TENSILE FAILURE PHENOMENA IN CARBON FIBRES

Haruki Okuda1,2,3,*, Robert J. Young2, Fumihiko Tanaka1, Jun Watanabe1,3 and Tomonaga

Okabe3

1 Composite Materials Research Laboratories, TORAY Industries, Inc., 1515 Tsutsui,

Masaki-cho, Iyo-gun, Ehime, 791-3193, Japan

2 School of Materials and National Graphene Institute, University of Manchester, Oxford

Road, Manchester, M13 9PL UK

3 Department of Aerospace Engineering, Tohoku University, 6-6-01, Aoba-yama, Aoba-ku,

Sendai, Miyagi, 980-8579, Japan

ABSTRACT

In order to clarify the effect of nanostructure upon the tensile strength of

polyacrylonitrile (PAN)-based carbon fibres, experimental as well as theoretical studies have

been performed. A new technique for the quantitative evaluation of the high strength region

has been developed by combining the loop test with Raman spectroscopic measurements to

overcome uncertainties in fibre stress, which have been the major drawback of the

conventional loop test. The tensile strength at gauge lengths of a few tens of µm was

successfully evaluated and a tensile strength as high as 13 GPa was observed experimentally

for commercially-available PAN-based carbon fibres, showing their potential high tensile

strengths. The strength distributions were found to be highly uniform in the high strength

region, represented by Weibull shape parameters of ~20. A tensile strength model that can

reasonably account for the effect of the nanostructures has been proposed, suggesting there is

considerable scope for further improvements in the tensile strength of PAN-based carbon

fibres.

1. INTRODUCTION

PAN-based carbon fibres are now widely used in a broad range of applications that

include sporting and leisure, aerospace, industrial and, most recently, automotive [1] due to

their high Young’s modulus and excellent tensile strength. In order to maximise these

mechanical properties, a deep understanding of the effect of the fibre structure upon its

* Corresponding author. Tel: +81-89-960-3839. E-mail: Haruki_Okuda@nts.toray.co.jp (Haruki Okuda)

1

properties is necessary so that precise control of the basic fibre structure becomes possible. In

particular, a clearer insight into the tensile failure mechanism is of particular importance

since improvements in tensile strength of carbon fibres have opened up new fields of

application continuously.

Extensive studies on the tensile strength of carbon fibres have already been

undertaken mainly from statistical [2-6] as well as fracture mechanics viewpoints [7-11].

These studies have shown that the flaws, which are thought to exist statistically within fibres,

have a major influence upon single fibre tensile strength of carbon fibres. With respect to the

nanostructure, on the other hand, we have recently demonstrated that the PAN-based carbon

fibres can be regarded as nanocomposites, in which crystallites are dispersed in the matrix

made of materials with more disordered structure (which has been called as “amorphous” in

the referred papers) [12-14]. Loidl et al. suggested earlier the importance of this disordered

structure [15,16]. Moreover, complex nanostructures have been observed in a number of

electron microscope analyses [17-23]. Considering that these nanostructures are

heterogeneous, it is likely that flaw reduction will eventually lead to the single fibre tensile

strength being controlled essentially by these nanostructures. Some single fibre fracture

surfaces have been reported as showing no specific features near their fracture origins [24].

Therefore, a clear understanding of the potential effect of nanostructure upon single fibre

tensile strength of carbon fibres is becoming more important than ever.

With the intention to clarify the effect of nanostructure upon tensile strength, several

models were proposed as early as in the 1970s. Local damage such as crystallite shear failure

[25], tensile failure of wrinkled crystallites [26], rupture of the basal planes within misaligned

crystallites (“Reynolds & Sharp model”) [27,28] and yielding of the disordered structure [29]

were assumed to be the strength-determining factor in the absence of large flaws. It has been

understood, however, that it is flaws that play the dominant role in controlling the tensile

strength in the range discussed in these studies. For this reason, better knowledge of the high

strength region is essential.

Reducing the gauge lengths in the single fibre tensile test is the most straightforward

approach to evaluate the high tensile strength region. It has, however, been understood that

the minimum accessible gauge length is limited by clamping effects [30]. In the case of

typical combinations of carbon fibres and matrices, this effect becomes significant below 5-

10 mm. Gauge length can be further reduced using the single fibre fragmentation test (SFFT),

where the tensile strength distribution can be estimated from the fragmentation behaviour of

fibres stressed in the matrix polymer using a shear-lag theory [32,33]. We have recently

2

applied the SFFT successfully to a range of PAN-based carbon fibres [14,24]. There is still,

however, a limitation such that a strong fibre/matrix interface is necessary to accurately

evaluate the high strength region by avoiding slippage between fibre and matrix.

The single fibre loop test is a unique method that enables accessing much shorter

gauge lengths [34-41]. Due to the localised tensile stress near the top surface of the looped

fibre, the measured tensile strengths become higher than those evaluated with the tensile test.

However, the unrealistic assumptions in the calculation of the tensile stress based on the

beam bending theory make these test methods only semi-quantitative. In fact, a shift of the

neutral plane has been observed using micro-beam X-ray diffraction (XRD) [42], which can

affect the tensile stress at the loop top proportionally. Single fibre flexural tests are more

straightforward yet may suffer from essentially the same problem [43]. In addition to these

conventional techniques, artificial notches have been used to estimate the tensile strengths in

the absence of flaws [9,10]. They drastically reduce test lengths to the order of notch tip

radius (<100 nm), while the estimated strength is largely dependent upon the elastic constant

used in the calculation. Quantitative understanding of the high strength region is therefore

still a challenge.

