Weibull statistics

Let us say that a series of identical samples are tested to failure. From such tests we can obtain the fraction of identical samples, each of volume that survives when loaded to a given stress, a. Let us call this According to the Weibull distribution, this survival probability is given by 

We show in Section 10.5 how the Weibull modulus can be obtained graphically by using a double log plot of such an equation that gives a straight line of slope (3. 

Usually, ceramics fail as soon as a crack starts to propagate. Therefore, their strength is determined by the stress value at which the first, and thus critical, 

To describe the probability of failure, a statistical approach is needed that takes into account the statistical distribution of the density and size of the cracks [104]. 

If we consider a component with homogeneous stress distribution and known number of defects, the size distribution of the defects can be used to determine the failure probabihty. This is equal to the probability that at least one crack has the critical crack length. As the critical crack determines failure, only the largest defects are relevant. 

The probability of finding a large defect eventually becomes smaller with increasing defect size therefore, different defect size distributions will look similar in the relevant region. The details of the defect size distribution are thus not important. 

Because the number of defects differs between different components, the defect density distribution must also be taken into account by using it to accumulate the probability of failure for all possible numbers of defects. 

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Electrical endurance data, including that for plastic insulation, are frequently analysed using a Weibull statistical technique (see Section 8.17). Three ageing models are used in the literature thermodynamic [29], space charge life [30] and electrokinetic endurance [31]. As summarised in [32], they share some similarities in that they assume that there are several stages in the ageing process and that the dominant one involves the rupture and repair of chemical bonds within the plastic. The space charge life and electrokinetic... 

Weibull statistics are widely used in analysing the lifetime of components ( lifing ). Based on probability theory, they apply the formula ... 

The Survivor function, F(t), is the number of elements of the statistical sample which have not failed or lost their function at time t and are still working. In the case of a Weibull statistic model, the survivor function is given by the relation ... 

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

Following Huttinger (1990), we can correlate the modulus and strength of carbon fiber to its diameter. We make use of Weibull statistics to describe the mechanical properties of brittle materials (see Chapter 10). Brittle materials show a size effect, i.e. the experimental strength decreases with increasing sample size. This is demonstrated in Fig.8.10 which shows a log-log plot of Young s modulus as a function of carbon fiber diameter for three different commercially available carbon fibers. The curves in Fig. 8.10 are based on the following expression ... 

The MORs of a series of cylindrical samples (/ = 25 mm and diameter of 5 mm) were tested and analyzed using Weibull statistics. The average strength was 100 MPa, with a Weibull modulus of 10. Estimate the stress required to obtain a survival probability of 95 percent for cylinders with diameters of 10 mm but the same length. State all assumptions. 

In this paper the Weibull theory is applied to very small specimens. The analysis follows the ideas presented in [13]. The relationships between flaw population, size of the fracture initiating flaw and strength are discussed. It is shown that a limit for the applicability of the classical fracture statistics (i.e. Weibull statistics based on the weakest link hypothesis) exists for very small specimens (components). 

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... 

However, this is not true for very small specimens. Here the flaw densities become so high that interaction between flaws becomes possible. Then Weibull statistics predicts too high a strength i.e. there exists an upper limit of strength. 

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. J. DeSalvo, Theory and Structural Design Applications of Weibull Statistics, Report WANL-TME-2688, Westinghouse Electric Corporation, 1970. 

CARES (Ceranfics Analysis and Reliability Evaluation of Structures) is a public-domain program from the National Aeronautic and Space Agency (NASA) that incorporates Weibull statistics. The program was formally known by the less friendly acronym SCARE (Structural Ceranfics Analysis and Reliability Evaluation). 

The following considerations and assumptions apply to the use of Weibull statistics ... 

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 ... 

Another feature of this material is its narrow distribution of the fracture strength. When the strength distribution was expressed in Weibull statistics, the Weibull modulus was 46 - substantially higher than the value of 26 obtained for a conventional, self-reinforced silicon nitride. Thus, the seeded and tape-cast silicon nitride showed a synergistic improvement in all of the important fracture attributes, such as fracture strength, fracture toughness, and strength stability (Weibull modulus). 

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