Composites Science and Technology 68 (2008) 3373–3379

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Composites Science and Technology

journal homepage: www.elsevier .com/ locate/compsci tech

Modeling of damage sensing in fiber composites using carbon nanotube networks

Chunyu Li *, Tsu-Wei Chou 1

Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA

a r t i c l e i n f o

Article history:Received 30 April 2008Received in revised form 1 September 2008Accepted 9 September 2008Available online 25 September 2008

Keywords:A. Carbon nanotubeA. CompositesB. Electrical conductivityC. Damage sensingC. Modeling

0266-3538/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.compscitech.2008.09.025

* Corresponding author. Tel.: +1 302 831 6541.E-mail addresses: Chunyulee@hotmail.com (C. Li),

1 Tel.: +1 302 831 1550; fax: +1 302 831 3619.

a b s t r a c t

This paper presents the modeling of damage sensing in [0�/90�]s cross-ply glass fiber composites usingembedded carbon nanotube network. The wavy nanotubes are distributed in the polymer matrixbetween fibers and their contact resistances are modeled considering the electrical tunneling effect.The effective electrical resistance of the percolating nanotube network is calculated by considering nano-tube matrix resistors and employing the finite element method for electrical circuits. The entire deforma-tion process of the composite, from initial loading to final failure, is simulated by using the finite elementmethod for two-dimensional stress analyses. The deformation and damage induced resistance change isidentified in each loading step. The results demonstrate that the current simulation model captures theessential parameters affecting the electrical resistance of nanotube networks, which can serve as an effi-cient tool for structural health monitoring of fiber composites.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

The failure of a fiber composite is usually a complex processwhich may involve an accumulation of microscopic damage,including fiber fracture, fiber matrix interfacial debonding, matrixcracking and delamination. The concept of damage sensing in com-posites based upon the carbon fiber reinforcements was pioneeredby Schulte and co-workers (see Refs. [1–4]). The basic idea of thisapproach is in the use of conductive carbon fibers as the electriccurrent carrier and the measurement of resistivity change in thefiber direction due to fiber breakages or in the transverse directiondue to the separation of fiber contacts [5]. However, the techniqueis not applicable to composites with non-conductive fibers such asglass and ceramic fibers. Even in carbon fiber composites, somematrix-dominated damage may not be detected by the fiber net-work. Thus, more versatile technique is needed for in situ damagesensing of fiber-reinforced composites, and carbon nanotubes(CNTs) turn out to have great potential for such applications [6,7].

CNTs possess exceptionally high stiffness and strength [8,9] aswell as high electrical and thermal conductivities [10,11]. Theunique mechanical and physical properties of CNTs combined withtheir high aspect ratio (length/diameter) and low density havebrought about extensive research in creating composite materialsystems to exploit these properties [7,8]. Besides the effort inutilizing nanotubes as passive reinforcement to tailor toughness,impact resistance, vibration damping, electrical conductivity and

ll rights reserved.

chou@udel.edu (T.-W. Chou).

thermal conductivity, much attention has also been devoted todeveloping sensors and actuators using CNTs [12,13].

An interesting recent development in nanocomposites is the useof CNTs as multi-functional reinforcements where they serve asstrain or damage sensors. Wood et al. [14] and Zhao et al. [15,16]examined the Raman spectral shifts of carbon nanotubes embed-ded in a polymer matrix due to the elastic strain in the nanotubes,and proposed the use of nanotubes as microscale strain sensors formeasuring the strain fields around defects or fibers. Dharap et al.[17] investigated resistance-based CNT strain sensors by using thinfilms of randomly oriented CNTs. Zhang and co-workers [18]reported that multi-walled CNT reinforced composites can beutilized as strain sensors and suggested that the instantaneouschange in resistance with strain can be utilized for self-diagnosticsand real-time health monitoring. Kang et al. [19,20] utilized single-walled CNT/polymer composite films for strain sensing and sug-gested that a sensor network attached to the surface of a structuralcomponent could enable structural health monitoring. Someresearchers [21–23] have also proposed using CNT composites aschemical sensors based on their changes in electrical resistivity.The principle behind these examples of CNT-based composite sen-sors is the sensing of the change in volume resulted from chemical,thermal or mechanical loading.

