PSI - Issue 72

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

553

- n ′ : Cyclic strain-hardening exponent; - ∆σ : Stress range; - ∆σ∆ε P : Plastic strain range; - E : Elastic modulus; - σ m : Mean stress; and, - ∆W t : Total strain energy range.

Correia et al. (2017) introduced a generalisation of the probabilistic fatigue model proposed by Castillo and Fernández-Canteli (2001), which accommodates various local fatigue parameters, as illustrated in Figure 5, and given by p=1 - exp{ - [ log(N f N 0 ⁄ )∙log(ψ ψ 0 ⁄ ) - λ δ ] β } (19) log(N f N 0 ⁄ )∙log(ψ ψ 0 ⁄ )≥λ (20) where N 0 and ψ 0 are normalisation values for the lifetime threshold and fatigue limit, respectively, and λ , β and δ are the parameters of the Weibull distribution.

1

p=0

B = Log N 0

Weibull or Gumbel distributions

p=0.05

Log 

p=0.5

probability, p [-]

p=0.95

C = log  0

0

v =(log N - B )(log ψ - C )

LogN f

Fig. 5. Generalised probabilistic fatigue fields allowing various fatigue parameters (adapted from Correia et al., 2017).

3. Non-linear accumulation fatigue model proposed by Huffman & Beckman A non-linear fatigue damage model using the Coffin-Manson constants for materials was suggested by Huffman and Beckman (2013). In this model, the total accumulated damage in a material is calculated as the sum of the damage increments caused by individual events. This relationship can be expressed as: D T =∑ D i 2N T i=1 (21) where is the state of damage, is the damage increment of event , 2 is the total number of damaging events in reversals, and is increment number. Each damage event is interpreted as a crack growth process – an approach previously applied in fatigue modelling – and is described by a hyperbolic sine function: d d N a ∝sinh( 2N k 2N f ρ) (22) In this equation, / represents the damage rate associated with the current load. The term, 2 , “effective age” of a specimen, refers to the number of load reversals required at a specific loading condition to reach the current state of damage ( ). Additionally, 2 indicates the number of reversals at that loading level that would lead to the