PSI - Issue 75

Felix-Christian Reissner et al. / Procedia Structural Integrity 75 (2025) 382–391 Felix-Christian Reissner / Structural Integrity Procedia 00 (2025) 000–000

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2. Composition of S-N Models

2.1. Deterministic S-N Models

In engineering literature, S-N models are usually specified as fatigue-life models log 10 ( ˆ N ) = g (log 10 ( S a )) where the estimated fatigue life ˆ N (usually number of cycles) is a function of the load amplitude S a (usually stress amplitude). In this paper, the bilinear Basquin model log 10 ( ˆ N ) =   N k − k 1 � log 10 ( S a ) − log 10 ( S a , k )  , if S a ≥ S a , k N k − k 2 � log 10 ( S a ) − log 10 ( S a , k )  , if S a < S a , k (1) consisting of two linear Basquin models, see Basquin (1910), is used. In Eq. 1, the index k indicates the knee point and k i are the slopes before and after the knee point. This model has exactly one deterministic result. In fatigue en gineering, it is usually assumed that a deterministic result has a survival probability of P S = 50 %. The specification of a fatigue-life model takes into account that the fatigue life is the dependent variable and the load amplitude is the independent variable. It can be viewed as a mathematical description of the relation between fatigue life and load in real world fatigue experiments. However, the relationship can be reversed. In that case, the relation is specified as a load model log 10 ( ˆ S a ) = g (log 10 ( N )), where the load amplitude is a function of the fatigue-life. Linear models like the Basquin model can easily be reversed, as shown in Eq. 2: log 10 ( ˆ S a ) =   S a , k − 1 k 1 � log 10 ( N ) − log 10 ( N k )  , if N ≤ N k S a , k − 1 k 2 � log 10 ( N ) − log 10 ( N k )  , if N > N k (2) If the load model is specified, the dependent and independent variables are switched and the physical relation is lost. However, there are advantages if the load model is used instead of the fatigue-life model. Meeker et al. (2024) show, on the basis of empirical S-N data, that the variance of load amplitude stays more stable than the corresponding variance observed in fatigue-life. This is especially the case if not just the high cycle fatigue (HCF) regime is taken into account. Therefore, if the load model is used, there is less to no need to model a non-constant variance that depends on the fatigue-life. In addition, this reduces the risk of numerical problems if the load model has a horizontal asymptote or a very narrow slope in the long-life fatigue (LLF; N > N k ) regime, which can lead to large discrepancies between the model and the data. Castillo (2009) introduces compatibility conditions for the use of load and fatigue-life models. In addition, Castillo et al. (1985) presents the Castillo model which is specified as load and fatigue-life model. In Meeker et al. (2024), the topic of specifying a load model is discussed in detail. For the following investigations, the bilinear Basquin model as shown in Eq. 1 and 2 is used. To incorporate variability, the deterministic S-N model in this paper is extended using a log-normal error term. Which results in the statistical S-N model log 10 (ˆ x ) = g � log 10 ( y )  + σ x , log ε (3) with ε ∼ N (0 , 1). Depending on the model orientation, x and y represent either load amplitude or fatigue life. This formulation leads to well-known expressions for the cumulative distribution function (CDF) F ( x ) and the probability 2.2. Statistical S-N Models