In this study, we introduce an experimental technique based on a novel combination

of the loop test and Raman spectroscopic measurements. The tensile stresses at the loop-top

surface are evaluated using Raman band shifts. Using this new technique, single fibre tensile

strengths at short gauge lengths, or in the high strength region, have been characterised

quantitatively for a series of PAN-based carbon fibres. We then discuss the tensile failure

mechanisms from a nanostructural viewpoint.

2. EXPERIMENTAL

2.1 Materials

A series of PAN-based carbon fibres that are intended to represent a wide range of

different nanostructures were used (Table 1). Young’s modulus was varied mainly by

changing the maximum heat treatment temperature. Either no treatment or a mild surface

treatment was performed after carbonisation or graphitisation to avoid any possible effect

from changes in the fibre surface structure. All the individual fibres have essentially circular

cross sections.

3

Table 1: The physical and the mechanical properties for the carbon fibres studied. Lc, La and

π002 are the sizes of the crystallites in the thickness, the longitudinal direction, respectively,

and the orientation parameter of the crystallites, which were measured with XRD [44].

Material

s

Young’s

Modulus

/GPa

Density

/gcm-3

Diameter

/µm

Lc

/nm

La

/nm

π002

/-

CF1 55 1.57 6.6 1.4 1.3 0.815

CF2 240 1.78 5.8 1.5 1.9 0.822

T800S 294 1.80 5.5 1.9 2.7 0.821

CF3 294 1.73 5.6 2.4 3.6 0.845

CF4 380 1.80 5.4 3.7 7.2 0.883

CF5 440 1.85 5.3 4.7 10.0 0.904

2.2 Methods

In order to evaluate the tensile stress at the loop top quantitatively, we undertook

micro-Raman measurements at the loop top and the Raman band shifts were converted to the

corresponding stress using an independently-obtained band-shift/stress relationship. Loop

tests were performed under the microscope objective connected to a Renishaw 1000

spectrometer. A single fibre of about 10 cm in length was taken from fibre bundle and the

both ends were passed through a small hole in a polyethylene terephthalate (PET) thin film to

form a fibre loop (Fig. 1a). Both the fibre ends were then attached onto a micrometer head

using adhesive tapes. The loop axis was carefully aligned parallel to the incident beam axis

by adjusting the relative position of the micrometer head against the hole in the PET film so

that the misalignment angle with the incident beam axis was less than 5º. The loop diameter

was made smaller stepwise by moving the micrometer head, which enables both the fibre

ends pulled simultaneously, maintaining the location of the fibre surface being irradiated

almost unchanged throughout a series of measurements. Raman spectra of the loop-top

surface were then recorded for each step. Similarly, Raman spectra were recorded during

single fibre tensile tests with short gauge lengths (L = 2 mm) to extract the band-shift/stress

relationship. For both tests, 3-5 single fibres were used for each fibre type.

4

A Renishaw 1000 spectrometer equipped with a He-Ne laser source (wavelength λ =

633 nm) was used to obtain Raman spectra. The laser power was set below 3 mW in order to

avoid any damage associated with heating. An objective magnification of 50 was chosen,

which gives beam spot diameter of less than 2 µm (Fig. 1b) [46]. The polarisation direction

of the incident beam was set parallel to the fibre axis and no analyser was used. The

penetration depth of the excitation beam into the carbon fibre was estimated to be around 10

nm, given 2.3% of the incident/scattered light is absorbed by each graphitic layer [45].

Therefore, using micro-Raman measurement for evaluating the stress state at the loop-top

surface could be justified (Fig. 1c). The fibre diameter was measured from micrographs

recorded using a Zeiss EVO60 scanning electron microscope (SEM) with an accelerating

voltage of 12 keV, and the scale was calibrated using a copper fine grid as a standard. The

Raman spectra were fitted with four Gaussian functions and a linear baseline to extract the

peak-top wavenumbers for the G and D bands. Band shifts ΔG (ΔD) were calculated as ωG -

ωG0 (ωD - ωD

0), where ω and ω0 denotes the peak-top wavenumber under stress and without

stress, respectively. The band-shift/stress relationships for the G and D bands were fitted with

quadratic functions of the tensile stress, which were used to convert the shifts of both bands

at the top of the loop to tensile stress.

Fractographic analyses were performed using the fibre fragments which were

recovered after loop tests. In order to enable fibre fragments to be collected, a drop of

glycerol was added on the loop-top before testing and the loop test was carried out as normal.

Fibre ends were collected, washed thoroughly with water and then air-dried. The fracture

surfaces were observed using an FEI Quanta 650 SEM with an accelerating voltage of 12

keV.

5

Figure 1: (a) Experimental geometry of the loop test. Special care was taken to align the loop

axis parallel to the incident beam axis for accuracy. (b) Top view of the looped fibre. The

beam spot diameter is less than 2 µm [46]. (c) Side view. The penetration depth was

estimated to be less than 10 nm [45].

