MATERIAI NE EGWEERI ELSEVIER Materials Science and Engineering A238(1997)336-342 Size effect on tensile strength of carbon fibers Tetsuya Tagawa, Takashi Miyata Department of Materials Science and Engineering, Nagoya Unirersity, Chikusa-ku, Nagoya 464-01, Japan Abstract Tensile strength of carbon fibers exhibits statistical Weibull type distribution and significant size dependence. In the present work, ten types of polyacrylonitrile based and mesophase-pitch based carbon fiber monofilaments were tested for two or three different gauge lengths. Size effect in both axial and radial directions were analyzed based on the two parameters Weibull statistics. It was found that the size effect in axial direction was almost similar for all fibers tested. This result suggests that the tensile strength obtained for a certain gauge length is a meaningful measure as a representative strength of the fiber strength. In radial direction, the size effect of the tensile strength was larger than that in axial direction. The tensile strength of the carbon fibers seemed to have unisotropic statistical characteristics. Size dependence in diameter was numerically simulated with an assumption of unisotropic distribution of Reynolds-Sharp type defects. o 1997 Elsevier Science S.A Keywords: Size effect; Carbon fibers; Tensile strengt 1. Introduction work. the tensile tests of mono-filaments for various types of carbon fibers were performed, the statistical Carbon fiber is one of the high performance fibers size effects of the tensile strength were investigated on employed in the advanced composites. In the case of the basis of the Weibull statistics the composites reinforced by carbon fibers, continuous fibers are used and fiber volume fraction usually ex ceeds 40%. Therefore, characteristics of the fiber 2 Experimental procedure strength is the most influential factor on the strength of the composites. At the composites strength design, the 2.1. Carbon fibers fiber strengths evaluated from the fibers strands with a certain gauge length are usually used. Tensile strength Four types of polyacrylonitrile(PAn) based and six of carbon fibers, however, shows a large scatter and types of mesophase-pitch(MP)based carbon fibers markable size dependence according to the weakest were used for the tensile tests. Their tensile propertie link analogy [1-3]. Although the statistical characteri and diameter shown by the manufacturers are listed tic of fiber strength will give significant influences on Table I. The first term of the material code shows the the strength will give significant influences on the types of precursor and in the second term, the letters A, strength of composites, the statistical behaviors of fiber B, C,.. show the difference of the fiber manufacturer strength have never been systematically characterized and the numbers refer to the elastic modules the yet. Supposing that, for instance the size effect of the tensile properties of these fibers are from 2.5 to 5.6GPa strength is different for each fiber, the tensile strength in tensile strength and from 160 to 600 GPa in elastic evaluated for a certain gauge length will be lacking in a modulus. These properties belong to all types of the generality as a measure of fiber strength. In the present high performance carbon fiber, that are called as Hs high strength), IM(intermediate modulus)and HM Corresponding author. Tel: +81 52 7893577: fax: +81 52(high modulus). All fibers have approximately circular 7893236. 0921-5093/97/S17.00 0 1997 Elsevier Science S.A. All rights reserved PIS0921-5093097100454-1
Materials Science and Engineering A238 (1997) 336–342 Size effect on tensile strength of carbon fibers Tetsuya Tagawa *, Takashi Miyata Department 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 present work, ten types of polyacrylonitrile based and mesophase-pitch based carbon fiber monofilaments were tested for two or three different gauge lengths. Size effect in both axial and radial directions were analyzed based on the two parameters Weibull statistics. It was found that the size effect in axial direction was almost similar for all fibers tested. This result suggests that the tensile strength obtained for a certain gauge length is a meaningful measure as a representative strength of the fiber strength. In radial direction, the size effect of the tensile strength was larger than that in axial direction. The tensile strength of the carbon fibers seemed to have unisotropic statistical characteristics. Size dependence in diameter was numerically simulated with an assumption 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 fibers employed in the advanced composites. In the case of the composites reinforced by carbon fibers, continuous fibers are used and fiber volume fraction usually exceeds 40%. Therefore, characteristics of the fiber strength is the most influential factor on the strength of the composites. At the composites strength design, the fiber strengths evaluated from