Nicalon fibers are one of the most attractive ceramic fibers for reinforcing high temperature structural composites. Experiment show that the diameter variation (from 8 to 22 {mu}m) in a tow of commercial Nicalon fibers has an effect on statistical strength distribution of Nicalon fibers. Therefore, an appropriate characterization of the statistical distribution of fiber strength, capable of accounting for the effect of diameter variation, is required. The two-parameter single-modal Weibull distribution is found inadequate for characterizing the strength of Nicalon fibers. This work demonstrates that a 3-parameter modified Weibull distribution can successfully characterize the strength of Nicalon fibers; this distribution yields a higher {beta} value, which indicates less scatter in fiber strength than with the 2-parameter single modal Weibull distribution. It more accurately treats the strength variation caused by diameter variation. It is also much easier to use than the 4-parameter bimodal. Moreover, it is seen to more accurately treat the strength variation caused by diameter variation than the single modal 2-parameter analysis

Blumenthal, William R.

Butt, Darryl P.

Lowe, Terry C.

Stout, Michael G.

Taylor, Seth T.

Zhu, Yuntian T.

UNT Digital Library

A uthor(s): Presented at: To be submitted to: ON THE STATISTICAL STRENGTH OF NICALON FIBERS AND ITS CHARACTERIZATION 0 I Yuntian T. Zhu, Seth T. Taylor, Michael G. Stout, Darryl P. Butt, William R. Blumenthal, Terry C. Lowe Ceramic Science and Engineering RF 71 Los Alamos National Laboratory. an alfirmative actionlequal opportunity employer, is operated by the University of California lor the U.S. Department Of Energy under contract W-7405-ENG-36. B acceptance of this article. the publisher recognizes that the U.S. Government retains a nonexclusive, royalty- free license to publish or reproduce the puglished form of this contribution, or to allow others to do so, for US.  Government purposes. The Los Alamos National Laboratory requests that the publisher identify this article as work performed under the auspices of the U.S. Department of Energy. This is a preprint 01 a paper intended for publication in a joumal or proceedings. Because changes may be made before publication, this preprint is made available with the understanding that i t  will not be cited or reproduced without the permission of the author. ! c DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original dOCUment. I ON THE STATISTICAL STRENGTH OF NICALON FIBERS AND ITS CHARACTERIZATION Yuntian T. Zhu, Seth T. Taylor, Michael G. Stout, Darryl P. Butt, William R. Blumenthal, and Terry C. Lowe Mail Stop G755, Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA ABSTRACT Nicalon fibers are one of the most attractive ceramic fibers for reinforcing high temperature structural composites. Experimental results show that the diameter variation (from 8 to 22 pm) in a tow of commercial Nicalon fibers has an effect on the statistical strength distribution of Nicalon fibers. Therefore, an appropriate characterization of the statistical distribution of fiber strength, capable of accounting for the effect of diameter variation, is required. The two-parameter single-modal Weibull distribution is found inadequate for characterizing the strength of Nicalon fibers. In this work, we demonstrate that a three-parameter modified-Weibull distribution can successfully characterize the strength of Nicalon fibers. The three-parameter modified-Weibull distribution yields a higher p value, which indicates less scatter in fiber strength than would be found using the two parameter single-modal Weibull distribution. It more accurately treats the strength variation caused by diameter variation. In addition, it is also much easier to use than the four-parameter bi-modal Weibull distribution. Moreover, it is seen to more accurately treat the strength variation caused by diameter variation than the single modal two parameter analysis. INTRODUCTION NicalonTM silicon carbide fiber has attracted great interest in recent years due to its potential as a reinforcement for advanced composite materials. The strength of reinforcing fibers is one of the most prominent mechanical properties that governs the strength and toughness of composites, and its characterization is essential for