Composites: Part B 41 (2010) 33–41

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Composites: Part B

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Measuring mechanical properties of micro- and nano-fibers embeddedin an elastic substrate: Theoretical framework and experiment

Guoxin Cao a, Xi Chen a,*, Zhi-Hui Xu b, Xiaodong Li b

a Nanomechanics Research Center, School of Engineering and Applied Sciences, Mail Code 4709, Columbia University, New York, NY 10027-6699, USAb Department of Mechanical Engineering, University of South Carolina, 300 Main Street, Columbia, SC 29208, USA

a r t i c l e i n f o

Article history:Received 12 February 2009Accepted 7 March 2009Available online 17 March 2009

Keywords:A. FibresB. Mechanical propertiesC. Computational modelingD. Nanoindentation

1359-8368/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.compositesb.2009.03.002

* Corresponding author. Tel.: +1 212 854 3787; faxE-mail address: xc2107@columbia.edu (X. Chen).

a b s t r a c t

We propose to measure the elastoplastic properties of micro- and nano-fibers by a normal indentationtechnique in which the vertically aligned fibers are embedded in an elastic matrix. Measurements aretaken at two different indentation depths, which represent different levels of the matrix effects and leadto the establishment of two independent equations that correlate the fiber/matrix properties with theindentation responses. Effective reverse analysis algorithms are proposed, and by following which thedesired fiber properties can be determined from a sharp indentation test. Comprehensive analysis is alsocarried out to verify the effectiveness and error sensitivity of the presented method. The extracted mate-rial properties agree well with those measured from the parallel experiments on human hair and glassfibers.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Usefulness of normal indentation for measuring mechanicalproperties of fiber

Micro- and nano-fibers have been studied intensively due totheir wide applications in tissue engineering [1], biomaterial [2],filter media [3], reinforced composites [4], and micro/nano-elec-tro-mechanical systems [5]. The fibers can be either natural or syn-thesized: examples of the natural fibers include plant fiber,lignocellulosic fiber, and hair, and manmade fibers can be synthe-sized from polymers, carbon, glass, and metal, which can be eitherhomogeneous or in the form of an annular ‘‘layered” coaxial com-posite column.

In practical applications, fibers experience various mechanicalloads that may cause permanent deformation and/or malfunctiondue to failure; therefore, characterizing the mechanical propertiesof fibers is the first step of evaluating their structural integrity. Fornatural fibers, such as human hair, its mechanical properties canprovide useful information to diagnose hair disorders and to eval-uate the response to therapeutic regimens [6], thus important forbiomedical purposes. Moreover, the variation of hair’s mechanicalproperties can help to evaluate the effect of different cosmeticproducts (such as shampoo and conditioner), which has importantindustrial applications [7,8]. Therefore, it is very important to char-acterize the mechanical properties of a single fiber – although thepresent study is motivated by the importance of mechanical prop-

ll rights reserved.

: +1 212 854 6267.

erties of human hair, the fibers studied in this paper are ratherbroad and in general can be applied to polymer, glass, ceramic,or high-strength metal fibers.

Conventionally, the mechanical properties of fibers can bemeasured by tensile test [9,10], bending tests [11], or resonantfrequency [12]; however, handling and manipulating micro- andnano-fibers are usually difficult and time consuming [13]. Amongalternative techniques, nanoindentation is perhaps the most con-venient, simplest, and fastest way of measuring the mechanicalproperties of material at small scale [14,15]. Since nanoindenta-tion is a compressive test, the fiber must be backed up by a sub-strate or matrix. Conventionally, the fiber would lie on (or gluedto) a flat substrate, and the indenter tip would penetrate the lat-eral surface of fiber (cylindrical surface). However, such lateralindentation is difficult to extract the mechanical properties offiber, due to the curved lateral surface of fiber and complicateeffect of substrate [13]. In addition, the lateral surface of some fi-bers (e.g. hair) is not smooth where the roughness comes fromnumerous asperities, which act like a compliant ‘‘shell” uponindentation and lead to a larger contact area than that resultedfrom a perfectly smooth lateral surface [16] – such a wronglymeasured contact area, plus the uncertainties of adhesion forcedue to asperities, can lead to a significant error of the measuredhardness and stiffness of the fiber during a conventional lateralindentation experiment [17].

In order to circumvent these difficulties, an indentation testalong the axial direction and on the cross-section of the fiber (nor-mal indentation) is desired. In order to avoid buckling, the fiberneeds to be embedded in and supported by a large matrix (e.g.epoxy); multiple fibers can be aligned in the matrix as long as their

34 G. Cao et al. / Composites: Part B 41 (2010) 33–41

separation is sufficiently large such that they do not affect eachother upon indentation. Once the cross-section of the compositeis polished, a normal nanoindentation test on the cross-section ofthe fiber can be carried out (see the schematic in Fig. 1(a)) – suchtechnique has been applied to measure the contact stiffness of hu-man hair and optical fiber [7,18].