the fibers strands with a certain gauge length are usually used. Tensile strength of carbon fibers, however, shows a large scatter and remarkable size dependence according to the weakest link analogy [1–3]. Although the statistical characteristic of fiber strength will give significant influences on the strength will give significant influences on the strength of composites, the statistical behaviors of fiber strength have never been systematically characterized yet. Supposing that, for instance, the size effect of the strength is different for each fiber, the tensile strength evaluated for a certain gauge length will be lacking in a generality as a measure of fiber strength. In the present work, the tensile tests of mono-filaments for various types of carbon fibers were performed, the statistical size effects of the tensile strength were investigated on the basis of the Weibull statistics. 2. Experimental procedure 2.1. Carbon fibers Four types of polyacrylonitrile (PAN) based and six types of mesophase-pitch (MP) based carbon fibers were used for the tensile tests. Their tensile properties and diameter shown by the manufacturers are listed in Table 1. The first term of the material code shows the types of precursor and in the second term, the letters A, B, C, … show the difference of the fiber manufacturer and the numbers refer to the elastic modules. The tensile properties of these fibers are from 2.5 to 5.6 GPa in tensile strength and from 160 to 600 GPa in elastic modulus. These properties belong to all types of the high performance carbon fiber, that are called as HS (high strength), IM (intermediate modulus) and HM (high modulus). All fibers have approximately circular cross-sections. * Corresponding author. Tel.: +81 52 7893577; fax: +81 52 7893236. 0921-5093/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PII S0921-5093(97)00 454-1
T. Tagawa, T. Miyata/ Materials Science and Engineering 4238(1997)336-342 Table l Mechanical properties of carbon fibers tested CF code Precursor Average diameter day (um) Youngs modulus E(GPa) Tensile strength or (MPa) 40 56871 275 3040 3326 MP.B50 PAN-C25 PAN PAN-D40 PAN 412 3168 PAN-E25 PAN PAN-E30 PAN 4.8 5590 MP, mesophase pitch; PAN, polyacrylonitrile 2. 2. Tensile tests 3. Results and discussion For each type of carbon fibers, 40 tensile tests were 3.. Statistical distribution of tensile strength and their carried out for one gauge length and gauge lengths size effect adopted in test were 6, 15 and 25 mm. Fiber samples randomly selected from the fiber bundle were fixed by Fig 2 shows some results as Weibull distributions on an adhesive on the slotted paper holders. The speci- the tensile strength tested for 15 mm gauge length en with the paper holder was mounted to the tensile Single modal Weibull distribution can be approximately applied to each results except PAN-C25. The results of apparatus and started extending after burning off the PAN-C25 shows the deviation from a linear relation in paper frame. Samples were extended to failure at the rate of 0. 1 mm min-I by the rotation of a differential the low strength range, showing the mixed modal or micrometer head with a constant rotating motor. The multi-modal Weibull distribution. The Weibull shape parameters obtained are summarized in Fig 3. The two fracture load was evaluated with the load cell of 100g broken lines show the upper and lower limits of the in capacity. The tensile apparatus and the specimen set up are shown in Fig. 1. Before the tensile test, the 0.99 PAN-E30 PAN-E2 cross sectional area was evaluated from the diameter measured by the scanning electron microscope for 0=4907MPa Oo=4907MPa h sample. The results for each series of tests were nalyzed based on the two parameters Weibull statis Load cell 0.01 Adhesio 0.005 IPa o=5530MF Oo=5000MPa Differential micrometer head off- Gauge Carbon 0.05 0.0l 0.005 (a) Tensile test apparatus b)Specimen set up Fracture stress, Of, MPa Fracture stress, Of, MPa Fig. I. Tensile test apparatus of mono-filament and specimen set Fig. 2. Examples of Weibull distributie tensile strength for (a)Tensile test apparatus, (b)specimen se
T. Tagawa, T. Miyata / Materials Science and Engineering A238 (1997) 336–342 337 Table 1 Mechanical properties of carbon fibers tested Average diameter dav CF code (mm) Young’s modulus E (GPa) Tensile strength sf Precursor (MPa) MP-A15 MP 8.5 157 2540 MP-A30 MP 8.6 275 3315 MP-A50 MP 8.8 490 3040 MP-A50% MP 6.7 487 3830 MP-A60 MP 7.1 597 3326 MP-B50 MP 8.3 504 4021 PAN-C25 PAN 6.1 235 3825 PAN-D40 PAN 5.4 412 3168 PAN-E25 PAN 6.4 230 4810 PAN-E30 PAN 4.8 294 5590 MP, mesophase pitch; PAN, polyacrylonitrile. 