3. RESULTS

3.1 Validation of the quantitative loop test

First of all, the tensile stresses at the loop top were evaluated from the observed

Raman band shifts. The Raman spectra taken at the loop top of a T800S single fibre are

shown in Fig. 2a. Shown on the each spectrum is the ratio d/W, the fibre diameter d divided

by the loop diameter W, which was used throughout this study for indicating the bending

strain irrespective of the differences in fibre diameter. Each spectrum shows the G (1595

cm--1) and D (1340 cm-1) bands clearly, which are normally observed for PAN-based carbon

fibres [12]. Peak positions downshifted monotonically with increasing d/W, showing that

loop-top surface was under tensile stress. Fig. 2b shows the G band shift (ΔG) during loop

test. Only the results for T800S and CF5, which represent a high-strength and high-modulus

type, respectively, are shown for the sake of clarity. It was found that the G band shift of

T800S exhibited almost linear behaviour against d/W, whereas CF5 showed noticeable

nonlinearity. On the other hand, nonlinear behaviour against the tensile stress that was

obtained from the tensile test was generally smaller (Fig. 2c), suggesting this nonlinear

behaviour of CF5 against d/W is associated with the deviation from the ideal beam. It should

also be noted that, although both the loop and tensile tests were performed using multiple

single fibres for each fibre type, the results fall reasonably on the same line. This implies that

individual fibres are almost identical in terms of the elastic properties as well as their

nanostructures. The G band shifts in the loop test (Fig. 2b) were then converted to the

corresponding tensile stress using the band-shift/stress relationship (Fig. 2c). In addition, we

found that the D band can also be used to evaluate the tensile stress (see the supplementary

information). In this study, therefore, we used the average values of the stress that were

evaluated using both the G and D bands to improve accuracy.

The tensile stresses σ at the loop top are plotted against d/W in Fig. 2d. The solid

straight lines are the calculated results based on the beam bending theory (eq.1) [34] using the

manufacturer’s values of the Young’s moduli (YM, Table 1) as E.

6

σ =1.07E(d/W) (1)

This is the standard procedure that has widely been adopted in the preceding loop test work

[34-41]. Upon comparing to the observed stress values, we found that this calculation

overestimates the tensile stress at the loop top. This is qualitatively consistent with the

experimental results using micro-beam XRD [42], where the neutral plane of the looped fibre

was observed to shift outwards. This neutral plane shift leads to lower stresses at the loop-top

surface. This shift will be affected by compressive failure at the inner surface [35], stress-

strain nonlinearity in tension [47] as well as in compression [13].

7

Figure 2: (a) Raman spectra measured at the loop top of a T800S single carbon fibre for

different d/W values. Downshift of the peak positions shows the loop-top surface is under

tensile stress. (b) G band shifts against d/W for T800S (high-strength type) and CF5 (high-

modulus type). (c) G band shifts during the tensile test plotted against tensile stress. Solid

lines are the quadratic fitting curves that are used to convert the band shift to the

corresponding tensile stress. (d) Tensile stress at the loop-top plotted against d/W (fibre

diameter/ loop diameter). Straight lines are the calculated results using the manufacturer’s

values of the Young’s moduli (solid line, Table 1) and the initial moduli E0 (dashed line)

[12].

8

On the other hand, assuming that neither compressive failure nor intensive nonlinear

elasticity occur at the small strain, it is expected that beam bending theory can be reasonably

applied to the carbon fibres. The use of the initial moduli E0 [12] instead of Young’s moduli

resulted in dashed lines in Fig. 2d, shows good agreement with the observed initial slopes.

This, in turn, suggests that the stress at the loop top is reasonably well evaluated with the

Raman measurements. The slope of the observed curve appears to be almost constant for

T800S, whereas more nonlinear behaviour can be seen in the case of CF5. This may reflect

differences in compressive behaviour and stress-strain nonlinearity. While it is very

challenging to take all these factors into account to estimate the correct tensile stress, the use

of the Raman technique is much more straightforward. It is therefore demonstrated that this

present method gives a practical route to the quantitative evaluation of local stress in the loop

test.

3.2 Fracture surfaces in the loop test

In order to evaluate tensile strength from the loop test, initiation of loop fracture needs

to be tensile failure at the outer surface of the looped fibre. The fibres examined in this

present study have improved tensile strengths compared with those in earlier studies [35,37].

It is therefore possible that the critical failure mode may be by compressive or shear failure

rather than tensile failure from critical flaws. We thus examined the fracture surfaces after

loop tests. In Fig. 3a, a typical pair of fracture surfaces for T800S is shown with one of the

surfaces mirrored for the sake of comparison. There are overall similarities in terms of

surface topologies, which includes a smooth compressive failure pattern (lower part), a

tensile failure pattern (upper part) and the position of the fracture origin (red arrow) as well as

a step-like feature (middle part). This step-like feature, which was absent in earlier studies

[35,37], but often observed in this study, may be formed by local shear failure, which is

expected to occur exclusively near the centre of the bent beam. The fracture surfaces

examined showed no distinct architecture such as the flaws or voids near the fracture origin

(Fig. 3b). Similarities in topological features were generally observed irrespective of the fibre

type. Considering this as well as the fact that the tensile failure, once initiated, does not stop

until final failure, we concluded that the loop breakup is still initiated by the tensile failure.

