Composites Science and Technology 85 (2013) 142–147

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Composites Science and Technology

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Review

Young’s modulus prediction of long fiber reinforced thermoplastics

0266-3538/$ - see front matter � 2013 Published by Elsevier Ltd.http://dx.doi.org/10.1016/j.compscitech.2013.06.009

⇑ Corresponding author.E-mail address: fgaresci@unime.it (F. Garesci).

F. Garesci a,⇑, S. Fliegener b

a DIECII, Department of Electronic Engineering, Industrial Chemistry and Engineering, University of Messina, C.da di Dio, Italyb Fraunhofer Institute for Mechanics of Materials IWM, Freiburg, Germany

a r t i c l e i n f o

Article history:Received 27 March 2013Received in revised form 30 May 2013Accepted 9 June 2013Available online 27 June 2013

Keywords:A. Glass fibersB. Mechanical propertiesC. Elastic properties

a b s t r a c t

The aim of this paper is to provide analytical models able to predict the elastic properties of long fiberreinforced thermoplastics (LFT) in dependence of microstructural parameters such as the fiber volumecontent (vf), the fiber orientation distribution (FOD) and the fiber length distribution (FLD). The analyticalpredictions are compared to the experimental stiffness values from tensile tests on the composite mate-rial showing a good agreement. The FLD in terms of the probability density distribution as function of thefiber aspect ratio has been computed by an automated fiber separation and image analysis tool. On theother hand, the FOD in terms of the probability density distribution as function of fiber in-plane orienta-tion was identified by using an image correlation procedure based on computer tomography scans ofcharacteristic LFT specimens. Our analysis shows a good performance to predict Young’s modulus ofLFT due to incorporation of the FOD as well as the FLD into the calculations.

� 2013 Published by Elsevier Ltd.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1422. Experimental methods and materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

2.1. Tensile tests to determine the elastic stiffness of the LFT material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1432.2. Microstructure analysis by means of computer tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1432.3. Automated fiber separation and analysis procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3. Analytical procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

3.1. Halpin–Tsai theory to evaluate the properties of the transversely isotropic composite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443.2. Fiber orientation distribution function to determine the Advani–Tucker tensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443.3. Mean Tensor averaging procedure to predict the elastic properties of LFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4. Fit functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.1. Curve fitting procedure for FOD (fiber orientation distribution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.2. Curve fitting procedure for FLD (fibers length distribution). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

4.2.1. Results obtained by using the recovered functions for FOD and FLD and by using the mean tensor averaging procedure . . . . 1465. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

1. Introduction

In the last years, the demand of high performing and economi-cally producible materials for automotive applications is growing.LFT are a promising solution meeting the criteria cited above, inparticular they provide a good specific stiffness and strength if isthe part design is adapted to the respective load cases. The efficient

value of the material properties such as elastic stiffness andstrength varies in a broad range up to 250% [18,19]. Those valuesdepend on the process-driven microstructural properties such asthe fiber volume content (vf), the fiber orientation distribution(FOD) and the fiber length distribution (FLD) and they must be con-sidered in order to fully exploit the lightweight potential of LFT. Inthis work, the elastic stiffness of LFT is derived from the character-istic values of the microstructure by means of analytical models.There are several procedures in literature for predicting theproperties of composite materials in dependence of their

