Availableonlineatwww.sciencedirect.com CARBON ELSEVIER Carbon42(2004)715-725 ww. elsevier. com/locate/carbon The tensile behavior of carbon fibers at high temperatures upto2400°C Cedric Sauder, Jacques Lamon, Rene Pailler Laboratoire des Composites Thermostructuraux, UMR 5801: CNRS-Snecma-CEA-UBl, Domaine Universitaire 3. Allee de la boetie. 33600 Pessac. Fr Received 17 June 2003: accepted 20 November 2003 abstract The tensile behavior of four different brands of carbon fibers(a rayon-based, a PAN-based, and 2 pitch-based fibers) has been investigated at various ratures up to 2400C. The tests were carried out using an original fiber testing apparatus. Various mechanical properties including strength and Youngs modulus, as well as Weibull statistical parameters were extracted from test data. Typical tensile behaviors were evidenced such as an essentially linear elastic behavior at room temperature and intermediate temperatures up to 1400-1800C, then a nonlinear elastic delayed response at higher temperatures and ultimately an inelastic esponse with permanent deformations at very high temperatures. Such unusual nonlinear responses for homogeneous materials were related to structure and texture features at the nanometer scale that were described through an X-ray diffraction technique C 2004 Elsevier Ltd. All rights reserved Keywords: A Carbon fibers, Carbon/carbon composites; D. Mechanical properties, Texture 1. Introduction fter a few minutes under zero load. This strain recovery was attributed to the reverse rotation of graphitic layers One of the main difficulties with C/C composites, is Mostovoi et al. [2] measured the mechanical proper- the lack of data on their constituent properties at high ties of a PAN-based carbon fiber(VMN-RK high tem- temperatures near the service conditions (up to 3000 perature carbon fiber) in the temperature range 20-2000 C). Such information is required for C/C composite C. Strength increased first by about 40% from room performance predictions as well as for design with re- temperature to 1500oC and then it dropped at 2000C. pect to end use. Today, only a limited amount of work Youngs modulus temperature dependence was similar on the mechanical properties of carbon fibers at high to that one reported by Bacon and Smith [1]. Plastic temperatures is reported in the literature. deformations were noticed at 2000 oC but this feature Bacon and Smith [1] determined mechanical proper- was not discussed ties in the temperature range 20-1900C for a rayon Tanabe et al. [3] examined a pitch-based carbon fiber based carbon fiber (VYB105 and VYB70 from Union (HM 70 from Petoca Co. )in the temperature range 20- Carbide Corp ) heat treated at 1900C for 5 min. 1300C. Neither strength nor Youngs modulus signif- Strength data displayed a pronounced maximum at icant temperature dependence was observed in this temperatures around 1700-1800C and then a steep narrow temperature range. rop at higher temperatures. Youngs mo In all the above mentioned tests, temperature was creased slightly as temperature increased to 1500C and uniform over the entire gauge length(hot grip tech then it dropped substantially. A marked nonlinear nique). The tests were performed using original appa stress-strain response was observed at these latter tem- ratuses. However, a major shortcoming is that since only peratures. A large part of deformations was recovered one fiber was generally tested at each temperature, the may Corresponding author. Tel. +33-5-5684-4700: fax: +33-5-5684- more, the nonlinear behavior at high temperatures was 1225 not investigated nor discussed with respect to structural E-inail address: lamon(alcts. ul-bordeaux fr( J. Lamon) features 0008-6223/S- see front matter 2004 Elsevier Ltd. All rights reserved doi:10.10l6 carbon2003.11020
The tensile behavior of carbon fibers at high temperatures up to 2400 C Cedric Sauder, Jacques Lamon *, Rene Pailler Laboratoire des Composites Thermostructuraux, UMR 5801: CNRS––Snecma––CEA––UB1, Domaine Universitaire, 3, Allee de la Boetie, 33600 Pessac, France Received 17 June 2003; accepted 20 November 2003 Abstract The tensile behavior of four different brands of carbon fibers (a rayon-based, a PAN-based, and 2 pitch-based fibers) has been investigated at various temperatures up to 2400 C. The tests were carried out using an original fiber testing apparatus. Various mechanical properties including strength and Young’s modulus, as well as Weibull statistical parameters were extracted from test data. Typical tensile behaviors were evidenced such as an essentially linear elastic behavior at room temperature and intermediate temperatures up to 1400–1800 C, then a nonlinear elastic delayed response at higher temperatures and ultimately an inelastic response with permanent deformations at very high temperatures. Such unusual nonlinear responses for homogeneous materials were related to structure and texture features at the nanometer scale, that were described through an X-ray diffraction technique. 2004 Elsevier Ltd. All rights reserved. Keywords: A. Carbon fibers, Carbon/carbon composites; D. Mechanical properties, Texture 1. Introduction One of the main difficulties with C/C composites, is the lack of data on their constituent properties at high temperatures near the service conditions (up to 3000 C). Such information is required for C/C composite performance predictions as well as for design with respect to end use. Today, only a limited amount of work on the mechanical properties of carbon fibers at high temperatures is reported in the literature. Bacon and Smith [1] determined mechanical properties in the temperature range 20–1900 C for a rayonbased carbon fiber (VYB105 and VYB70 from Union Carbide Corp.), heat treated at 1900 C for 5 min. Strength data displayed a pronounced maximum at temperatures around 1700–1800 C and then a steep drop at higher temperatures. Young’s modulus decreased slightly as temperature increased to 1500 C and then it dropped substantially. A marked nonlinear stress–strain response was observed at these latter temperatures. A large part of deformations was recovered after a few minutes under zero load. This strain recovery was attributed to the reverse rotation of graphitic layers. Mostovoi et al. [2] measured the mechanical properties of a PAN-based carbon fiber (VMN-RK high temperature carbon fiber) in the temperature range 20–2000 C. Strength increased first by about 40% from room temperature to 1500 C and then it dropped at 2000 C. Young’s modulus temperature dependence was similar to that one reported by Bacon and Smith [1]. Plastic deformations were noticed at 2000 C but this feature was not discussed. Tanabe et al. [3] examined a pitch-based carbon fiber (HM 70 from Petoca Co.) in the temperature range 20– 1300 C. Neither strength nor Young’s modulus significant temperature dependence was observed in this narrow temperature range. In all the above mentioned tests, temperature was uniform over the entire gauge length (hot grip technique). The tests were performed using original apparatuses. However, a major shortcoming is that since only one fiber was generally tested at each temperature, the sample size may not be statistically sufficient. Furthermore, the nonlinear behavior at high temperatures was not investigated nor discussed with respect to structural features. * Corresponding author. Tel.: +33-5-5684-4700; fax: +33-5-5684- 1225. E-mail address: lamon@lcts.u-bordeaux.fr (J. Lamon). 0008-6223/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2003.11.020 Carbon 42 (2004) 715–725 www.elsevier.com/locate/carbon
