PSI - Issue 72

José A.F.O. Correia / Procedia Structural Integrity 72 (2025) 547–556

549

A

1

Double Linear damage Rule

Linear damage Rule

B β 2 = n 2 / N F2 , Remaining Life Ratio

(N int,1 /N F1 ; N prop,2 /N F2 )

C

0

1

β 1 = n 1 / N F1 , Fatigue Life Ratio

Fig. 1. Double linear damage rule applied to two-level loading sequences (adapted from Correia, De Jesus, et al., 2016).

A key issue with theories on fatigue damage accumulation is that they are often developed in a deterministic framework, despite the recognised probabilistic nature of fatigue lifetimes. While there have been some attempts in the literature to address fatigue from a probabilistic standpoint, these efforts primarily rest on linear damage summation methods (Pinto et al., 2010; Fernández-Canteli et al., 2014; Correia et al., 2015; Correia et al., 2016). Preliminary research on a probabilistic approach to non-linear cumulative fatigue damage was conducted by Correia, De Jesus, et al. (2016). This approach combines the Double Linear Damage Rule (DLDR) with the P-S-N Weibull fields derived for constant amplitude loading. For high-low (H-L) and low-high (L-H) loading sequences, the DLDR can be expressed as follows: { 1, =0.35( 1, 2, )  ( ) 2, =0.65( 1, 2, )  ( ) ; 1, 2, ≤ 1 ( − ) (5) and { 1, =1−0.65( 2, 1, )  ( ) 2, =1−0.35( 2, 1, )  ( ) ; 1, 2, ≥ 1 ( − ) (6) where, 1, and 2, are the fractions of lives spent at each load level related to a probability ,  ( ) is the probabilistic distribution function using the Weibull distribution derived from the experimental data, 1, and 2, are the numbers of cycles to failure associated to a probability based on the Castillo & Fernández-Canteli fatigue model (Castillo and Fernández-Canteli, 2009), using the values from the low and high strain ranges, respectively, given by 1, = [ + + (− (1− )) 1⁄ (∆ )− ] (7)