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(a)Amount /10080604020ARTICLE IN PRESS+ MODEL42 G. Gouadec, Ph. Colomban / Progress in Crystal Growth and Characterization of Materialsxx (2007) 1e56IIIIII00 10 20 30 40 50 60 70IVVSiCCarbon(b)Width D / cm -1300200100DTO30252015Width TO / cm -1(c)04000 10 20 30 40 50 60 701040E / GPa300200100302010H / GPa00 10 20 30 40 50 60 70Position / µm0Fig. 24. (a) Raman detection of carbon (intensities of D, G and D 0 peaks are added) and SiC (intensities of TO and LOpeaks are added) as a function of the position along the fibre radius (l laser ¼ 632.8 nm). A 100% corresponds to themaximum intensity detected for each phase. (b) Bandwidths obtained after spectra fitting for D carbon peak(l laser ¼ 632.8 nm) and SiC TO modes (three separate scans; l ¼ 514.5 nm). (c) Young’s modulus and ‘‘Berkovich’shardness’’ (data from Mann et al [489]). Reprinted from Ref. [38, pp. 1249e1259] with permission from Elsevier.5.4.3. Young’s modulus and ‘‘Raman microextensometry’’At the macroscopic scale, fibres present the same tendency as their bonds to strengthen incompression and soften in tension [453]. Hooke’s law being nothing but the macroscopic manifestation of bond stiffness, one may indeed expect to find a relationship linking E to k b (introducedin Eq. (37)) via S 3 .The simplest expression modelling a ‘‘realistic’’ bond potential V(l b ) would be thefollowing:Vðl b Þ¼ k b2 ðl b l 0 Þ 2 þ k0 b6 ðl b l 0 Þ 3 ; k 0 < 0 ð54ÞPlease cite this article in press as: G. Gouadec, Ph. Colomban, Prog. Cryst. Growth Charact. Mater. (2007),doi:10.1016/j.pcrysgrow.2007.01.001
ARTICLE IN PRESS+ MODELG. Gouadec, Ph. Colomban / Progress in Crystal Growth and Characterization of Materialsxx (2007) 1e56ν D / cm -113571347NLMairredHi– air716661Linewidth / cm -14313371.2 1.7 2.2 2.7Strength / GPa56Annealed FibersCrystallineAs-receivedAmorphousFig. 25. Raman shift and linewidth of the carbon D mode (l ¼ 514.5 nm) plotted as a function of tensile strengths foundin Berger et al. [383] or Kumagawa et al. [370] for heat-treated SiC fibres. Reprinted from Ref. [38, pp. 1249e1259]with permission from Elsevier.With this form, the quasi-harmonic approximation (Eqs. (37) and (38)) leads to:sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinðl b Þf k b 1 þ k0 bðl b l 0 Þk bð55Þwhere k 0 b is expected to be much smaller than k so, as long as l b remains close to l 0 (small strainassumption), a Taylor expansion is justified:pnðl b Þwffiffiffiffi k 0bk b þ pffiffiffiffiðl b l 0 Þ ð56Þk bnðl b2 Þnðl b1 Þwp k0b ffiffiffiffi ðl b2 l b1 Þ ð57Þk bUpon comparison with Eq. (50), this is equivalent to:S 3 b ¼ k0 b l 0p100 ffiffiffiffi < 0 ð58Þk bHere again, we find that the energy shift for a strained bond can be expected to be proportionalto the bond deformation. Besides, in such isotropic structures as most nanophased fibresare, the ‘‘bond compression model’’ predicts a Young’s modulus in the following form [490]:E ¼ X bondsl 2 b k b9ð59ÞFor fibres with a single (or dominant) type of bonds, k b and kb 0 have unique values and, accordingto Eqs. (58) and (59), S 3 b must then be proportional to E 1/2 and the same will apply toS 3 (see discussion of Eqs. (49) and (50)). Fig. 26 plots S 3 G coefficient measured in different carbonand ‘‘first generation’’ SiC fibres as a function of Young’s modulus square root. The proportionand the distribution of the carbon and SiC phases differ in the fibres and, yet, a globalclassification is possible. Note that a direct proportionality between S 3 and E has been reportedPlease cite this article in press as: G. Gouadec, Ph. Colomban, Prog. Cryst. Growth Charact. Mater. (2007),doi:10.1016/j.pcrysgrow.2007.01.001
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