Fiedler and co-workers [5] were the first to propose the concept ofconductive modification with nanotubes for both strain and damagesensing. Because CNTs possess higher electrical conductivity thancarbon fibers, it is expected that the sensitivity of changes in electri-cal resistance resulted from strain or damage could be enhanced.Thostenson and Chou [6] concluded that the change in the size ofreinforcements, from conventional micron-sized fiber reinforce-ment to carbon nanotubes with nanometer-level diameters, enables

3374 C. Li, T.-W. Chou / Composites Science and Technology 68 (2008) 3373–3379

unique opportunity for the creation of multi-functional in situ sens-ing capability. They demonstrated that the percolating networks ofCNTs are remarkably sensitive to the onset of matrix-dominated fail-ure and can detect the progression of damage. More recently, Parket al. [24] evaluated the inherent sensing of CNT/epoxy compositesusing an electro-mechanical testing technique and concluded thatuniform dispersion and interfacial adhesion are key factors forimproving sensing performance.

Although some modeling work has been carried out on the elec-trical resistance-based damage detection in carbon fiber composites[25,26], there has been no report on the modeling of nanotube net-work for damage sensing of fiber composites. Given the potentialapplications of multi-functional nanocomposites, an in-depthunderstanding of the key factors controlling the effectiveness ofcarbon nanotube network for damage sensing is indispensable.This paper reports our studies on the computational modeling ofa carbon nanotube network embedded in a glass fiber compositewith a particular focus on the variation of electrical resistance withdamage evolution.

2. Nanotube network in fiber-reinforced composites

In the recent experimental work of Thostenson and Chou [6],they activated some of the typical fiber composite failure modesand correlated them to the electrical resistance measurementsfrom distributed carbon nanotube sensor networks. Among thepatterns of damage evolution in unidirectional and cross-ply com-posites, the matrix cracking in the 90� layer of the [0�/90�]s lami-nate exhibited a progressive accumulation of damage and isparticularly interesting from both the experimental and analyticalmodeling point of view. Thus the cross-ply configuration is chosenas the model system for our simulation research.

In the following study, we assume that the CNTs form a percola-tion network in the [0�/90�/90�/0�] laminate of glass fiber-reinforcedepoxy composite. The focus of the simulation is on the demonstra-tion of the interaction between the CNT network and transversecracks in the 90� plies. For simplicity of the simulation, we adopt atwo-dimensional model as shown in Fig. 1. This two-dimensionalmodel may inevitably increase the contact between nanotubescomparing with an actual nanocomposite, in which the conductivity

Fig. 1. Nanotube network in the [0�/90�/90�/0�] composite laminate.

network is three-dimensional in nature and nanotubes are notlimited to a planar distribution. But this undesirable effect is some-what reduced in our simulations by assuming the gaps of inter-nanotube contacts are statistically distributed in a rather largerange. The middle section shows two layers of 90� fibers, while thetwo sections on the sides represent the 0� fiber layers. The fiberdiameter is assumed to be 3.15 lm in order to keep the total fibervolume fraction of 55%, which is a in the range commonly used forstructural composites. The multi-walled carbon nanotubes in thesimulation have a diameter of 15 � 25 nm and a length in the rangeof 1.0 � 1.5 lm. The carbon nanotubes are allowed to penetrate intothe inter-fiber matrix region in the 90� layers. To avoid the complex-ity in dealing with overlapping nanotubes and fibers, the two 0�layers are replaced by their effective medium and consequentlythe 0� fibers are not shown in Fig. 1. The CNTs naturally assume athree-dimensional network. But for simplicity of simulation, weonly consider a small two-dimensional model with limited size of30 � 30 lm. Here, the assumed nanotube size is reasonable. Butthe fiber size is smaller than that of typical E-glass fibers, which havediameters in the range of 8 � 15 lm. We initially attempted to sim-ulate the problem with 10 lm E-glass fibers inside a 100 � 100 lmspecimen. But the task in FEM meshing turned out to be ratherdaunting because of the huge number of nanotubes distributed inthe composite. The purpose of the simulation is just to demonstratethe effect of the essential factors contributing to the nanocompositeelectro-mechanical behavior in damage sensing. The current modelsize is adequate in demonstrating the general characteristics ofelectrical resistance in the carbon nanotube network.

