J Am. Ceram. Soc., 78 171 1965-68 (1995)

Tensile Strength of Sic Fibers

Frank W. Zok,' Xiuyen Chen, and Christian H. Weber',? 93 106 Materials Department, University of California, Santa Barbara, California

An experimental investigation has been conducted on the effects of gauge length on the tensile strength of SIC fibers. The results show that the overall strength distribution cannot be described solely in terms of the two-parameter Weibull function. The overall distribution is found to be consistent with two concurrent flaw populations, one of them being characteristic of the pristine fibers, and the other characteristic of the additional flaws introduced into the fiber during processing of the composite.

I. Introduction

HE axial fracture, creep, and fatigue properties of fiber- T reinforced metal and ceramic matrix composites are domi- nated by the strength distribution of the fibers."' The focus of the present study is on the strength characteristics of S i c fibers, extracted from a metal matrix composite. The study augments concurrent work on the creep, fatigue, and fracture properties of the composite.'w'2 One of the main features of this study involves the effects of gauge length on the fiber strength distri- bution. Such effects are important in using experimental data obtained from afixed gauge length (typically on the order of a few centimeters) in predicting the mechanical response of the composite, particularly when the relevant gauge length in the composite is smaller than that used to characterize the fibers.' It

J. Lamon--contributing editoi

Manuscript No. 193012. Received November 28, 1994; approved April 17,1995. Supported by the Defense Advanced Research Projects Agency through the Univer-

'Member, American Ceramic Society. +Currently at Corporate Technology, Raychem Corporation, Menlo Park, CA

sity Research Initiative under ONR Contract N-0014-92-J-1808.

94025-1 164.

Ti Ti6 TiB2 C Sic - c

is also demonstrated that conducting tests over a range of gauge lengths provides access to a broader range of strengths, yielding a more thorough description of the strength distribution. One of the key results emanating from this work is that the strength distribution of the S ic fibers cannot be described in terms of a single flaw population, characterized by the two-parameter Weibull function, despite the fact that the data sets obtained from fixed gauge lengths appear to be consistent with this function. The implications of such effects in the use and inter- pretation of fiber tests are briefly discussed.

11. Theoretical Background

The strength distribution of brittle ceramic fibers can be described using weakest-link statistics. A convenient empirical description of this distribution is the Weibull function

where P, is the cumulative failure probability up to a stress, o; L is the length of the fiber; L, is a reference length, taken to be l m ; o0 is the reference strength; and m is the Weibull modulus. The radius, R, of all fibers is assumed to be the same.

The parameters characterizing the strength distribution (ao and m ) are usually obtained experimentally by conducting a series of N tensile tests on individual fibers with a fixed gauge length, L. The strength data are then sorted in ascending order and the failure probability corresponding to each strength obtained using the mean rank

*Other estimators of failure probability yield essentially the same results, provided N is sufficiently large.

(a) (b) Fig. 1. SEM micrographs showing (a) the C/TiB, coating in the composite, and (b) spalling of the coating following matrix dissolution.

1965

1966 Communications of the American Ceramic Society Vol. 78, No. 7

Stress, (J (MPa)

(4

"-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

In ((J / GPa)

Fig. 2. of -1 h).

Strength distributions obtained from fibers of different gauge lengths: (a) linear coordinates, (b) logarithmic coordinates (dissolution time

1 P, = - N S 1

where i is the ranking of the data. The parameters in Eq. (1) are obtained from a least-squares fit of Eq. (1) to the experimental data.

111. Experiments

The motivation for the experimental part of this program stems from concurrent studies on the tensile strength'' and creep response"," of fiber-reinforced Ti matrix composites. The composite system of interest is a Ti-6A1-4V alloy matrix reinforced unidirectionally with continuous S i c (Sigma fibers, British Petroleum) fibers. Prior to composite consolidation, the fibers had a dual coating, consisting of an inner C layer, -0.5 ym thick, and an outer TiB, layer, -1 pm thick (Fig. l(a)). During consolidation, the TiB, reacts with the Ti alloy to form a layer of TiB needles, -0.7 ym thick. The fibers were extracted from the composite by dissolving the matrix in concentrated HF acid. Typically, the dissolution was completed within -45 to 60 min. Extreme care was taken in handling the extracted fibers, such that no new flaws were introduced. Some of the extracted fibers were examined in a scanning electron microscope. Over much of the fiber surface, the C/TiB, coating was found to have completely spalled off (Fig. l(b)). Spalling is presumed to be a result of the residual stresses present in the

c c - -

1 -

0 -

-1 -

-2

-3

4 -

-

-

A 0

i -1 -51 ' ' I ' ' I ' I ' ' I I

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

In (0 / GPa)

Fig. 3. (25 mm gauge length).

Effects of dissolution time on the fiber strength distribution

coating, coupled with the relaxation of these stresses following matrix dissolution. The effects of the HF acid on the fiber strength distribution were examined by allowing some of the fibers to soak in the acid following dissolution for a period of 60 h. The diameter of the fibers was extremely uniform, with an average value d = 100 pm.