C. Sauder et al. / Carbon 42(2004)715-725 Carbon fibers can be produced essentially from three the P100 fiber which was received as-treated(the fiber is recursor materials: rayon, polyacrylonitrile(PAN) and heat-treated at a temperature above 2400c by the pitch(isotropic or anisotropic). Their properties depend supplier) all the fibers were heat treated for 2 h at a mainly on the precursor material, but they can be im- temperature above 2000C so that they remain stable proved through high temperature treatments, possibly during the tensile tests. under load. In the past, the structure of different carbon The characteristics that are summarized in Table I fibers has been studied extensively and different struc- indicate that a wide spectrum of fibers was covered tural models were proposed [4-12]. It was observed that Except for the P100 fiber, these characteristics were most carbon fibers consist of two-dimensional graphite measured in the laboratory. The technique described in crystals oriented preferentially along the fiber axis. The the next section was used. Density was measured using relationship between the structure and the mechanical an helium pycnometer. Fiber diameters were estimated properties of both PAN-and pitch-based carbon fibers from scanning electron micrographs of fiber cross sec- has been studied in detail [10]. Models for the inter- tions and also using in situ laser diffractometry(the laser pretation of the relationship between Youngs modulus was mounted on the testing machine and the angular distribution of layer normals have been proposed [12, 13]. Equations combine the elastic con- stants of individual particles with definite integrals of 2.2. Tensile tests he angular distribution of layer normals. They can b used for the interpretation of fiber stiffening observed The fibers were taken from tows. Graphite grips were at room temperature. This effect is attributed to a ffixed to sample ends using a carbon based cement gradual straightening of the crystalline regions along (gauge length 50 mm). Tensile tests were performed the loading direction. This interpretation is consistent using an original device described in details elsewhere with the correlation between preferred orientation and [15]. An electric current is applied to the test specimen Youngs modulus for a variety of rayon-based carbo under secondary vacuum(residual pressure <10-3 Pa), so that temperatures as high as 3000C can be generated The dual intent of this paper is, therefore, to deter- uniformly along the gauge length. Computatio mine the mechanical properties and to investigate the tensile behavior of various carbon fibers with respect to conductivity showed that the temperature gradient temperature and to fiber structure and texture. less than 2.5oC between the surface and the heart of fibers, at 2000C. Temperatures at the surface of fibe were measured using a bichromatic pyrometer. Tem- perature profiles were uniform over a domain longer 2. Materials and structural characterizations than 95% of gauge length. It thus appeared that the grips remained at room temperature during the tests. 2.1. Description of materials Then, the loading frame compliance was not affected. The displacement of grips was measured by a sensor Four of carbon fibers were investigated: whose sensitivity was less than 0 I um. All the tests were XNO5. P P100 and FC2 fibers. The PANEX performed in the following temperature ranges: 1000- 33 fiber ZOLTEK Corp )is produced from a 2000C for the XNO5, PANEX 33 et FC2 fibers and PAN precursor, the XN05 fiber(from Nippon Graphite 1000-2400C for the P100 fiber. Most of the tests were Fiber Co. )from an isotropic pitch precursor, the P100 carried out at a slow displacement rate(0.5 mm/min) fiber(from AMOCO Performance Products Co Then. the influence of strain rate was examined merly Union Carbide Co. )from an anisotropic the following displacement rates: 0.05 and 5 mm/min precursor and the FC2 fiber(from SNECMA PRO- Strains were derived from grip displacement taking inte PULSION SOLIDE) from a rayon precursor. Except account deformations of the loading frame. The system Table I Main characteristics of as-received fibers at room temperature PANEX 33 Precursors Anisotropic pitch Isotropic pitch D s modulus(GPa) hs in MPa (Lo 50 mm) 2100 1.37 l81 2.15 iameter(um)
Carbon fibers can be produced essentially from three precursor materials: rayon, polyacrylonitrile (PAN) and pitch (isotropic or anisotropic). Their properties depend mainly on the precursor material, but they can be improved through high temperature treatments, possibly under load. In the past, the structure of different carbon fibers has been studied extensively and different structural models were proposed [4–12]. It was observed that most carbon fibers consist of two-dimensional graphite crystals oriented preferentially along the fiber axis. The relationship between the structure and the mechanical properties of both PAN- and pitch-based carbon fibers has been studied in detail [10]. Models for the interpretation of the relationship between Young’s modulus and the angular distribution of layer normals have been proposed [12,13]. Equations combine the elastic constants of individual particles with definite integrals of the angular distribution of layer normals. They can be used for the interpretation of fiber stiffening observed at room temperature. This effect is attributed to a gradual straightening of the crystalline regions along the loading direction. This interpretation is consistent with the correlation between preferred orientation and Young’s modulus for a variety of rayon-based carbon fibers [14]. The dual intent of this paper is, therefore, to determine the mechanical properties and to investigate the tensile behavior of various carbon fibers with respect to temperature and to fiber structure and texture. 