Waviness is a prevailing feature of CNTs in a composite. In a CNT/fiber hybrid composite, where the fibers occupy a significant portionof the volume, the nanotubes are infused into the gaps betweenneighboring fibers and may assume a larger degree of waviness.Wavy CNTs dispersed in a matrix also tend to have more contactpoints than straight nanotubes, which could have a considerableeffect on the electric conductivity due to the dominant role ofcontact resistance [27]. Thus, the nanotube waviness needs to beconsidered in order to simulate the composites in a more realisticmanner. The method for generating wavy nanotubes is based on aversatile approach proposed by the authors for composites contain-ing multiple fillers of arbitrary shapes [28], which are approximatedby polygons. The wavy nanotubes so generated are then placed inrandom locations in the designated area. The orientation of ananotube in the 90� ply may need to be adjusted for avoiding over-lapping with the fiber cross-section. If a nanotube cannot fit into adesignated position after a certain number of adjustments, it isabandoned and a new nanotube is selected for a new position. Theprocess continues until the designated volume fraction of nanotubesis reached. The contact points between two neighboring nanotubesare determined by following the method described in Ref. [27].

Based on the knowledge of contact points, the nanotube clus-ters can be identified. The next step is to check the existence ofspanning cluster. In some cases, there is no spanning cluster, whilein others more than one spanning cluster has formed. The percola-tion probability at a specific nanotube volume fraction can bedetermined by the percentage of the number of times that at leastone spanning cluster has occurred out of the total number ofMonte Carlo simulations. Here, we only need to select one percola-tion nanotube network for studying the effect of damage on elec-trical conductivity of the network. Fig. 2 gives the percolatingnanotube network in the composite shown in Fig. 1.

3. Electrical resistance of nanotube network

The electrical resistance of a percolating nanotube networkcomes from two sources, i.e., the intrinsic resistance of nanotubes

0 2 4 6 8 10 12 14 16 18 200

1

2

3

4

5

6

7

8

9

10

Log[Rc (ohms)]

Prob

abilit

y (%

)

Fig. 3. The distribution of contact resistance Rc.

Fig. 4. A schematic diagram of resistor network (Yellow: nanotube resistance;Brown: Contact resistance; the inset showing the pseudo 3D model of contactresistance [26]). (For interpretation of the references in colour in this figure legend,the reader is referred to the web version of this article.)

Fig. 2. A percolating nanotube network in the fiber composite.

C. Li, T.-W. Chou / Composites Science and Technology 68 (2008) 3373–3379 3375

and the contact resistance at nanotube junctions. Thus the electri-cal conductivity of the nanotube network strongly depends on themorphology of nanotube network and the number of contactpoints. The electric conductivity of individual carbon nanotubesis in the order of 104 � 107 S/m. But the contact resistance is rathercomplicated and depends on nanotube diameter, tunneling gap atcontact points and matrix material filling the tunneling gap. Arecent study of the authors indicated that the contact resistanceplays a dominant role in the electric conductivity of nanotube-based composites [26]. It was concluded that the thickness of insu-lating film that fills the tunneling gap at a contact point needs to beless than 1.8 nm in order for the electric tunneling to take place.

Depending on a number of contributing factors, the contactresistance between carbon nanotubes in composites could varyin a wide range, from 102 kX to 1016kX. Here, different approachesare adopted for dealing with the two possible nanotube contactconfigurations. One configuration is the overlapping contact. Inthis case, it is often difficult to determine the thickness of an insu-lating film and, hence, the precise value of the contact resistance.Here, we assume that the thicknesses of insulating films follow anormal distribution in the range of 0 � 1.8 nm, and the corre-sponding distribution of contact resistances is calculated usingthe method introduced in Ref. [26]. The distribution of contactresistance resulted from the normal distribution of insulating filmsis shown in Fig. 3, which is similar to the contact resistance distri-bution adopted in Ref. [29]. The lower bound of contact resistanceis taken as 100 kX; which is the lowest contact resistance betweennanotubes assuming no insulating film. The other nanotube con-tact configuration is the in-plane contact, where two neighboringnanotubes are not overlapping but are situated close enough topermit electrical tunneling. In this case, based on the tunnelinggap size, the contact resistance at a specific contact point isapproximately calculated using the formula given in Ref. [26].

The calculation of the electrical current flowing through thepercolating nanotube network can be carried out by using thefinite element method [30]. We assume that the nanotube segmentbetween any two contact points is represented by a resistor. Theresistance of this resistor is calculated based on the Ohms law. Theelectrical conductivity rcnt of CNTs is taken as 106 S/m. The resis-tance at a contact point, which is assumed to follow the distribution

in Fig. 3, is also represented by a resistor. Thus the nanotube networkis replaced by a resistor network. Although the nanotube network ofthe model composite is two-dimensional, the resistor network isactually located in a three-dimensional space (see Fig. 4). Fortu-nately, only one unknown, i.e. electrical potential, needs to deter-mined at each node. Assuming that the two opposite sides of themodel composite are connected to superconducting electrodes, avoltage is applied to the superconducting electrodes by assigningthe electric potential at one side to be zero and at the opposite sideto be unity. The electric potential distribution at each node of theresistor network can be obtained by solving a system of algebraicequations [31].