9

Figure 3: (a) SEM images for a pair of fracture surfaces of T800S. One of the images is

mirrored for the sake of comparison. Complementary topological features can be seen at

these fibre ends. (b) Magnified view of the failure initiation site, which shows the absence

any kind of distinct features such as flaws or voids.

3.3 Evaluation of the high-strength region

The tensile strength σf was then evaluated by converting the value of d/W at failure in

the loop test to the corresponding tensile stress σ using the stress-d/W relationship shown in

Fig. 2d. Fig. 4 shows a Weibull plot of the tensile strengths σf for T800S. If we ignore the two

weakest data points (the reason for which will be discussed later), we find that the

experimental data follow a Weibull distribution reasonably well

ln {−ln (1−F ) }=m ln( σ f

σ0) (2)

where F, σ0 and m denotes the failure probability, the Weibull scale parameter and the

Weibull shape parameter, respectively. The Weibull scale parameter σ0 and shape parameter

m were found to be 11.5 GPa and 21, respectively. This shape parameter is significantly

larger than ones (m~4) that have been obtained using conventional tensile tests [14,24]. This

means that the tensile strength distribution becomes narrower with increasing tensile strength.

This is essentially the same trend as that has been observed using the SFFT [24]. The

existence of different strength distributions depending upon the strength range tells us that

10

care has to be taken when the high strength region is probed by extrapolating the

experimental results at long gauge lengths [31]. It is important to note that a tensile strength

of as high as 13 GPa can be achieved at shorter gauge lengths. Considering that this figure is

about twice the corresponding unidirectional composite strength (6 GPa), corrected for the

fibre volume fraction, we can expect that there is a considerable scope for the improvement in

the tensile strength by eliminating the large flaws.

Figure 4: Weibull plot of the tensile strength of T800S obtained with the loop test. Symbols:

experimental results, solid line: Weibull distribution fitting (excluding the two weakest data

points).

In order to have a clearer insight into the failure phenomena in the high strength

region, we analysed the experimental results from a statistical as well as a fracture mechanics

viewpoint. For brittle materials, which include carbon fibres, gauge lengths have a strong

influence upon the tensile strengths, since the largest flaw in a given length determines the

fibre tensile failure. Although fibres break in a non-uniform stress field in the loop test, it is

thought possible to estimate the effective gauge lengths Leff using certain statistical treatments

[48]. Here, Leff is the gauge length at which the tensile test of the same fibre gives the same

tensile strength as that obtained with the loop test. Once Leff is identified, the tensile strength

evaluated using the loop test can be normalised to any length scale and, thus, the whole range

11

of the tensile strength distribution can be clarified. The average Leff for T800S was estimated

to be 42 µm (see the supplementary information). By normalising the length to 1 mm, the

tensile strengths obtained using the loop test and the conventional tensile test (L=10, 50 mm)

can be plotted in the same Weibull plot (Fig. 5). Interestingly, the lower data points of the

loop test in Fig. 4 were found to fall on the almost same line as that for the tensile tests with

no special adjustment. This may support the validity of the statistical treatments. In addition,

as shown in the supplementary information, the number of fibre breaks in the SFFT [24] was

reproduced using this tensile strength distribution in Fig. 5 reasonably well. This suggests

that the high strength region observed using the loop test starts to determine the

fragmentation behaviour of single fibres embedded in a matrix polymer at short gauge

lengths.

Figure 5: Normalised Weibull plots for T800S. Diamonds, triangles: SFT with the gauge

length of 50 mm and 10 mm, respectively. Filled circles: loop test. The average gauge length

of the loop test was estimated to be 42 µm.

The fracture origin is assumed to be a Griffith crack, which has an ideally sharp tip.

Although the initial crack size is too small to be identified using SEM, it could be roughly

estimated using the Griffith equation

σ =KIC/F(πc)-0.5 (3)

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Using 1.1 MPa m0.5 and 0.773 for the mode I fracture toughness KIC and the geometrical

correction factor F, respectively [14], the crack size c was estimated to be c.a. 5 nm for the

Weibull scale parameter σ0 of 11.5 GPa. This value is expected to be the lower bound as the

actual fracture origin might not be a Griffith crack. Nevertheless, considering that the basic

structural unit size within carbon fibres is on the nanometre-scale, e.g. the size of crystallite

Lc is 2 nm for T800S [24], it is likely that the high strength region that is accessed using the

loop test is controlled by the nanostructure.

3.4 Effect of structure development upon the high strength region

The loop test was applied for a series of PAN-based carbon fibres in order to examine

the dependence of the high strength region on structure (Table 2, Fig. 6). The tensile strength

measured using the loop test followed the Weibull distribution regardless of the fibre type.

The observed Weibull shape parameters m at the equivalent gauge length of c.a. 50 µm were

in the range of 15-23 except for CF1, which showed a smaller value. We have reported

similar behaviour, as mentioned earlier, for T800S using the SFFT [24] and a similar trend

has been also observed for the high modulus type fibre M60J using the single fibre flexural

test [43]. This result clearly suggests that the steep strength distribution in the high strength

region is a common feature for PAN-based carbon fibres. The lack of a clear trend in m

except for CF1 suggests that the strength distribution in this high strength region is

independent of the nanostructure once it has developed to a certain level. On the other hand,

the Weibull scale parameter σ0 shows a rapid increase until it peaks for T800S and then

decreases slowly, suggesting a strong influence of structure development during heat

treatments.

