Standard Active Last Updated: Jul 10, 2018
ASTM C1239-13(2018)

Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Ceramics

Significance and Use

5.1 Advanced ceramics usually display a linear stress-strain behavior to failure. Lack of ductility combined with flaws that have various sizes and orientations leads to scatter in failure strength. Strength is not a deterministic property, but instead reflects an intrinsic fracture toughness and a distribution (size and orientation) of flaws present in the material. This practice is applicable to brittle monolithic ceramics that fail as a result of catastrophic propagation of flaws present in the material. This practice is also applicable to composite ceramics that do not exhibit any appreciable bilinear or nonlinear deformation behavior. In addition, the composite must contain a sufficient quantity of uniformly distributed reinforcements such that the material is effectively homogeneous. Whisker-toughened ceramic composites may be representative of this type of material.

5.2 Two- and three-parameter formulations exist for the Weibull distribution. This practice is restricted to the two-parameter formulation. An objective of this practice is to obtain point estimates of the unknown parameters by using well-defined functions that incorporate the failure data. These functions are referred to as “estimators.” It is desirable that an estimator be consistent and efficient. In addition, the estimator should produce unique, unbiased estimates of the distribution parameters (6). Different types of estimators exist, including moment estimators, least-squares estimators, and maximum likelihood estimators. This practice details the use of maximum likelihood estimators due to the efficiency and the ease of application when censored failure populations are encountered.

5.3 Tensile and flexural test specimens are the most commonly used test configurations for advanced ceramics. The observed strength values are dependent on test specimen size and geometry. Parameter estimates can be computed for a given test specimen geometry ( m^, ^σθ), but it is suggested that the parameter estimates be transformed and reported as material-specific parameters ( m^, ^σ0). In addition, different flaw distributions (for example, failures due to inclusions or machining damage) may be observed, and each will have its own strength distribution parameters. The procedure for transforming parameter estimates for typical test specimen geometries and flaw distributions is outlined in 8.6.

5.4 Many factors affect the estimates of the distribution parameters. The total number of test specimens plays a significant role. Initially, the uncertainty associated with parameter estimates decreases significantly as the number of test specimens increases. However, a point of diminishing returns is reached when the cost of performing additional strength tests may not be justified. This suggests that a practical number of strength tests should be performed to obtain a desired level of confidence associated with a parameter estimate. The number of test specimens needed depends on the precision required in the resulting parameter estimate. Details relating to the computation of confidence bounds (directly related to the precision of the estimate) are presented in 9.3 and 9.4.


1.1 This practice covers the evaluation and reporting of uniaxial strength data and the estimation of Weibull probability distribution parameters for advanced ceramics that fail in a brittle fashion (see Fig. 1). The estimated Weibull distribution parameters are used for statistical comparison of the relative quality of two or more test data sets and for the prediction of the probability of failure (or, alternatively, the fracture strength) for a structure of interest. In addition, this practice encourages the integration of mechanical property data and fractographic analysis.

1.6 The values stated in SI units are to be regarded as the standard per IEEE/ASTM SI 10.

1.7 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.

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Book of Standards Volume: 15.01
Developed by Subcommittee: C28.01
Pages: 18
DOI: 10.1520/C1239-13R18
ICS Code: 81.060.99