Weibull ceramic

The Kawakita compaction equation is another equation which is often used for ceramic powder pressing. It can be derived by considering that compaction is similar to packing by tapping, where the compaction pressure, P, is directly substituted for the number of taps, N, in the analysis in Section 13.5.1. The Kawakita equation is a special case, where the value of m in the Weibul distribution function for tapping is 1. In the Kawakita equation, the compaction, C, defined as the relative reduction in volume is given by [72]... 

The mechanical performance of femoral heads and acetabular cups made from alumina have been the subject of intense research and development effort as it is crucial for the longevity of the endoprosthetic hip implant. Hence, the failure probability of these ceramic construction parts has been investigated and expressed by the Weibull probability density function. As safe design of... 

One can describe the strength distribution of a ceramic in a variety of formalisms. The one most widely used today is the Weibull distribution This two-parameter semiempirical distribution is given by... 

This is an important result since it indicates that the survival probability of a ceramic depends on both the volume subjected to the stress and the Weibull modulus. Equation (11.30) states that as the volume increases, the stress level needed to maintain a given survival probability has to be reduced. This can be seen more clearly by equating the survival probabilities of two types of specimens — test specimens with a volume and component specimens with volume Tcomp- Equating the survival probabilities of the two types of samples and rearranging Eq. (11.30), one can easily show that... 

The brittle nature of ceramics together with the stochastic nature of finding flaws of different sizes, shapes, and orientations relative to the applied stress will invariably result in some scatter to their strength. According to the Weibull distribution, the survival probability is given by... 

Ceramic Mean fracture stress Weibull modulus... 

The service conditions are geometrically identical to the test conditions and impose a stress of 300 MPa. By constructing Weibull graphs with S — /2 for mean fracture stress or any other method, decide which ceramic will be more reliable and compare the probabilities of failure at 300 MPa. At what stress would the two ceramics give equal performance ... 

The Weibull distribution is the state of the art statistics in the mechanical design process of ceramic components [1 - 3]. Strength testing and data evaluation are standardised. A sample of at least 30 specimens has to be tested. The range of measured failure probabilities increases with the sample size [3, 13] and is - for a sample of 30 specimens - very limited (it is between 1/60 and 59/60). To determine the design stress, the measured data have to be extrapolated with respect to the volume and to the tolerated failure probability. This often results in a very large extrapolation span [3]. 

Fig. 1 a) Strength data of a silicon nitride ceramic tested in four point bending (4PB) in a Weibull plot and b) the relative frequency distribution of flaw sizes. The data points were determined by fracture experiments. 

A further and very important consequence of the Weibull distribution is the size effect, i.e. the mean strength decreases with increasing specimen size. This is the most important consequence of fracture statistics for designing with ceramics. 

R. Danzer, P. Supancic, J. Pascual, and T. Lube, Fracture Statistics of Ceramics - Weibull Statistics and Deviations from Weibull Statistics, Engineering Fracture Mechanics, 74, 2919-2932, (2007). 

R. Danzer, Some Notes on the Correlation between Fracture and Defect Statistics Are Weibull Statistics Valid for Very Small Specimens , J. Eur. Ceram. Soc., 26, 3043-3049, (2006). 

G. D. Quinn, "Weibull Strength Scaling for Standardized Rectangular Flexure Specimens", Journal of the American Ceramic Society, vol. 86, pp. 508-510, 2003. 

The strength variability of ceramic materials can be evaluated using Weibull stahshc, which is based on the weakest-link theory, where the more severe flaw results in fracture propagation and determine the strength [69]. The Weibull two-parameter distribution is given by [14] ... 

Lamon J. Ceramics reliability statistical analysis of multiaxial failure using the Weibull approach and the multiaxial elemental strength model. Journal of the TVmeiican Ceramic Society 1990 73(8) 2204-2212. 

ASTM C 1239-07 Standard practice for reporting uniaxial strength data and estimating Weibull distribution parameters for advanced ceramics. West Conshohocken ASTM International 2007. 

C1239-95 Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Ceramics... 

Flaws dominate the mechanical properties of ceramics. They determine how we test them and how we design components from them. Flaws are also the reason why ceramics are stronger in compression than tension. In this chapter we described the methods used to measure mechanical properties of ceramics. The important ones are bend testing, compression testing, and indentation. To determine the mechanical properties of small volumes we use nanoindentation. This technique is especially important for thin hlms, surfaces, and nanomaterials. An understanding of statistics is particularly important when using ceramics in load-bearing applications. The Weibull approach is the one most widely used for ceramics. 

This has been demonstrated for a commercial glass ceramic [38], and for alumina [39]. It has been suggested that the microstructure and the defect populations themselves may vary throughout the volume of a component, causing variations in strength unpredictable by Weibull analysis [40,41]. An attempt to verify the minimal specimen size required for strength determination was carried out lately [42]. 

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