Although the normal indentation approach has overcome thedisadvantages of surface curvature and roughness encountered inthe lateral indentation on fibers, the influence of matrix emergesas the primary issue of affecting the normal indentation measure-ment. The finite stiffness of matrix makes the ‘‘matrix effect” prom-inent in the measured indentation force–displacement behavior,that is, the measured property is a mixture of the fiber and matrixproperties even at relatively shallow penetration depths [19].Hence, the classic indentation theory [14] based on bulk materials(which is also the default option on many commercial nanoinden-ters) can cause error if one would use that to extract the fiber prop-erties without careful considerations. In previous experiments ofthe normal indentation on fibers [7,18], the important contributionof matrix compliance was not considered although in some casesthe indentation depths were on the same order of fiber/hair thick-ness, which may result in error of the reported fiber propertiessince it is expected that at such deep penetration the matrix effectmay not be negligible.

P

2α

R

Fiber Matrix

δ

P

2α

R

Fiber Matrix

δ

Inde

ntat

ion

load

, P

Indentation Depth, δ

Loading

Unloading

Pm

δm

(P2,δ2)

(P1,δ1)

(P3,δ3)Loading

Unloading

Pm

(P2,δ2)

(P1,δ1)

(P3,δ3)Loading

Unloading

Pm

(P2,δ2)

(P1,δ1)

(P3,δ3)

a

b

Fig. 1. Schematic of (a) conical indentation on a fiber/matrix system and (b)indentation load–depth ðP—dÞ curve obtained from indentation test.

While the substrate effect on the thin film indentation problemhas been studied extensively [19–21], the study of the matrix effectof normal indentation on a fiber bounded by a matrix is still lacking.To fulfill the potential of the normal indentation technique, a theo-retical framework needs to be established to understand and thensubtract off the matrix effect from the measurement, and to obtainthe intrinsic mechanical properties of the micro- and nano-fiber. Inaddition, the developed method can be extended to measure themechanical properties of fibers in a fiber-reinforced compositematerial, without separating fiber from the matrix – such conve-nience in sample preparation may be important in tissue engineer-ing, where it is very time consuming and challenging to extract asingle fiber from its matrix.

1.2. Basic principle of normal indentation on fiber: using the matrixeffect

With reference to Fig. 1(a), a fiber of radius R is embedded in anotherwise semi-infinite matrix, and we assume perfect bondingbetween the fiber and matrix. During a normal indentation teston the center of the cross-section of the fiber (Fig. 1(a)), the rela-tionship between indentation load ðPÞ and indentation depth ðdÞcan be continuously measured with the penetration of a hardindenter tip into the specimen (Fig. 1(b)). Note that due to thecomplicated stress and strain fields resulting from finite deforma-tion, the P—d curve is only implicitly related with the material elas-toplastic properties as well as the material/system structure (e.g.the presence of interface in the fiber/matrix system) – the func-tional relationships need to be established such that through areverse analysis, the most important and intrinsic elastic and plas-tic parameters of the fiber can be derived from the experimentaldata [15].

Although theoretically, one could keep the indentation depthmuch smaller than the fiber radius, so as to avoid the matrix effectand then apply the conventional closed-form formulae based onthe Oliver–Pharr method [14], such approach might encounter dif-ficulties for micro- and nano-fibers for the following reasons: (1)very shallow nanoscale indentation experiment often relies on astate-of-the-art commercial nanoindenter that is not accessibleto many users, it is more desirable to develop a technique thatcould work for a much less expensive microindenter; (2) the ma-trix/substrate effect on elastic property measurement persists atvery small depth especially when the matrix is compliant[19,20], and thus even smaller indentation depth may be required;(3) if a nanoscale indent is required to avoid the matrix effect in avery thin fiber, uncertainties such as surface roughness, adhesion,irregular indenter geometry and size effects may become domi-nant [21] which makes the modeling very hard. To circumventsuch nanoscale effects, an indentation test with moderate penetra-tion depth is needed (so as to apply the well-developed continuummechanics approach), which would inevitably induce the matrixeffect. The primary goal of this paper is then to understand the ma-trix effect and obtain the intrinsic fiber properties with the pres-ence of (or assistance of) the matrix effect.

Due to the lack of analytical solution of the indentation probleminvolving finite deformation and two-phase materials, numericalsimulations based on the finite element method (FEM) will be ap-plied to establish explicit relationships between the mechanicalproperties of the fiber and matrix (such as the elastic modulusand yield stress) and the indentation response (such as the shapefactors of the P—d curve). Inspired by the study of the substrate ef-fect of nanoindentation on thin films [19–21], the contribution ofmatrix is enhanced with increasing d in the present study. Thatis, the curvature of the measured P—d curve is different at distinctindentation depths, and the matrix effect can provide sufficientinformation for determining the elastoplastic properties of fiber.

G. Cao et al. / Composites: Part B 41 (2010) 33–41 35

In the following, we will first vary the fiber and matrix propertiesin a wide range, such that during the forward analysis with FEMsimulations, the explicit relationships can be established betweenthe material properties and the indentation response at differentrepresentative indentation depths (which correspond to differentmatrix effects). An effective reverse analysis is established suchthat, once the matrix modulus is known or measured, the elasticmodulus and yield stress of the fiber can be readily identifiedfrom the measured indentation P—d curve. The robustness ofthe technique is verified through numerical indentation experi-ments and an error sensitivity analysis. In addition, parallelexperiments on human hair and glass fibers are carried out toexamine the proposed indentation method, and the results (ex-tracted fiber properties) agree well with the reference properties.The framework proposed in this study may provide an effectiveapproach to measure the elastoplastic properties of micro- andnano-fibers.

