Weibull statistics failure probability

The probability of fiber failure is described by Weibull statistics. The probability of failure within a gauge section of length L is given by... 

Brittle materials fail by unstable crack propagation from a pre-existing flaw. The size, orientation, and shapes of the flaws in a given flaw population determine the variability of the failure stress the statistical failure probability, F, first formulated by Weibull [34] and described by the Weibull function ... 

The investigation of failures of manufactured components and systems, especially in the electronics and aerospace industries, has generated a variety of statistical models on which data analysis may be based. Each model uses a specific distibution of failure probabilities, and it is important to select a model that matches the actual distribution inherent in the product concerned. In the case of dielectric breakdown, where a large number of quite different modes of failure are known to occur, sometimes even together, the application of a particular statistical failure model must be approached with great caution. Nevertheless, one treatment, based on a Weibull distribution of failure probability, has taken root, and is most generally used in practice. For a dielectric, the Weibull failure probability function has the form... 

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]. 

Bazant formulated a statistical theory of fracture for quasibrittle materials [5, 23, 24]. He assumed that there exist several hierarchical orders which each can be described by parallel and serial linking of so-called representative volume elements (RVEs). For large specimens (and low probability of failures) the fracture statistics is equal to the Weibull statistics, i.e. if the specimens size is larger than 500 to 1000 times of the size of one RVE. In the actual case this is similar to the diameter of the critical flaw. For smaller specimens the volume effect disappears and the fracture... 

Actually, it was Leonardo da Vinci (1452-1519) who first noticed that the strength of steel wire increased for decreasing wire length. Indeed, assuming uniform stresses in test pieces having different volume but the same failure probability, Weibull statistics predicts a strength ratio ... 

Modeling. In order to carry out the analysis of the nature of the operational phenomena in facilities and equipment, it is very useful to use statistics as a support for the quantification of the parameters. The phenomena s historical behavior is characterized based on operation and failure periods that have occurred since the commissioning time. The conditions that characterize the equipment operational time data are so numerous that it is not possible to say when exactly the next failure will occur. However, it is possible to express which will be the probability that the equipment is in operation or out of service at any given time. These times are associated with a cumulative distribution function of the random variable, which is defined as the addition of the probabilities of possible values of the variable that are lower or equal to a preset value. The mentioned random variable is constituted by the operating times and downtime of equipment or system in a given period. For its parameterization Weibull distribution is very appropriate as it is very effective and relatively simple to use in the reliability evaluation of a system by quantifying the probability of failure in the performance of the system s duties from the failure probabilities of its components based on the operation times. There are three different parameters ... 

A statistical definition of brittleness can be formulated in terms of the Weibull distribution of fracture probability for a material (Derby et al., 1992). The Weibull modulus m (see Eq. 2) can range from zero (totally random fracture behaviour, where the failure probability is the same at all stresses, equivalent to an ideally brittle material) to infinity (representing a precisely unique, reproducible fracture stress, equivalent to an ideally non-brittle material). 

Future work must address two areas to provide the foundation for statistically based analyses of high-cycle CF (as well as environmental LCF and FCP). For simple laboratory conditions, the Weibull analysis of mechaniccil HCF failure probability [82] must be extended to include CF. Second, variable load, temperature, and environment chemistry histories are likely to be complex in applications and significantly affect CF Hfe. Such history effects have not been studied. The scaling of Basquin relationship data to predict the Ufe of a structure is qualitative and uncertain. Either the local strain approach to CF crack formation/eeurly growth life or the fracture mechanics analysis of CF propagation provide a better foundation for life prediction and failure analysis. 

K-S Test. The Kolmogorov-Smirnov goodness of fit test assesses the ability of a probability distribution calculated from WEiBULL STATISTICS (q.v.) to fit the experimental data. It and the A-D (Anderson-Darling) test (which is more sensitive to discrepancies at low and high probabilities of failure) are used as part of the CARES (q.v.) computer program for failure prediction. For a discussion of... 

