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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 typically leads to large scatter in failure strength. Strength is not a deterministic property but instead reflects the intrinsic fracture toughness and a distribution (size and orientation) of flaws present in the material. This standard is applicable to brittle monolithic ceramics which fail as a result of catastrophic propagation of flaws. Possible rising R-curve effects are also not considered, but are inherently incorporated into the strength measurements.

5.2 Two- and three-parameter formulations exist for the Weibull distribution. This standard is restricted to the two-parameter formulation.

5.3 Tensile and flexural test specimens are the most commonly used test configurations for advanced ceramics. Ring-on-ring and pressure-on-ring test specimens which have multi-axial states of stress are also included. Closed-form solutions for the effective volume and effective surfaces and the Weibull material scale factor are included for these configurations. This practice also incorporates size scaling methods for C-ring test specimens for which numerical approaches are necessary. A generic approach for arbitrary shaped test specimens or components that utilizes finite element analyses is presented in Annex A3.

5.4 The fracture origins of failed test specimens can be determined using fractographic analysis. The spatial distribution of these strength controlling flaws can be over a volume or an area (as in the case of surface flaws). This standard allows for the conversion of strength parameters associated with either type of spatial distribution. Length scaling for strength controlling flaws located along edges of a test specimen is not covered in this practice.

5.5 The scaling of strength with size in accordance with the Weibull model is based on several key assumptions (5). It is assumed that the same specific flaw type controls strength in the various specimen configurations. It is assumed that the material is uniform, homogeneous, and isotropic. If the material is a composite, it is assumed that the composite phases are sufficiently small that the structure behaves on an engineering scale as a homogeneous and isotropic body. The composite must contain a sufficient quantity of uniformly-distributed, randomly-oriented, reinforcing elements such that the material is effectively homogeneous. Whisker-toughened ceramic composites may be representative of this type of material. This practice is also applicable to composite ceramics that do not exhibit any appreciable bilinear or nonlinear deformation behavior. This standard and the conventional Weibull strength scaling with size may not be suitable for continuous fiber-reinforced composite ceramics. The material is assumed to fracture in a brittle fashion, a consequence of stress causing catastrophic propagation of flaws. The material is assumed to be consistent (batch to batch, day to day, etc.). It is assumed that the strength distribution follows a Weibull two parameter distribution. It is assumed that each test piece has a statistically significant number of flaws and that they are randomly distributed. It is assumed that the flaws are small relative to the specimen cross section size. If multiple flaw types are present and control strength, then strengths may scale differently for each flaw type. Consult Practice C1239 and the example in 9.1 for further guidance on how to apply censored statistics in such cases. It is also assumed that the specimen stress state and the maximum stress are accurately determined. It is assumed that the actual data from a set of fractured specimens are accurate and precise. (See Terminology E456 for definitions of the latter two terms.) For this reason, this standard frequently references other ASTM standard test methods and practices which are known to be reliable in this respect.

5.6 Even if test data has been accurately and precisely measured, it should be recognized that the Weibull parameters determined from test data are in fact estimates. The estimates can vary from the actual (population) material strength parameters. Consult Practice C1239 for further guidance on the confidence bounds of Weibull parameter estimates based on test data for a finite sample size of test fractures.

5.7 When correlating strength parameters from test data from one specimen geometry to a second, the accuracy of the correlation depends upon whether the assumptions listed in 5.5 are met. In addition, statistical sampling effects as discussed in 5.6 may also contribute to variations between computed and observed strength-size scaling trends.

5.8 There are practical limits to Weibull strength scaling that should be considered. For example, it is implicitly assumed in the Weibull model that flaws are small relative to the specimen size. Pores that are 50 μm (0.050 mm) in diameter are volume-distributed flaws in tension or flexural strength specimens with 5 mm or greater cross section sizes. The same may not be true if the cross section size is only 100 μm.

**1. Scope**

1.1 This standard practice provides methodology to convert fracture strength parameters (primarily the mean strength and the Weibull characteristic strength) estimated from data obtained with one test geometry to strength parameters representing other test geometries. This practice addresses uniaxial strength data as well as some biaxial strength data. It may also be used for more complex geometries proved that the effective areas and effective volumes can be estimated. It is for the evaluation of Weibull probability distribution parameters for advanced ceramics that fail in a brittle fashion. Fig. 1 shows the typical variation of strength with size. The larger the specimen or component, the weaker it is likely to be.

Geometries and Stress Distributions

Annex A3

1.5 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.

1.5.1 The values stated in SI units are in accordance with IEEE/ASTM SI 10.

1.6 *This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use.*

**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

C1161 Test Method for Flexural Strength of Advanced Ceramics at Ambient Temperature

C1211 Test Method for Flexural Strength of Advanced Ceramics at Elevated Temperatures

C1239 Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Ceramics

C1273 Test Method for Tensile Strength of Monolithic Advanced Ceramics at Ambient Temperatures

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

C1323 Test Method for Ultimate Strength of Advanced Ceramics with Diametrally Compressed C-Ring Specimens at Ambient Temperature

C1366 Test Method for Tensile Strength of Monolithic Advanced Ceramics at Elevated Temperatures

C1499 Test Method for Monotonic Equibiaxial Flexural Strength of Advanced Ceramics at Ambient Temperature

E6 Terminology Relating to Methods of Mechanical Testing

E456 Terminology Relating to Quality and Statistics

**ICS Code**

ICS Number Code 81.060.30 (Advanced ceramics)

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**DOI:** 10.1520/C1683-10R15

**Citation Format**

ASTM C1683-10(2015), Standard Practice for Size Scaling of Tensile Strengths Using Weibull Statistics for Advanced Ceramics, ASTM International, West Conshohocken, PA, 2015, www.astm.org

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