predicting the mechanical properties of composites during composite design. The strength of Nicalon fibers is size dependent and scattered, which is typical for ceramic fibers. In addition, a tow of commercial Nicalon fibers DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. typically exhibits a range of fiber diameters. For example, the Nicalon (Sic) fiber spool used in this study contains fibers with diameters ranging from 8 to 22 pm. The large diameter variation of Nicalon fibers complicates the characterization of fiber strength. Although the strength and fracture behavior of Nicalon fibers have been studied extensively in recent years [l-61, no thorough study has been done to investigate the effect of fiber diameter variation on the characterization of their statistical strength. Traditionally, the single-modal Weibull distribution has been used to characterize brittle ceramic fibers for their strength distribution and dependence on gauge length. However, the Weibull distribution cannot account for the effect of fiber diameter on fiber strength. It cannot differentiate the strength variation caused by the fiber diameter change from that caused by the real strength scatter. As a result, the Weibuli distribution has been found invalid for characterizing the strength of fibers with varying diameters [7]. In this work, extensive mechanical testing has been conducted on Nicalon fibers, and a three-parameter modified Weibull distribution proposed previously by us [7, 81 has been successfully used to characterize the strength of Nicalon fibers. It will be shown that the modified Weibull distribution is much more accurate in characterizing fiber strength than the single-modal two parameter Weibull distribution, and is also much easier to use than the bi-modal four parameter Weibull distribution. EXPERIMENTAL PROCEDURE 1 1  attached to the grips of a fiber testing machine (MicroPull  Science,  Thousand Oaks, CA). Fibers were tested at three different gauge lengths (10, 25 and 50 mm), with data sets for each gauge length consisting of about 50 individual fiber tests. All of these tests were conducted in mineral oil to permit consistent recovery of the fractured f i b e r  e n d s  f o r  fractographic analysis and diameter measurement on a JEOL 6300FVX S c a n n i n g  E lec t ron  Individual NicalonTM fibers (Dow Corning) were glued at the ends onto stiff DaDer mounts. Mounted fibers were secured into an alignment jig which was then ._ z 35 n 2 30 Q, 25 L Y- 20 0 15 10 3 5  z 0 5 E 8 9 1011 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2  fiber diameter (pm) Fig. 1 Distribution of fiber diameters. Microscope (SEM). Fibers breaking outside of the gauge length, or at the extreme ends of the fiber, were discarded. Peak load was used to calculate fiber strength. EXPERIMENTAL RESULTS Figure 1 shows the d is t r ibu t ion  of f iber  diameters for all samples tested. Individual fiber diameters vary significantly within the tow, ranging from approximately 8 to 22 pm. From this distribution, the average fiber diameter is calculated as 16.0 pm. Several  different flaw populat ions,  such as  individual pores, pore clusters, granular defects, and surface flaws, have been found to cause failure of the Nicalon fibers (see reference 9 for more information). Figure 2 shows the measured fiber strength vs. fiber diameter for all three gauge lengths (10, 25, and 50 mm). It can be seen that fiber Fig. 2 The measured fiber strength value plotted against fiber diameter for all three gauge lengths (10,25, and 50 mm). strength increases with decreasing fiber diameter. CHARACTERIZATION OF THE STRENGTH OF NICALON FIBERS Strength Characterization Using Weibull Distribution Weibull statistics assumes that the probability of encountering a critical flaw, and hence the probability of fiber failure, is proportional to fiber volume for fibers failing due to volume flaws [lo]. This yields the Weibull distribution in the form: where F(o) is the probability of a fiber having a strength less than or equal to o. Note (T, and p, are Weibull scale and shape parameters related to volume, and V is fiber volume. The average fiber strength as a function of fiber diameter can be obtained from Eq. 1 as: (2) Equation 2 can be rewritten as 1 ln?F=lnK,--lnV (3) P V  3 I '  1 '  ' I " " I  1 '  ' ' 4  I n 