In reality, fibre properties are statistically distributed. This is true for their geometry (length and diameter), but also, especially in the case of ceramic fibres, for their mechanical properties that are distributed according to Weibull statistics (see section 7.3). Non-ceramic fibres are also usually not identical since they may contain surface defects, for instance. Because of this statistical distribution of their properties, not all fibres fail simultaneously even in a homogeneously loaded composite. Instead, the weakest fibre will fail first. Due to the volume effect (see section 7.3.1), the failure probability of a long fibre is greater than that of a short one. 

It may seem surprising that the notch support factor of brittle materials, like cast iron, is rather large (see figure 10.36). This is due to the fact that cracks in cast iron start at the graphite particles which act as inner defects and are statistically distributed. It is rather improbable that the crack that determines failure behaviour (the largest crack) is situated exactly at the notch root where the stress concentration becomes important. This is analogous to the dependence of the failure probability on the material volume according to the Weibull statistics (see section 7.3). This notch support in brittle materials also occurs under static loads. 

For the quantitative comparison of the critical stress for maincrack formation under thermal shock, oth, with mechanical loading, oc, the volume effect of strength should be considered. According to Weibull statistics, the cumulative probability of failure of brittle materials is written in the following simplified form. 

In Weibull statistics, the failure probability function, P a) is given as... 

The statistical description of the strength was introduced by Weibull. In his model the assumption is made that failure is due to sudden catastrophic growth of pre-existing defects corresponding to local failure stresses. Failure at the most serious defect, i.e. the defect with the lowest fracture stress, leads to immediate failure of the fiber. It is further assumed that the defects are uniformly distributed throughout the fiber. The cumulative failure probability function, P, which represents the fraction of fibers that fail at or below a stress cr is, according to Weibull, given by... 

Although the TOFC state-of-the-art anode-supported cells possess a world record mechanical average strength of about 400 MPa [10], (as measured in round robin tests in previous EU projects), statistical analysis of the strength behaviour bears evidence that most cells may behave unreliably under real dynamic operation conditions. Expressed according to Weibull statistical analysis (failure probability as function of applied mechanical stress), the Weibull modulus is in the range of 8-10 (see Fig. 9). 

Tertiary failure processes akin to this have been modeled by Phoenix and co-workers13-15 in the context of epoxy matrix composites. Indeed, they show that such tertiary failure can occur even when the fiber strength is statistical in nature. This mechanism will not be pursued further in this chapter but some other basic results considered on the assumption are that when there is a sufficient spread in fiber strengths, such tertiary failures can be postponed well beyond the occurrence of first fiber failure or indeed eliminated completely. Thus, attention will be focused on fibers which obey the classical Weibull model that the probability of survival of a fiber of length L stressed to a level oy is given by... 

A statistical analysis of the liber tensile strength values determined on a series of fiber samples can be easily made by using the two-parameter Weibull distribution described above. Using the form of the Weibull expression given in Eq. (10.2), we can write the probability of failure F(o) of the liber at a stress a, as... 

Unlike that for the classical linear responses of such solids, the extreme nature of the breakdown statistics, nucleating from the weakest point of the sample, gives rise to a non-self-averaging property. We will discuss these distribution functions F a), or F(/), or F E) giving the cumulative probability of failure of a disordered sample of linear size L. We show that the generic form of the function F a) can be either the Weibull (1951) form... 

Weibull developed his statistical theory of brittle fracture on the basis of the weakest link hypothesis, i.e. the specimen fails if its weakest element fails [6, 7], In its simplest form and for an uniaxial homogenous and tensile stress state, ct, and for specimens of the volume, F, the so called Weibull distribution of the probability of failure, F, is given by ... 

Prepare Prediction of an unobserved random variable is a fundamental problem in statistics. The aim of this paper is to construct lower (upper) prediction limits under parametric uncertainty that are exceeded with probability 1—a (a) by future observations or functions of observations. The prediction limits depend on early-failure data of the same sample from the two-parameter Weibull distribution, the shape and scale parameters of which are not known. 

The statistical strength of fibers is usually described as follows. If the probability of failure, F(a), of a single fiber unit element is a Weibull distribution... 

Outline Life of product population is predicted with the help of Weibull analysis in which a statistical distribution is attempted to fit into life data from a representative sample of units. Then same data set can be used for estimation of important life parameters/cbaracteristics such as reliability or probability of failure at a specific time, the mean life, and the failure rate. For Weibull data analysis, the following information are required ... 

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