F. Garesci, S. Fliegener / Composites Science and Technology 85 (2013) 142–147 143

microstructure. Tucker et al. [1] provide a good overview on themicromechanical models used to calculate the stiffness of short fi-ber composites. This traditional approach combines several meth-ods such as the Eshelby equivalent inclusion, the Mori–Tanakamodels and the Halpin–Tsai equations [2] to predict the elasticproperties of short fiber reinforced plastics in dependence of bothfiber orientation and volume content. Other authors [3–5] use aunidirectional short fiber composite and improve the microme-chanical models by taking into account the spatial distribution ofthe reinforced fibers. In this way, it is also possible to predict thelocal thermo-mechanical properties excellently. Other studies fo-cus on the closure approximations [6–8] for evaluating the ther-moelastic properties of short fiber composite to optimize theinjection molding process. A recent study [9] is based on the elasticprediction of the correct experimental fiber length distribution(FLD) and the correct fiber orientation distribution (FOD). The fiberdimension could be a relevant aspect for understating the elasticproperties of composites. Simple models are based on the predic-tion of spherical particles [12] or short fiber composite, but re-cently studies focused on not aligned long fiber composite [13]and classic theory applied to new materials like carbon nanotubes[14] and nanocomposite [15] are very promising in term of under-stating the elastic properties. Other authors focused their method-ology in the effect of the interfacial bonding conditions only toevaluate the effective properties of the composite [10,11]; whileothers used finite element method in order to investigate howtwo of the most widespread models (i.e. Tandon–Weng and Hal-pin–Tsai) are suitable to predict either the stiffness of short fibercomposite [21] and a composite by applying the orientation aver-aging scheme (several hundred multi-fiber with different orienta-tion states [22]). In Ref. [12] the two previous approaches aremodified in two relevant steps. Firstly, by incorporating an exper-imental FLD as well as a fiber orientation distribution (FOD) andsecondly by evaluating the Mori–Tanaka model to obtain the stiff-ness matrix of the aligned fiber composite containing the previousFLD. In this article, the elastic stiffness of a commercial long fiberreinforced thermoplastic has been investigated by using a hybridprocedure based on a theoretical approach describing the trans-versely isotropic equivalent composite. The model uses the exper-imental data of the fiber orientation and length distributions. Tovalidate the procedure the predicted stiffness values are comparedwith experimental values from tensile tests. The scenario of theexperimental framework presented in this manuscript is differentfrom the works that studied short fiber composite (aspect rationear the value of 20); in fact, in our experimental data the smallestvalue measured of aspect ratio is 27. In addition, if we consideredthe long fiber studies, most of them do not use directly experimen-tal FLD but the corrected FLD data or the Monte Carlo method. Theadvantage of our procedure is due to the direct use of the experi-mental data and thus to predict the stiffness properties of LFTrather quickly. The working scheme is given by three main steps:

1. Experimental methods and materials1.1 Tensile tests to determine the elastic stiffness of the LFT

material.1.2 Microstructure analysis by means of computer tomogra-

phy to analyze the fiber orientation distribution (FOD).1.3 Automated fiber separation and analysis procedure to

investigate the fiber length distribution (FLD).2. Analytical procedure:

2.1 Halpin–Tsai theory to evaluate the properties of the trans-versely isotropic composite.

2.2 Fiber orientation distribution function to determine thetensor Rank 2 aij and the tensor Rank 4 aijkl after Advani,Tucker [16].

2.3 Mean tensor averaging procedure to predict the elasticproperties of LFT.

3. Fit functions:3.1 Curve fitting procedure for fiber orientation

distribution(FOD);3.2 Curve fitting procedure for fiber length distribution (FLD).

2. Experimental methods and materials

The LFT material was produced by a so-called direct LFT pro-duction route, where the continuous fiber roving are introduceddirectly into a double screw extruder and are broken into fiberfragments by the overlapping edges of the contra-rotatingscrews. The resulting material is characterized through a broadFLD reaching from a large amount of fiber fragments well below1 mm until very long fibers up to 50 mm. The matrix consists ofpolypropylene resin (DOW� C711–70RNA) with some stabilizersand coupling agents and is merged with 30 mass% glass fibers(TufRov� 4575) to form the LFT composite. To simulate differentmaterial states in terms of a locally varying microstructure,3 mm thick plates with outer dimensions of 400 � 400 mm wereproduced by compression molding. The LFT-D strand as it comesout of the extruder was placed asymmetrically in the mold toseparate the so-called press region near the strand inlayposition from the flow-region, where a lateral flow is responsiblefor a higher degree of fiber orientation compared to the press-region.

2.1. Tensile tests to determine the elastic stiffness of the LFT material

The tensile stiffness of specimens with different orientation rel-ative to the fiber mean orientation or flow direction, but similar fi-ber volume fraction and fiber length distribution was determinedwith a Hegewald & Peschke ‘‘Inspekt 100’’ testing machine. Thespecimen geometry was chosen according to DIN EN ISO 3167.The tests were performed at a strain rate of approximately0.00022 (mm/s) to a maximum stress of 10 MPa in order avoidany damage. Three loading–unloading cycles per specimen wereperformed. The stiffness is taken as the mean value of the threeloading cycles. At least three specimens per orientation weretested. The stiffness values for E11 and E22 are 7.83 and 3.25 GPaand 6.49 and 3.77 GPa for flow and press region LFT (30 mass%/13 volume-% fiber fraction) respectively.