13

Table 2: Weibull parameters σ0 and m characterised using the loop test. The effective gauge

length Leff was estimated as described in section 3.3. The details can be obtained in the

supplementary information.

Scale parameter

σ0 /GPa

Shape parameter

m /-

Effective gauge

length Leff /µm

CF1 3.2 6 61

CF2 10.3 15 53

T800S 11.5 21 42

CF3 10.9 15 46

CF4 9.4 15 68

CF5 9.2 23 73

Figure 6: Weibull plot of the tensile strength σf measured using the loop test. Lines are the

least square regression lines (The lowest two data points for T800S were ignored).

It is also worth mentioning that the shorter gauge length that becomes accessible

using the quantitative loop test (Table 2) is comparable to the critical length for carbon fibres

in typical epoxy matrices [49]. This means that clarifying the effect of nanostructure upon

14

this high strength region is important not only from fundamental viewpoint but also for

practical reasons.

4. DISCUSSION

PAN-based carbon fibres, being a sort of hard carbons, have complex internal

nanostructures: crystallites and the remainder (“disordered region”). Crystallites are made

with stacked graphitic planes being well oriented in the fibre axis direction. The disordered

region, on the other hand, would be a complex mixture of sp2 and sp3 carbons, although its

detailed structure has not yet been determined. Taking into account that the graphitic planes

extended out from the crystallites may gradually loose its stacking and more defects may be

introduced, we suppose that the defective, non-stacked graphitic layers are the major

constituent of this disordered region. On that basis, it is also likely that there is no clear

boundary between the crystallites and the disordered region. With respect to the tensile

failure mechanism in the absence of large flaws (=Griffith cracks), as discussed in the

introduction, earlier analyses have focused mainly on the crystallites [25-28] and little

attention has been paid to the potential contribution of this disordered region [29]. However,

considering that the crystallites would be stiffer and, most probably, stronger than the

disordered region in the fibre axis direction, it appears to be probable that the crystallites

work as a source of stress concentration which leads to local failure in the surrounding

structure. Local damage thus formed may readily accumulate to a certain size which fulfils

Griffith criterion [50], since the critical crack size that accounts for the observed strength

range (~12GPa) is only a few times larger than the basic structural unit size (section 3.2). We

therefore hypothesised that the disordered region controls the tensile strength of the PAN-

based carbon fibres in the high strength region.

To test this hypothesis, we applied a simple stress criterion for failure as

σ app=σ d

α (4)

where σapp, σd and α are the applied macroscopic tensile stress at fibre failure, the inherent

tensile strength of the disordered region and the stress concentration factor due to the

nanostructural inhomogeneity, respectively. Since the crystallites and the disordered region

would be smoothly connected each other in PAN-based carbon fibres, stress concentration

factor cannot be easily estimated. Nevertheless, it is likely that the highest tensile stress in the

disordered region, at given applied tensile stress, is somehow related to the stress

15

concentration factor which is calculated for more simplified case such as the ellipse floating

in the matrix with clear boundary. Cross section of the crystallites in the PAN-based carbon

fibres may be approximated as an ellipse with the aspect ratio r= La/Lc. Stress concentration

factors near the poles of the ellipse have been studied by Nisitani [53] and we adopted their

result given as

α '=(γ +1 ) [( γ+1 ) r+( γ+3 ) ]

8 γ (5)

where α’ and r are the stress concentration factor for the simplified geometry and the aspect

ratio of the crystallites, respectively. γ is a parameter related to bulk Poisson’s ratio and take a

value around 2, either for the plane strain or the plane stress condition [53, 54], if we use 0.3

[60] as the Poisson’s ratio for the carbon fibres used in this study. Introducing an adjustment

factor k which translate the calculated stress concentration factor for the simplified geometry

α’ to that for the actual nanostructure α (=kα’), upon substituting eq.5 into eq.4, the following

equation can be derived.

σ d=σ app3

16(3 r+5 ) k (6)

The inherent tensile strength of the disordered region σd for the carbon fibres used in this

study were then calculated substituting σapp (=σ0), r and k into eq.6. Several arbitrary values of

k were used as the exact degree of stress fluctuation in the actual nanostructure has not yet

been determined. The calculated values of σd show an asymptotic behaviour such that it

initially increases (CF1 T800S) and then reaches an almost constant value of 9, 19 and 37

GPa for k=0.5, 1 and 2, respectively (Fig.7 and Table 3). It is important to note that this

asymptotic behaviour cannot be explained with the existing models, which focus on the

crystallite failure [25-28]. Quite interestingly, this behaviour is almost identical to the trend

of the Young’s moduli of the disordered region Ed which has been estimated in our previous

work [12]. Further investigations, which include molecular dynamics simulation [57,58],

clear understanding of the actual stress distribution in nanostructure and the direct

observation of the resulting tensile failure event, are scope of our future study. Nevertheless,

from this semi-quantitative agreement, we conclude that it is more probable that the

disordered region rather than the crystallites determines the tensile strength of the PAN-based

carbon fibres in the absence of large flaws.