2. Computation method

2.1. Modeling and basic assumptions

For the fiber/matrix system show in Fig. 1(a), the fiber is simpli-fied as an elastic-perfectly plastic material. Such assumption isvalid for ceramic/glass and high-strength metal fibers [22–26], inaddition, most polymer materials have negligible hardening effect[27,28]; moreover, for these materials their tensile and compres-sive behaviors are almost symmetric, allowing the determinationof fiber’s elastoplastic properties from indentation test (which iscompression-dominant). The matrix is homogeneous and isotropic,and it is assumed to deform elastically – such assumption is validfor the model epoxy matrix (which is also used in parallel experi-ments described below) and for the range of fiber properties andindentation depths studied in this paper (see below), where ourFEM simulation results verified that at the maximum penetration,the peak Mises stress induced in epoxy is insufficient to cause ma-trix yielding.

Since the matrix is much larger than the fiber, the matrix’sYoung’s modulus, Em, can be easily measured by using the uniaxialtension test. Alternatively, the Oliver–Pharr method [14] may beemployed to obtain the stiffness of matrix through indentation(where the matrix can be taken as a bulk material) which is alsoused in our parallel experiment in this paper. If the matrixmechanical property is known, there are only two unknown mate-rial parameters: the fiber’s Young’s modulus, Ef and the fiber’syield stress, rf , which can be mathematically determined by usingtwo independent equations. Meanwhile, more equations may beemployed and such redundancy may help to improve the numeri-cal reverse analysis result through optimization. The Poisson’sratio, normally regarded as a minor factor during indentation[19], is set to be 0.3 for both fiber and matrix.

To simplify the analysis, the indenter tip is assumed as a rigidcone with a half apex angle a ¼ 70:3�, which has the similar ratioof cross-section area to indentation depth as the widely used Ber-kovich indenter [15]. It has been reported that the conical indenta-tion has almost the same loading and unloading curves (with lessthan 1% difference in general) as the pyramidal indentation as longas both types of indenters have the same ratio of cross-section areato indentation depth [15]; therefore, conical indenter will be usedin this study to take advantage of axisymmetry (see Section 2.3).Coulomb’s friction law is applied between indenter and fiber sur-face with a friction coefficient of 0.1. The friction is a minor factorin indentation as long as the friction coefficient is small, which hasbeen verified by our FEM analysis and other groups [15]. We as-sume perfect bonding between the fiber and matrix during theexperiment.

2.2. Basic functional forms based on dimensional analysis

Fig. 1(b) shows a typical P—d curve, where Pm and dm are themaximum indentation load and indentation depth, respectively.From the principle of dimensional analysis, the dimensionless cur-vature of the loading curve can be expressed as:

P

rf d2 ¼ U

Ef

rf;

Ef

Em;dR

� �ð1Þ

which may be regarded as shape factors reflecting the matrix effectat various penetrations (e.g. d1; d2; d3). To solve the fiber propertiesðEf ;rf Þ, the required two independent equations can be provided bythe distinct matrix effects at different indentation loads/depthswith ðP1; d1Þ and ðP2; d2Þ, see Fig. 1(b) where d1 ¼ dm:

P1

rf d21

¼ F1Ef

rf;

Ef

Em

� �ð2Þ

P2

rf d22

¼ F2Ef

rf;

Ef

Em

� �ð3Þ

The dimensionless indentation depths d1=R and d2=R are chosensuch that their corresponding matrix effects are distinct(d1=R ¼ 0:25 and d2=R ¼ 0:075 in this paper, see Section 3.1 for jus-tification). F1 and F2 are dimensionless functions that can be deter-mined from extensive FEM simulations in forward analysis, seeSection 3.1. Practically, one could use more data points (such asðP3; d3Þ in Fig. 1(b)) to establish additional (redundant) functionalforms of a different matrix effect, which can help to improve theaccuracy of the reverse analysis, discussed in Section 3.2.4.

Note that with the current normal indentation framework(where the fiber is elastic-perfectly plastic), we do not requirethe use of unloading P—d curve from the experiment. In fact thisis advantageous based on the following reasons:

(1) Since we do not have to perform unloading at the maximumpenetration (d1=R ¼ 0:25 in this study), the fiber radius does notneed to be pre-determined accurately and a moderate scatter offiber radius is allowed. One can indent sufficiently deep to makesure that the maximum depth is above 1/4 of the radii of all fibers,without the need to withdraw the indenter at a particular moment.This extends the applicability of the proposed technique and wouldbe particular welcome for automated test.