Tensile tests were conducted on individual fibers using a dedicated fiber testing machine (Micropull Science). The fiber ends were attached to cardboard tabs (designed specifically for use with the fiber testing machine) using an epoxy adhesive. To ensure good alignment, the fiber ends were placed into straight shallow grooves that had been stamped along the center line of the tabs. The fibers were loaded using a computer-controlled linear stepper motor, and the load monitored using a 100 N load cell.

Four fiber gauge lengths were used: 5 , 12.7, 25.4, and 265 mm. For each gauge length, 50 tests were performed. The strength data for each gauge length were sorted and the failure probability evaluated using the mean rank estimator (Eq. (2)). The values of un and m were then evaluated from a least-squares fit of Eq. (1) to the data.

IV. Experimental Results

The strength distributions obtained from each of the four gauge lengths are shown in Fig. 2 (both in linear and logarith- mic coordinates). For comparison, the results for pristine fibers with a 25 mm gauge length are also shown. The latter results were derived from >lo00 fiber tests conducted by the fiber manufacturer. As expected, the strength decreases with increased gauge length. This trend is manifested as shifts in the failure probability curves. It should also be noted that the strengths of the extracted fibers are substantially lower than those of the pristine fibers for the same gauge length, indicating that additional flaws are introduced into the fibers during com- posite processing.

The effects of the HF acid on the strength distribution for a 25 mm gauge length are shown in Fig. 3. Evidently, there is no significant change in the fiber strength following the additional -59 h soak in HF. This result is consistent with fractographic examinations of the fibers which show that fracture generally originates from the tungsten core, not from the fiber surface."

The values of uo and m determined from each data set are shown in Fig. 4. Evidently, both of these parameters depend on the fiber gauge length: decreasing as length increases. The dependencies of un and m on gauge length suggest that the measured distributions are not consistent with a single flaw population, characterized by the two-parameter Weibull func- tion. In contrast, the low values of x2 obtained from the least- squares fit for each data set (typically -0.02 to 0.07) suggest

July 1995 Communications of the American Ceramic Society 1967

1.8

c c

m Q

p!

2

a, 0

1.5 p! a, a, -

1.4 c c p!

1.3 2

1.2

Gauge Length, L (mm) Gauge Length, L (mm)

Fig. 4. standard errors, calculated from the least-squares fit of the Weibull function.

Effects of fiber gauge length on (a) the apparent Weibull modulus, m, and (b) the apparent reference strength, u". The bars represent the

that the individual data sets are consistent with a Weibull distribution.

To elucidate the origin of these seemingly contradictory con- clusions, the data sets were combined in such a way that direct comparisons could be made between the results from various gauge lengths. For this purpose, it is useful to rewrite Eq. (2) as

(3)

The form of this equation suggests that the data corresponding to different gauge lengths should collapse onto a single curve when the results are plotted as In In [l/(l - P,)] ~ In (L/L,) vs In u. Figure 5 shows such a plot. In this form, the data do collapse onto essentially a single curve, suggesting that they are indeed derived from the same overall strength distribution. However, the overall distribution does not appear to be consis- tent with the two-parameter Weibull distribution. The observa- tion that the data are consistent with the Weibull function for a fixed gauge length is a result of the limited range of strengths accessed by that gauge length. The variations in m and uo with gauge length are manifestations of the various strength regimes accessed by the different gauge lengths, each giving values of m and uo that represent an approximation to the distribution over a limited strength range.

In In []/(I - P,)] - In (L/L,) = m In u - m In uo

h

7

Y

K T- v

-

2

0 :I I

4.3 K

-2 - - I -

-4 I I I I I I ,

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

In (0 /GPa)

Fig. 5. The strength distributions from Fig. 2, replotted using a parameter that combines the failure probability with the fiber gauge length (see Eq. 3)). Also shown for comparison are the data for the pristine fibers (dashed line) and the prediction based on two concurrent flaw populations (solid line).

V. Analysis The strength distributions in Fig. 2 can be rationalized in

terms of a modified form of the Weibull function. For this purpose, it is assumed that there are two concurrent flaw popu- lations, each characterized by the Weibull distribution. The total survival probability at a prescribed stress u and gauge length L is then the product of the survival probabilities from each of the two flaw populations. This result can be expressed as

(4)

where m, and u, are the Weibull parameters that characterize the two populations.

The utility of this approach in describing the strength distri- bution of the S ic fibers was assessed in the following way. In the high-strength regime, the Weibull modulus and reference strength were taken to be equal to the values characteristic of the pristine fibers (m, = 18 and ul = 3.2 GPa) determined from tests on fibers with a 25.4 mm length. The Weibull modulus and reference strength for the other flaw population were evaluated by a least-squares fit of Eq. (4) to the data corresponding to the 25.4 mm gauge length (the same as that used to characterize the pristine fibers). The results of this fit are m, = 4.3 and u2 = 1.3 GPa. Using these values, the failure probabilities for the other gauge lengths were calculated and compared with the experimental data. The comparisons are shown in Fig. 6. The predictions are in good agreement with the results for L = 265 mm; for shorter gauge lengths (5 and 12.7 mm), they tend to overestimate the strengths somewhat, though the discrepan- cies are typically <lo%.

