J. Am. Ceram. Soc., 80 [6] 1447–52 (1997) Analysis of Size Dependence of Ceramic Fiber and Whisker Strength Yuntian T. Zhu, * William R. Blumenthal, * Seth T. Taylor, and Terry C. Lowe Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 Benlian Zhou Institute of Metal Research, Chinese Academy of Sciences, Shenyang, China The strengths of ceramic fibers and whiskers have been with a diameter range from 8 to 22 m. Many ceramic whisk- ers, such as Al 2 O 3 whiskers, 8,10,11 Si 3 N 4 whiskers, 12 and SiC observed to increase with decreasing fiber diameter and length. Typically, both surface flaws and volume flaws exist whiskers, 13 also exhibit significant diameter variations. There- fore, a good characterization of the statistical distribution of in ceramic fibers and whiskers, which makes it impossible to characterize the strength dependence of both the diame- fiber and whisker strength, capable of accounting for the size effect of diameter variation, is needed. At present, there is no ter and the length with a single-modal Weibull distribution function. Our data also show that the single-modal Weibull satisfactory theory available which has this capability. After surveying 700 pieces of literature on size effect on distribution is inadequate to characterize the strength of fibers with varying diameters even in the case of a constant materials strength published from 1859 to 1975, Harter 14 con- cluded that the statistical theory alone cannot explain all the fiber length. In addition, experimental data also show that, for sapphire whiskers whose surface flaws were removed size-effect phenomena that have been observed. He listed three major competing theories which were first enumerated by by chemical polishing, the whisker strength has a much stronger size dependence on diameter than predicted by the Davidenkov: 15 (1) the statistical theory, which explains the size single-modal Weibull function, which indicates that factors effect by the probability of encountering a critical-size flaw, other than those characterized by the Weibull function also (2) the energy theory, which states that the effect of the stored play a role in the strength of sapphire whiskers. In this energy in the specimen–machine system increases with increas- paper, the factors affecting the strengths of ceramic fibers ing specimen dimension, and (3) the technological theory, and whiskers are analyzed in terms of Weibull statistics, which ascribes the size effect to the unequal treatment condi- fracture mechanics, and flaw size density variation with vary- tions caused by the size difference. Volkov 16 proposed that the ing fiber diameters. A three-parameter modified Weibull above three theories must be combined in order to attain a distribution, which combines the above strength-affecting comprehensive theory which will explain all the observed factors, is proposed to characterize both the diameter and phenomena. the length dependence for ceramic fibers and whiskers with Batdorf 17 proposed that the fracture statistics should be based or without surface flaws. Characterization of the strength on a proper consideration of three elements—extreme value data of sapphire whiskers and Nicalon SiC fibers with vary- statistics (e.g., weakest-link theory), fracture mechanics, and ing diameters shows the validity of the modified Weibull materials structure. Statistical theories incorporating these distribution function. elements are as yet in their infancy. Batdorf 17 believes that such theories offer the greatest long-range promise for future progress. I. Introduction For brittle ceramic materials, the majority of mathematical C ERAMIC fibers and whiskers have been increasingly used as descriptions of fracture statistics are based on weakest-link reinforcements for advanced composite materials. 1–4 The theory (WLT). 18,19 Examples of the weakest-link theories 18 are mechanical properties of these reinforcements significantly the Weibull theory, 20 the three extreme-value distributions, 21,22 affect the strength of composite materials. In order to make full and flaw density distributions. 23,24 The Weibull distribution has use of the reinforcing potential of ceramic fibers and whiskers become most popular, due to its mathematical simplicity and in composite design, it is essential to understand and accurately relatively good success in its application. 18 The single-modal characterize their mechanical properties. Brittle fractures have Weibull distribution 20 is based on the assumption that the proba- been observed in most advanced ceramic fibers and whiskers bility of finding flaws of critical size is proportional to the under tensile stress. It is a well-known fact that the strengths of volume of material tested if the strength is controlled by volume ceramic fibers and whiskers are size-dependent. 5 As the diame- flaws, or to surface area if the strength is controlled by surface ter or gauge length decreases, the strength of ceramic fibers and flaws. It has been widely used to characterize brittle ceramic whiskers increases. 6–9 In addition, experimental strengths of fibers for their strength distribution and strength dependence on brittle ceramic fibers and whiskers display a range of values for gauge length. 25–28 However, as it will be demonstrated later in a given configuration. this paper, the single-modal Weibull distribution is inappropri- Some commercial ceramic fibers and whiskers may have a ate to describe the statistical strength of ceramic fibers with range of fiber diameters. For example, a tow of the Nicalon varying diameters. For example, our experimental results on (SiC) fibers (manufactured by Dow Corning) contains fibers Nicalon fibers with a constant gauge length of 30 mm, but a varying diameter from 8 to 22 m, yield a significant inconsis- tency in the Weibull shape parameter  when characterized Michael Thouless—contributing editor using the single-modal Weibull distribution. Another example which demonstrates the inadequacy of the single-modal Weibull distribution is that the strength of sapphire whiskers Manuscript No. 192069. Received January 26, 1996; approved November 1, 1996. with polished surfaces 29 increases with decreasing fiber diame- Supported in part by the Director’s Postdoctoral Fellowship and the Laboratory ter faster than would be predicted by the Weibull distribution. Directed Research and Development Office of Los Alamos National Laboratory. * Member, American Ceramic Society. For the sapphire whiskers whose surface flaws are removed by 1447