(2) In a nanoindentation test, stress relaxation is often observedat the maximum load when unloading just occurs, since both thestrain and the strain rate are very large just beneath the indenter(and the loading rate is usually finite during the experiment).The sudden withdraw of indenter may cause oscillations in themeasurement and holding at the maximum load may also lead touncertainty in the measured quantities. In addition, the initial por-tion of unloading is prone to thermal drift. All these factors requireextra attention during the collection of unloading data [29–31] inparticular if the indentation depth is very small.

(3) In many established theoretical and numerical indentationtechniques, a rigid indenter is used [14,15,19–21,25] whereas inpractice, a diamond indenter tip is employed. It is well known thatthe unloading behavior is strongly affected by the compliance ofindenter tip [14]. Therefore, in order to apply the developed inden-tation method based on the rigid indenter, a real P—d curve mea-sured from experiment must be converted to that measured by arigid indenter – for example, our previous work on thin film inden-tation [21] utilizes the unloading work in the indentation analysis,which requires the full information of the unloading P—d curvewith a rigid indenter, and if the diamond tip contribution is notproperly removed from the experimental data, plus any of theabove-mentioned error due to the nonelastic effects in the unload-ing P � d curve, a large error may occur based on error sensitivityanalysis [21] and an careless user may make mistakes [32]. Since

Fig. 2. The surfaces of fitting functions ðf1; f2Þ obtained from the forward analysis.

36 G. Cao et al. / Composites: Part B 41 (2010) 33–41

the diamond tip compliance has much less influence on the loadingP—d curve (especially at moderate indentation depth adopted inthis paper), it is more convenient to circumvent such difficultyby using the arguably more reliable loading P—d curve measuredfrom an experiment.

2.3. Finite element analysis

The FEM simulations are carried out using commercial codeABAQUS [33]. The fiber is assumed to be a perfect cylinder andthe indentation load is applied along the center axis of fiber. Withthese assumptions, the fiber and matrix can be simplified as a 2Daxisymmetric system instead of using three-dimensional modeland the computational cost can be significantly reduced (note thatif the fiber’s cross-section significantly deviates from circular, or ifthe indentation load is severely off from the fiber’s center axis,then 3D model/analysis must be carried out). The fiber is meshedwith about 20,000 4-node axisymmetric elements with reducedintegration, and the matrix includes about 15,000 elements withthe same type. The mesh density is designed such that the meshis more refined near to the indenter tip along both the axial andthe radial directions. In order to accurately capture the contact ra-dius, the mesh size is refined in the contact area to make sure morethan 200 fiber elements are in contact with the indenter tip whend2=R ¼ 0:075. The matrix radius is set to be 50 times larger than thefiber radius to simulate the semi-infinite ‘‘substrate”, and the fiberlength is taken as 50 times of the fiber radius – these dimensionsensures that the matrix is semi-infinite. The degree of freedomalong axial direction of nodes on the bottom of the fiber and matrixis fixed to represent the boundary condition when the sample ismounted on a rigid substrate.

In the present numerical experiments, the ratio of Ef =rf is var-ied from about 20 to 3000 to cover a wide range of engineering/biomaterials during the forward analysis, including human hairand optical fiber. Although epoxy is selected as a representativematrix material in the parallel experiment (the typical epoxy mod-ulus is about 1–5 GPa), the ratio of Ef =Em is also varied in a widerange from about 2 to 150, such that the established frameworkmay be applied to a wide combinations of fiber/matrix systems.

2.4. Experiment

To verify the methodology developed in this study, nanoinden-tations were performed on the cross-section of human hair andglass fiber using a Triboscope (Hysitron Inc.) in conjunction witha Veeco Dimension 3100 AFM system (Veeco Metrology Group).The human hair and glass fiber were embedded into a specialepoxy (302-3M, Epoxy Technology Inc.), which produces no heator thermal expansion during cure and thus does not create anystress into the fibers. The sample was then cut with a cutting sur-face perpendicular to the fiber using a water-cooled, low speed dia-mond saw and mechanically ground and polished using abrasiveand powders down to 50 nm. Nanoindentations were performedon the polished cross-sections of human hairs and glass fibersusing different indentation forces, with the indentation along theaxial direction of the fibers. For the determination of referencemechanical properties of the fibers, small peak indentation forces(250 lN for human hair and 400 lN for glass fiber) were first usedto generate a shallow penetration depth into the fibers (d=R ¼0:004 for human hair and d=R ¼ 0:01 for glass fiber) so that the ma-trix effects can be ignored. The indentation loading curves at mod-erate depths were then used as an input to derive the mechanicalproperties of fiber using the established framework, with thedetailed results given in Section 4. Since our method requires thematrix modulus to be known a priori, nanoindentations were alsocarried out on the epoxy matrix with a nominal peak force of

100 lN to determine its mechanical properties. For all the indenta-tion tests, the peak indentation force was applied in 5 s and thentotally released in the same time after 5 s holding at the peak forceexcept a fast unloading (2 s) for epoxy matrix to minimize thecreep effect (note that the technique proposed in this paper fordetermining fiber properties does not require unloading).

