PSI - Issue 72

554 José A.F.O. Correia / Procedia Structural Integrity 72 (2025) 547–556 failure of the specimen, =1 . Finally, is a parameter that adjusts the damage rate based on the present loading conditions. The damage associated with a specific load can be normalised using the following expression: D i = ∑ sinh( 2 2 N N k f ρ) N k=T 1+1 - ∑ sinh( 2 2 N N k f ρ) N k=T 1 ∑ sinh( 2 2 N N k f ρ) j N = f 1 ≈ ∫ sinh( 2 2 N N k f ρ)dN k NN T T+1 ∫ sinh( 2 2 N N f j ρ) 1 N f dN j (23) This equation represents a continuous approximation obtained by replacing a sum, which is mathematically more rigorous but complex to apply, with an integration that facilitates numerical analysis. As with damage increments, the total damage, a known value, results from the cumulative sum of these damages and can be expressed as follows: D T = ∑ sinh( 2 2 N N k f ρ) N k=T 1+1 ∑ sinh( 2 2 N N k f ρ) j N = f 1 ≈ ∫ sinh( 2 2 N N k f ρ)dN k 1 NT ∫ sinh( 2 2 N N f j ρ) 1 N f dN j (24) In this equation, is the number of equivalent cycles that, if applied at a constant load level, would lead to the current damage state. By integrating this expression, solving for , and considering load reversals instead of full cycles, it is possible to estimate how many reversals of a given amplitude would be required to achieve the observed damage: 2N T =( 2N f ρ )×cosh - 1 (D T (coshρ - cosh( 2 ρ N f ))+cosh( 2 ρ N f )) (25) Huffman and Beckman also proposed an expression for the parameter given by ρ=log(2N f γ ) (26) where a possible formulation for from the same work is γ= - 2ε c f ' (27) where and ′ are the Coffin-Manson parameters for that material. At the beginning of the simulation, an initial damage value (possibly non-zero) is assumed. The first load reversal is taken from the history, and the corresponding damage is calculated based on the current damage state, the applied load and the fatigue life at constant strain amplitude. This value is added to the total accumulated damage. The process is repeated for each new load reversal: the history is traversed, the damage is recalculated based on the new strain amplitude and the new damage state, until the total accumulated damage reaches or exceeds the failure limit, =1 . 4. The proposed methodology The proposed methodology for the probabilistic stress-life prediction of components and connections at variable amplitude loading, based on non-linear damage and probabilistic local fatigue modelling, is given by the following steps (Figure 6): - A probabilistic field must be derived for material fatigue data under constant amplitude loading using the generalised probabilistic fatigue model proposed by Correia et al. (2017), allowing various fatigue parameters based on stress, strain, and energy methods (Step 1). - Predict the fatigue life resistance of components and connections using both fatigue phases (crack initiation and propagation) by the application of the Neuber rule (analytical method) or the finite element method