2. Materials and structural characterizations 2.1. Description of materials Four brands of carbon fibers were investigated: XNO5, PANEX 33, P100 and FC2 fibers. The PANEX 33 fiber (from ZOLTEK Corp.) is produced from a PAN precursor, the XN05 fiber (from Nippon Graphite Fiber Co.) from an isotropic pitch precursor, the P100 fiber (from AMOCO Performance Products Co., formerly Union Carbide Co.) from an anisotropic pitch precursor and the FC2 fiber (from SNECMA PROPULSION SOLIDE) from a rayon precursor. Except the P100 fiber which was received as-treated (the fiber is heat-treated at a temperature above 2400 C by the supplier) all the fibers were heat treated for 2 h at a temperature above 2000 C so that they remain stable during the tensile tests. The characteristics that are summarized in Table 1, indicate that a wide spectrum of fibers was covered. Except for the P100 fiber, these characteristics were measured in the laboratory. The technique described in the next section was used. Density was measured using an helium pycnometer. Fiber diameters were estimated from scanning electron micrographs of fiber cross sections and also using in situ laser diffractometry (the laser was mounted on the testing machine). 2.2. Tensile tests The fibers were taken from tows. Graphite grips were affixed to sample ends using a carbon based cement (gauge length 50 mm). Tensile tests were performed using an original device described in details elsewhere [15]. An electric current is applied to the test specimen under secondary vacuum (residual pressure < 103 Pa), so that temperatures as high as 3000 C can be generated uniformly along the gauge length. Computations of temperature distributions for various values of thermal conductivity showed that the temperature gradient is less than 2.5 C between the surface and the heart of fibers, at 2000 C. Temperatures at the surface of fibers were measured using a bichromatic pyrometer. Temperature profiles were uniform over a domain longer than 95% of gauge length. It thus appeared that the grips remained at room temperature during the tests. Then, the loading frame compliance was not affected. The displacement of grips was measured by a sensor whose sensitivity was less than 0.1 lm. All the tests were performed in the following temperature ranges: 1000– 2000 C for the XNO5, PANEX 33 et FC2 fibers and 1000–2400 C for the P100 fiber. Most of the tests were carried out at a slow displacement rate (0.5 mm/min). Then, the influence of strain rate was examined at the following displacement rates: 0.05 and 5 mm/min. Strains were derived from grip displacement taking into account deformations of the loading frame. The system Table 1 Main characteristics of as-received fibers at room temperature Fibers FC2 PANEX 33 P100 XN05 Precursors Rayon PAN Anisotropic pitch Isotropic pitch Young’s modulus (GPa) 33 300a 690a 53 Strengths in MPa (L0 ¼ 50 mm) 720 2110 2100a;b 980 Density 1.37 1.81 2.15a 1.63 Mean diameter (lm) 6.5 7 11a 10 aFrom supplier. b L0 ¼ 25 mm. 716 C. Sauder et al. / Carbon 42 (2004) 715–725
C. Sauder et al. Carbon 42(2004)715-725 (d) 8=9041 zHz=u>H 中( degrees) Fig. 1 Schematic diagram of crystallite orientation with diagram of some X-ray diffraction o scan profiles 1(o): (a)E=210 GPa, (b)E=350 GPa, (c)E=490GPa,(d)E=690GPa[19 compliance was estimated at room temperature using the mean distance between two successive layers. La, Le, the conventional calibration technique [15, 16 and dooz were determined using the Scherrer equation The stress-strain behavior at high temperatures was from the positions of the diffraction maxima and the determined on batches of five test specimens. Statistical width at half-maximum intensity of the(002)and distributions of strength data were determined on bat 10)peaks. The microvoid content in the fibers was ches of 25 test specimens, for the FC2, PANEX 33 and derived from the structural parameters using the fol- XN05 fibers at room temperature, and for the FC2 and lowing equatio PANEX 33 fibers at high temperatures. The well-known Weibull,s model was used for the P=l description of the statistical distributions of failure strengths. Failure probability under a uniform stress where Pr is the fiber density, Pg is the graphite density state is given by (2.26 g/cm)and dg=0.3354 nm for graphite crystal P=1-exp(v/Vo)(o/oo)" The orientation distribution of graphitic planes was where m is the shape parameter (often referred to as determined by X-ray analysis of the fibers, as described Weibull modulus), Go is the scale factor, v is the fiber by Ruland [19]. It was obtained from an azimuthal volume under stresses, Vo is a reference volume (o=1 spread of the 002 reflection arc, which provides mm ), o is the applied stress. indication of the preferred orientation of the basic Probabilities of failure were determined using ranking structural units relative to fiber axis as illustrated on Fig statistics[17]. Ordering the failure data from smallest to 1. The orientation parameter Z is the full width at half largest and assigning a ranking number i, the probabil- the maximum intensity obtained in the azimuthal scan ities of failure were then assigned by the following measured in degrees. Fiber texture was described using relationship the distribution of the intensity of scattering at angle 0.5)/N, p(1(o)), and using the orientation parameter Z where N is the total number of specim 3. Results The statistical parameters were then extracted from strength distributions using the conventional Weibull 3.1. Stress-strain behavior under a slow strain rate linear regression estimator [17] The tensile stress-strain curves exhibit temperature 23. Structural characterization of fibers dependent features(Fig. 2) The common parameters used in the description of The stress-strain relation is essentially linear carbon structures Le, La, and dooz were determined using room temt perature to temperatures as high as X-ray diffraction on ground-up samples of the fibers. La and Le are the dimensions of crystallites (La is the A more and more pronounced nonlinear deformation longitudinal whereas Le is the lateral extension). door is is then observed at higher temperatures