3. Results and discussion

3.1. Forward analysis

Unlike in a homogeneous bulk material where the loading cur-vature C ¼ P=d2 is a constant during sharp indentation [15], thestress field and plastic flow beneath the indenter tip will be dis-turbed by the presence of the matrix in a fiber/matrix system,and thus the loading curvature is dependent with the indentationdepth (Fig. 1(b)). For a compliant matrix, Ef =Em > 1, the loading curva-ture continuously decreases with the increase of indentation depth;whereas for a stiff substrate, Ef =Em < 1;C increases with the increas-ing d. In this paper, we choose d1=R ¼ 0:25 and d2=R ¼ 0:075 – for thewide material space adopted in this paper, these two indentationdepths yield sufficiently distinct matrix effects. By varying Ef =rf

from 50 to 3000, and Ef =Em from 2 to 150, the normalized loadingcurvatures P1

rf d21

and P2rf d

22

are measured from the numerical indenta-

tion experiments, based on which the dimensionless functions F1

and F2 (Eqs. (2) and (3)) are fitted.Due to the wide range of material properties involved, it is more

practical to change Eqs. (2) and (3) to logarithmic form:

lnP1

rf d21

!¼ f1ðn;gÞ ð4Þ

lnP2

rf d22

!¼ f2ðn;gÞ ð5Þ

where n ¼ lnðEf =rf Þ and g ¼ lnðEf =EmÞ. The fitting error is less than1% for the material space considered, with details given in theAppendix A.

The functions f1 and f2 are visualized as two three-dimensionalcontinuous surfaces in Fig. 2: their difference denotes the distinctmatrix effect at d1=R ¼ 0:25 and d2=R ¼ 0:075. The normalizedloading curvatures increase as Ef =rf decreases or Ef =Em increases.The presence of matrix has a remarkable effect on the normalizedindentation load when Ef =rf is large (i.e. more plastic materials),whereas the matrix effect is smaller for materials with smallerEf =rf (more elastic materials). At the maximum penetration, theplastic flow is still constrained in the small fiber and thus the plas-

0 200 400 600 800 10000

20

40

60

80

100

E /σ

E

mf /E

f f

Fig. 3. The comparison between the input material properties (end) and identifiedmaterial properties (head) from the reverse analysis of numerical indentationexperiments: the fiber/matrix combinations are varied in a large range.

0 50 100 150 2001

2

3

4

5

1

2

3

4

5

E

mf /E

E /σf f

Fig. 4. The comparison between the input material properties (end) and identifiedmaterial properties (head) from the reverse analysis of numerical indentationexperiments: specified for human hair/epoxy composite. The red arrow is based ontwo functions (4) and (5), and black arrow is based on three functions (4), (5), and(11). (For interpretation of color mentioned in this figure the reader is referred tothe web version of the article.)

G. Cao et al. / Composites: Part B 41 (2010) 33–41 37

tic fibers are more sensitive to the presence of matrix. In addition, alarger difference between the fiber/matrix elastic moduli will takemore advantage of the matrix effect. These two functions relate theindentation response with the fiber elastoplastic properties as wellas fiber/matrix elastic mismatch; once calibrated via FEM, they canbe employed in the reverse analysis for determining the fiber prop-erties, discussed next.

3.2. Reverse analysis

3.2.1. General procedureFor an elastoplastic fiber embedded in an elastic matrix with

known matrix modulus, in order to extract the intrinsic materialparameters of the fiber, ðEf ;rf Þ, the following procedures are car-ried out:

(1) A normal indentation test is performed (as sketched inFig. 1(a)), and two different loading curvatures, P1=d

21 ¼ 16P1=R2

and P2=d22 ¼ 178P2=R2are measured.

(2) During the reverse analysis, the postulated fiber propertiesðEf ;rf Þ are varied over a large range and substituted into Eqs. (4)and (5) along with the measured indentation data – the combina-tion of ðEf ;rf Þ that minimizes the total error in Eqs. (4) and (5) rep-resents the identified elastoplastic properties of the fiber. In otherwords, the identified ðEf ;rf Þ will have the least square deviation e:

e ¼ P1

rf d21

� f1Ef

rf;

Ef

Em

� � !2

þ P2

rf d22

� f2Ef

rf;

Ef

Em

� � !2

ð6Þ

A practical issue is that multiple local minima may occur during thenumerical reverse analysis procedure. However, according to ourrecent work [15] the solution of film indentation with moderatelydeep penetration depth is unique and we expect such conclusionis transferable to fiber normal indentation. Therefore, if one indeedencounters multiple local minima, the corresponding candidatematerial properties are substituted back into Eqs. (4) and (5) toexamine if the theoretical loading curvatures agree with those mea-sured – finally, the correct solution of fiber properties should beable to reproduce the loading P—d curve measured from experi-ment. Such numerical iteration does not need the performance ofany additional FEM simulations.

3.2.2. Effectiveness of the reverse analysisThe effectiveness of the reverse analysis procedure is first

examined by numerical indentation tests covering a wide rangeof fiber material parameters. In the numerical experiment, the fiberproperties ðEf ;rf Þ used in simulation (termed as the input proper-ties, or true solution) are different than those employed in the for-ward analysis (Section 3.1). The fiber is embedded in an epoxymatrix with a representative reference Young’s modulus of 2 GPa(which does not equal to the real material used in parallel experi-ment elaborated in Section 4; here we purposely use a differentmatrix modulus than the one used in experiment to test therobustness of our proposed technique). From the data obtainedfrom such numerical test, the reverse analysis is carried out follow-ing the procedures described in Section 3.2.1, and the identifiedelastoplastic properties of fiber are compared with the input prop-erties in Fig. 3.