2.2. Tensile tests For each type of carbon fibers, 40 tensile tests were carried out for one gauge length and gauge lengths adopted in test were 6, 15 and 25 mm. Fiber samples randomly selected from the fiber bundle were fixed by an adhesive on the slotted paper holders. The specimen with the paper holder was mounted to the tensile apparatus and started extending after burning off the paper frame. Samples were extended to failure at the rate of 0.1 mm min−1 by the rotation of a differential micrometer head with a constant rotating motor. The fracture load was evaluated with the load cell of 100 g in capacity. The tensile apparatus and the specimen set up are shown in Fig. 1. Before the tensile test, the cross sectional area was evaluated from the diameter measured by the scanning electron microscope for each sample. The results for each series of tests were analyzed based on the two parameters Weibull statistics. 3. Results and discussion 3.1. Statistical distribution of tensile strength and their size effect Fig. 2. shows some results as Weibull distributions on the tensile strength tested for 15 mm gauge length. Single modal Weibull distribution can be approximately applied to each results except PAN-C25. The results of PAN-C25 shows the deviation from a linear relation in the low strength range, showing the mixed modal or multi-modal Weibull distribution. The Weibull shape parameters obtained are summarized in Fig. 3. The two broken lines show the upper and lower limits of the Fig. 2. Examples of Weibull distributions on tensile strength for GL=15 mm. Fig. 1. Tensile test apparatus of mono-filament and specimen set up. (a) Tensile test apparatus, (b) specimen set up
338 T. Tagawa, T. Miyata/ Materials Science and Engineering 4238(1997)336-342 12 simulation with assumption of constant shape parameter, 4 O GL=6mm 042 Lower limit PAN-C25 G L=6mm 1000 MP based fibers Pan based fibers Diameter, um Fig. 3. Weibull shape parameters for fibers tested. Fig. 5. Dependence of tensile strength on fiber diameter for PAN Monte-Carlo simulation for 40 samples with an as- sumption of a constant shape parameter four. The cally linear relationship in Fig. 4 and the inverse of the shape parameters for all fibers and gauge lengths are slope should be equal to the Weibull shape parameter included in the range of the Monte-Carlo simulation The inverse values of the slopes for the most of the results. The Weibull shape parameter indicates the de- fibers tested approximately coincident with the results gree of a scatter on the strength and a smaller shape in Fig 3 except for MP-A50 and PAN-C25 parameter indicates a larger scatter of the strength. The The size effect of the strength was observed not only value of four for shape parameters is smaller than the in the axial direction but also in the radial direction of one of the bending strength in engineering ceramic fiber. Fig. 5 shows the dependency of the tensile materials. The Weibull shape parameter also has the strength on fiber diameter for PAN-C25 tested in 6 mm physcial meaning as the factor of size effect on the gauge length. The inverse of the slope in Fig. 5 is 0.42 strength. The results shown in Fig. 3 suggest that the size effect on the tensile strength seems to be similar for any carbon fibers independent on the precursor and the strength level. This similarity of the size effect supports Weibull that the relative strengths between any carbon fibers can be kept in any fiber length. Fig. 4 shows the 6/m,G dependency of the tensile strength (Weibull scale arameter)on the gauge length for several fibers. The Log day Weibull statistics with two parameters gives theoreti Log d d 10000 (a) Experimental dependency b) Distribution of equivalent on diameter strength for average diameter 5000 ￥。= △PANC2 ●MPA50 1000 (c)Dependency on gauge length (d)Optimum md due to convergent 15 Gauge length, mm Fig. 6. Procedure for correction of anisotropy in tensile strength. (a) Experimental dependency on diameter, (b) of equivalent Fig. 4. Dependence of tensile strength(Weibull scale parameter strength for average diameter, (c) dependency on gauge length, (d) optimum a due to convergent calculation
338 T. Tagawa, T. Miyata / Materials Science and Engineering A238 (1997) 336–342 Fig. 3. Weibull shape parameters for fibers tested. Fig. 5. Dependence of tensile strength on fiber diameter for PANC25. Monte-Carlo simulation for 40 samples with an assumption of a constant shape parameter four. The shape parameters for all fibers and gauge lengths are included in the range of the Monte-Carlo simulation results. The Weibull shape parameter indicates the degree of a scatter on the strength and a smaller shape parameter indicates a larger scatter of the strength. The value of four for shape parameters is smaller than the one of the bending strength in engineering ceramic materials. The Weibull shape parameter also has the physcial meaning as the factor of size effect on the strength. The results shown in Fig. 3 suggest that the size effect on the tensile strength seems to be similar for any carbon fibers independent on the precursor and the strength level. This similarity of the size effect supports that the relative strengths between any carbon fibers can be kept in any fiber length. Fig. 4 shows the dependency of the tensile strength (Weibull scale parameter) on the gauge length for several fibers. The Weibull statistics with two parameters gives theoretically linear relationship in Fig. 4 and the inverse of the slope should be equal to the Weibull shape parameter. The inverse values of the slopes for the most of the fibers tested approximately coincident with the results in Fig. 3 except for MP-A50 and PAN-C25. The size effect of the strength was observed not only in the axial direction but also in the radial direction of fiber. Fig. 5 shows the dependency of the tensile strength on fiber diameter for PAN-C25 tested in 6 mm gauge 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 equivalent strength for average diameter, (c) dependency on gauge length, (d) optimum md due to convergent calculation. Fig. 4. Dependence of tensile strength (Weibull scale parameter) on gauge length
T. Tagawa, T. Miyata/ Materials Science and Engineering 4238(1997)336-342 339 0.99PANC25 PAN-C2S should be equal to a half value of m. The result in Fig 5 seems to obey Eq(3). However, precise value of 0.90 G L=lSmm is not given due to insufficient number of data. For the 0.50 correction of the size effect in radial direction semi-nu merical method schematically shown in Fig. 6 was carried out. The value of ma is arbitrarily assumed like the result in Fig. 5. Equivalent tensile strength for the 0. average diameter could be calculated in each sample 0.05 from the Eq (3)using the or the 0.02 distribution of an equivalent tensile strength for each 0.01 gauge length tested, Weibull parameters were obtained Based on Weibull statistics, Weibull shape parameter, 0.005 m=4.0 m, for a certain gauge length, I, should be equal to the Oo=5530MPa Oo=4722MPa inverse of the slope in the relation between logarithmic 000100001000 5000100 Weibull scale parameter, or and logarithmic gauge Fracture stress, Of MPa Fracture stress of gth, I,(the value of m in Eq. (2)) (a)without correction (b) with correction the deviation between m, and m for all gauge lengths was defined as an error factor err(ma), which was a Fig. 7. Weibull distributions of tensile strength with and function of md. The optimum value of ma was calcu correction for size effect in diameter. (a) without correction lated for the minimum value of err(ma) by Newton correction Rapson method. These corrections were examined for that is one tenth of the value in Fig 4, that indicates the mp-a50 and PAN-c25 of which diameters were unisotropic statistical distribution of the streng varied in wide range. One example of the corrected Therefore, the results in Figs. 2 and 3 involve the results in shown in Fig. 7. This figure shows a Weibull effect in diameter of fibers distribution of the tensile strength of PAN-C25 tested at 15 mm gauge length without and with this correc- 3. 2. Anisotropy on size effect tion. Correction of the size effect in diameter leads to higher linearity and the single modal Weibull distribu- The tensile strengths obtained in the present work are tion. The variation of Weibull shape parameter with not evaluated under constant volume in spite of testing and without this correction is not so large and mainly at a constant gauge length. This is due to the variation the Weibull scale parameter is varied by the correction. f the diameter even in one fiber. It is necessary of Fig.8 shows the dependence of Weibull scale parameter separate the size effect in radial direction in order to on the gauge length without and with the correction strictly discuss the unistropy in size dependency of the Discrepancy of the size effect in Fig. 4 with the shape tensile strength parameter m for the fibers PAN-C25 and MP-A50 can A commonly used form of the two parameters be explained by the consideration of the size effect in Weibull distribution for a fiber of a constant diamete F(o)=l-exp[-1(o oor" where F(o) is the probability of failure up to a stress level o, I is the gauge length. o and m are the scale 5000 parameter and the shape parameter, respectively. For a certain probability, the fracture strength, ar is given as function of of gauge length, I 2000 ●MP-A50 ●MPA50 O PAN-C25 O PAN-C25 For a diameter, similar equation with Eqs. (I)and(2) can be assumed for a given gauge length lo as log ar=-1/ma log d+ constant(I=lo) Eqs.(2)and (3)give the theoretical size dependencies of the tensile strength for the axial direction and the radial direction, respectively. If the defects that control the Fig. 8. Dependence of tensile strength without correction of size effect in diameter. (a) without correction. tensile strength exist isotropically in the materials, my b)with correction