2.2. Microstructure analysis by means of computer tomography

Microstructure analysis has been performed on specimens fromdifferent regions of a 3 mm thick LFT-plate. Because of the thincross section, the fibers are aligned in planes parallel to the speci-men surface and the out-of-plane component of the fiber orienta-tion is negligibly small. The specimens for computer tomographicanalysis have been prepared from similar regions as the specimensfor mechanical testing were taken from. The tomography scanshave been performed on a Phoenix ‘‘nanome|x 180NF’’ with a voxelsize of 5–8 lm, so the complete plate thickness could be analyzedwith one scan. The data was analyzed by the plug in ‘‘Directional-ity’’ for ImageJ [20], after converting the 3D voxel data to a stack of2D images parallel to the surface. The 2D images parallel to thesurface 2D orientation histograms have been extracted for eachof the stack. The resulting orientation distribution for the wholestack has been generated by summing up the single histogramsand re-normalizing. Flow region specimens (Fig. 1a) were takenfrom a region of approx. 100 mm distance from the LFT strand in-lay position, so a relatively high amount of fiber alignment resultsfrom the lateral flow field inside the mold. A low degree of fiber

Fig. 1. Experimental Ct scan for flow (a) and press (b) region.

144 F. Garesci, S. Fliegener / Composites Science and Technology 85 (2013) 142–147

waviness can be observed in a characteristic CT scan of a flow re-gion specimen. However, the scan shows only a section of approx.2.8 mm edge length due to resolution limits of the scanning device.An exemplary CT scan of a press region specimen (Fig. 1b) exhibitsremarkable deviation from the flow region structure which can beexplained by a different flow field during specimen fabrication andthus a significantly lower degree of fiber alignment.

2.3. Automated fiber separation and analysis procedure

The FLD was determined with a characteristic LFT specimen of asize of approx 70 � 40 � 3 mm3. The specimen was first inciner-ated and then dispersed in a dilute solvent to separate the fibers.Finally, the fibers were analyzed by the image analysis software‘‘FASEP�’’. The procedure was carried out by the company ‘‘xyzhigh precision’’ [17].

3. Analytical procedure

3.1. Halpin–Tsai theory to evaluate the properties of the transverselyisotropic composite

The Halpin Tsai equations are used to predict the mechanicalproperties of continuous long fiber composites [2]:

Pc

Pm¼ 1þ fgVf

1� gVfð1Þ

where Pc and Pm are the composite properties (E11, E22, G12) and thematrix properties respectively, f is a measure of reinforcementgeometry and loading conditions and g is a fiber parameter com-puted as:

g ¼ Pf =Pm � 1Pf =Pm þ f

ð2Þ

For continuous, cylindrical fiber reinforced composites thegeometry factor is given as:

fE11 ¼ 2 � AR þ 40V10f

fE22 ¼ 2þ 40V10f

fG12 ¼ 1þ 40V10f

ð3Þ

If the fibers are aligned along 1-direction and the behaviors inthe 2-direction and 3-direction are identical, the composite canbe treated as transversely isotropic. Under this hypothesis the fol-lowing identities hold: E22 = E33, m12 = m13, G12 = G13 and the follow-ing relationship is verified:

G23 ¼ E22=2ð1þ m23Þ ð4Þ

In this scenario, the independent constants for the transverselyisotropic material are only five (i.e. E11, E22, G12, m12, m23) and theCompliance Matrix [S] can be expressed by:

½S� ¼

1=E11 m21=E22 �m31=E33 0 0 0�m12=E11 1=E22 �m32=E33 0 0 0�m13=E11 �m23=E22 1=E33 0 0 00 0 0 1=G23 0 00 0 0 0 1=G13 00 0 0 0 0 1=G12

2666666664

3777777775

ð5Þ

The stiffness matrix [C] is the inverse of [S] matrix.

½C� ¼

C11 C12 C13 0 0 0C21 C22 C23 0 0 0C31 C32 C33 0 0 00 0 0 C44 0 00 0 0 0 C55 00 0 0 0 0 C66

2666666664

3777777775

ð6Þ

The stiffness matrix for the transversely isotropic material iscompletely defined if every parameter of matrix and fiber is knownand all terms inside the matrix simply depend on the aspect ratioand the volume fraction values.