16

Figure 7: Estimated inherent tensile strength of the disordered region σd for several values of

k (left axis). Estimated Young’s moduli of the disordered region Ed [12] (right axis).

Table 3: Aspect ratio of the crystallites r,(=La/Lc), stress concentration factor α and estimated

inherent tensile strength of the disordered region σd. Estimated Young’s moduli for the

disordered region Ed [12] are shown for the sake of comparison.

Aspect ratio

r

(=La/Lc)

SCF

α

(for k=1)

Inherent tensile strength

of the disordered region

σd /GPa (for k=1)

Young’s moduli for

the disordered region

Ed /GPa [12]

CF1 0.9 1.5 4.1 43

CF2 1.3 1.7 16.9 159

T800S 1.4 1.7 19.5 200

CF3 1.5 1.8 18.7 193

CF4 1.9 2.0 18.5 211

CF5 2.1 2.1 18.9 212*1

*1 Interpolated using the values for M40S and M50S in the previous study [12].

The presence of the peak in the observed tensile strength σ0 with fibre modulus (Table

1 and 2) could now be interpreted in relation to the nanostructure development. Both the

crystallites and the disordered region develop up to certain point, and then the disordered

region attains an almost constant structure while the crystallites continue to grow, increasing

17

the degree of stress concentration. In conclusion, this new interpretation of carbon fibre

fracture has important implications with respect to the possibilities for the further

improvements in tensile strength of the PAN-based carbon fibres. It is particularly

encouraging that, considering that the crystallites already possess well-ordered graphitic

structure, there would be a considerable scope for improving single fibre tensile strength by

precise control of the more disordered region.

5. CONCLUSIONS

In order to obtain a clearer insight into the tensile strength of the PAN-based carbon

fibres, we have studied the high strength region and the following results were obtained.

- It has been shown that the loop test can be used as a quantitative tool for evaluating high

strength region of carbon fibres by calibrating with Raman spectroscopic measurements.

- Commercial high performance carbon fibres (T800S) showed tensile strengths as high as

13 GPa at short gauge lengths. This demonstrates the considerable potential of current

carbon fibres.

- A tensile strength model that takes into account the fibre nanostructure has been

proposed. It has been suggested that the tensile strength in the absence of large flaws is

controlled by the disordered region rather than the crystallites. Fine tuning of the fibre

nanostructure is believed to be the key to further improvements in the tensile strength of

carbon fibres.

REFERENCES

[1] Holmes M. Global carbon fibre market remains on upward trend. Reinf Plast

2014;58:38–45.

[2] Moreton R. The effect of gauge length on the tensile strength of R.A.E. carbon fibres.

Fibre Sci Technol 1969;1:273–84. (doi:10.1016/0015-0568(69)90012-8)

[3] El Asloun M, Donnet JB, Guilpain G, Nardin M, Schultz J. On the estimation of the

tensile strength of carbon fibres at short lengths. J Mater Sci 1989;24:3504–10.

(doi:10.1007/BF02385732)

[4] Tagawa T, Miyata T. Size effect on tensile strength of carbon fibers. Mater Sci Eng A

1997;238:336–42.

[5] Pickering KL, Murray TL. Weak link scaling analysis of high-strength carbon fibre.

Compos Part A Appl Sci Manuf 1999;30:1017–21.

18

[6] Naito K, Yang JM, Tanaka Y, Kagawa Y. The effect of gauge length on tensile

strength and Weibull modulus of polyacrylonitrile (PAN)- and pitch-based carbon

fibers. J Mater Sci 2012;47:632–42. (doi:10.1007/s10853-011-5832-x)

[7] Whitney W, Kimmel RM. Griffith equation and carbon fibre strength. Nature Physical

Science 1972;237(75):93–94. (doi:10.1038/physci237093a0)

[8] Honjo K. Fracture toughness of PAN-based carbon fibers estimated from strength–

mirror size relation. Carbon N Y 2003;41:979–84. (doi:10.1016/S0008-

6223(02)00444-X)

[9] Shioya M, Inoue H, Sugimoto Y. Reduction in tensile strength of polyacrylonitrile-

based carbon fibers in liquids and its application to defect analysis. Carbon N Y

2013;65:63–70. (doi:10.1016/j.carbon.2013.07.102)

[10] Ogihara S, Imafuku Y, Yamamoto R, Kogo Y. Direct evaluation of fracture toughness

in a carbon fiber. ICCM17, 2009.

[11] Kant M, Penumadu D. Fracture behavior of individual carbon fibers in tension using

nano-fabricated notches. Compos Sci Technol 2013;89:83–8.

(doi:10.1016/j.compscitech.2013.09.020)

[12] Tanaka F, Okabe T, Okuda H, Ise M, Kinloch IA, Mori T, Young RJ. The effect of

nanostructure upon the deformation micromechanics of carbon fibres. Carbon

2013;52:372–378. (doi:10.1016/j.carbon.2012.09.047)

[13] Tanaka F, Okabe T, Okuda H, Kinloch IA, Young RJ. The effect of nanostructure

upon the compressive strength of carbon fibres. J Mater Sci 2012;48:2104–2110.