T. Tagawa, T. Miyata / Materials Science and Engineering A238 (1997) 336–342 339 Fig. 7. Weibull distributions of tensile strength with and without correction for size effect in diameter. (a) Without correction, (b) with correction. 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 the correction of the size effect in radial direction, semi-numerical method schematically shown in Fig. 6 was carried out. The value of md is arbitrarily assumed like the result in Fig. 5. Equivalent tensile strength for the average diameter could be calculated in each sample from the Eq. (3) using the assumed value of md. For the distribution of an equivalent tensile strength for each gauge length tested, Weibull parameters were obtained. Based on Weibull statistics, Weibull shape parameter, mi for a certain gauge length, li should be equal to the inverse of the slope in the relation between logarithmic Weibull scale parameter, s0i and logarithmic gauge length, li , (the value of m in Eq. (2)). The summation of the deviation between mi and m for all gauge lengths was defined as an error factor err(md), which was a function of md. The optimum value of md was calculated for the minimum value of err(md) by NewtonRapson method. These corrections were examined for the MP-A50 and PAN-C25 of which diameters were varied in wide range. One example of the corrected results in shown in Fig. 7. This figure shows a Weibull distribution of the tensile strength of PAN-C25 tested at 15 mm gauge length without and with this correction. Correction of the size effect in diameter leads to higher linearity and the single modal Weibull distribution. The variation of Weibull shape parameter with and without this correction is not so large and mainly the Weibull scale parameter is varied by the correction. Fig. 8 shows the dependence of Weibull scale parameter on the gauge length without and with the correction. Discrepancy of the size effect in Fig. 4 with the shape parameter m for the fibers PAN-C25 and MP-A50 can be explained by the consideration of the size effect in that is one tenth of the value in Fig. 4, that indicates unisotropic statistical distribution of the strength. Therefore, the results in Figs. 2 and 3 involve the size effect in diameter of fibers. 3.2. Unisotropy on size effect The tensile strengths obtained in the present work are not evaluated under constant volume in spite of testing at a constant gauge length. This is due to the variation of the diameter even in one fiber. It is necessary of separate the size effect in radial direction in order to strictly discuss the unistropy in size dependency of the tensile strength. A commonly used form of the two parameters Weibull 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 stress level s, l is the gauge length. s0 and m are the scale parameter and the shape parameter, respectively. For a certain probability, the fracture strength, sf is given as a 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 of the tensile strength for the axial direction and the radial direction, respectively. If the defects that control the tensile strength exist isotropically in the materials, md Fig. 8. Dependence of tensile strength on gauge length with and without correction of size effect in diameter. (a) Without correction, (b) with correction
T. Tagawa, T. Miyata/ Materials Science and Engineering 4238(1997)336-342 dency on the diameter should be two times larger than 10000 PAN-C25 the slope of the dependency on the gauge length. How Dependency ever,the dependency on the diameter in Fig. 9 is ten length times larger than the dependency on the gauge length Based on the weakest link analogy, the higher strength should be obtained in the smaller volume of the materi- als. In the carbon fiber, however, the unisotropic size dependency of the tensile strength. It is well known that 000 the strength of the carbon fiber shows the unisotropic cs due to the alignment of graphite crys talline. The results in Fig. 9 suggest that the statistical Dependency characteristics of the tensile strength can also show on diameter ignificant anistropy 1000 3.3. Structure and distribution of defects 0.30.40.5 It was confirmed that the tensile strength in the carbon fibers almost obeys the single modal Weibull Fig 9. Volume dependence of tensile strength distribution. This suggests that the fracture of a carbon fiber may be controlled by single mechanism. Suppe diameter. At this correction the optimum ma value is ing that the same fracture sources are isotropically 0. 46 for PAN-C25 and this value also agrees well with distributed in fibers, the unisotropic size dependence of the dependency of the results in Fig. 5. the tensile strength should not be observed. Considering With above correction, unistropic size dependence the results of Fig 9, it can be suggested that the density can be strictly divided into the axial direction and the of the fracture source is lower at the center of the fiber radial direction of the fiber. These two size dependen- and or the strength of the fracture source is higher at cies are shown in Fig. 9 for PAN-C25 as the volume the center of the fiber dependency of the tensile strength. The dependency or On the fracture mechanisms of the carbon fibers it the gauge length in Fig. 9 is converted o was proposed that the misorientation of the graphite dency