compliance was estimated at room temperature using the conventional calibration technique [15,16]. The stress–strain behavior at high temperatures was determined on batches of five test specimens. Statistical distributions of strength data were determined on batches of 25 test specimens, for the FC2, PANEX 33 and XN05 fibers at room temperature, and for the FC2 and PANEX 33 fibers at high temperatures. The well-known Weibull’s model was used for the description of the statistical distributions of failure strengths. Failure probability under a uniform stress state is given by P ¼ 1 exp½ðV =V0Þðr=r0Þ m; ð1Þ where m is the shape parameter (often referred to as Weibull modulus), r0 is the scale factor, V is the fiber volume under stresses, V0 is a reference volume (V0 ¼ 1 mm3), r is the applied stress. Probabilities of failure were determined using ranking statistics [17]. Ordering the failure data from smallest to largest and assigning a ranking number i, the probabilities of failure were then assigned by the following relationship: Pi ¼ ði 0:5Þ=N; ð2Þ where N is the total number of specimens. The statistical parameters were then extracted from strength distributions using the conventional Weibull linear regression estimator [17]. 2.3. Structural characterization of fibers The common parameters used in the description of carbon structures Lc, La, and d002 were determined using X-ray diffraction on ground-up samples of the fibers. La and Lc are the dimensions of crystallites (La is the longitudinal whereas Lc is the lateral extension). d002 is the mean distance between two successive layers. La, Lc, and d002 were determined using the Scherrer equation from the positions of the diffraction maxima and the width at half-maximum intensity of the (0 0 2) and (1 1 0) peaks. The microvoid content in the fibers was derived from the structural parameters using the following equation: Vp ¼ 1 qfd002 qgdg ; ð3Þ where qf is the fiber density, qg is the graphite density (¼ 2:26 g/cm3) and dg ¼ 0:3354 nm for graphite crystals [18]. The orientation distribution of graphitic planes was determined by X-ray analysis of the fibers, as described by Ruland [19]. It was obtained from an azimuthal spread of the 002 reflection arc, which provides an indication of the preferred orientation of the basic structural units relative to fiber axis as illustrated on Fig. 1. The orientation parameter Z is the full width at half the maximum intensity obtained in the azimuthal scan, measured in degrees. Fiber texture was described using the distribution of the intensity of scattering at angle /ðIð/ÞÞ, and using the orientation parameter Z. 3. Results 3.1. Stress–strain behavior under a slow strain rate The tensile stress–strain curves exhibit temperature dependent features (Fig. 2): • The stress–strain relation is essentially linear from room temperature to temperatures as high as 1200 C. • A more and more pronounced nonlinear deformation is then observed at higher temperatures. Fig. 1. Schematic diagram of crystallite orientation with diagram of some X-ray diffraction / scan profiles Ið/Þ: (a) E ¼ 210 GPa, (b) E ¼ 350 GPa, (c) E ¼ 490 GPa, (d) E ¼ 690 GPa [19]. C. Sauder et al. / Carbon 42 (2004) 715–725 717
71 C. Sauder et al. Carbon 42(2004)715-72 (a)FC2 (b)XN05 1000°c 1400°C 1800°c 三 0 (c) PANEX 33 (d)P100 2400 2000 2500 1600 1200 1500 1600°c 1800°C-20 04 Strain E(%) Strain E(s) Fig. 2. Evolution of stress-strain curves obtained at various temperatures for all the fibers of this study: (a)FC2,(b)xN05, (c) PANEX 33 and (d)P100. The associated strains are also larger and larger The transition from linear to nonlinear deformation occurs at temperatures depending on tested fiber 1200-1400 oC for the Fc2 fibers. 1000-1200 oC for the xnos fibers 1600-1800 oC for the panex 33 -e- FC2 fiber fibers and 2000-2200 oC for the p100 fibers xNo5 fiber r - PANEX 33 fiber It is also worth mentioning that a slight increase in the fiber Youngs modulus was observed on the PANEX 33 fibers at room temperature. The data were fitted satisfactorily by the empirical equation a= Eo(1+fa)E, 0.4 4008001200160020002400 where Eo is the initial tangent modulus Table 1),f is a measure of the degree of nonlinearity (f= 23) Fig 3. Dependence of relative initial Youngs modulus E(T)/Eo The influence of temperature on Youngs modulus is (Eo=E(4C)on temperature shown in Fig 3. Young's modulus decreases gently in a first step, and then rather steeply at temperatures above ≈1000° for the Fo2 fiber and≈1200° for the other The failure data determined at various temperatures fibers. This trend is similar to that one reported in the e summarized in Table 2. Fig. 4 shows that the literature [1-3] strength of FC2 fibers increases first slightly as temper
• The associated strains are also larger and larger. • The transition from linear to nonlinear deformation occurs at temperatures depending on tested fiber: 1200–1400 C for the FC2 fibers, 1000–1200 C for the XN05 fibers, 1600–1800 C for the PANEX 33 fibers and 2000–2200 C for the P100 fibers. It is also worth mentioning that a slight increase in the fiber Young’s modulus was observed on the PANEX 33 fibers at room temperature. The data were fitted satisfactorily by the empirical equation r ¼ E0ð1 þ f eÞe; ð4Þ where E0 is the initial tangent modulus (Table 1), f is a measure of the degree of nonlinearity (f ¼ 23). The influence of temperature on Young’s modulus is shown in Fig. 3. Young’s modulus decreases gently in a first step, and then rather steeply at temperatures above 1000 C for the FC2 fiber and 1200 C for the other fibers. This trend is similar to that one reported in the literature [1–3]. The failure data determined at various temperatures are summarized in Table 2. Fig. 4 shows that the strength of FC2 fibers increases first slightly as temper- 0 200 400 600 800 1000 1200 0123456 Strain ε (%) Stress (MPa) 24°C 1000°C 1200°C 1400°C 1600°C 1800°C 2000°C 0 150 300 450 600 750 0 1.5 3 4.5 6 7.5 9 Strain ε (%) Stress (MPa) 24°C 1000°C 1200°C 1400°C 1600°C 1800°C 2000°C 0 500 1000 1500 2000 2500 3000 3500 01234 Strain ε (%) Stress (MPa) 24°C 1000°C 1200°C 1400°C 1600°C 1800°C 2000°C 0 400 800 1200 1600 2000 2400 0 0.2 0.4 0.6 0.8 Strain ε (%) Stress (MPa) 24°C 1600°C 1800°C 2000°C 2200°C 2400°C (a) FC2 (c) PANEX 33 (b) XN05 (d) P100 Fig. 2. Evolution of stress–strain curves obtained at various temperatures for all the fibers of this study: (a) FC2, (b) XN05, (c) PANEX 33 and (d) P100. 0.4 0.5 0.6 0.7 0.8 0.9 1 0 400 800 1200 1600 2000 2400 Temperature (°C) E/Eo FC2 fiber XN05 fiber PANEX 33 fiber P100 fiber Fig. 3. Dependence of relative initial Young’s modulus EðT Þ=E0 (E0 ¼ E (24 C)) on temperature. 718 C. Sauder et al. / Carbon 42 (2004) 715–725
