Weibull statistics for subcritical crack growth

If a ceramic can fail by subcritical crack growth, its life time when loaded with a certain stress can be calculated from equation (7.2). In section 7.2.6, we used a deterministic approach to do so, but by now we have learned that the failure stress values scatter and the probability of failure follows a Weibull distribution Pf( r). The time to failure, depending on the failure stress, must therefore also be distributed stochastically. 

The failure probability after a certain time t can be calculated when equation (7.2) is solved for a, putting the result into equation (7.6)  

The Weibull modulus m characterises the failure type of the ceramic [2]. If m 1, failure usually occurs shortly after applying the load (so-called infant failure, see figure 7.13). If m = 1, the probability of failure is the 

To measure the failure probability, a large number of experiments are performed on identical specimens, measuring the failure stress (Tj or failure time ti of each. In both cases, the determination of the parameters ( 7o and m or to and m, respectively) is done in the same way. In the following, we use the example of the failure stress. 

One method to determine the distribution of failure stresses is to divide the stress region containing the failure stress values into intervals of width Act as shown in figure 7.14. For each stress interval i, we count the number of specimens that failed at stress values within it. The probability that another specimen will also fail in this stress interval is given by rii/N. Normalising this by the interval width Aa yields the discrete probability density /  

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