(doi:10.1007/s10853-012-6984-z)

[14] Tanaka F, Okabe T, Okuda H, Kinloch IA, Young RJ. Factors controlling the strength

of carbon fibres in tension. Composites: Part A 2014;57:88–94.

(doi:10.1016/j.compositesa.2013.11.007)

[15] Loidl D, Peterlik H, Müller M, Riekel C, Paris O. Elastic moduli of nanocrystallites in

carbon fibers measured by in-situ X-ray microbeam diffraction. Carbon NY

2003;41:563–70. (doi:10.1016/S0008-6223(02)00359-7)

[16] Loidl D, Peterlik H, Paris O, Müller M, Burghammer M, Riekel C. Structure and

mechanical properties of carbon fibres: a review of recent microbeam diffraction

studies with synchrotron radiation. J Synchrotron Radiat 2005;12:758–64.

(doi:10.1107/S0909049505013440)

[17] Bennett SC, Johnson DJ, Johnson W. Strength-structure relationships in PAN-based

carbon fibres. J Mater Sci 1983;18:3337–47. (doi:10.1007/BF00544159)

19

[18] Guigon M, Oberlin A, Desarmot G. Microtexture and structure of some high tensile

strength, PAN-base carbon fibres. Fibre Sci Technol 1984;20:55–72.

(doi:10.1016/0015-0568(84)90057-5)

[19] Guigon M, Oberlin A, Desarmot G. Microtexture and structure of some high-modulus,

PAN-base carbon fibres. Fibre Sci Technol 1984;20:177–98. (doi:10.1016/0015-

0568(84)90040-X)

[20] Oberlin A. Carbonization and graphitization. Carbon NY 1984;22:521–41.

(doi:10.1016/0008-6223(84)90086-1)

[21] Oberlin A, Bonnamy S, Lafdi K. Structure and texture of carbon fibers. In: Donnet JB,

editor. Carbon fibers. New York: Mercel Dekker Inc.; 1989. p. 85–159.

[22] Johnson DJ. Structure-property relationships in carbon fibres. J Phys D Appl Phys

1987;20:286–91. (doi:10.1088/0022-3727/20/3/007)

[23] Johnson DJ. Structural studies of PAN-based carbon fibers. In: Thrower PA, editor.

Chemistry and physics of carbon, vol.20. New York: Marcel Dekker; 1987. p. 1–58.

[24] Watanabe J, Tanaka F, Okuda H, Okabe T. Tensile strength distribution of carbon

fibers at short gauge lengths. Adv Compos Mater 2014;23(5-6):535–550.

(doi:10.1080/09243046.2014.915120)

[25] Cooper GA, Mayer RM. The strength of carbon fibres. J Mater Sci 1971;6(1):60–67.

(doi:10.1007/BF00550292)

[26] Stewart M, Feughelman M. The fracture mechanism of carbon fibres. J Mater Sci

1973;8(8):1119–1122. (doi:10.1007/BF00632763)

[27] Reynolds WN, Sharp JV. Crystal shear limit to carbon fibre strength. Carbon

1974;12(2):103–110. (doi:10.1016/0008-6223(74)90018-9)

[28] Reynolds WN, Moreton R, Worthington PJ. Some factors affecting the strengths of

carbon fibres [and discussion]. Philos Trans R Soc A Math Phys Eng Sci

1980;294:451–61. (doi:10.1098/rsta.1980.0054. 1016/j.carbon.2014.12.067)

[29] Tyson CN. Fracture mechanisms in carbon fibres derived from pan in the temperature

range 1000-2800 degrees C. J Phys D Appl Phys 1975;8(7):749–758.

(doi:10.1088/0022-3727/8/7/007)

[30] Phoenix S, Sexsmith R. Clamp effects in fiber testing. J Compos Mater 1972;6:322.

[31] Chae HG, Newcomb B, Gulgunje PV, Liu Y, Gupta KK, Kamath MG, et al. High

strength and high modulus carbon fibers. Carbon NY 2015;93:81–7.

(doi:10.1016/j.carbon.2015.05.016)

20

[32] Okabe T, Takeda N. Elastoplastic shear-lag analysis of single-fiber composites and

strength prediction of unidirectional multi-fiber composites. Composites: Part A

2002;33:1327–35. (doi:10.1016/S1359-835X(02)00170-7)

[33] Okabe T, Takeda N. Estimation of strength distribution for a fiber embedded in a

single-fiber composite: experiments and statistical simulation based on the elasto-

plastic shear-lag approach. Compos Sci Technol 2001;61:1789–800.

(doi:10.1016/S0266-3538(01)00080-X)

[34] Sinclair D. A bending method for measurement of the tensile strength and Young’s

modulus of glass fibers. J Appl Phys 1950;21(5):380. (doi:10.1063/1.1699670)

[35] Jones WR, Johnson JW. Intrinsic strength and non-Hookean behaviour of carbon

fibres. Carbon 1971;9(5):645–655. (doi:10.1016/0008-6223(71)90087-X)

[36] Thorne DJ. Carbon fibres with large breaking strain. Nature 1974;248(5451):754–756.

(doi:10.1038/248754a0)

[37] Da Silva JLG, Johnson DJ. Flexural studies of carbon fibres. J Mater Sci

1984;19(10):3201–3210. (doi:10.1007/BF00549805)

[38] Krucinska I. Evaluation of the intrinsic mechanical properties of carbon fibres.