on the fiber volume under a constant average crystal layers controlled the fracture [4, 5). Reynolds and diameter and the dependency on the fiber diameter is Sharp [5] have proposed that the fracture of the carbon shown as a dependency for the volume under the fibers initiated at the crystal misorientation region onstant gauge length of 6 mm. In the case of the around some inclusions or voids. They also showed tatistically isotropic materials, the slope of the depen- that the fracture strength, or was given as follow: r=oJ(sinφ:cosφ) 1.0 where as is the shear strength of the graphite interlayer and is the misorientation angle from the fiber axis On the micorstructure of the carbon fibers it has been shown that the graphitization varies along the radial direction and proceeds around that fiber surface for both Pan based fibers [6,7] and MP based fibers [8] Guigon et al. [9] observed the graphite layers of PAN Case I based fiber by a high-resolution transmission electron 日: Case Il microscope and showed that the radius of curved layers was around 45 nm near the fiber surface and was 20 nm near the center. Considering above works, that fracture go n length Experiment misorientation of graphite layers around inclusions or E Dependency(in Fig9) voids and the misorientation angles and their density 0. are varied from the surface to the center 0.30.40.5 The misorientation of the graphite layers is assumed Volume, x10mm the unisotropic size dependency of the tensile strength Fig. 10. Simulation results for volume dependency of normalized ig. 8 are calculatively investigated through
340 T. Tagawa, T. Miyata / Materials Science and Engineering A238 (1997) 336–342 Fig. 9. Volume dependence of tensile strength. dency on the diameter should be two times larger than the slope of the dependency on the gauge length. However, the dependency on the diameter in Fig. 9 is ten times larger than the dependency on the gauge length. Based on the weakest link analogy, the higher strength should be obtained in the smaller volume of the materials. In the carbon fiber, however, the unisotropic size dependency of the tensile strength. It is well known that the strength of the carbon fiber shows the unisotropic characteristics due to the alignment of graphite crystalline. The results in Fig. 9 suggest that the statistical characteristics of the tensile strength can also show significant unistropy. 3.3. Structure and distribution of defects It was confirmed that the tensile strength in the carbon fibers almost obeys the single modal Weibull distribution. This suggests that the fracture of a carbon fiber may be controlled by single mechanism. Supposing that the same fracture sources are isotropically distributed in fibers, the unisotropic size dependence of the tensile strength should not be observed. Considering the results of Fig. 9, it can be suggested that the density of the fracture source is lower at the center of the fiber and/or the strength of the fracture source is higher at the center of the fiber. On the fracture mechanisms of the carbon fibers, it was proposed that the misorientation of the graphite crystal layers controlled the fracture [4,5]. Reynolds and Sharp [5] have proposed that the fracture of the carbon fibers initiated at the crystal misorientation region around some inclusions or voids. They also showed that the fracture strength, sf was given as follow: sf=ss/(sin f · cos f) (4) where ss is the shear strength of the graphite interlayer and f is the misorientation angle from the fiber axis. On the micorstructure of the carbon fibers, it has been shown that the graphitization varies along the radial direction and proceeds around that fiber surface for both PAN based fibers [6,7] and MP based fibers [8]. Guigon et al. [9] observed the graphite layers of PAN based fiber by a high-resolution transmission electron microscope and showed that the radius of curved layers was around 45 nm near the fiber surface and was 20 nm near the center. Considering above works, that fracture of a carbon fiber can be assumed to be governed by the misorientation of graphite layers around inclusions or voids and the misorientation angles and their density are varied from the surface to the center. The misorientation of the graphite layers is assumed as a defect controlling the tensile strength. Hereafter, the unisotropic size dependency of the tensile strength shown in Fig. 8 are calculatively investigated through the varition of the defect density and the strength along diameter. At this correction the optimum md value is 0.46 for PAN-C25 and this value also agrees well with the dependency of the results in Fig. 5. With above correction, unistropic size dependence can be strictly divided into the axial direction and the radial direction of the fiber. These two size dependencies are shown in Fig. 9 for PAN-C25 as the volume dependency of the tensile strength. The dependency on the gauge length in Fig. 9 is converted on the dependency on the fiber volume under a constant average diameter and the dependency on the fiber diameter is shown as a dependency for the volume under the constant gauge length of 6 mm. In the case of the statistically isotropic materials, the slope of the depenFig. 10. Simulation results for volume dependency of normalized strength