C. Sauder et al. Carbon 42(2004)715-725 Table 2 Influence of tture on strength data for paneX 33 and Fc2 fibers Temperature(C) PANEX 33 fiber FC2 fiber Go(MPa) GR(MPa) MPa) 2107(405) 0.71(0.13) 719(157) 2.16(045) 2292(410)0.80(0.14) 789(103) 2.39(0.31) 0.86(0.18)5 756(118)232(0.38) 2386(507)0.85(0.18) 825(116)2.680.36) 2359(520) 0.91(0.23) 82l(114) 5.4l(1.75) 79b 857(6l1) .53(0.44) 605(72) 12(6.7 ND ND 2348(443)3.71(1.84)59° 850 127(6.1) ND: nondetermined, O standard error b Result to take with care because of the nonlinear elastic tensile curve to the above mentioned transition temperatures, iden PANEX 33 fiber tified at a slow strain rate Figs. 7 and 8 show the presence of residual defor- 62500 mations at zero load. It can be noticed from Fig. 7 that the residual strains were relieved after 30 min under zero 2000 load, at the temperatures of 1600C for the FC2 fiber, and 1800C for the PanEX 33 fiber. This complete recove indicative of a delayed elastic 1000 coelastic type behavior. At higher temperatures, only a partial recovery is observed(Fig. 8), indicating the presence of permanent deformations. This feature is typical of a viscoplastic behavior. 4008001200160020002400 For the XNO5 fiber, nonlinear elasticity appeared at 400C, and permanent deformations were detected at Fig. 4. Dependence of average failure stress for FC2 and PANEX 33 temperatures above 1800C. For the P100 fiber, non linear elasticity was observed above 1800C, and per manent residual deformations were not detected in the range of temperatures that was examined (<2400C) ature increases, and then drops at temperatures above It is worth mentioning that the elastic-inelastic transition temperatures mentioned above for the FC2 1600C. For the PANEX 33 fibers, the strength increase and the paneX 33 fibers coincide to the temperatures is more significant. A peak is reached at 1800C. It can at which a strength drop was observed( Fig 4) be noticed from Table 2 that the strain-to-failure in- creases substantially. The statistical parameters do not In summary, all the carbon fibers examined in the exhibit a significant dependence on temperature, indi present paper exhibited three different stress-strain cating that no significant change occurred in the popu behaviors depending upon temperature lations of fracture inducing flaws 4 The above trend in strength seems to be at variance a purely elastic behavior at low temperatures h that observed on most materials which a viscoelastic behavior(delayed elasticity) at interme- generally a fracture stress decrease as temperature in diate temperatures, creases, and steep strength drops at fragile-ductile a viscoplastic behavior (inelastic deformations)at high temperatures. transition The transition temperatures are reported in Table 3. 3. 2. Stress-strain behavior under increasing strain rates It can be noticed that they depend upon fiber. Figs. 5 and 6 show the influence of loading rate on the 3.3. Description of the structure and the texture of fibers tress-strain behavior at various temperatures. It can be noticed that nonlinearity is enhanced by decreasing strain rates above 1400 oC for the Fc2 fiber. and above e The parameters describing the structure and the tex- ire of fibers are summarized in Table 4. The fibers are 1600C for the PANEX 33 fiber. Below these temper- put according to increasing Young's modulus It can be atures, the behavior is essentially linear elastic and noticed that the respective parameters vary in a logical insensitive to strain rate. These temperatures correspond way with respect to fibers stiffness
ature increases, and then drops at temperatures above 1600 C. For the PANEX 33 fibers, the strength increase is more significant. A peak is reached at 1800 C. It can be noticed from Table 2 that the strain-to-failure increases substantially. The statistical parameters do not exhibit a significant dependence on temperature, indicating that no significant change occurred in the populations of fracture inducing flaws. The above trend in strength seems to be at variance with that observed on most materials, which show generally a fracture stress decrease as temperature increases, and steep strength drops at fragile–ductile transition. 3.2. Stress–strain behavior under increasing strain rates Figs. 5 and 6 show the influence of loading rate on the stress–strain behavior at various temperatures. It can be noticed that nonlinearity is enhanced by decreasing strain rates above 1400 C for the FC2 fiber, and above 1600 C for the PANEX 33 fiber. Below these temperatures, the behavior is essentially linear elastic and insensitive to strain rate. These temperatures correspond to the above mentioned transition temperatures, identified at a slow strain rate. Figs. 7 and 8 show the presence of residual deformations at zero load. It can be noticed from Fig. 7 that the residual strains were relieved after 30 min under zero load, at the temperatures of 1600 C for the FC2 fiber, and 1800 C for the PANEX 33 fiber. This complete recovery is indicative of a delayed elastic response: viscoelastic type behavior. At higher temperatures, only a partial recovery is observed (Fig. 8), indicating the presence of permanent deformations. This feature is typical of a viscoplastic behavior. For the XN05 fiber, nonlinear elasticity appeared at 1400 C, and permanent deformations were detected at temperatures above 1800 C. For the P100 fiber, nonlinear elasticity was observed above 1800 C, and permanent residual deformations were not detected in the range of temperatures that was examined ( 6 2400 C). It is worth mentioning that the elastic–inelastic transition temperatures mentioned above for the FC2 and the PANEX 33 fibers coincide to the temperatures at which a strength drop was observed (Fig. 4). In summary, all the carbon fibers examined in the present paper exhibited three different stress–strain behaviors depending upon temperature: • a purely elastic behavior at low temperatures, • a viscoelastic behavior (delayed elasticity) at intermediate temperatures, • a viscoplastic behavior (inelastic deformations) at high temperatures. The transition temperatures are reported in Table 3. It can be noticed that they depend upon fiber. 3.3. Description of the structure and the texture of fibers The parameters describing the structure and the texture of fibers are summarized in Table 4. The fibers are put according to increasing Young’s modulus. It can be noticed that the respective parameters vary in a logical way with respect to fibers stiffness: Table 2 Influence of test temperature on strength data for PANEX 33 and FC2 fibers Temperature (C) PANEX 33 fiber FC2 fiber rR (MPa) eR (%) m r0 a (MPa) rR (MPa) eR (%) m r0 a (MPa) 24 2107 (405) 0.71 (0.13) 5.1 660 719 (157) 2.16 (0.45) 5.1 230 1000 2292 (410) 0.80 (0.14) 6.5 950 789 (103) 2.39 (0.31) 8.3 390 1200 2401 (540) 0.86 (0.18) 5.2 890 756 (118) 2.32 (0.38) ND ND 1400 2386 (507) 0.85 (0.18) 5.4 810 825 (116) 2.68 (0.36) ND ND 1600 2359 (520) 0.91 (0.23) 5.4 800 821 (114) 5.41 (1.75) 7.9b 390 1800 2857 (611) 1.53 (0.44) 5.6b 1020 605 (72) 12 (6.7) ND ND 2000 2348 (443) 3.71 (1.84) 5.9b 850 371 (39) 12.7 (6.1) 10b 200 ND: nondetermined, () standard error. a V0 ¼ 1 mm3. b Result to take with care because of the nonlinear elastic tensile curve. 0 500 1000 1500 2000 2500 3000 3500 0 400 800 1200 1600 2000 2400 Temperature (°C) Average failure stress σ R (MPa) FC2 fiber PANEX 33 fiber Fig. 4. Dependence of average failure stress for FC2 and PANEX 33 on temperature. C. Sauder et al. / Carbon 42 (2004) 715–725 719