3.2. Fiber orientation distribution function to determine the Advani–Tucker tensors

In the earliest work, to study the elastic properties of a compos-ite taking into account the fiber orientation, each fiber was mod-elled as a rigid cylinder with fixed length and diameter [16]. If thefiber concentration is spatially uniform, a spherical coordinate sys-tem W(#, u) that represents the probability of the fiber orientationinside the unit sphere can be introduced. W is normalized and sym-metric function. In addition, a vector p describing the position of thefiber in the spherical coordinate system can be defined as follows:I

dp ¼Z 2p

/¼0

Z p

#¼0sin#d#du: ð7Þ

The tensor Rank 2 and the tensor Rank 4 are:

aij ¼I

pipjWðpÞdp

aijkl ¼I

pipjpkplWðpÞdpð8Þ

being i, j, k, l = 1,2,3. For a bi-dimensional material, where all fibersare located in the 1–2 plane (as it is the case for the investigated

150 100 50 50 100 150degree

0.02

0.01

0.01

0.02

Residual

150 100 50 50 100 150degree

0.005

0.010

0.015

0.020

0.025FOD

Fig. 2. Flow region: comparison and error variations (in the insertion) betweenexperimental data and the FOD fitting functions and error variations.

F. Garesci, S. Fliegener / Composites Science and Technology 85 (2013) 142–147 145

LFT), the orientation state is planar and the distribution functionWu(u) depends only on u and the tensor Rank 2 and the tensorRank 4 are:

aij ¼Z 2p

0pipjWuðpÞdp ð9Þ

aijkl ¼Z 2p

0pipjpkplWuðpÞdp ð10Þ

Starting from the experimental data, it is not possible to describeand absolutely quantify the orientation of each fiber inside the com-posite, in this sense a probability distribution function to describethe spatial or plan probability distribution of the fibers is used.

3.3. Mean Tensor averaging procedure to predict the elastic propertiesof LFT

Basically, the properties of the composite are related to theprobability distribution function W(p) and to the properties ofthe transversely isotropic tensor T(p) by:

hTi ¼I

TðpÞWðpÞdp ð11Þ

where hTi denotes the orientation average property. The tensorT(pi,j,k,l) is given by:

Ti;j;k;l ¼ B1ðpipjpkplÞ þ B2ðpipjdkl þ pkpldijÞ þ B3ðpipkdjl

þ pipldjk þ pjpkdilÞ þ B3ðdij þ dklÞ þ B5ðdikdjl þ dildjkÞ ð12Þ

where djl denotes the Kronecker delta function and the scalar Bi

parameters are depending on the stiffness matrix elements (Eq.(6)) as shown:

B1 ¼ C11 þ C22 � 2C12 � 4C66

B2 ¼ C12 � C23

B3 ¼ C66 þ12ðC23 � C22Þ

B4 ¼ C23

B5 ¼12ðC22 � C23Þ

ð13Þ

By applying the Eqs. (12) and (13), it is possible to evaluate theelastic properties of the LFT composite weighted by the orientationdistribution function. It is also important to remark that the fibershave a uniform length and constant aspect ratio (fiber length/fiberdiameter) because the volume fraction vf is constant.

4. Fit functions

4.1. Curve fitting procedure for FOD (fiber orientation distribution)

From microstructure analysis (CT-scans) we obtain the fiber ori-entation distribution (FOD) in terms of probability density distri-bution as function of the in-plane fiber orientation for flow andpress regions (Fig. 2). As described in Section 2.2, the FOD functionWu is to be symmetric and normalized. Despite different functionscan be applied, not all of them satisfy the two criteria. For example,the Gaussian function (blue1 curve in Fig. 2) is normalized and sym-metric but does not reproduce the experimental data; on the otherhand the Gauss function (red curve) is a symmetric function suitableto fit the experimental data but it is not a normalized function. Withthis in mind, we selected the trigonometric function y = a + bcos (x + c) (orange curve in Fig. 2). The function is symmetric and

