Statistical approach to fatigue crack growth under random loads considering loading sequence effects

This paper presents a model for statistical analysis of crack growth under stationary, gaussian random loading processes including the loading sequence effect. The model defines and incorporates an equivalent closure stress, which is included into the fatigue crack growth law through the effective stress intensity factor. The equivalent closure stress for each loading process is obtained from the probability density function of peaks, p(S), the crack growth properties of the material, the specimen geometry and one of the existing analytical approaches to the closure stress produced by an overload. An equivalent overload stress, S-ol, is obtained using the following scheme. First, an initial value of this overload stress, S-q, is assumed. From p(S), the average number of cycles, N-p, between two consecutive equivalent overloads is obtained as the inverse to the probability of having a peak equal or higher than S-q. Using S-q as a representative overload stress, the value of the closure stress, S-cli, produced by such an overload is obtained analytically using one of the known expressions of the closure stress as a function of the overload stress and the stress ratio, R. From this S-cli value, a linear law of variation of the closure stress, S-cl, while the crack is growing through the overload plastic zone, r(ol), is proposed. The number of cycles, N-pi, to grow the crack through this zone is obtained from a crack growth rate equation using effective stresses and considering a constant stress amplitude between overloads equal to the root mean square of the peaks in the load history, S-rms, and the varying S-ol. The value S-q that yields N-p = N-pi is adopted as representative of the overload stresses produced during the load history, S-ol, and the corresponding N-p value as the representative number of cycles between overloads. Knowing N-p, a representative closure stress, S-clr, is obtained as the constant closure stress value with which the increase of the crack length is equal to r(ol) in N-p cycles. By doing the same process for different crack lengths, a, it is possible to fit an expression of the representative closure stress as a function of a for the loading process under analysis, S-clr(a). Finally, an estimation of the number of cycles to growth the crack from the initial to the final crack length is obtained using the equation: N = (a0)integral(af) da/(-infinity)integral(infinity) p(S)f(a)(S - S-clr(a))(m) dS The results of the model summarized above as well as those of a cycle-by-cycle simulation are compared to the average life of more than twenty specimens of a 2024-T351 aluminium alloy.