13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- Figure 3: Crack propagation data of 42CrMo4 and the adaption by the Forman/Mettu-equation 3.2. Statistical analysis The statistical analysis of the crack propagation data is divided in three steps. First for each stress ratio quantile curves are calculated for different probabilities of survival. Next, for each probability of survival the quantile curves are fitted by the Forman/Mettu-equation to obtain the fitting coefficients. Those are statistically analyzed in the last step to obtain their distribution functions. To determine quantile curves the logarithm is taken from the crack propagation data. Afterwards the co-domain of the crack propagation data is divided into intervals. For each interval a polynomial regression function and a confidence interval for a pretended confidence probability are calculated. Therewith and for each domain discrete values are calculated for the mean and the upper and lower bound of the confidence interval. By use of interpolation functions the transition between the domains is smoothed and equidistant spaced values are calculated. Taking the antilogarithm the three quantile curves are obtained e.g. for a stress ratio R = 0,1, Figure 4. This procedure is also applied to the crack propagation data of the other stress ratios. Every quantile curve corresponds to a probability of survival P. Taking the quantile curves for one probability of survival the fitting coefficients of the threshold value and the Forman/Mettu-equation are determined by use of the adaption program. For the statistical analysis of the crack propagation data of 42CrMo4 the probabilities of survival P = 5 %, P = 50 % and P = 95 % are used. The corresponding fitting coefficients are listed in Table 1. As can be seen the analytical crack propagation curve is shifted to top left with increasing probability of survival. In the last step every fitting coefficient is statistically analyzed. This contains the choice of a distribution function and the calculation of the mean value and the standard deviation. For all fitting coefficients a normal distribution is used to describe the scattering, except of the coefficient C, Table 1. Here, a logarithmic normal distribution leads to the best results.
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