PSI - Issue 72

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

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1. Introduction In 1924, Palmgren (Palmgren, 1924) introduced the concept of the Linear Damage Rule (LDR) to assess the lifetime of roller bearings. Nearly twenty years later, in 1945, Miner (Miner, 1945) formalised the first mathematical equation to evaluate fatigue damage accumulation in materials and components. This equation is expressed as =∑ = 1 (1) where represents the fatigue damage, is the number of cycles at applied stress range, and is the number of cycles to failure at his stress range. The Miner’s model is commonly cited in engineering regulations as the Palmgren -Miner damage rule, honouring its proponents. It is well- established that Miner’s model does not address the effects of sequential loading. When these sequential effects are significant, non-linear damage accumulation models become necessary. Due to these shortcomings and the limitations of the Linear Damage Rule (LDR) model, Marco and Starkey (Marco and Starkey, 1954) introduced the first non-linear load-dependent damage theory in 1954. The theory, referred to as the Marco Starkey theory, is defined by a power relation: = ∑ = 1 =∑ ( ) = 1 (2) In this equation, represents a variable quantity associated with the th loading level (load function). When is equal to 1 , it indicates that this theory corresponds to a specific case of the linear damage rule, known as the Palmgren-Miner rule. In 1960, Grover (Langer, 1937; Grover, 1961) proposed a two-stage linear damage approach that considered stress cycle ratios for two distinct stages in the fatigue damage process under constant amplitude loading. This approach differentiates between the damage due to crack initiation, denoted as = , and the damage due to crack propagation, denoted as =(1−  ) . Here, represents a life fraction factor for the crack initiation stage. In a study conducted from 1966 to 1981, Manson and Halford introduced the Double Linear Damage Rule (DLDR) (Manson and Halford, 1981). This rule facilitates the evaluation of cumulative fatigue damage for high-low (H-L) and low-high (L-H) loading sequences, as illustrated in Figure 1. The coordinates of the knee points on the DLDR lines are defined by

{ 1 =0.35( 1 2 ) 0.25 2 =0.65( 1 2 ) 0.25 ; { 1 =1−0.65( 2 1 ) 0.25 2 =1−0.35( 2 1 ) 0.25 ;

1 2 ≤ 1 ( − )

(3)

1 2 ≥ 1 ( − )

(4) where 1 and 2 are the fractions of lives spent at each load level, 1 and 2 are the number of cycles to failure for lower (L) and higher (H) fatigue lives. In Figure 1, the linear damage rule can be understood by examining the connection between points and . In contrast, the double linear damage rule is illustrated by the connection among points , and . The coordinates of point are determined by the relationship between ,1 1 ⁄ and ,2 2 ⁄ .