For a typical resistor element i–j, the elemental matrix repre-senting the relation between the current (I) entering the elementat the ends and the end voltage (V) is

Iei

Iej

8><>:

9>=>; ¼ ½K

eij�

Vi

Vj

� �¼ 1

Re

1 �1�1 1

� �Vi

Vj

� �: ð1Þ

According to the Kirchhoff’s current law, a system of algebraicequations can be assembled for the entire network:

I ¼ KV; ð2Þ

where V ¼ fV1;V2; :::;VngT stands for the nodal voltages, I ¼fI1; I2; :::; IngT is the vector of external input current at the nodes.The global coefficient matrix is obtained from

3376 C. Li, T.-W. Chou / Composites Science and Technology 68 (2008) 3373–3379

K ¼Xm

e¼1

½Keij�; ð3Þ

where m is the number of resistors in the network.After applying the boundary conditions to Eq. (2), the electrical

potential at each node can be obtained. The current flowingthrough each resistor element is then determined by

Ie ¼ ðVi � VjÞ=Re; ð4Þ

and the total current passing through the network can be obtainedby summing up the currents in the resistors directly connected toone side of boundaries. The effective resistance of the entire perco-lating network is then obtained from

Reff ¼ ðV top � VbottomÞ=Itotal: ð5Þ

Fig. 5. Finite element mesh of the carbon nanotube/fiber cross-ply composite.

4. Damage evolution in composites

As indicated in the introduction of the model composite ofFig. 1, the 0� plies are replaced by their effective media, of whichthe elastic modulus and tensile strength are computed from therule-of-mixtures. The total number of nanotubes placed in themodel composites is about 750. The elastic and strength propertiesas well as volume fractions of the constituents are given in Table 1.The damage evolution of the model composite is simulated by fix-ing the lower boundary while applying a uniform displacement onthe upper end. This configuration of deformation is equivalent toan isostrain axial loading condition of a specimen with a lengthof twice of that of Fig. 1.

It is well known in the short fiber composite literature, for in-stance, the presence of stiff, discontinuous fillers tends to inducehigh concentration of strains in the matrix material with lowermodulus at the ends of the discontinuous reinforcements [32,33].In the fiber/nanotube/polymer hybrid composite, the degrees ofdeformation of the matrix, nanotube and fiber are very different.Relatively larger deformation developed in the polymeric matrixdue to its lower modulus. Furthermore, the CNTs in this hybridcomposite, in spite of their low volume fraction, can also signifi-cantly affect the local stress and strain distributions at the tube-ends because of their high modulus. Thus, it is for the same reasonas in short fiber composites that high strain concentration occursboth at nanotube tips as well as in the inter-fiber matrix regionwhere a unidirectional continuous fiber composite is subjected totransverse loading.

To simulate the damage evolution of fiber-reinforced compos-ites with randomly distributed CNTs, we employ the commercialfinite element method software ANSYS [34]. The coordinates ofwavy nanotubes, glass fibers and matrix are generated by in-housesoftware and then exported into ANSYS. The nanotubes with over-lapping contacts are assumed to be directly connected because ofthe two-dimensional nature of the model. The meshing is con-ducted by first manually assigning the material numbers to thefour types of constituents and then automatically performed bythe ANSYS software with mixed triangular and quadrilateral ele-ments. The final mesh (shown in Fig. 5) is composed of about357,000 elements and 360,000 nodes.

Table 1Material properties of composite constituents

Young’s modulus (GPa) Tensile stre

0� composite ply 39 1.08Epoxy Matrix[35] 3.7 0.09E-glass fiber[34] 73 3.45Carbon nanotube[7,8] 800 30.0

The nanotube volume fraction refers to the entire composite specimen.

