Size effect on tensile strength of carbon fibers

Materials Science and Engineering A238 (1997) 336–342

Size effect on tensile strength of carbon fibers

Tetsuya Tagawa *, Takashi MiyataDepartment of Materials Science and Engineering, Nagoya Uni6ersity, Chikusa-ku, Nagoya 464-01, Japan

Received 30 April 1997

Abstract

Tensile strength of carbon fibers exhibits statistical Weibull type distribution and significant size dependence. In the presentwork, ten types of polyacrylonitrile based and mesophase-pitch based carbon fiber monofilaments were tested for two or threedifferent gauge lengths. Size effect in both axial and radial directions were analyzed based on the two parameters Weibullstatistics. It was found that the size effect in axial direction was almost similar for all fibers tested. This result suggests that thetensile strength obtained for a certain gauge length is a meaningful measure as a representative strength of the fiber strength. Inradial direction, the size effect of the tensile strength was larger than that in axial direction. The tensile strength of the carbonfibers seemed to have unisotropic statistical characteristics. Size dependence in diameter was numerically simulated with anassumption of unisotropic distribution of Reynolds-Sharp type defects. © 1997 Elsevier Science S.A.

Keywords: Size effect; Carbon fibers; Tensile strength

1. Introduction

Carbon fiber is one of the high performance fibersemployed in the advanced composites. In the case ofthe composites reinforced by carbon fibers, continuousfibers are used and fiber volume fraction usually ex-ceeds 40%. Therefore, characteristics of the fiberstrength is the most influential factor on the strength ofthe composites. At the composites strength design, thefiber strengths evaluated from the fibers strands with acertain gauge length are usually used. Tensile strengthof carbon fibers, however, shows a large scatter andremarkable size dependence according to the weakestlink analogy [1–3]. Although the statistical characteris-tic of fiber strength will give significant influences onthe strength will give significant influences on thestrength of composites, the statistical behaviors of fiberstrength have never been systematically characterizedyet. Supposing that, for instance, the size effect of thestrength is different for each fiber, the tensile strengthevaluated for a certain gauge length will be lacking in agenerality as a measure of fiber strength. In the present

work, the tensile tests of mono-filaments for varioustypes of carbon fibers were performed, the statisticalsize effects of the tensile strength were investigated onthe basis of the Weibull statistics.

2. Experimental procedure

2.1. Carbon fibers

Four types of polyacrylonitrile (PAN) based and sixtypes of mesophase-pitch (MP) based carbon fiberswere used for the tensile tests. Their tensile propertiesand diameter shown by the manufacturers are listed inTable 1. The first term of the material code shows thetypes of precursor and in the second term, the letters A,B, C, … show the difference of the fiber manufacturerand the numbers refer to the elastic modules. Thetensile properties of these fibers are from 2.5 to 5.6 GPain tensile strength and from 160 to 600 GPa in elasticmodulus. These properties belong to all types of thehigh performance carbon fiber, that are called as HS(high strength), IM (intermediate modulus) and HM(high modulus). All fibers have approximately circularcross-sections.

* Corresponding author. Tel.: +81 52 7893577; fax: +81 527893236.

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T. Tagawa, T. Miyata / Materials Science and Engineering A238 (1997) 336–342 337

Table 1Mechanical properties of carbon fibers tested

Average diameter dav (mm) Young’s modulus E (GPa)CF code Tensile strength sf (MPa)Precursor

8.5 157MP-A15 2540MP275 33158.6MPMP-A30

MP 8.8 490 3040MP-A5038304876.7MPMP-A50%

597 3326MP-A60 MP 7.1504 4021MP-B50 MP 8.3

6.1 235 3825PANPAN-C25PAN 5.4 412 3168PAN-D40

230 4810PAN-E25 PAN 6.44.8 294PAN-E30 PAN 5590

MP, mesophase pitch; PAN, polyacrylonitrile.

2.2. Tensile tests

For each type of carbon fibers, 40 tensile tests werecarried out for one gauge length and gauge lengthsadopted in test were 6, 15 and 25 mm. Fiber samplesrandomly selected from the fiber bundle were fixed byan adhesive on the slotted paper holders. The speci-men with the paper holder was mounted to the tensileapparatus and started extending after burning off thepaper frame. Samples were extended to failure at therate of 0.1 mm min−1 by the rotation of a differentialmicrometer head with a constant rotating motor. Thefracture load was evaluated with the load cell of 100 gin capacity. The tensile apparatus and the specimen setup are shown in Fig. 1. Before the tensile test, thecross sectional area was evaluated from the diametermeasured by the scanning electron microscope foreach sample. The results for each series of tests wereanalyzed based on the two parameters Weibull statis-tics.