In Fig. 3, for each fiber material, the end of the arrow shows thevalue of input parameters, and the head of arrow represents theidentified fiber properties from the reverse analysis. For the exam-ples shown in Fig. 3, the maximum deviation of the identified Ef =Em

is about 14%, and that occurs when both Ef and rf are large. Themaximum deviation of the extracted Ef =rf is about 6%, which hap-pens to a sample with both relatively small values of Ef and rf .Overall, the identified fiber elastoplastic properties match wellwith the true solution.

Besides numerical verification over a wide material space, wealso focus on human hair/epoxy system as it is used in parallelexperiments. The reported elastoplastic properties of human hairvary from [7,34,35]: Ef ¼ 5—8 GPa and rf ¼ 30—90 MPa. Withinsuch range, four different human hair examples are selected withinput properties ðEf ¼5 GPa;rf ¼40 MPaÞ;ðEf ¼5GPa;rf ¼80 MPaÞ;ðEf ¼8 GPa;rf ¼40 MPaÞ, and ðEf ¼8 GPa;rf ¼80 MPaÞ, respectively.The comparisons between the identified elastoplastic properties(head) and true solutions (end) are plotted in Fig. 4 as red arrows.The maximum deviations of Ef =Em and Ef =rf are about 8% and 3%,respectively. Thus, the proposed normal indentation techniquescan be effectively employed to measure the elastoplastic propertiesof human hair.

38 G. Cao et al. / Composites: Part B 41 (2010) 33–41

3.2.3. Error sensitivity of the reverse analysisIn a numerical indentation test, the measured data ðP1; P2;

d1; d2Þ cannot coincide exactly with the fitting functions (Fig. 2,or Eqs. (4) and (5)) established during the forward analysis, whichwill cause certain numerical error in the reverse analysis. Inaddition, small errors might also occur in the measurement ofðP1; P2; d1; d2Þ during instrumented indentation experiments (dueto either random noise or ‘‘systematic” calibration errors), whichwill also affect the accuracy of the fiber properties extracted fromthe reverse analysis. The error sensitivity, i.e. how sensitive theaccuracy of the reverse analysis is to these errors, is very impor-tant for examining the robustness of the proposed indentationframework.

A technique initial reported by Prchlik [36] is expanded to ourelastoplastic fiber/elastic matrix system. The following equationscan be obtained by differentiating the fitting functions (4) and (5):

dP1

P1� drf

rf� 2dd1

d1¼ @ðf1ðEf =rf ; Ef =EmÞÞ

@ðEf =rf Þd

Ef

rf

� �

þ @ðf1ðEf =rf ; Ef =EmÞÞ@ðEf =EmÞ

dEf

Em

� �ð7Þ

dP2

P2� drf

rf� 2dd2

d2¼ @ðf2ðEf =rf ; Ef =EmÞÞ

@ðEf =rf Þd

Ef

rf

� �

þ @ðf2ðEf =rf ; Ef =EmÞÞ@ðEf =EmÞ

dEf

Em

� �ð8Þ

Here, each differential term represents the error of that variable.The error of the substrate modulus is not considered since it isassumed known in this paper. From Eqs. (7) and (8), the variations(errors) of the fiber properties ðEf ;rf Þ can be explicitly expressed asa function of the perturbations (errors) of indentation measure-ments ðP1; P2; d1; d2Þ:

dEf

Ef¼ a1

dP1

P1þ a2

dP2

P2þ a3

dd1

d1þ a4

dd2

d2ð9Þ

drf

rf¼ b1

dP1

P1þ b2

dP2

P2þ b3

dd1

d1þ b4

dd2

d2ð10Þ

The dimensionless coefficients ai; biði ¼ 1—4Þ can be used torepresent the error sensitivity of a material parameter correspond-ing to a particular error of one indentation data. For example, if therelative error of the measured maximum indentation loadP1; dP1=P1 ¼ 1%, the corresponding error of the identified Ef fromthe reverse analysis will be a1%. In other words, a1 is the error sen-sitivity factor of Ef corresponding to the perturbation of P1. If thecoefficients ðai; biÞ are small, then the identified fiber elastoplasticproperties are relatively insensitive to the error of the indentationmeasurements, and thus the reverse analysis is robust.

We illustrate the error sensitivities of two types of importantfibers: human hair and glass fiber. The representative materialproperties for the human hair are Ef ¼ 8 GPa and rf ¼ 80 MPa,and the typical elastoplastic parameters of a glass fiber areEf ¼ 80 GPa and rf ¼ 1000 MPa. Based on the results of computederror sensitivity factors given in Table 1, since these coefficients arerelatively small, the proposed technique can be considered as quiterobust against the perturbation of experimental error for these twocategories of applications.