T. Tagawa, T. Miyata/ Materials Science and Engineering 4238(1997)336-342 (a) MP-A50 5 (b)MP-B50 5um (c)PAN-C25 5um (d)PAN-E25 5um Fig. Il. SEM micrographs of tensile fracture surface the radial direction of the fiber. Supposing that the Case I. Both the density and the angle of the misori- defect density and the strength are constant along the entation layers are constant along the radical direction axial direction, the cumulative fracture probability of of the fiber. the fiber under a stress, o can be given as follows: Case Il. Only the misorientation density continuously and linearly increases from the center of the surface F(o,d)=1-exp -2r P(o, r)' r /DD(r)di Case Ill. Only the misorientation angle nd linearly increases from the center to the surface In case Il, the variation of the density was assumed where d is a fiber diameter and I is a length. DD(r)is from 10 um-3(center)to 10 um-3(surface). In case the defect density at a distance, r from the center and Ill, the variation of the angle was assumed from 0 P(o, r) is a fracture probability of one defect (center) to 40%(surface). These values were assumed stress, o at a distance, r. P(o, r) is assumed as a with reference to previous work on the microstructures distribution The mean value of the function of [9]. The fracture strength can be evaluated as an ex is given by Eq. (4)and the standard deviation is given pected value of Eq.(5) for a certain probability. The as 15% of the mean value. Thus, the dependency of expected strength according to Eq (5)for 50% proba- P(o, r)on r is defined by the dependency of the misori bility are shown in Fig. 10 as a function of the sample entation angel on r. The numerical simulations were volume. The expected strengths are normalized by the erformed for three cases as follows value for the volume of 10-3 mm. The experimental
T. Tagawa, T. Miyata / Materials Science and Engineering A238 (1997) 336–342 341 Fig. 11. SEM micrographs of tensile fracture surface. the radial direction of the fiber. Supposing that the defect density and the strength are constant along the axial direction, the cumulative fracture probability of the 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) dr n (5) where d is a fiber diameter and l is a length. DD(r) is the defect density at a distance, r from the center and P(s, r) is a fracture probability of one defect under a stress, s at a distance, r. P(s, r) is assumed as a normal distribution. The mean value of the function of P(s, r) is given by Eq. (4) and the standard deviation is given as 15% of the mean value. Thus, the dependency of P(s, r) on r is defined by the dependency of the misorientation angel f on r. The numerical simulations were performed for three cases as follows. Case I. Both the density and the angle of the misorientation layers are constant along the radical direction of the fiber. Case II. Only the misorientation density continuously and linearly increases from the center of the surface. Case III. Only the misorientation angle continuously and linearly increases from the center to the surface. In case II, the variation of the density was assumed from 104 mm−3 (center) to 105 mm−3 (surface). In case III, the variation of the angle was assumed from 0 (center) to 40° (surface). These values were assumed with reference to previous work on the microstructures [9]. The fracture strength can be evaluated as an expected value of Eq. (5) for a certain probability. The expected strength according to Eq. (5) for 50% probability are shown in Fig. 10 as a function of the sample volume. The expected strengths are normalized by the value for the volume of 10−3 mm3 . The experimental
T. Tagawa, T. Miyata/ Materials Science and Engineering 4238(1997)336-342 results in Fig. 9 are also shown as bands in Fig. 10. As 4. Conclusions the carbon fibers are continuously manufactured, the defect distribution along the fiber length could be al- The statistical distributions of the tensile strength in lost constant. The simulation of case I, in which the carbon fibers were investigated for 10 years of various defect density is independent on the location, seems to fibers. The following conclusions were obtained correspond to the dependency on the fiber length. The (1)Weibull shape parameters in axial direction show simulated dependency of the strength on a sample almost constant value of four, irrespective of the car- volume in case Ill is much larger than the result in case bon precursor and the strength level. This result sup- Il.The large dependency on a sample volume in case ports that the tensile strength obtained for a certain Ill qualitatively describe the experimental results. The