Derivation of basic elasticity equations in2D orthogonal curvilinear system

Based on continuum mechanics (Green and Zerna,1992; Malvern, 1969), basic elasticity equations for smallstrain condition including equilibrium, stress–displace-ment relationship, and constitutive law are derived for2D orthogonal curvilinear system as follows.

The 2D curvilinear system we employ is shown inFig. 4A. Point (r,s) in r–s curvilinear system can be repre-sented by a vector ~r of R� ðR� rÞ sinða� s

RÞiþ ðR� rÞcosða� s

RÞj in x–y Cartesian coordinate system. Base vectors~gi;~gi and metric tensors gij; gij are defined as

~gi¼~r;i; ~gi ~gj¼dij¼1ði¼ jÞ0ði–jÞ

; ~gi ~gj¼gij; ~g

i ~gj¼gij: ð19AÞ

In detail,

~g1 ¼@~r@r¼ sin a� s

R

� �i� cos a� s

R

� �j;

~g2 ¼@~r@s¼ R� r

Rcos a� s

R

� �iþ R� r

Rsin a� s

R

� �j;

ð20AÞ

~g1 ¼ sin a� sR

� �i� cos a� s

R

� �j;

~g2 ¼ RR� r

cos a� sR

� �iþ R

R� rsin a� s

R

� �j;

ð21AÞ

g11 ¼ ~g1 ~g1 ¼ 1; g22 ¼ ~g2 ~g2 ¼R� r

R

� �2

;

g12 ¼ g21 ¼ ~g1 ~g2 ¼ 0; g11 ¼ ~g1 ~g1 ¼ 1;

g22 ¼ ~g2 ~g2 ¼ RR� r

� �2

; g12 ¼ g21 ¼ ~g1 ~g2 ¼ 0: ð22AÞ

The Christoffel symbols of the second kind

Ckij ¼ Ck

ji ¼ ~gk ~gi;j; ð23AÞwhere

~g1;1 ¼ 0; ~g1;2 ¼~g2;1 ¼ �1R

cos a� sR

� �i� 1

Rsin a� s

R

� �j;

~g2;2 ¼R� r

R2 sin a� sR

� �i� R� r

R2 cos a� sR

� �j: ð24AÞ


Thus,

Ck11 ¼ 0;C1

12 ¼ C121 ¼ 0; C2

12 ¼ C221 ¼

1r � R

;

C122 ¼

R� r

R2 ; C222 ¼ 0: ð25AÞ

Static equilibrium without body force is

r s ¼ r ðsij~gi~gjÞ ¼ sij:i~gj ¼ 0; i:e:; sij

:i

¼ sij;i þ skjCi

ki þ sikCjki ¼ 0: ð26AÞ

In r-direction j = 1, we have equilibrium:

s11;1 þ s21

;2 þ s11ðC111 þ C2

12Þ þ s21ðC121 þ C2

22Þ þ s11C111

þ s12C121 þ s21C1

12 þ s22C122 ¼ 0: ð27AÞ

In s-direction j = 2, we have:

s12;1 þ s22

;2 þ s12ðC111 þ C2

12Þ þ s22ðC121 þ C2

22Þ þ s11C211

þ s12C221 þ s21C2

12 þ s22C222 ¼ 0: ð28AÞ

We have to transform contravariant components to physi-cal components as follows.

s11 ¼ rr ; s12 ¼ s21 ¼ RR� r

srs; s22 ¼ R2

ðR� rÞ2rs: ð29AÞ

The equilibrium equations represented by physical compo-nents are thus expressed as follows.

@rr

@rþ R

R� r@srs

@sþ rs � rr

R� r¼ 0 in r-direction: ð30AÞ

@srs

@rþ R

R� r@rs

@s� 2

R� rsrs ¼ 0 in s-direction: ð31AÞ

As for the strain–displacement relations under a smallstrain condition, based on

cij ¼12ðvijj þ vjjiÞ; ð32AÞ

where vijj ¼ vi;j � Crijvr we have

c11¼v1j1¼@v1

@r�C1

11v1�C211v2;

c12¼c12¼v1j2

¼12

@v1

@s�C1

12v1�C212v2þ

@v2

@r�C1

21v1�C221v2

� �

c22¼@v2

@s�C1

22v1�C222v2:

ð33AÞ

Again, we have to transform covariant components tophysical components as follows.

v1 ¼ ur ; v2 ¼R� r

Rus; c11 ¼ er; c12 ¼ c21

¼ R� rR

ers; c11 ¼R� r

R

� �2

es: ð34AÞ

Therefore, the final form of the strain–displacement rela-tion is

er ¼@ur

@r;

ers ¼12

RR� r

@ur

@sþ R� r

R@us

@rþ us

R

� �¼ 1

2crs;

es ¼R

R� r@us

@s� ur

R� r:

ð35AÞ

The constitutive law does not change with coordinate sys-tem. Therefore, for an isotropic material with no Poissoneffect, we still have

er ¼rr

E; es ¼

rs

E; and crs ¼

srs

Gð36AÞ

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