Table 1The coefficients in Eqs. (9) and (10).

i= 1 2 3 4

Representative human hair ai �0.09993 1.45607 0.19986 �2.91214bi 0.01821 1.28227 �0.03642 �2.56453

Representative glass fiber ai �0.07248 1.28186 0.14495 �2.56372bi 0.03352 1.04815 �0.06703 �2.09629

3.2.4. Redundancy analysisIn the above analysis, two independent functions representing

different matrix effects are employed to solve the fiber elastoplas-tic properties via a reverse analysis. Theoretically, the accuracy ofthe method can be further improved by the redundancy, i.e. byemploying more similar functions in the forward and reverse anal-yses. With reference to Fig. 1(b), one can choose one more datapoint at ðP3; d3Þ, with d3=R ¼ 0:15. And the third dimensionlessfunction can be described as:

lnP3

rf d23

!¼ f3ðn;gÞ ð11Þ

Note that this function is not completely independent with Eqs. (4)and (5). The function f3 is also given in Appendix A. In the reverseanalysis, Eq. (6) can be rewritten as:

e ¼ P1

rf d21

� f1Ef

rf;

Ef

Em

� � !2

þ P2

rf d22

� f2Ef

rf;

Ef

Em

� � !2

þ P3

rf d23

� f3Ef

rf;

Ef

Em

� � !2

ð12Þ

With one more functional constraint, the least square deviationmay coincide better with the real material properties.

For the four hair examples used in Section 3.2.2, the redundancyanalysis (Eq. (12)) is applied to recalculate the fiber propertiesfrom the numerical indentation experiments, and the comparisonsbetween the identified parameters vs. input values are shown inFig. 4 as black arrows. With the addition of the redundant equation,there is no significant improvement of the deviations of the identi-fied Ef =Es and Ef =rf . Therefore, in practical applications, the twofunctions (Eqs. (4) and (5)) and their reverse analysis (Eq. (6)) pro-posed in Sections 3.2.1 and 3.2.2 may be sufficient.

4. Experiment on glass fiber and human hair

The parameters and reference properties of glass fiber and hu-man hair are shown in Table 2, whose radius was measured frommicroscopy and mechanical properties were measured from care-ful shallow nanoindentation tests (Section 2.4), and used as bench-mark for examining the indentation technique proposed herein.The representative AFM images of indents on glass fiber andhuman hair are given in Fig. 5. Next, we measure the modulusand yield strength of these fibers by carrying out deep indentationtests on the cross-sections of the fibers embedded in an epoxy ma-trix, and then use the framework proposed above to deduce thefiber’s elastoplastic properties via the reverse analysis. The Young’smodulus of the epoxy matrix was Em ¼ 5:2 GPa, measured alsofrom the shallow nanoindentation test (Section 2.4).

Fig. 6(a) and (b) show the indentation load–depth ðP—dÞ curveson three distinct glass fibers and human hairs, respectively. By in-put the shape factors of loading P—d curves at d=R ¼ 0:075 and0:25 into the functions built in the forward analysis, their elasto-plastic properties can be identified from the reverse analysis:Ef ¼ 65:9—72:8 GPa and rf ¼ 3:2—3:8 GPa for glass fibers; andEf ¼ 7:62—7:95 GPa and rf ¼ 0:077—0:079 GPa for human hair.

Table 2The parameters and properties of human hair and glass fiber used in experiments.

RadiusR ðlmÞ

ReferenceEf (GPa)

Referencerf (GPa)

Reverseanalysis Ef (GPa)

Reverse analysisrf (GPa)

Human hair 40 7.93 0.08 7.79 0.078Glass fiber 5 73.8 3.23 69.4 3.5Epoxy – 5.2 – – –

Fig. 5. (a) AFM image of an indent on a glass fiber; (b) AFM image of an indent on ahuman hair.

0

0.1

0.2

0.3

0.4

0.5

0.6

Experiments

Human HairIn

dent

atio

n lo

ad (N

)

Indentation Depth (μm)0 2 4 6 8 10

0

0.1

0.2

0.3

0.4

0.5

0.6

Experiments

Inde

ntat

ion

load

(N)

Indentation Depth (μm)

δ/R = 0.075

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Inde

ntat

ion

load

(N)

Indentation Depth (μm)

Experiments

Glass Fiber

0 0.25 0.5 0.75 1 1.250

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Inde

ntat

ion

load

(N)

Indentation Depth (μm)

Experiments

δ/R = 0.075

δ/R = 0.25

δ/R = 0.25

a

b

Fig. 6. Indentation load–depth ðP—dÞ curves for (a) glass fibers with epoxy matrix;(b) human hair with epoxy matrix; the data is used in reverse analysis to obtainglass fiber and human hair properties in Table 2.

G. Cao et al. / Composites: Part B 41 (2010) 33–41 39

These results are within the ranges reported in the literature[7,34,35]. The difference between the average identified propertiesand the reference properties is less than 8% for glass fiber and lessthan 4% for human hair, respectively, and thus the proposed inden-tation technique is essentially robust. It is quite interesting that theyield strength of human hair is comparable with some of the stron-gest polymers used in industry [37], whose mechanism is worthinvestigating in the future. We note that although in our currentstudy, the reference fiber properties can be measured from shallownanoindentations because the glass fiber and human hair em-ployed in this study were not very thin, the proposed indentationtechnique which relies on the matrix effect at moderately deepindentation depth should have wider applications especially incases where shallow nanoindentations are not reliable or available(as mentioned in Section 1.2).