ge length can be generalized as a measure of the experimental dependency on the fiber diameter, how- fiber strength ever, is larger than the simulation result in case Ill. This (2) Unisotropic size effect on strength was observed probably comes from the assumption of constant in The size dependency of strength in the radial direction layer shear strength, o, in Eq (4). At the misorientation is ten times larger than the dependency in and axial region, the graphitization might be relatively low and direction the interlayer shear strength seems to be smaller than ()Size dependence in diameter of the fiber can be that of the ideal crystalline qualitatively explained by numerical statistic simulation According to the Reynolds-Sharp mechanism, the with the assumption of a certain distribution of the misorientation angle o could be larger around larger misorientation angle inclusion or larger voids. The unisopropic character of the volume dependency of the tensile strength is ex pected to relate with the size distribution of inclusions References or voids along the radial direction of the fiber. [S. Chwastoak, J.B. Barr, R. Didchenko, Carbon 17(1979) 4. Fractography 2J B. Jones, J. Barr, R. Smith, J. Mater. Sci. 15(1980)2454 Some fracture surfaces observed by scanning electron lte, F Girot, Y Le Petitcorps, Mater. Sci microscope are shown in Fig. 11. Based on the above ngA135(1991)59-63 discussions. the fracture could initiate from some of 4J. D.H. Hughes. J. Phys. D: Appl. Phys. 20(1987)276- second particles of voids. However, any defects or 5w.N. Reynolds, J.V. Sharp, Carbon 12(1974)103-110 6KJ. Chen, RJ. Diefendorf, Proc. 16th Biennial Conf.on second particles could not be observed on the fracture Carbon, Am. Carbon Soc. (1983)490-491 surface within the magnification of Sem. These frac [7S.C. Bennett, D.J. Johnson, Carbon 17(1979)25-39. correspond to the [8 M. Inagaki, M. Endo, A. Oberlin, M. Nakamizo microstructure. Nano-scopic observations will be neces Hishiyama, H. Fujimaki, Tanso, 99(1979)130-137 sary to investigate the defects themselves controlling the 9 M. Guigon, A. Oberlin, G. Desarmot, Fiber Sci. Technol. 20 tensile strength of the carbon fiber (1984)177-185
342 T. Tagawa, T. Miyata / Materials Science and Engineering A238 (1997) 336–342 results in Fig. 9 are also shown as bands in Fig. 10. As the carbon fibers are continuously manufactured, the defect distribution along the fiber length could be almost constant. The simulation of case I, in which the defect density is independent on the location, seems to correspond to the dependency on the fiber length. The simulated dependency of the strength on a sample volume in case III is much larger than the result in case II. The large dependency on a sample volume in case III qualitatively describe the experimental results. The experimental dependency on the fiber diameter, however, is larger than the simulation result in case III. This probably comes from the assumption of constant interlayer shear strength, ss in Eq. (4). At the misorientation region, the graphitization might be relatively low and the interlayer shear strength seems to be smaller than that of the ideal crystalline. According to the Reynolds-Sharp mechanism, the misorientation angle f could be larger around larger inclusion or larger voids. The unisopropic character of the volume dependency of the tensile strength is expected to relate with the size distribution of inclusions or voids along the radial direction of the fiber. 3.4. Fractography Some fracture surfaces observed by scanning electron microscope are shown in Fig. 11. Based on the above discussions, the fracture could initiate from some of second particles of voids. However, any defects or second particles could not be observed on the fracture surface within the magnification of SEM. These fracture appearances seem to correspond to the transverse microstructure. Nano-scopic observations will be necessary to investigate the defects themselves controlling the tensile strength of the carbon fiber. 4. Conclusions The statistical distributions of the tensile strength in carbon fibers were investigated for 10 years of various fibers. The following conclusions were obtained. (1) Weibull shape parameters in axial direction show almost constant value of four, irrespective of the carbon precursor and the strength level. This result supports that the tensile strength obtained for a certain gauge length can be generalized as a measure of the fiber strength. (2) Unisotropic size effect on strength was observed. The size dependency of strength in the radial direction is ten times larger than the dependency in and axial direction. (3) Size dependence in diameter of the fiber can be qualitatively explained by numerical statistic simulation with the assumption of a certain distribution of the misorientation 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 (in Japanese). [9] M. Guigon, A. Oberlin, G. Desarmot, Fiber Sci. Technol. 20 (1984) 177–185.