3. Results and discussion

3.1. Statistical distribution of tensile strength and theirsize effect

Fig. 2. shows some results as Weibull distributions onthe tensile strength tested for 15 mm gauge length.Single modal Weibull distribution can be approximatelyapplied to each results except PAN-C25. The results ofPAN-C25 shows the deviation from a linear relation inthe low strength range, showing the mixed modal ormulti-modal Weibull distribution. The Weibull shapeparameters obtained are summarized in Fig. 3. The twobroken lines show the upper and lower limits of the

Fig. 2. Examples of Weibull distributions on tensile strength forGL=15 mm.

Fig. 1. Tensile test apparatus of mono-filament and specimen set up.(a) Tensile test apparatus, (b) specimen set up.

T. Tagawa, T. Miyata / Materials Science and Engineering A238 (1997) 336–342338

Fig. 3. Weibull shape parameters for fibers tested. Fig. 5. Dependence of tensile strength on fiber diameter for PAN-C25.

Monte-Carlo simulation for 40 samples with an as-sumption of a constant shape parameter four. Theshape parameters for all fibers and gauge lengths areincluded in the range of the Monte-Carlo simulationresults. The Weibull shape parameter indicates the de-gree of a scatter on the strength and a smaller shapeparameter indicates a larger scatter of the strength. Thevalue of four for shape parameters is smaller than theone of the bending strength in engineering ceramicmaterials. The Weibull shape parameter also has thephyscial meaning as the factor of size effect on thestrength. The results shown in Fig. 3 suggest that thesize effect on the tensile strength seems to be similar forany carbon fibers independent on the precursor and thestrength level. This similarity of the size effect supportsthat the relative strengths between any carbon fiberscan be kept in any fiber length. Fig. 4 shows thedependency of the tensile strength (Weibull scaleparameter) on the gauge length for several fibers. TheWeibull statistics with two parameters gives theoreti-

cally linear relationship in Fig. 4 and the inverse of theslope should be equal to the Weibull shape parameter.The inverse values of the slopes for the most of thefibers tested approximately coincident with the resultsin Fig. 3 except for MP-A50 and PAN-C25.

The size effect of the strength was observed not onlyin the axial direction but also in the radial direction offiber. Fig. 5 shows the dependency of the tensilestrength on fiber diameter for PAN-C25 tested in 6 mmgauge length. The inverse of the slope in Fig. 5 is 0.42

Fig. 6. Procedure for correction of anisotropy in tensile strength. (a)Experimental dependency on diameter, (b) distribution of equivalentstrength for average diameter, (c) dependency on gauge length, (d)optimum md due to convergent calculation.

Fig. 4. Dependence of tensile strength (Weibull scale parameter) ongauge length.


Fig. 7. Weibull distributions of tensile strength with and withoutcorrection for size effect in diameter. (a) Without correction, (b) withcorrection.

should be equal to a half value of m. The result in Fig.5 seems to obey Eq. (3). However, precise value of md

is not given due to insufficient number of data. For thecorrection of the size effect in radial direction, semi-nu-merical method schematically shown in Fig. 6 wascarried out. The value of md is arbitrarily assumed likethe result in Fig. 5. Equivalent tensile strength for theaverage diameter could be calculated in each samplefrom the Eq. (3) using the assumed value of md. For thedistribution of an equivalent tensile strength for eachgauge length tested, Weibull parameters were obtained.Based on Weibull statistics, Weibull shape parameter,mi for a certain gauge length, li should be equal to theinverse of the slope in the relation between logarithmicWeibull scale parameter, s0i and logarithmic gaugelength, li, (the value of m in Eq. (2)). The summation ofthe deviation between mi and m for all gauge lengthswas defined as an error factor err(md), which was afunction of md. The optimum value of md was calcu-lated for the minimum value of err(md) by Newton-Rapson method. These corrections were examined forthe MP-A50 and PAN-C25 of which diameters werevaried in wide range. One example of the correctedresults in shown in Fig. 7. This figure shows a Weibulldistribution of the tensile strength of PAN-C25 testedat 15 mm gauge length without and with this correc-tion. Correction of the size effect in diameter leads tohigher linearity and the single modal Weibull distribu-tion. The variation of Weibull shape parameter withand without this correction is not so large and mainlythe Weibull scale parameter is varied by the correction.Fig. 8 shows the dependence of Weibull scale parameteron the gauge length without and with the correction.Discrepancy of the size effect in Fig. 4 with the shapeparameter m for the fibers PAN-C25 and MP-A50 canbe explained by the consideration of the size effect in

that is one tenth of the value in Fig. 4, that indicatesunisotropic statistical distribution of the strength.Therefore, the results in Figs. 2 and 3 involve the sizeeffect in diameter of fibers.