5. Conclusions

In this paper, we propose a framework of indentation technique,which measures the elastoplastic properties of a fiber embedded inan elastic matrix via normal indentation. The method is applicableto a range of polymer, glass, ceramic, and metal fibers. The matrixeffect is investigated during the forward analysis and the intrinsicfiber properties are extracted via a reverse analysis. The proposedmethod focuses on studying the matrix effect, utilizing such effect,

and finally subtracting off such effect to obtain the fiber’s mechan-ical properties. During the analysis, only the loading curve of anindentation test is employed, which may circumvent the commonsources of errors encountered during the unloading measurementand thus increases the potential applicability.

From the numerical forward analysis over a large space of mate-rial properties, the relationships between the material propertiesand the indentation parameters are established by using extensivefinite element simulations. Based on these relationships, after anindentation test is carried out on the sample fiber embedded inan elastic matrix, the data at two different indentation depthscan be measured that correspond to different matrix effects, andthen the fiber properties can be determined from the reverse anal-ysis. The effectiveness of the reverse analysis is verified by both thenumerical experiments spanning a wide range of fiber/matrix com-binations and those of the human hair/epoxy system. In addition,an error sensitivity analysis is carried out, which shows that thecurrent method is essentially insensitive to the perturbations of

Table A1The fitting parameters in Eqs. (14) and (16).

Coefficients in Eq. (14) Coefficients in Eq. (16) Coefficients in Eq. (17)

p1 3.18880 q1 �5:19590 � 1012 o1 4.37145p2 �1.01670 q2 2:35174 � 1012 o2 �1.30194p3 �3.22878 q3 9:40246 � 1012 o3 �6.48993p4 �0.02700 q4 �2:12310 � 1012 o4 �0.05701p5 �0.39386 q5 �6:91708 � 1012 o5 �0.01768p6 0.30349 q6 �0:17615 � 1012 o6 0.63007p7 1.02616 q7 �0:35666 � 1012 o7 3.25744p8 �0.02821 q8 �0:11329 � 1012 o8 0.00009p9 �0.14069 q9 �0:06559 � 1012 o9 0.01213p10 0.08576 q10 0:30213 � 1012 o10 �0.02115p11 0.54120 q11 0:34787 � 1012 o11 �0.27911

40 G. Cao et al. / Composites: Part B 41 (2010) 33–41

the measured indentation load and depth, and thus it is quiterobust for determining the elastoplastic properties of micro- andnano-fibers. We carried out experiments on both glass fibers andhuman hair, and the effectiveness of the proposed indentationtechnique is validated.

In the future, the present work may be expanded to inhomoge-neous coaxial fibers (e.g. multi-layer optical fibers or coated hair) –again, the different indentation depths will induce the differentmulti-layer effects and those effects are quite controllable (asopposed to the conventional lateral indentation where the outmostlayer is overly dominant during the test). Another future researchdirection is to incorporate the strain hardening effect of the fiberand/or matrix, as well as possible debonding between the fiberand matrix. The framework can also be extended to the fiber-rein-forced composites if the fiber spacing is sufficiently apart; other-wise, the interaction among the fibers needs to be considered,which is still possible [23] by using more complicated formulations.

Acknowledgements

The work of G.C. and X.C. was supported by NSF CMS-0407743and CMMI-0643726. Z.H.X. and X.L. were supported by NSF EPS-0296165 and CMMI-0653651.

Appendix A

With n � lnðEf =rf Þ and g ¼ lnðEf =EmÞ, the dimensionless func-tions of the normalized indentation curvatures are:

lnP1

rf d21

!¼ f1ðn;gÞ ð13Þ

with

f1ðn;gÞ¼p1þp3 lnðnÞþp5 lnðgÞþp7ðlnðnÞÞ

2þp9ðlnðgÞÞ2þp11 lnðnÞlnðgÞ

1þp2 lnðnÞþp4 lnðgÞþp6ðlnðnÞÞ2þp8ðlnðgÞÞ

2þp10 lnðnÞlnðgÞð14Þ

and

lnP2

rf d22

!¼ f2ðn;gÞ ð15Þ

with

f2ðn;gÞ ¼q1 þ q3nþ q5gþ q7n

2 þ q9g2 þ q11ng1þ q2nþ q4gþ q6n

2 þ q8g2 þ q10ngð16Þ

The coefficients in (14) and (16) are given in Table A1.The redundant Eq. (11) can be fitted as:

f3ðn;gÞ ¼o1 þ o3 lnðnÞ þ o5gþ o7ðlnðnÞÞ2 þ o9g2 þ o11 lnðnÞg1þ o2 lnðnÞ þ o4lgþ o6ðlnðnÞÞ2 þ o8g2 þ o10 lnðnÞg

ð17Þ

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