1 For interpretation of color in Fig. 2, the reader is referred to the web version othis article.

f

normalized and it reproduces with a good agreement the experimen-tal data, in fact it does not underestimate the presence of fibers farfrom the 1-direction. The shape of the trigonometric function wellreplicates the shape of the experimental data and shows the lowestvalue for the quadratic error function (Fig. 2). The aij tensors are fi-nally obtained by applying Eq. (9) and after choosing the trigono-metric functions to model the flow and the press region. Their

values are:0:718 �0:066 0�0:066 0:282 0

0 0 0

0@

1A and

0:594 �0:005 0�0:005 0:406 0

0 0 0

0@

1A

respectively. The FOD function was used in the computation. There-fore, we evaluate Young’s modulus value of the composite startingfrom the properties of transversely isotropic material and by usingthe tensor procedure and the recovered FOD function. We focusedon the flow region; this region is the part of the specimen wherethe highest degree of fiber alignment can be observed. Based onthe information from Section 1.3, the Halpin–Tsai theory whichwas developed for ideal straight fibers can most likely be appliedfor flow region LFT while it must be rated as critical to model pressregion specimens.

4.2. Curve fitting procedure for FLD (fibers length distribution)

From specimen incineration and image analysis we can deter-mine the fiber length distribution (FLD) in terms of probabilitydensity distribution versus aspect ratio. The experimental fiber dis-tribution as function of the aspect ratio is characterized by twomain regions, see purple line in Fig. 3, in the first region achievedfor values smaller than 500 the probability decreases strongly from15.77% to 0.93%, in the second region scattered data are observed.For this reason, for this experimental data the distribution func-tions (logarithmic) used in [11] are not suitable. Our investigationsindicate as more appropriate curves the ones shown in Fig. 3: (i)the exponential function (the red curve: y = ea+bx) and (ii) thehyperbolic function (the orange curve y = a/x). The experimentalrange of the aspect ratio is between 27 and 3340. We can observethat: both red and orange curves in Fig. 3 are in good agreement tothe experimental data for low values of aspect ratio (smaller than200); the percentage amount of the fibers with aspect ratio lessthan 200 is 37.24% and 50.18% respectively. In particular, the32.8% of the fibers have an aspect ratio value smaller than 120and the 15.77% of the fibers have aspect ratio equal to 27. For high-er values of aspect ratio scattered data are observed. In this condi-tion, two main mistakes could be occur, (i) to underestimate theE11 value if we focus on the lower aspect ratio or (ii) to exclude ahigh amount of fibers if we focus on the higher aspect ratio values.Furthermore the orange curve follows well the experimental datafor high values of aspect ratio. All curves are fitted with data inside

0 500 1000 1500 2000 2500 3000 3500AR0

5

10

15

20Probability

Fig. 3. Comparison between experimental data (dark purple) and several fit curves.(For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

Table 1Fiber (f) and matrix (m) properties.

Ef 72.00 (GPa) Fiber Young’s modulusmf 0.22 Fiber Poisson’s ratiodf 0.017 (mm) Fiber diameterEm 1.40 (GPa) Matrix Young’s modulusmm 0.35 Matrix Poisson’s ratio

0 200 400 600 800 1000AspectRatio0

2

4

6

8

10E11 [GPa]

____12%____13%

____14%

____15%

____16%

____17%

____18%

Fig. 4. E11 as function of the aspect ratio computed by taking into account thetensor procedure for DOF at different volume fraction and experimental values(purple points). (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)

0 200 400 600 800 1000AspectRatio0

2

4

6

8

10

12

14E11 [GPa]

Fig. 5. E11 as function of the aspect ratio achieved for a transversely isotropicmaterial (thick blue line), by using tensor procedure (dashed line) and by using therecovered fit function (continuous line) for a fixed volume fraction value (13%). (Forinterpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

Table 2Analytically predicted E11 values for flow region LFT atdifferent volume fraction values.

Volume fraction (%) E11 (GPa)

12 8.0613 8.5414 9.0515 9.5616 10.0517 10.5818 11.09

146 F. Garesci, S. Fliegener / Composites Science and Technology 85 (2013) 142–147

the experimental range of AR. Outside this range, it does not makesense to recover the FLD, because values lower than 27 correspondto fiber fragments with low load-carrying ability and values higherthan 2500 are less relevant for our investigation as the elastic stiff-ness has almost reached its saturation level for AR above 1000.

