Modelling curved S–N curves

Abstract

A B S T R A C T The fatigue limit distribution is estimated using fatigue data and under the assumption that the fatigue limit is random. The stress levels for the broken and unbroken specimens are used. For the broken specimen the number of cycles to failure is also used. By combining the finite life and fatigue limit distribution it is possible to get the probability of not surviving a certain life. This probability is used to estimate a curved S-N curve by using the method of likelihood. The whole S-N curve is estimated at the same time. These curves show the predictive life given a certain stress level. The life and the quantile of the fatigue limit distribution are also predicted by using profile predictive likelihood. In this way the scatter around the S-N curve as well as the uncertainty of the S-N curve are taken into account.Keywords fatigue limit; finite life; predictive profile likelihood; staircase test. N O M E N C L A T U R E1 = indicator function C = set of censored observations e yx = n x f , g = density function l y = likelihood function L p , L ps = profile predictive likelihood L p ,L ps = normalized profile predictive likelihood N f = finite life n ro = run-out level n x = life given a stress level S q = probability S = stress level S ∞ = fatigue limit U = set of uncensored observations x = ln e (S) y = ln e (N ) y ro = ln e (n ro ) Z = random variable θ = parameter µ = mean of fatigue limit distribution ν (x) = mean of logarithm of the finite life distribution σ 2 = variance of fatigue limit distribution τ 2 = variance of logarithm of the finite life distribution = standard normal cumulative distribution · = estimate Correspondence: Sara Lorén.

Fitting fatigue data with a bi‐conditional model

“…Again, Φ is either the normal or the SEV CDF, and, for the sake of clarity, the choice can be different for F end and F exi , leading to four combinations. Although it is the result of different assumptions and framework, this resulting CDF is very similar to the one presented in Loren et al 15 (see also Loren 21 ) and to the case of 'one failure mode due to one cause with fatigue limit' in Paolino et al 16 The marginal PDF of the BCM is the partial derivative of F N f with respect to N ( Fig. 4f):…”

“…or, in other words, when the applied stress is lower than the fatigue limit at the (randomly) chosen probability (15). In case of applications where censored data are involved, a further effort must be employed in order to properly take them into account.…”

“…However, many models like the Random Fatigue Limit Model 13 or the one proposed by Castillo and Fernandez-Canteli, 14 although built on strong statistical basis, tie a fairly strong bond between the High (sometimes the Low) and Very High Cycle Fatigue behaviour, implicitly assuming similarities in the probabilistic outcome of different damaging mechanisms. A more disjointed approach is proposed by Loren et al 15 and Paolino et al, 16,17 whose models allow for a different degree of scatter in strength between the finite life region and the fatigue limit.…”

Analysis of selected mathematical models of high-cycle S-N characteristics

“…Also, considering that these values are random, it should be assumed that the likelihood of fracture will be equal to the likelihood of fulfilling function (5) and function (6). With this assumption, the probability of fracture for the limited and unlimited fatigue life scope may be expressed as follows (LORÉN, LUNDSTRÖM 2005):…”

Experimental Method for Plotting S-N Curve with a Small Number of Specimens

“…(11) and (13). With this assumption, the probability of failure for limited and unlimited fatigue life regime can be expressed as follows, [16]: (14) where Φ is the normal distribution function and q is the probability of specimen failure.…”