The criterion for damage initiation is set by the use of principaltensile stress. For identifying the location of potential damage, themaximum principal stress in each element is traced in every loadstep in the simulation process. If the maximum principal stressof an element is equal to or larger than the tensile strength ofthe corresponding material of that element, the critical elementis identified and assigned as a death element and deactivated inthe next load step. In most cases, there could be a few up to manyelements becoming death elements at one load step. The death ele-ments give rise to stress redistribution, which may result in signif-icant stress concentrations. These stress concentrations contributeto the deterioration of the load-carrying capacity as well as thedamage progression of the composite.

The strain concentrations at nanotube ends are depicted inFig. 6, which shows some close-up views of damage evolution.Here, at point A, two adjacent nanotubes are in contact with eachother. As the applied strain increases, the two nanotube ends moveapart, resulting in high strain concentration in the matrix andeventually leading to microcracks along the nanotube/matrixinterface. Also, at point B, the matrix strain concentration at thenanotube end initiates nanoscopic crack at e0 ¼ 0:500%, whichevolves into a microscopic matrix crack at e0 ¼ 1:167%. At this le-vel of applied strain, another site of damage initiation is noticed atpoint C.

Fig. 7 displays the overall view of damage evolution in the com-posite with increasing applied strain. Multiple damage spots arevisible at the strain level of about 0.433%. These damage spotsare located in the inter-fiber matrix region where the strain con-centration is high, as shown by the strain contours in Fig. 7(a).

ngth (GPa) Poisson’s ratio (GPa) Volume fraction (%)

0.280.35 43.150.23 54.800.20 2.05

Fig. 6. A close-up view of damage evolution in the vicinity of nanotube ends.

C. Li, T.-W. Chou / Composites Science and Technology 68 (2008) 3373–3379 3377

With increasing tensile strain, the number of damage spots also in-creases and the size of damage spots expands, as shown inFig. 7(b). When the applied strain reaches 1.167% (Fig. 7(c)), itcan be seen that the damage spots merge to form microscopiccracks, which extend through the entire width of 90� plies. Thesecracks could further propagate into the 0� plies, eventually result-ing in the failure of the cross-ply composite.

5. Damage sensing by electrical resistance method

The effective resistance of a composite can be changed when itis deformed under applied loading. Several factors may contributeto the electrical resistance change. First, when a fiber/nanotube/polymer hybrid composite is deformed, the nanotube length anddiameter will alter, resulting in the change of nanotube intrinsicresistance, and hence, the effective resistance of the nanotube net-work. However, this resistance change is expected to be negligiblebecause of the extremely small elastic deformation in nanotubes.

The second and more important factor contributing to the resis-tance change of the composite is the contact resistance. Under ap-plied load, the thickness of the insulating matrix film betweenadjacent nanotubes may be changed considerably. The contactresistance increases dramatically with the increase in insulatingfilm thickness [26]. The matrix damage also contributes to thechange of contact resistance. It has been shown in Fig. 6 that thematrix damage first appears at low applied strain at the nanotubecontact points where there are high strain concentrations. Fig. 8illustrates the effect of damage evolution on the change of contactresistance. The damage spot assumes the form of a nanoscopic voidand its formation give rise to the increase of the opening gap at the

contact area where electrical tunneling takes place and thusincreases the contact resistance. With the evolution of damage,the electrical tunneling at a damage area can be eventually cutoff, resulting in significant change of the effective resistance ofthe entire composite. The real effect of the damage is in changingthe contact resistance.

Fig. 9 displays the load–strain relationship of the cross-ply com-posite. A ‘‘knee” is visible in the curve at about 0.433% strain whichis an indication of significant damage accumulation in the compos-ite. This is evident from the multiple damage spots in the 90� pliesof Fig. 7(a). The ultimate load level of the composite occurs atabout 1.2% applied tensile strain, when multiple microcracks haveformed in the 90� plies and also extended into the 0� plies. Theload-carrying capacity of the composite drops rapidly after 1.2%applied strain and eventually the composite fails at about 1.5%applied strain when microcracks propagate through the 0� plies.

Fig. 9 also shows the change of electrical resistance with appliedtensile strain. It can be seen that the resistance change is very smallin the beginning. But significant change occurs around 0.4% strain,which roughly corresponds to the knee point in the load-straincurve. A careful browsing of the image of strain distribution at thisstage (Fig. 7(a)) reveals that some damage spots are located right inthe percolation path and two of the conducting branches have beencut off. The resistance change reflects the cutoff of current path dueto matrix cracking. It should be noted that several small damageareas actually appeared earlier when the loading level was lower,but the resulting resistance change is insignificant because thelocations of these damages are not on the percolation path.