Compos Sci Tech 1991;41(3):287–301. (doi:10.1016/0266-3538(91)90004-9)

[39] Trinquecoste M, Carlier JL, Derré A, Delhaès P, Chadeyron P. High temperature

thermal and mechanical properties of high tensile carbon single filaments. Carbon

1996;34(7):923–929. (doi:10.1016/0008-6223(96)00052-8)

[40] Fukuda H, Yakushiji M, Wada A. A loop test to measure the strength of

monofilaments used for advanced composites. Adv Compos Mater 1999;8(3):281–

291. (doi:10.1163/156855199X00272)

[41] López Jiménez F, Pellegrino S. Failure of carbon fibers at a crease in a fiber-

reinforced silicone sheet. J Appl Mech 2012;80(1):011020.

[42] Loidl D, Paris O, Burghammer M, Riekel C, Peterlik H. Direct observation of

nanocrystallite buckling in carbon fibers under bending load. Phys Rev Lett

2005;95(22):225501. (doi:10.1103/PhysRevLett.95.225501)

[43] Naito K, Tanaka Y, Yang JM, Kagawa Y. Flexural properties of PAN- and pitch-based

carbon fibers. J Am Ceram Soc 2009;92(1):186–192. (doi:10.1111/j.1551-

2916.2008.02868.x)

[44] Takaku A, Shioya M. X-ray measurements and the structure of polyacrylonitrile- and

pitch-based carbon fibres. J Mater Sci 1990;25:4873–9. (doi:10.1007/BF01129955)

21

[45] Nair RR, Blake P, Grigorenko AN, Novoselov KS, Booth TJ, Stauber T, et al. Fine

structure constant defines visual transparency of graphene. Science 2008;320:1308.

(doi:10.1126/science.1156965)

[46] Li Z, Kinloch IA, Young RJ, Novoselov KS, Anagnostopoulos G, Parthenios J, et al.

Deformation of wrinkled graphene. ACS Nano 2015;9(4):3917-25.

(doi:10.1021/nn507202c)

[47] Reder C, Loidl D, Puchegger S, Gitschthaler D, Peterlik H, Kromp K, et al. Non-

contacting strain measurements of ceramic and carbon single fibres by using the laser-

speckle method. Compos Part A Appl Sci Manuf 2003;34:1029–33.

(doi:10.1016/S1359-835X(03)00240-9)

[48] Matthewson MJ, Kurkjian CR, Gulati ST. Strength measurement of optical fibers by

bending. J Am Ceram Soc 1986;69:815–21. (doi:10.1111/j.1151-

2916.1986.tb07366.x)

[49] van den Heuvel PWJ, Peijs T, Young RJ, Heuvel P. Analysis of stress concentrations

in multi-fibre microcomposites by means of Raman spectroscopy. J Mater Sci Lett

1996;15:1908–11. (doi:10.1007/BF00264093)

[50] Zweben C, Rosen BW. A statistical theory of material strength with application to

composite materials. J Mech Phys Solids 1970;18:189–206. (doi:10.1016/0022-

5096(70)90023-2)

[51] Wang S, He Y, Huang H, Zou J, Auchterlonie GJ, Hou L, et al. An improved loop test

for experimentally approaching the intrinsic strength of alumina nanoscale whiskers.

Nanotechnology 2013;24:285703. (doi:10.1088/0957-4484/24/28/285703)

[52] Lee C, Wei X, Kysar JW, Hone J. Measurement of the elastic properties and intrinsic

strength of monolayer graphene. Science 2008;321:385–8.

(doi:10.1126/science.1157996)

[53] Nisitani H. Method of approximate calculation for interference of notch effects and its

application. Bull JSME 1968;11:725–38. (doi:10.1299/jsme1958.11.725)

[54] Pilkey WD, Pilkey DF. Peterson's stress concentration factors. John Wiley & Sons,

2008.

[55] Pyrz R. Correlation of microstructure variability and local stress field in two-phase

materials. Mater Sci Eng A 1994;177:253–9. (doi:10.1016/0921-5093(94)90497-9)

[56] Sun CJ, Saffari P, Sadeghipour K, Baran G. Effects of particle arrangement on stress

concentrations in composites. Mater Sci Eng A 2005;405:287–95.

(doi:10.1016/j.msea.2005.06.032)

22

[57] Ito A, Okamoto S. Using molecular dynamics to assess mechanical properties of PAN-

based carbon fibers comprising imperfect crystals with amorphous structures. Int J

Mech Aero Ind Mechatron Eng 2013;7:771-776.

[58] Penev ES, Artyukhov VI, Yakobson BI. Basic structural units in carbon fibers:

atomistic models and tensile behavior. Carbon N Y 2014;85:72–8.

(doi:10.1016/j.carbon.2014.12.067)

[59] Shioya M, Takaku A. Characterization of crystallites in carbon fibres by wide-angle

X-ray diffraction. J Appl Crystallogr 1989;22:222–30.

(doi:10.1107/S0021889888014256)

[60] Krucinska I, Stypka T. Direct measurement of the axial poisson's ratio of single carbon

fibres. Compos Sci Technol 1991;41:1–12. (doi:10.1016/0266-3538(91)90049-U)

23