720 C. Sauder et al. Carbon 42(2004)715-72 (a)T=1400C b)T=1600°C 1000 0%/ min 800 600 --10%/min 00 400 0.1%/min Strain E(%) Strain E(% (c)T=1800C (d)T=2000C s300 -10%/min 至s苏 00 1%/min 0.1%/min 100 0.1%/min Strain E(%) Strain E(%) Fig. 5. Evolution of tensile curves of FC2 fiber for various strain rates (0. 1%o, I% and 10% per min) at high temperatures. · L and l. increase, fibers (Table 4), it seems logical to conjecture that the dooz decreases FC2 fiber possesses the smallest crystallites, and the · Z decreases highest degree of isotropy This trend indicates the presence of crystallites with an increasing size, separated by a shorter distance between two successive layers and with a more pronounced ori 4. Discussion entation in the fiber axis direction This reflects an increasing degree of anisotropy. It is illustrated by Fig In order to understand the main features of the 9. The distributions of intensity of scattering 1() show stress-strain behavior at high temperatures, the stress that fiber XN05 is the most isotropic, whereas fiber state induced by the anisotropic structure of fibers was P100, as reported in the literature, is the most aniso- determined using equations of the theory of elasticity for anisotropic solids Because of a weak response to X-ray neither La nor Let us consider a crystallite oriented at angle with the distribution in 1(o)could be determined for the FC2 respect to fiber axis as shown on Fig. 10. Deformation fiber. However, from the comparison of the available parallel to fiber axis(e)is related to deformations along arameters(Le, p, E) to the data pertinent to the other crystallites axes by the following equation
• Lc and La increase, • d002 decreases, • Z decreases. This trend indicates the presence of crystallites with an increasing size, separated by a shorter distance between two successive layers and with a more pronounced orientation in the fiber axis direction. This reflects an increasing degree of anisotropy. It is illustrated by Fig. 9. The distributions of intensity of scattering Ið/Þ show that fiber XN05 is the most isotropic, whereas fiber P100, as reported in the literature, is the most anisotropic. Because of a weak response to X-ray neither La nor the distribution in Ið/Þ could be determined for the FC2 fiber. However, from the comparison of the available parameters (Lc; Vp; E) to the data pertinent to the other fibers (Table 4), it seems logical to conjecture that the FC2 fiber possesses the smallest crystallites, and the highest degree of isotropy. 4. Discussion In order to understand the main features of the stress–strain behavior at high temperatures, the stress state induced by the anisotropic structure of fibers was determined using equations of the theory of elasticity for anisotropic solids. Let us consider a crystallite oriented at angle / with respect to fiber axis as shown on Fig. 10. Deformation parallel to fiber axis (e) is related to deformations along crystallites axes by the following equation: (a) T=1400˚C (b) T=1600˚C (c) T=1800˚C (d) T=2000˚C 0 200 400 600 800 1000 0123 Stress (MPa) 0 200 400 600 800 1000 02468 Strain ε (%) Strain ε (%) Strain ε (%) Strain ε (%) Stress (MPa) 0 100 200 300 400 500 600 02468 Stress (MPa) 0 100 200 300 400 500 0 2.5 5 7.5 10 Stress (MPa) 10% / min 1% /min 0.1% /min 10% / min 1% /min 0.1% /min 10% / min 1% /min 0.1% /min 10% / min 1% /min 0.1% /min Fig. 5. Evolution of tensile curves of FC2 fiber for various strain rates (0.1%, 1% and 10% per min) at high temperatures. 720 C. Sauder et al. / Carbon 42 (2004) 715–725
(a)T=1400°C b)T=1600°C 3600 10%/min 10%e/min 1%/min 0.1%/min 0.1%0/min I800 三1800 600 0 0.4 12 Strain(%) (c)T=1800°C (d)T=2000°C 3000 3000 是1800 200 1200 10%/ min 600 0.1%/min 0.5 6. Evolution of tensile curves of PANEX 33 fiber for various strain rates(0.1%,1% and l0% per min) at high temperatures. a)T=1600°C b)T=1800C ress strain oops En吧 Fig. 7. Recovery under zero load after unloading for FC2 fiber at (a)1600C and(b)1800C. E=axcs2φ+ Ey sin2φ+2 Eyy cosφsinp (5)ey= Er, Ex and Exr are deformations along crystallite axes (Fig.10) (6) Ox, ar and txy are the local, normal and shear stress components(Fig. 10). Ex is the modulus in the direction
e ¼ eX cos2 / þ eY sin2 / þ 2eXY cos / sin /: ð5Þ eX , eX and eXY are deformations along crystallite axes (Fig. 10): eX ¼ rX EX m rY EY ; ð6Þ eY ¼ rY EY m rX EX ; ð7Þ eXY ¼ sXY 2GXY : ð8Þ rX , rY and sXY are the local, normal and shear stress components (Fig. 10). EX is the modulus in the direction (a) T=1400˚C (b) T=1600˚C 0 600 1200 1800 2400 0 0.2 0.4 0.6 0 600 1200 1800 2400 3000 3600 0 0.4 0.8 1.2 Stress (MPa) (c) T=1800˚C (d) T=2000˚C 0 600 1200 1800 2400 3000 0 0.5 1 Strain (%) Strain (%) Strain (%) Strain (%) Stress (MPa) Stress (MPa) 1% / min 1.5 0 600 1200 1800 2400 3000 0 12 Stress (MPa) 10% / min 0.1% / min 1% / min 1% / min 10% / min 0.1% / min 1% / min 10% / min 0.1% / min 1% / min 10% / min 0.1% / min 3 Fig. 6. Evolution of tensile curves of PANEX 33 fiber for various strain rates (0.1%, 1% and 10% per min) at high temperatures. 0 100 200 300 400 500 600 01234 Strain (%) Stress strain loops Stress strain loop after 10 min without load Tensile curve after 30 min without load 0 100 200 300 400 500 600 700 800 0 0.5 1 1.5 2 2.5 3 3.5 Strain (%) Stress strain loops Tensile curve after 30 min without load Stress (MPa) Stress (MPa) (a) T= 1600˚C (b) T= 1800˚C Fig. 7. Recovery under zero load after unloading for FC2 fiber at (a) 1600 C and (b) 1800 C. C. Sauder et al. / Carbon 42 (2004) 715–725 721
C. Sauder et al. Carbon 42(2004)715-72 (a)T=1800C b)T=2000C 2800 c二 2000 00.2040.608 Strain(%) Fig. 8. Recovery under zero load after unloading for PANEX 33 fiber at(a) 1800C and(b)2000 Typical stress-strain behaviors and transition temperatures observed with the fibers tested FC2(°C) XNo5(°C) P00(° Linear elastic deformation 4-1200 24-1400 24-1200 24-1800 Viscoelastic behavior 200-1600 l400-1800 1200-1800 >1800 Table 4 Parameters describing the structure of fibers d oo(n Le (nm) La(nm) z(°) 0.356 35.8 PANEX 33 0.343 183 0.339 Weak response to hape of yarns( twisted) not compatible with test normal to the c-axis, Ey is the modulus parallel to thi (10) axis, Vxy the Poissons' ratio. Gxy is the modulus for shear between the planes oriented normal to the c-axis. x=cosφsinφ G cos Introducing Eqs. (6(11)into(5)leads to the following equation for Youngs modulus of the volume element considered above XNO5 PANEX 33 sin2φ(sin2φ-vcos2) Rearranging Eq(13)gives E(中 s4中 φ Fig. 9. Distribution of intensity of scattering /(o)for the carbon fibers +sin2pcos2φ indicating the degree of E
