Issue 30

D.S. Paolino et alii, Frattura ed Integrità Strutturale, 30 (2014) 417-423; DOI: 10.3221/IGF-ESIS.30.50

the 10 parameters are substituted with their estimates, Eq. (6) provides the relationship between x and y when the probability of failure equals  , which is the definition of the  -th quantile S-N curve. Transition stress may vary from one specimen to another and, in a statistical framework, each specimen can be considered as representative of a particular quantile of the transition stress distribution. Similarly, a particular quantile S-N curve hides out each specimen. Therefore, for a given specimen, both the quantile S-N curve and the quantile of the transition stress distribution are uniquely determined. In particular, let the specimen be representative of the  -th quantile S-N curve (i.e.,

t t X x 

  ) and of the  -th quantile of the transition stress distribution (i.e.,

F

). If the stress amplitude equals the



Y

,

x x  ), then the fatigue life of the specimen corresponds to the transition life

transition stress of the specimen (i.e., if



, t

y y 

of the specimen (i.e., then

). Thus, Eq. (6) becomes:



, t









   

   

   

  





, x b  t Y surf a    

, x b  t Y int a    

y

y

  

 





x





t

t

,

,

Y surf

Y int







t

X

,













Φ

Φ

Φ

1

(7)

  

l

 

 

 

 

X

Y surf

Y int

l

where   denotes a parameter estimate,

y

 is the  -th quantile of the transition life distribution,  is the  -th quantile of the transition stress distribution.

, t

x

, t

   

 



x



t

X

,

Φ

It must be noted that the term

   in Eq. (7) is almost equal to one since, according to the hypotheses stated

l



X

l



X must be larger than

l X (i.e.,

x

in the definition of the unified statistical model [8], t

). By taking into account





t

X

,

l



1     



x



t

X

,

Φ

that

, Eq. (7) can be reformulated as follows:

l



 

 

X

l





   

   



, x b  t Y int a    

y



t

,

Y int

Φ

 

Y int





(8)









   

Φ         

   





, x b  t Y int a    

, x b  t Y surf a    

y

y





t

t

,

,

Y int

Y surf

1 Φ 

 

 

Y int

Y surf

Eq. (8) provides an implicit relationship between , t y  the statistical distribution of the transition life.

and  and, consequently, permits the numerical computation of

N UMERICAL EXAMPLE

A

n experimental dataset taken from the literature [13] is analyzed in order to show main characteristics of the statistical distribution of the transition life. The selected experimental data [13] are obtained by testing Ti-6Al-4V titanium alloy specimens and are shown in Fig. 2. Estimates of the parameter involved in the model given in Eq. (6) can be computed by applying the Maximum Likelihood Principle to the experimental data. Results, obtained with a code developed in Matlab®, are given in the following list:



Y surf    a

 

 



b

100.20

33.26 11.67

0.4639 0.3280

Y surf

Y surf

  



 



 



a

b

40.36

  

Y int

Y int

Y int

(9)

   

t    X   

 

 

2.8192 2.7200

0.0023 0.0059

X

t



X

X

l

l

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