0  = 2.7717 - 0.19199 InV 2 -1 o~ 6 7 8 9 10 InV Fig. 3 Fitting Eq. 3 (single modal Weibull distribution) to the experimental fiber strength data. Fitting Eq. 3 to the experimental strength data for Nicalon fibers (see Fig. 3) yields K, = 15.986 and p, = 5.209, from which we also obtain o, = 17.370. Rearranging Eq. 1, we can obtain another way to calculate the Weibull parameters: In In 1 -In V = -& hav + Pv l n o  1 - F(o) Equation 4 is the most common form used to calculate Weibull parameters. F(o) can be calculated from experimental data as [ 101 z F ( 6 , )  = - N + l  where N is the total number of tests, and the observed fiber strength values, o, ... oN, are ranked in an ascending order. Plotting the experimental data of In In{ l/[l-F(oJ]} - In Vi against In oi for Nicalon fibers and fitting Eq. 4 to the plot (see Fig. 4) yields p, = 6.022 and o, (4) In In{l/[l-F( o)]} - InV = - 15.954 + 6.0219 In 2 -6.0 e - 5 -9.0 g -12.0 Ls. P Y - S - -1 5.0 -0.5 0.0 0.5 1.0 1.5 2.0 Ino Fig. 4 Fitting Eq. 4 (derived from the single modal Weibull distribution) to the strength data. = 14.144. It is obvious that the Weibull parameters obtained from Eq. 3 and Eq. 4 are not consistent, although both equations are derived from Eq. 1. This suggests that the single modal Weibull distribution is inadequate for characterizing the strength of Nicalon fibers with varying diameters. Strength Characterization Using Modified Weibull Distribution We have proposed a three-parameter modified Weibull distribution [1,8] in the form: The average fiber strength as a function of fiber diameter and length can be derived from Eq. 6 as [ 111: where r( 1+1/p,) is a gamma function. Eq. 7 can also be written as where the values of K, m and n can be obtained by fitting Eq. 8 to experimental data, and the parameters for the modified Weibull distribution can be subsequently calculated as h = P,n and K Bo = r(1+ W P  ) The following two equations can be derived from Eq. 8 InZ?+nlnd = InK-mlnL (12) I n 5  + mln L = In K - nlnd (13) Assuming an initial value for n, Eq. 12 can be fitted to experimental strength data for Nicalon fibers to obtain a value of rn. The m value can then be used to fit Eq. 13 to experimental data to yield a new n value. The above calculation is iterated if the rn or n value is different from the previous m or n value, until consistent values are obtained. Table I1 lists the rn and n values during an iteration. The initial n value was assumed to be 0.7. Figures 5a and 5b show the final round of curve fittings, which yielded m = 0.14909, n = 0.56433, and K =  23.8456. The modified Weibull distribution parameters can be calculated using Eqs. 9-1 1 as p17 = 6.707, h = 3.875, and cro = 25.696. Note that the above calculation was done using basic spreadsheet software. The calculation is much simpler than that necessary for the bi-modal Weibull distribution, which requires more advanced software or computer programs. 3.0 - Ino +rnlnL = 3.1761 - 0.56433 Ind (a) 1 2.0 E b = 1.0 - + 0 0.0 2.00 2.25 2.50 2.75 3.00 3.25 Ind I Ino+nlnd=3.1761 -0.14909InL tb) 1 2.0 2.5 3.0 3.5 4.0 InL Fig. 5 a) Fitting to Eq. 12 yields an n value of 0.56433, while b) fitting to Eq. 13 yields an rn value of 0.14909. Both fits yield a K value of 23.8456. Table I1 The rn and n values during the iteration I it as obtained using Eq. 13 I m as obtained using Eq. 12 I I ,, I I n ,  = 0.56439 I m, = 0.14909 I We can also derive another way to obtain the Weibull parameters 0, and pl7 by rearranging Eq. 6 as . In In 1 -InLdh =-/3,1no,+PpIna 1 - F(o) Fitting Eq. 14 to the experimental data (Fig. 6) using h = 3.875 yields pp = 6.713 and o0 = 25.967. It is obvious that the pp and o0 values obtained from Eq. 14 are very consistent with that obtained from Eq. 12 and 13. Equations 12, 13 and 14 are all derived from Eq. 6. Therefore, they should yield consistent pF and 0, values if Eq. 6 is valid. The fact that -5 Inln{l/[l - F( o)u - Ind t875 = - 21.872 + 6.713 In -0.5 0 0.5 1 1.5 2 Ino Fig. 6 Fitting Eq. 14 to experimental strength data for Nicalon fibers. very consistent pp and o0 values have been obtained validates the three-parameter m o d i f i e d  W e i b u l l  distribution. It is clear that the three-parameter modified-Weibull distribution can satisfactorily characterize the strength of Nicalon fibers, while the single-modal