normal to the c-axis, EY is the modulus parallel to this axis, mXY the Poissons’ ratio. GXY is the modulus for shear between the planes oriented normal to the c-axis. rX ¼ r cos2 /; ð9Þ rY ¼ r sin2 /; ð10Þ sXY ¼ r cos / sin /; ð11Þ m ¼ mXY : ð12Þ Introducing Eqs. (6)–(11) into (5) leads to the following equation for Young’s modulus of the volume element considered above: 1 Eð/Þ ¼ 1 EX cos2 /ðcos2 / m sin2 /Þ þ 1 EY sin2 /ðsin2 / m cos2 /Þ þ 1 GXY 1 2 sin 2/: ð13Þ Rearranging Eq. (13) gives 1 Eð/Þ ¼ 1 EX cos4 / þ 1 EY sin4 / þ sin2 / cos2 / 1 GXY m EX m EY : ð14Þ 0 400 800 1200 1600 2000 2400 0 0.5 1 1.5 2 2.5 Strain (%) Strass strain loops Stress strain loop after 30min without load Tensile curve after 20 min without load 0 400 800 1200 1600 2000 2400 2800 0 0.2 0.4 0.6 0.8 1 1.2 Strain (%) Stress strain loops Stress strain loops after 30 min without load (a) T= 1800˚C (b) T= 2000˚C Stress (MPa) Stress (MPa) Fig. 8. Recovery under zero load after unloading for PANEX 33 fiber at (a) 1800 C and (b) 2000 C. Table 3 Typical stress–strain behaviors and transition temperatures observed with the fibers tested FC2 (C) PANEX 33 (C) XN05 (C) P100 (C) Linear elastic deformation 24–1200 24–1400 24–1200 24–1800 Viscoelastic behavior 1200–1600 1400–1800 1200–1800 >1800 Viscoplastic behavior >1600 >1800 >1800 >2400 Table 4 Parameters describing the structure of fibers d002 (nm) Lc (nm) La (nm) Z () Vp (%) E FC2 0.356 1.4 –a –b 35.8 33 XN05 0.345 3.5 6.4 30 25.7 53 PANEX 33 0.343 4.2 7 19 18.3 300 P100 0.339 10.6 26 6.4 4.1 690 aWeak response to X-ray. b Shape of yarns (twisted) not compatible with test. 0 0.2 0.4 0.6 0.8 1 -60 -40 -20 0 20 40 60 Φ (Degrees) Relative intensity (I/Io) XNO5 PANEX 33 P100 Fig. 9. Distribution of intensity of scattering Ið/Þ for the carbon fibers indicating the degree of anisotropy. 722 C. Sauder et al. / Carbon 42 (2004) 715–725
C. Sauder et al. Carbon 42(2004)715-725 Txx: shears tress Ox =osin o Exy =o coso sin o Er fiber axis Fig. 10. Schematic diagram showing two neighbour layers(in the same BSU)in a carbon fiber and the stress state induced by a stress parallel to fiber It is interesting to note that this equation is different Table 5 from those proposed by a few authors(see for instance the following properties: [llD, but it can be compared to that one derived by Er= 1000 GPa, Ey =36 GPa [20] and Gx <5 GPa [lI] Kelly from energy considerations for single crystal Fiber Youngs modulus graphite [12]: the coefficients take identical values Prediction(GPa) Experiments(GPa) The fiber Youngs modulus can then be derived from XNO5 Eq(13)or(14), using the following mixture law rela PANEX P100 U(φ)E(φ)dφ小, (15) were observed at high temperatures cannot be attributed ges in crystallites or where u(o) is the volume fraction of crystallites with Basal plane spacing doo2 probably expands at high angleφ mperatures. Data on door at high temperatures are not Prediction of Young's modulus requires volume available for carbon fibers. dooz dependence on temper- fraction u(o). D()was estimated from the distributions ature is described by the following equation for graphite of the intensity of scattering at angle (I())(Fig. 9) [21] which was derived from a polynominal description of thermal expansion. This equation was shown to be predicted by equations. However, the discrepancy that independent of the graphite considered [22]. It may be can be noticed results from uncertainty in the estimation of u(o). The determination of a closed form description strated. oft(φ) from(φ中) is not straightforward, and it would (A)=d2(24)+91910-67+5310-°72.(16) fall out of the scope of this paper. It will be treated in a forthcoming paper Increases in dooz should induce a decrease in elastic I(o) was not determined at high temperatures. moduli Er and Gxr, then a decrease in E()according to However, on the basis of the known trends in the Eq (13), and finally a decrease in fiber Youngs modulus influence of heat treatments on the texture of carbon according to Eq. (15). This effect is in agreement with fibers, it may be thought that the angle distributions the observed dependence of fiber Youngs modulus on are not affected but instead that they could becom temperature. narrower at high temperatures. Thus, it seems logical Eqs.(5),(13)(15)predict a linear elastic behavior, or think that the fiber Youngs modulus decreases that a slight modulus increase if chain straightening takes
It is interesting to note that this equation is different from those proposed by a few authors (see for instance [11]), but it can be compared to that one derived by Kelly from energy considerations for single crystal graphite [12]: the coefficients take identical values. The fiber Young’s modulus can then be derived from Eq. (13) or (14), using the following mixture law relation: E ¼ Z p 2 p 2 vð/ÞEð/Þd/; ð15Þ where vð/Þ is the volume fraction of crystallites with angle /. Prediction of Young’s modulus requires volume fraction vð/Þ. vð/Þ was estimated from the distributions of the intensity of scattering at angle /ðIð/ÞÞ (Fig. 9). Table 5 shows that Young’s moduli were satisfactorily predicted by equations. However, the discrepancy that can be noticed results from uncertainty in the estimation of vð/Þ. The determination of a closed form description of vð/Þ from Ið/Þ is not straightforward, and it would fall out of the scope of this paper. It will be treated in a forthcoming paper. Ið/Þ was not determined at high temperatures. However, on the basis of the known trends in the influence of heat treatments on the texture of carbon fibers, it may be thought that the / angle distributions are not affected but instead that they could become narrower at high temperatures. Thus, it seems logical to think that the fiber Young’s modulus decreases that were observed at high temperatures cannot be attributed to changes in crystallites orientation. Basal plane spacing d002 probably expands at high temperatures. Data on d002 at high temperatures are not available for carbon fibers. d002 dependence on temperature is described by the following equation for graphite [21] which was derived from a polynominal description of thermal expansion. This equation was shown to be independent of the graphite considered [22]. It may be valid for carbon fibers, but this has not been demonstrated: d002 ðAÞ ¼ d002 ð24 CÞ þ 91:9106 T þ 5:3109 T 2 : ð16Þ Increases in d002 should induce a decrease in elastic moduli EY and GXY , then a decrease in Eð/Þ according to Eq. (13), and finally a decrease in fiber Young’s modulus according to Eq. (15). This effect is in agreement with the observed dependence of fiber Young’s modulus on temperature. Eqs. (5),(13)–(15) predict a linear elastic behavior, or a slight modulus increase if chain straightening takes y = a 0= d002 Y X b Fiber axis X = Y = σ.sin2 φ X σ σ σ σ σ σ σ σ σ σ σ σ σ σ Y y = τ τ τ XY : shears tress XY = σ.cosφ.sin φ Y Y X X X E E − − = X X Y Y Y E E = φ φ X Y x y 0 = XY = YX 2. XY XY GXY = ε ε ε υ υ υ υ υ σ.cos2φ Fig. 10. Schematic diagram showing two neighbour layers (in the same BSU) in a carbon fiber and the stress state induced by a stress parallel to fiber axis. Table 5 Young’s moduli predicted using Eq. (15), for the following properties: EX 1000 GPa, EY 36 GPa [20] and GXY < 5 GPa [11] Fiber Young’s modulus Prediction (GPa) Experiments (GPa) XN05 66 53 PANEX 315 300 P100 536 690 C. Sauder et al. / Carbon 42 (2004) 715–725 723
