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**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^, σ^_{θ}), but it is suggested that the parameter estimates be transformed and reported as material-specific parameters ( m^, σ^_{0}). 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. Scope**

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.

1.2 The failure strength of advanced ceramics is treated as a continuous random variable determined by the flaw population. Typically, a number of test specimens with well-defined geometry are failed under isothermal, well-defined displacement and/or force-application conditions. The force at which each test specimen fails is recorded. The resulting failure stress data are used to obtain Weibull parameter estimates associated with the underlying flaw population distribution.

1.3 This practice is restricted to the assumption that the distribution underlying the failure strengths is the two-parameter Weibull distribution with size scaling. Furthermore, this practice is restricted to test specimens (tensile, flexural, pressurized ring, etc.) that are primarily subjected to uniaxial stress states. The practice also assumes that the flaw population is stable with time and that no slow crack growth is occurring.

1.4 The practice outlines methods to correct for bias errors in the estimated Weibull parameters and to calculate confidence bounds on those estimates from data sets where all failures originate from a single flaw population (that is, a single failure mode). In samples where failures originate from multiple independent flaw populations (for example, competing failure modes), the methods outlined in Section 9 for bias correction and confidence bounds are not applicable.

1.5 This practice includes the following:

| Section |

Scope | 1 |

Referenced Documents | 2 |

Terminology | 3 |

Summary of Practice | 4 |

Significance and Use | 5 |

Interferences | 6 |

Outlying Observations | 7 |

Maximum Likelihood Parameter Estimators for Competing Flaw Distributions | 8 |

Unbiasing Factors and Confidence Bounds | 9 |

Fractography | 10 |

Examples | 11 |

Keywords | 12 |

Computer Algorithm MAXL | Appendix X1 |

Test Specimens with Unidentified Fracture Origins | Appendix X2 |

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

**2. Referenced Documents** *(purchase separately)* The documents listed below are referenced within the subject standard but are not provided as part of the standard.

**ASTM Standards**

C1145 Terminology of Advanced Ceramics

C1322 Practice for Fractography and Characterization of Fracture Origins in Advanced Ceramics

E6 Terminology Relating to Methods of Mechanical Testing

E178 Practice for Dealing With Outlying Observations

E456 Terminology Relating to Quality and Statistics

**ICS Code**

ICS Number Code 81.060.99 (Other standards related to ceramics)

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**DOI:** 10.1520/C1239

**Citation Format**

ASTM C1239-13, Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Ceramics, ASTM International, West Conshohocken, PA, 2013, www.astm.org

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