3.2. Unisotropy on size effect

The tensile strengths obtained in the present work arenot evaluated under constant volume in spite of testingat a constant gauge length. This is due to the variationof the diameter even in one fiber. It is necessary ofseparate the size effect in radial direction in order tostrictly discuss the unistropy in size dependency of thetensile strength.

A commonly used form of the two parametersWeibull distribution for a fiber of a constant diameter,d0 is

F(s)=1−exp[− l(s/s0)m] (1)

where F(s) is the probability of failure up to a stresslevel s, l is the gauge length. s0 and m are the scaleparameter and the shape parameter, respectively. For acertain probability, the fracture strength, sf is given asa function of gauge length, l.

log sf= −1/m · log l+constant (d=d0) (2)

For a diameter, similar equation with Eqs. (1) and (2)can be assumed for a given gauge length l0 as

log sf= −1/md · log d+constant (l= l0) (3)

Eqs. (2) and (3) give the theoretical size dependencies ofthe tensile strength for the axial direction and the radialdirection, respectively. If the defects that control thetensile strength exist isotropically in the materials, md

Fig. 8. Dependence of tensile strength on gauge length with andwithout correction of size effect in diameter. (a) Without correction,(b) with correction.


Fig. 9. Volume dependence of tensile strength.

dency on the diameter should be two times larger thanthe slope of the dependency on the gauge length. How-ever, the dependency on the diameter in Fig. 9 is tentimes larger than the dependency on the gauge length.Based on the weakest link analogy, the higher strengthshould be obtained in the smaller volume of the materi-als. In the carbon fiber, however, the unisotropic sizedependency of the tensile strength. It is well known thatthe strength of the carbon fiber shows the unisotropiccharacteristics due to the alignment of graphite crys-talline. The results in Fig. 9 suggest that the statisticalcharacteristics of the tensile strength can also showsignificant unistropy.

3.3. Structure and distribution of defects

It was confirmed that the tensile strength in thecarbon fibers almost obeys the single modal Weibulldistribution. This suggests that the fracture of a carbonfiber may be controlled by single mechanism. Suppos-ing that the same fracture sources are isotropicallydistributed in fibers, the unisotropic size dependence ofthe tensile strength should not be observed. Consideringthe results of Fig. 9, it can be suggested that the densityof the fracture source is lower at the center of the fiberand/or the strength of the fracture source is higher atthe center of the fiber.

On the fracture mechanisms of the carbon fibers, itwas proposed that the misorientation of the graphitecrystal layers controlled the fracture [4,5]. Reynolds andSharp [5] have proposed that the fracture of the carbonfibers initiated at the crystal misorientation regionaround some inclusions or voids. They also showedthat the fracture strength, sf was given as follow:

sf=ss/(sin f · cos f) (4)

where ss is the shear strength of the graphite interlayerand f is the misorientation angle from the fiber axis.On the micorstructure of the carbon fibers, it has beenshown that the graphitization varies along the radialdirection and proceeds around that fiber surface forboth PAN based fibers [6,7] and MP based fibers [8].Guigon et al. [9] observed the graphite layers of PANbased fiber by a high-resolution transmission electronmicroscope and showed that the radius of curved layerswas around 45 nm near the fiber surface and was 20 nmnear the center. Considering above works, that fractureof a carbon fiber can be assumed to be governed by themisorientation of graphite layers around inclusions orvoids and the misorientation angles and their densityare varied from the surface to the center.

The misorientation of the graphite layers is assumedas a defect controlling the tensile strength. Hereafter,the unisotropic size dependency of the tensile strengthshown in Fig. 8 are calculatively investigated throughthe varition of the defect density and the strength along

diameter. At this correction the optimum md value is0.46 for PAN-C25 and this value also agrees well withthe dependency of the results in Fig. 5.

With above correction, unistropic size dependencecan be strictly divided into the axial direction and theradial direction of the fiber. These two size dependen-cies are shown in Fig. 9 for PAN-C25 as the volumedependency of the tensile strength. The dependency onthe gauge length in Fig. 9 is converted on the depen-dency on the fiber volume under a constant averagediameter and the dependency on the fiber diameter isshown as a dependency for the volume under theconstant gauge length of 6 mm. In the case of thestatistically isotropic materials, the slope of the depen-

Fig. 10. Simulation results for volume dependency of normalizedstrength.