VI. Concluding Remarks The present results indicate that the strength distribution of

the S ic fibers extracted from the composite cannot be described solely in terms of a single flaw population, characterized by the two-parameter Weibull function. These conclusions would not have readily been reached by considering only the results for a fixed fiber gauge length, despite the relatively large number of tests performed at each gauge length. Performing tests over a range of gauge lengths provides access to a broader range of strengths and a more complete description of the overall distribution. In the present case, the distribution can be rational- ized in terms of two Weibull distributions, one being char- acteristic of the flaws in the pristine fibers, and the other charac- teristic of new flaws introduced during processing of the composite.P

Tonceivably, other variations on the two-parameter Weibull function could be used to rationalize this strength distribution (see Ref. 13 for an alternate approach).

Communications of the American Ceramic Society Vol. 78, No. 7

can thus be extrapolated to other parts of the distribution. Non- linearity in failure probability plots (indicating a breakdown in the two-parameter Weibull distribution) become evident only when the strength range is sufficiently large, as obtained from tests on varying fiber gauge lengths.

1 .o

0.8

0.6 Acknowledgment: The authors gratefully acknowledge the assistance of J. R. Porter in the initial work on fiber testing.

References 0.4 ‘M. Y. He, A. G. Evans, and W. A. Curtin, “The Ultimate Tensile Strength of

Metal and Ceramic Matrix Composites,”Acta Metall. Muter., 41, 871-78 (1993). *W. A. Curtin, “Theory of the Mechanical Properties of Ceramic-Matrix

Composites,”./. Am. Ceram. Soc., 74,283745 (1991). ‘W. A. Curtin, “Exact Theory of Fiber Fragmentation in a Single Filament

Composite,”J. Muter. Sci., 26, 5239-53 (1991). 4J. Neumeister, “A Constitution Law for Continuous Fiber Reinforced Brittle

0 1 2 3 4 5 Matrix Composites with Fiber Fragmentation and Stress Recovery,” .I. Mech.

‘G. J. DcSalvo, Theory and Strnctur-a1 Design Applications of Weibull Statis-

0.2

0.0

Phys. Solids, 41, 1383404 (1993).

tics; Ch. 2-3. Westinghousc Electric, Pittsburgh, PA, 1970. Stress, (r (MPa)

Fig. 6. Comparisons between the experimental and predicted strength distributions.

The results have implications concerning the development of testing procedures for characterizing fiber strength distribu- tions. It is clear that results obtained from a single fiber gauge length may not be adequate for predicting fiber bundle proper- ties when the selected gauge length is substantially different from the length governing fiber bundle behavior. This problem is exacerbated by the general observation that failure probabil- ity plots, in the coordinates of In In 1/P, vs In (T, usually appear nearly linear when the strength range is relatively small. This observation may lead to the erroneous conclusion that the data are consistent with a two-parameter Weibull distribution and

6D. P. Walls, G. Baa, and F, W. Zok, “Mode I Fatigue Cracking in a Fiber Reinforced Metal Matrix Composite,” Acta Metall. Muter., 41, 2061-71 (1993).

’6. N. Cox and D. B. Marshall, “Crack Bridging in the Fatigue of Fibrous Composites,” Fatigue Fract. Eng. Muter. Struct., 14, 847 (1991).

‘M. D. Sensmeier and P. K. Wright, “The Effects of Fiber Bridging on Fatigue Crack Growth in Titanium Matrix Composites”; pp. 441-57 in Fundamental Relationships between Microstructure and Mechanical Properties of Metal Matrix Composites. Edited by P. K. Liaw and M. N. Gungor. The Metallurgical Society, Warrendale, PA, 1990.

9M. McLean, “Creep Deformation of Metal-Matrix Composites,” Compos. Sci. Technol., 23, 37-52 (1985).

‘OZ. Z. Du and R. M. McMeeking, “Creep Models for Metal Matrix Compos- ites Reinforced with Long Brittle Fibers,” J. Mech. Phys. Solids, in press.

“C. H. Weber, X. Chen, S. J. Connell, and F. W. Zok, “On the Tensile Properties of a Fiber Reinforced Titanium Matrix Composite: I, Unnotched Behavior,” Acta Metall. Muter., 42,3443-50 (1994).

‘’C. H. Weber, Z. Z. Du, S. J . Connell, and F. W. Zok, “High Temperature Deformation and Fracture of a Fiber-Reinforced Titanium Matrix Composite,” submitted to Acta Metall. Mater.

I3K. K. Phani, “Strength Distribution and Gauge Length Extrapolation in 0 Glass Fiber,” J . Mater. Sci., 23, 1 189-94 (1988).