定价： 110元 / 折扣价： 94 元

定价： 114元 / 折扣价： 97 元

5.1 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 Weibull distribution 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, such as moment estimators, least-squares estimators, and maximum likelihood estimators. This practice details the use of maximum likelihood estimators.5.2 Tensile and flexural specimens are the most commonly used test configurations for graphite. The observed strength values depend on specimen size and test geometry. Tensile and flexural test specimen failure data for a nearly isotropic graphite (7) is depicted in Fig. 1. Since the failure data for a graphite material can be dependent on the test specimen geometry, Weibull distribution parameter estimates (m, Sc) shall be computed for a given specimen geometry.FIG. 1 Failure Strengths for Tensile Test Specimens (left) and Flexural Test Specimens (right) for a Nearly Isotropic Graphite (7)5.3 The bias and uncertainty of Weibull parameters depend on the total number of test specimens. Variability in parameter estimates decreases exponentially as more specimens are collected. However, a point of diminishing returns is reached where the cost of performing additional strength tests may not be justified. This suggests a limit to the number of test specimens for determining Weibull parameters to obtain a desired level of confidence associated with a parameter estimate. The number of specimens needed depends on the precision required in the resulting parameter estimate or in the resulting confidence bounds. Details relating to the computation of confidence bounds (directly related to the precision of the estimate) are presented in 8.3 and 8.4.1.1 This practice covers the reporting of uniaxial strength data for graphite and the estimation of probability distribution parameters for both censored and uncensored data. The failure strength of graphite materials is treated as a continuous random variable. Typically, a number of test specimens are failed in accordance with the following standards: Test Methods C565, C651, C695, C749, Practice C781 or Guide D7775. The load at which each specimen fails is recorded. The resulting failure stresses are used to obtain parameter estimates associated with the underlying population distribution. This practice is limited to failure strengths that can be characterized by the two-parameter Weibull distribution. Furthermore, this practice is restricted to test specimens (primarily tensile and flexural) that are primarily subjected to uniaxial stress states.1.2 Measurements of the strength at failure are taken for various reasons: a comparison of the relative quality of two materials, the prediction of the probability of failure for a structure of interest, or to establish limit loads in an application. This practice provides a procedure for estimating the distribution parameters that are needed for estimating load limits for a particular level of probability of failure.1.3 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.

定价： 590元 / 折扣价： 502 元 加购物车

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.1.1 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.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, health, and environmental practices and determine the applicability of regulatory limitations prior to use.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.

定价： 646元 / 折扣价： 550 元 加购物车

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.

定价： 646元 / 折扣价： 550 元 加购物车