It can also be seen in Fig. 9 that after the first significant step-increase, the resistance curve exhibits a period of relatively small

Fig. 7. Damage evolution in the composite under different imposed strains (a)e0 = 0.433%, (b) e0 = 0.500%, (c) e0 = 1.167% (Contours showing the first principalstrain).

Fig. 8. An illustration of electrical tunneling affected by the damage evolution.

0

1

2

3

4

5

6

7

8

0.0 0.3 0.6 0.9 1.2 1.51.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

Elec

trica

l res

ista

nce

(GΩ

) Resistance

Tensile strain (%)

Tens

ile lo

ad (x

1000

nN)

Load

Fig. 9. Variations of tensile load and electric resistance with applied tensile strain.

3378 C. Li, T.-W. Chou / Composites Science and Technology 68 (2008) 3373–3379

changes. The nearly flat resistance curve indicates no further cutoffof the percolation path. With the increase of loading level, theexisting damages are continuously expanded and new damagesinitiate. When these damage areas cut off one or more percolationbranches, the resistance again changes significantly. The finalsharp increase in resistance occurs when the load level of the

composite just passes its maximum. A closer examination ofFig. 7(c) indicates that all the percolation branches but one havebeen cut off at this stage. The last current carrying path (the left-most path in Fig. 2) is cut off at �1.2% strain when the compositeis close to complete failure.

C. Li, T.-W. Chou / Composites Science and Technology 68 (2008) 3373–3379 3379

6. Conclusion and discussion

The methodology developed in this study is capable of captur-ing the key contributing factors to composite electrical percolationand effective resistance: nanotube waviness, electrical contactresistance, nanotube spanning cluster and conductive backbone.A detailed analysis of the local strain/stress concentrations at theinter-fiber as well as inter-nanotube matrix regions is also per-formed. Then, a combination of the knowledge in load–strain andresistance–strain relations enables the modeling of sensing of theinitiation and evolution of damage in fiber composites. The capa-bility so developed parallel to the relevant experimental work onnanotube-based damage sensing further enhances the potentialof its use for structural health monitoring of composites.

The modeling work of nanotube-based damage sensing indi-cates that the technique is capable of detecting the onset of dam-age, which is vital for any tool of structural health monitoring.However, it should be noted that the electric percolation behavioris statistical in nature and the sensitivity of the nanotube networkstrongly depends on the nanotube distribution and the structure ofpercolation clusters. If the damage at the early stage happens to cutthe percolation path, the resistance change would be noticed early.If the damage at the early stage does not cut a percolation path, theresistance still changes but it may not be clearly noticeable. Basedon the analysis of percolating network, the fraction of nanotubes inthe percolation clusters could be just a small percentage of totalnanotubes in a composite, depending on the volume fraction ofnanotubes. Thus it is likely that the resistance change does not re-flect the damage occurrence in a timely manner. Such deficiency indamage sensing can be minimized by enhancing the uniform dis-persion of nanotubes and, hence, forming more percolation paths.

Finally, further comments are needed regarding the implicationon the modulus and strength of the fiber composite due to theaddition of carbon nanotubes for the purpose of damage sensing.Existing experimental work has demonstrated that the additionof a small amount of carbon nanotube to a polymeric matrix mate-rial can result in enhancements in its elastic modulus and strength.The presence of nanotubes in a matrix of lower modulus inevitablyinduces local stress concentration and thus, higher probability ofinitiation of flaws. However, it is well known that the ultimate fail-ure of laminated composites is controlled by the fibers. The pres-ence of nanotubes in a fiber composite may have influence onthe initiation of microcracks and the onset of nonlinear behaviorbut will not significantly degrade its strength. In the present simu-lation, for the purpose of convenience in numerical work the as-sumed nanotube volume fraction of 2.05% is much higher thanthe 0.1 � 0.5% wt.% used by the authors in their experimental work[7]. Furthermore, the local stress concentration at nanotube ends isexpected to be much higher in a 2D model comparing to that in a3D model with the same amount of fillers [32]. Lastly, the assumedcarbon nanotube Young’s modulus of 800 GPa ignores the largescattering in experimentally measured modulus values.

Acknowledgments

This work has been supported by the Air Force Office of ScientificResearch (AFOSR), Grant No. FA9550-06-1-0489 (Dr. Byung-Lip Lee,Program Director), and the Office of Naval Research (ONR), Grant No.N00014-07-1-0345 (Dr. Yapa Rajapakse, Program Director).

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