Weibull distribution is inappropriate for the task due to the large variation in Nicalon fiber diameter. In practice, the single-modal Weibull distribution is often used to characterize the strength of Nicalon fibers due to its simplicity, with most users unaware of the aforementioned complication associated with varying diameter. In addition to the inconsistency of /3 values obtained from the single- modal Weibull distribution, another error caused by using it is the low calculated p value (large strength scatter). The single modal Weibull distribution cannot take into account the effects of diameter variation on fiber strength. Instead, it treats these effects as scatter of fiber strength, which will erroneously result in larger strength scatter and consequently a smaller p value. For example, we have obtained a p of 5.21 using Eq. 3 and 6.02 using Eq. 4 (both equations are derived from the single-modal Weibull distribution). The p value for Nicalon fibers has been reported to be in the range of 4 to 6, which were obtained using the single- modal Weibull distribution [ 121. These values are below the P value obtained using the three-parameter modified-Weibull distribution, which is 6.7 1, because the modified Weibull distribution can, in fact, account for the effects of diameter variation on fiber strength. CONCLUSIONS , d . The three-parameter modified Weibull distribution has shown excellent consistency in characterizing the strength of Nicalon fibers with varying diameters. The single-modal Weibull distribution is found to be inadequate for characterizing the strength of Nicalon fibers with varying diameters, and is found to yield an erroneously low Weibull modulus, p. The present modified Weibull distribution is also mathematically much simpler than a multi-modal Weibull distribution. REFEFENCES 9 L. C. Sawyer, M. Jamieson, D. Brikowski, M. I. Haider and R. T. Chen, “Strength, Structure, and Fracture Properties of Ceramic Fibers Produced from Polymeric Precursors: I, Base-Line Studies,” J. Am. Ceram. Soc., 70 [l 11 798-810 (1987). C-H. Andersson and R. Warren, “Silicon Carbide Fibers and Their Potential for Use in Composite Materials. Part 1,” Composites, 15, 16-24 (1984). K. M. Prewo, “Tension and Flexural Strength of Silicon Carbide Fibre-Reinforced Glass Ceramics,” J. Mat. Sci., 21,3590-600 (1986). C. H. Andersson and R. Warren, “The Strength and Elastic Modulus of Multifilament Sic Fibbers,” pp. 1129-39 in Advances in Composite Materials. Edited by A. R. Bunsell, C. Bathias, A. Martrenchjar, D. Menkes, and G. Verchery, Pergamon Press, New York, 1980. G. Simon and A. R. Bunsell, “Mechanical and Structural Characterization of the Nicalon Silicon Carbide Fibre,” J. Mat. Sci., 19,3649-57 (1984). G. Simon and A. R. Bunsell, “Mechanical and Structural Study of High Performance Silicon Carbide Fibers,” Sci. Ceram. 12 - Ceramurgica s.r.l., Faenza, 647-54, (1984). Y. T. Zhu, W. R. Blumenthal, S. T. Taylor, T. C. Lowe and B. L. Zhou, “Analysis of Size Dependence of Ceramic-Fiber Strength,“ J. Am. Ceram. SOC. (in press). Y. T. Zhu, W. R. Blumenthal and B. L. Zhou, “Characterizing Size Dependence of Ceramic-Fiber Strength Using Modified Weibull Distribution,” pp. 493-97 in Micromechanics of Advanced Materials. Edited by S. N. G. Chu, P. K. Liaw, R. J. Arsenault, K. Sadananda, K. S. Chen, W. W. Gerberich, C. C. Chau, and T. M. Kung, TMS, Warrendale, PA (1995). S. T. Taylor, Y. T. Zhu, W. R. Blumenthal, M. G. Stout, D. P. Butt, and T. C. Lowe, “Characterization of Nicalon Fibers with Varying Diameters. Part I: Strength and Fracture Studies.” submitted to J.  Am. Ceram. Soc.. 10 S. B. Batdorf, “Fundamentals of the Statistical Theory of Fracture,” pp. 1-29 in Fracture Mechanics of Ceramics, vol. 3. Edited by R. C. Bradt, D. P. H. Hasselman, and F. F. Lange, Plenum Press, New York (1978). 11 R. S. Burington and D. C. May, Jr., Handbook of Probability and Statistics with Tables, 2nd ed., McGraw-Hill Book Company, New York, 1970, p. 109. 12 J. Lipowitz, “Polymer-Derived Ceramic Fibers,” Am. Ceram. SOC. Bull., 70, 1888-94 (1991). 

On the statistical strength of nicalon fibers and its characterization

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On the statistical strength of nicalon fibers and its characterization

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