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42) is more justiﬁed. 42) has been subsequently studied and successfully applied to diﬀerent cases of lifetime problems, such as, for instance, prestressing wires and strands of diﬀerent lengths, plain concrete, etc. (see Castillo et al. (1985)). 42) implies that: V = (log N − B)(g(Δσ) − C) ∼ W (λ, δ, β). 47) reveal that the probability of failure of a piece subject to a stress range Δσ during N cycles, depends only on the product β V = (log N − B)(g(Δσ) − C) or V = (log N − B) (g(Δσ) − C)γ , showing that V is useful to compare fatigue strength at diﬀerent, but constant, stress levels, and can be considered as a normalizing variable.
10 Appendix A: Derivation of the general model . 11 Appendix B: S-N curves for the general model E. Castillo, A. V. 2009 . . . . . . . . . . . . . . . . . . . 36 38 41 41 42 43 43 45 48 49 49 53 55 56 57 59 64 65 69 71 72 84 85 89 35 ¨ CHAPTER 2. 1 Introduction In the evaluation and prediction of the fatigue lifetime of machines and structures the role of mathematical and statistical models is crucial, due to the high complexity of the fatigue problem, in which the consideration of the stress range, stress level and the size eﬀect, together with an eﬃcient estimation of the corresponding parameters represents one of the most diﬃcult and attracting challenges, which have not yet been satisfactorily solved.
In the case of Fig. 3 we dealt with two minimum laws, one for N ∗ |Δσ ∗ and one for Δσ ∗ |N ∗ , which are associated with the weakest link principle, and here we are dealing with one minimum law N ∗ |a∗ , and one maximum law a∗ |N ∗ , because a∗ is the largest crack size. 6. 10: Illustration of the compatibility condition showing equal areas (probabilities) of the two intersecting densities. where qmax and qmin are distributions for maxima and minima, respectively. Taking into account that these distributions satisfy the condition (see Castillo (1988); Castillo et al.