4.2.1. Results obtained by using the recovered functions for FOD andFLD and by using the mean tensor averaging procedure

The last part of this work is related to the prediction of theproperties of LFT through the combination of the Halpin Tsai the-ory, the orientation distribution function, the mean tensor averag-ing procedure and the length distribution function.

– The first step is to evaluate the elastic properties of the trans-versely isotropic material from the Halpin–Tsai theory. Asknown, for a fixed fiber volume fraction, the elastic stiffness ofthe transversely isotropic composite in the 1 direction (E11) isexpressed as function of the aspect ratio and its value increaseshighly for low values of aspect ratio and up aspect ratio equal to200 reaches a constant value near. This behavior is independenton the value of fiber volume fraction. To achieve the curves weused for fibers and matrix the data shown in the appendix(Table 1) under application of equations from Eqs. (1)–(4).

– The second step is to apply the mean tensor averaging proce-dure to take into account the orientation distribution func-tion. The Elastic Modulus of the composite is evaluated byusing the Eq. (11) and the FOD identified from experimentaldata. As shown in Fig. 4 for volume fraction values between13% and 18%, E11depends on the aspect ratio up to 200, forlarger values the modulus does not change appreciably. InFig. 4, the purple points are the constant experimental value(see Section 1.1). From this point of view, it seems that thecomposite with similar mechanical properties has a volumefraction between 17% and 18%, which is well above the nom-inal value of 13% of the investigated material. However, tohave a full overview about the elastic modulus we have totake into account the amount of each fiber inside the com-posite that means that we have to weigh the Elastic modulusvalues on the FLD.

– The third step is to use the recovered fit function to achieve theexperimental distribution lengths data, for weighting the E11

obtained from the previous tensor procedure.

A recovered fit function for each E11 values is found. As anexample, Fig. 5 shows for vf = 13%, the E11 value for the followingcases: (i) a perfect fiber aligned (in 1-direction) composite (thickblue line) computed by using Halpin–Tsai equations; (ii) the ten-sor procedure explained in the second step (dashed blue line)computed by using the tensor procedure; and (iii) the recovereddistribution function of the previous function (continuous blueline).

Applying the procedure we obtain the E11 value for each volumefraction value as indicated in Table 2. By using this procedure it ispossible to evaluate the E11 value for a fixed volume fraction withthe experimental distribution length function. For vf = 13.29% theE11 = 8.70 GPa.

F. Garesci, S. Fliegener / Composites Science and Technology 85 (2013) 142–147 147

5. Conclusion

A hybrid methodology to predict Young’s modulus of a long fi-ber thermoplastic composite was developed in detail in this paper.Experimental data for fiber length distribution and for orientationdistribution was used to obtain the fitting functions and to com-bine the fitting functions with the Halpin Tsai equations. Our re-sults indicate that our approach can be used with success topredict the stiffness value for a fixed volume fraction, and to esti-mate the volume fraction using a fixed stiffness value. We concludethat: (a) if we do not take into account the experimental length dis-tribution we overrate the volume fraction; (b) to improve the goodagreement with the experimental results, we need more accurateexperimental data for FLD; (c) if our threshold is the experimentalvalue of stiffness (7.83 GPa) the obtained volume fraction assumesa lower value than the expected one (11.5%). To reach the point (b),we need to take into account a larger amount of fibers, in our caseit is around 50% between the AR range (27 < AR < 1000); for higherAR values, experimental measurements show scattered data so weneed more experimental investigations for improving this point.This could also explain the point (c): the detected volume fractionvalue is lower than the experimental one because if we used onlythe 50% of the total amount of the real fiber present in the speci-men we underestimate the presence of the fiber inside the com-posite. We focus on predicting Young’s modulus of LFT by using,as much as possible, the experimental data in terms of DOF andFLD. if we compared the value obtained with the experimentalone, we can say that the proposed methodology can be considereduseful for the purposes of the prediction of Young’s modulus.

References

[1] Tucker III Charles L, Liang Erwin. Stiffness predictions for unidirectional short-fiber composites: review and evaluation. Compos Sci Technol 1999;59():655–67.

[2] Halpin JC, Kardos JL. The Halpin–Tsai equations: a review. Polym Eng Sci1976;16(5):344–52.

