Probabilistics 3

“Probabilistics” 3

There is more to Monte Carlo simulation than replacing constants with probability densities.

Monte Carlo Modeling Specifics:

After fitting individual Paris equations to each of the 68 specimens, the mean and standard deviation for the individual Paris parameters, intercept, \(C\), and slope, \(m\), were computed. The well-known Paris model for fatigue crack growth is given in equation 1.

\[da/dN=10^C (\Delta K)^m  \tag{1} \]

where \(da/dN\) is the crack growth rate, in \(mm\) per cycle, and \(\Delta K\) is the applied stress intensity factor, in \(MPa \sqrt{m}\), given by equation 2.

\[\Delta K = 10^{-C} \Delta \sigma \sqrt{\pi a}f(a|geometry) \tag{2}\]

Here, \(\Delta \sigma\) is the testing stress range, \(\sigma_{max} – \sigma_{min}\), \(a\) is the crack length, and \(f( )\) is a function of the specimen (or component) geometry and crack length. Of course, when equation 1 is plotted on a log-log grid this is a straight line with intercept \(C\) and slope \(m\).

Assuming for the sake of simplicity that there was no variation in the starting crack size, the final crack size, or the test stress, the calculated cyclic lifetime can be computed from the individual Paris fits using equation 3.

\[da/dN = 10^C \Big(\Delta \sigma \sqrt{\pi a}f(a|geometry)\Big)^m \]

\[dN = da/ \bigg(10^C \Big(\Delta \sigma \sqrt{\pi a}f(a|geometry)\Big)^m \bigg) \]

\[N=\int_{a_0}^{a_{final}} 10^{-C}\Big(\Delta \sigma \sqrt{\pi a}f(a|geometry)\Big)^{-m} da  \tag{3} \]

In practice this integration is usually carried out numerically.

To conduct the usual MC simulation Ni is computed from \(h(C_i , n_i)\) where \(h( )\) is equation 3, and \(i\) ranges from 1 to say 1000 (or 10 000).

Many MC practitioners then calculate a mean and standard deviation for \(N\), or \(\log10(N)\), report the results and stop there, since there is nothing against which to compare the distribution of computed values for \(N_i\).

In this case, however, Virkler’s data show the observed distribution of actual specimen lives and thus provide a direct comparison for these calculations.