C. Sauder et al. Carbon 42(2004)715-72 place. The nonlinear deformation that was obtained above 1400-1800 C cannot be predicted by these τx(P00)t', cracking occurs. The behav. using the following equation derived from(11) lor is now inelastic as a result of the presence of cracks in t’= GEmIn(sinφcosφ), the planes oriented normal to the c-axis, and associated where min(sin o cos p)represents the minimum value of sliding friction. The following theoretical expression for sin o cos o. min(sin o cos o)=0.5 for a FC2 fiber r was proposed by Frenkel, on the basis of an atomistic whereas min(sin o cos )=0.4 for a PANEX 33fiber model [24] (Fig.9): r=Grb ≈500 MPa for PaNeX33(at2000°C) r'≈200 MPa for Fo2(at1800°C) where b is the distance between two atoms in the These latter t' estimates are consistent with the range of direction normal to the c-axis Eq. (17)shows that, when data obtained using Eq.(17). The analysis could be temperature increases, t' tends to decrease, since do refined with more accurate data for Gxy at high tem increases and Gxr decreases. As a consequence, the peratures. However, the use of Eq(21)seems to be elastic/inelastic transition appears to be favored by appropriate temperature increases. Eq. (I7)also indicates that t' should depend on the fiber structure through doo2. Thus, following ranking in t' can be proposed using Eq (17): 5.Conclusions The tensile stress-strain behavior and the main I(Pioo)>t (PANEX 33)>t'(XN05)>T"(FC2) mechanical properties of various carbon fibers were (18) measured at temperatures up to 2400C. The structure and the texture of fibers were described using various Then, txy increases with o according to Eq.(11). parameters determined through X-ray difiraction at Therefore higher txy values are expected in the isotropic room temperature. In particular, the orientation distri fibers for which a wider range of o values is covered. As butions of graphitic planes along the fiber axis provided a consequence, under a given applied stress o, the fol- also a pertinent description of the degree of fiber iso- ing can be established for t tropy/anisotropy
place. The nonlinear deformation that was obtained above 1400–1800 C cannot be predicted by these equations unless additional phenomena activated by temperature influence the strain components eY and eXY which contribute the major component of strain under a given fiber stress. There are many examples of materials manifesting a flattening of the stress–strain curve following the yield point, and a strain recovery (amorphous materials or textile fibers [23]). The recovery behavior following the release of stress is related to molecular mechanisms of deformation. The inelastic deformation is associated to shear strains that are produced as a switch of atoms occurs. Examination of Eqs. (5)–(15) suggests that the nonlinear deformation of carbon fibers at high temperatures is associated to increases in the shear stress component sXY , as a result of mechanisms activated by temperature. These mechanisms may be studied in terms of molecular deformations, as mentioned above. However, on the basis of stress-state considerations, the nonlinear deformation can be described as follows. Assuming that s0 XY ðT ; rÞ is the stress increment induced by the temperature activated mechanisms, the nonlinear deformation occurs under the action of sXY þ s0 XY ðT ; rÞ. When sXY þ s0 XY ðT ; rÞ s ðPANEX 33Þ > s ðXN05Þ > s ðFC2Þ: ð18Þ Then, sXY increases with / according to Eq. (11). Therefore higher sXY values are expected in the isotropic fibers for which a wider range of / values is covered. As a consequence, under a given applied stress r, the following ranking can be established for sXY : sXY ðP100Þ < sXY ðPANEX 33Þ < sXY ðXN05Þ < sXY ðFC2Þ: ð19Þ It can be concluded from (18) and (19) that the criterion for failure should be reached under the lowest temperature for the FC2 fiber, then under higher temperatures respectively for the XN05, for the PANEX 33 and fi- nally for the P100 fiber, ranking as follows: T ðFC2Þ < T ðXN02Þ < T ðPANEX 33Þ < T ðP100Þ: ð20Þ It is worth mentioning that this trend is in agreement with the temperatures reported in Table 3. It was pointed out previously that the transition temperatures coincide with the temperatures at which a strength decrease occurred for the FC2 and the PANEX 33 fibers. This may be attributed to a fiber embrittlement caused by the cracks created at the transition temperature. Numerous experimental values for the shear modulus have been reported. They are listed in [11]. In general, low values are found for pyrolitic carbon [25]. They are attributed to the effect of slip causing plastic creep [26] or the properties of glissile dislocations [27]. s was estimated from Eq. (17). s was obtained in the range 8– 400 MPa for 0:1 < GXY < 5 GPa [11] and b d002 ¼ 1 2 . For comparison purposes, s was also estimated from the elastic/inelastic transition stresses rE (Figs. 5 and 6), using the following equation derived from (11): s ¼ rE minðsin / cos /Þ; ð21Þ where minðsin / cos /Þ represents the minimum value of sin /cos /. minðsin / cos /Þ ¼ 0:5 for a FC2 fiber whereas minðsin / cos /Þ ¼ 0:4 for a PANEX 33 fiber since 0 < / < 30 (Fig. 9): s 500 MPa for PANEX 33 ðat 2000 CÞ; s 200 MPa for FC2 ðat 1800 CÞ: These latter s estimates are consistent with the range of data obtained using Eq. (17). The analysis could be refined with more accurate data for GXY at high temperatures. However, the use of Eq. (21) seems to be appropriate. 5. Conclusions The tensile stress–strain behavior and the main mechanical properties of various carbon fibers were measured at temperatures up to 2400 C. The structure and the texture of fibers were described using various parameters determined through X-ray diffraction at room temperature. In particular, the orientation distributions of graphitic planes along the fiber axis provided also a pertinent description of the degree of fiber isotropy/anisotropy. 724 C. Sauder et al. / Carbon 42 (2004) 715–725