PSI - Issue 66

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Alfonso Fernández-Canteli et al. / Procedia Structural Integrity 66 (2024) 296–304 A. Fernández-Canteli et al / Structural Integrity Procedia 00 (2025) 000–000

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∗ = , =1 − �− � − 0 � �

(2)

where the model parameters, according with Fig. 1, are: a₀ , the prospective initial crack location, identified as the location Weibull parameter; and , the scale and shape Weibull parameters, respectively; the experimental recorded cycles and , the total number of cycles up to failure.

: initial crack size 0 : retrospective initial crack location : critical crack specimen failure

0 : retrospective initial crack location � = /

0

1

(a) (b) Fig. 1. Schematic representation of the retro-extrapolation (REX) model [3,4]. (a) Non-normalized representation of the a − curve; (b) Representation of the normalized a − curve identified as a Weibull cdf. 2.2. Definition of a corrected crack size In order to avoid the contradictory asymptotic trend of the crack size, according to Expr. (2), an alternative definition of the crack growth equation, as a function of the number of cycles, is proposed. It follows the application of the Weibull cdf identification with the normalized ∗ − equation and substitution of Expr. (3) to recover the new equation, which could be interpreted as a four- parameter Weibull, see Kohout [8]: ∗ = − 0 − (3) Accordingly, the new expression for � becomes: � = =1 − �− � ∗ − 0∗ ∗ � ∗ � =1 − �− � � − −0 − 0∗ ∗ � ∗ � (4) 2.3. Analytical deduction of the / − curve, following the previous derivative / from Expr. (2) = ⎢⎢⎢⎡ ⎣ � � − 0 − − 0∗ ∗ � ∗ ⎦ ⎥ ⎥ ⎥ ⎤ ∗ ∗ �� − 0 − − 0∗ ∗ � ∗ −1 � − 0 � − 2 � (5)