[3] Hine Peter J, Lusti Hans Rudolf, Gusev Andrei A. Numerical simulation of theeffects of volume fraction, aspect ratio and fiber length distribution on theelastic and thermoelastic properties of short fiber composites. Compos SciTechnol 2002;62:1445–53.

[4] Hine Peter J, Lusti Hans Rudolf, Gusev Andrei A. On the possibility of reducedvariable predictions for the thermoelastic properties of short fibre composite.Compos Sci Technol 2004;64:1081–8.

[5] Lusti Hans Rudolf, Hine Peter J, Gusev Andrei A. Direct numerical predictionsfor the elastic and thermoelastic properties of short fibre composites. ComposSci Technol 2002;62:1927–34.

[6] Dray Delphine, Gilormini Pierre, Régnier Gilles. Comparison of several closureapproximations for evaluating the thermoelastic properties of an injectionmolded short-fiber composite. Compos Sci Technol 2007;67:1601–10.

[7] Agboola BabatundeO, Jack DavidA, Montgomery-Smith Stephen. Effectivenessof recent fiber-interaction diffusion models for orientation and the partstiffness predictions in injection molded short-fiber reinforced composites.Composites: Part A 2012;43:1959–70.

[8] Lee Doo Jin, Oh Hwajin, Song Young Seok, Youn Jae Ryoun. Analysis of effectiveelastic modulus for multiphased hybrid composite. Compos Sci Technol2012;72:278–83.

[9] Duan HL, Yi X, Huang ZP, Wang J. A unified scheme for prediction of effectivemoduli of multiphase composites with interface effects. Part I: Theoreticalframework. Mech Mater 2007;39:81–93.

[10] Duan HL, Yi X, Huang ZP, Wang J. A unified scheme for prediction of effectivemoduli of multiphase composites with interface effects: Part II—Applicationand scaling laws. Mech Mater 2007;39:94–103.

[11] Chin WK, Liu HT, Lee YD. Effect of fiber length and orientation distribution onthe elastic modulus of short fiber reinforced thermoplastics. Polym Compos1988;9:27–35.

[12] Nguyen BN, Bapanapalli SK, Holbery JD, Smith MT, Kunc V, Frame BJ, et al.Fiber length and orientation in long-fiber injection-molded thermoplastics –Part I: Modeling of microstructure and elastic properties. J Compos Mater2008;42(10):1003–29. http://dx.doi.org/10.1177/0021998308088606.

[13] Qiu YP, Weng GJ. On the application of Mori–Tanaka’s theory involvingtransversely isotropic spheroidal inclusions. Int J Eng Sci 1990;28(11):1121–37.

[14] Kundalwal SI, Ray MC. Effective properties of a novel composite reinforcedwith short carbon fibers and radially aligned carbon nanotubes. Mech Mater2012;53:47–60.

[15] Tsai Jia-Lin, Tzeng Shi-Hua, Chiu Yu-Tsung. Characterizing elastic properties ofcarbon nanotubes/polyimide nanocomposites using multi-scale simulation.Composites: Part B 2010;41:106–15.

[16] Advani SG, Tucker III CL. The use of tensors to describe and predict fiberorientation in short fiber composites. J Rheol 1987;31:751–84.

[17] Hoehn N. FASEP� fiber separation and length determination procedure.<http://www.fasep.de>, 2013-03-06.

[18] Troester S. Materialentwicklung und – CharakterisierungfuerThermoplastische Faserverbundwerkstoffe im Direktverfahren,Dissertation. Germany: University of Stuttgart; 2003.

[19] Seelig T, Latz A, Sanwald S. Modelling and crash simulation of long-fibre-reinforced thermoplastics. Bamberg, Germany: 7.LS-DYNA Anwenderforum;2008.

[20] Tinevez J. Directionality plugin for ImageJ. <http://fiji.sc/wiki/index.php/Directionality>, 2013-03-06.

[21] Gusev A, Lusti HR, Hine PJ. Stiffness and thermal expansion of short fibercomposites with fully aligned fibers. Adv Eng Mater 2002;4(12):927–31.http://dx.doi.org/10.1002/adem.200290007.

[22] Gusev A, Heggli M, Lusti HR, et al. Orientation averaging for stiffness andthermal expansion of short fiber composites. Adv Eng Mater2002;4(12):931–3. http://dx.doi.org/10.1002/adem.200290008.