Fig. 11. SEM micrographs of tensile fracture surface.

the radial direction of the fiber. Supposing that thedefect density and the strength are constant along theaxial direction, the cumulative fracture probability ofthe fiber under a stress, s can be given as follows:

F(s, d)=1−exp�

−2p& d/2

0

P(s, r) · r · l ·DD(r) drn

(5)

where d is a fiber diameter and l is a length. DD(r) isthe defect density at a distance, r from the center andP(s, r) is a fracture probability of one defect under astress, s at a distance, r. P(s, r) is assumed as a normaldistribution. The mean value of the function of P(s, r)is given by Eq. (4) and the standard deviation is givenas 15% of the mean value. Thus, the dependency ofP(s, r) on r is defined by the dependency of the misori-entation angel f on r. The numerical simulations wereperformed for three cases as follows.

Case I. Both the density and the angle of the misori-entation layers are constant along the radical directionof the fiber.

Case II. Only the misorientation density continuouslyand linearly increases from the center of the surface.

Case III. Only the misorientation angle continuouslyand linearly increases from the center to the surface.

In case II, the variation of the density was assumedfrom 104 mm−3 (center) to 105 mm−3 (surface). In caseIII, the variation of the angle was assumed from 0(center) to 40° (surface). These values were assumedwith reference to previous work on the microstructures[9]. The fracture strength can be evaluated as an ex-pected value of Eq. (5) for a certain probability. Theexpected strength according to Eq. (5) for 50% proba-bility are shown in Fig. 10 as a function of the samplevolume. The expected strengths are normalized by thevalue for the volume of 10−3 mm3. The experimental


results in Fig. 9 are also shown as bands in Fig. 10. Asthe carbon fibers are continuously manufactured, thedefect distribution along the fiber length could be al-most constant. The simulation of case I, in which thedefect density is independent on the location, seems tocorrespond to the dependency on the fiber length. Thesimulated dependency of the strength on a samplevolume in case III is much larger than the result in caseII. The large dependency on a sample volume in caseIII qualitatively describe the experimental results. Theexperimental dependency on the fiber diameter, how-ever, is larger than the simulation result in case III. Thisprobably comes from the assumption of constant inter-layer shear strength, ss in Eq. (4). At the misorientationregion, the graphitization might be relatively low andthe interlayer shear strength seems to be smaller thanthat of the ideal crystalline.

According to the Reynolds-Sharp mechanism, themisorientation angle f could be larger around largerinclusion or larger voids. The unisopropic character ofthe volume dependency of the tensile strength is ex-pected to relate with the size distribution of inclusionsor voids along the radial direction of the fiber.

3.4. Fractography

Some fracture surfaces observed by scanning electronmicroscope are shown in Fig. 11. Based on the abovediscussions, the fracture could initiate from some ofsecond particles of voids. However, any defects orsecond particles could not be observed on the fracturesurface within the magnification of SEM. These frac-ture appearances seem to correspond to the transversemicrostructure. Nano-scopic observations will be neces-sary to investigate the defects themselves controlling thetensile strength of the carbon fiber.

4. Conclusions

The statistical distributions of the tensile strength incarbon fibers were investigated for 10 years of variousfibers. The following conclusions were obtained.

(1) Weibull shape parameters in axial direction showalmost constant value of four, irrespective of the car-bon precursor and the strength level. This result sup-ports that the tensile strength obtained for a certaingauge length can be generalized as a measure of thefiber strength.

(2) Unisotropic size effect on strength was observed.The size dependency of strength in the radial directionis ten times larger than the dependency in and axialdirection.

(3) Size dependence in diameter of the fiber can bequalitatively explained by numerical statistic simulationwith the assumption of a certain distribution of themisorientation angle.

References

[1] S. Chwastoak, J.B. Barr, R. Didchenko, Carbon 17 (1979)49–53.

[2] J.B. Jones, J. Barr, R. Smith, J. Mater. Sci. 15 (1980) 2455–2465.

[3] J.J. Masson, K. Schulte, F. Girot, Y. Le Petitcorps, Mater. Sci.Eng. A135 (1991) 59–63.

[4] J.D.H. Hughes, J. Phys. D: Appl. Phys. 20 (1987) 276–285.[5] W.N. Reynolds, J.V. Sharp, Carbon 12 (1974) 103–110.[6] K.J. Chen, R.J. Diefendorf, Proc. 16th Biennial Conf. on

Carbon, Am. Carbon Soc. (1983) 490–491.[7] S.C. Bennett, D.J. Johnson, Carbon 17 (1979) 25–39.[8] M. Inagaki, M. Endo, A. Oberlin, M. Nakamizo, Y.

Hishiyama, H. Fujimaki, Tanso, 99 (1979) 130–137 (inJapanese).

[9] M. Guigon, A. Oberlin, G. Desarmot, Fiber Sci. Technol. 20(1984) 177–185.

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Size effect on tensile strength of carbon fibers