PSI - Issue 4

Shun-Peng Zhu et al. / Procedia Structural Integrity 4 (2017) 3–10

6

S.P. Zhu et al. / Structural Integrity Procedia 00 (2017) 000 – 000

4

2. Shaded areas equal     n f f d f N dN f D dD 

1. Known PDF of fatigue life

D C

Mean damage growth curve l 1

Damage, D

3. Transformed PDF of cumulative damage

O

N f

Number of loading cycles, n

Fig. 2 Format of one - to - one PDF transformation In Fig. 2, curve 1 l represents the trend line of mean cumulative damage in Eq. (4) at a given stress level S , which depicts a nonlinear relationship between the cumulative damage and n . Note that the initial variability of loading cycles equals zero and increases with n . Considering that cumulative damage at S and no initial damage, Eq. (4) can be rewritten as   a D n kn  (6) where   a m C k D S C  . Similarly, the cumulative damage   i D n under multi - level stresses can be expressed by the sum of i a i i k n with   i a m i C i k D S C  . According to the one - to - one PDF transformation [22], two aspects are needed to derive the distribution of cumulative damage   D n : (1) an accurate relationship between cumulative damage and loading cycles (Eq. (6) in this work); (2) the known PDF of loading cycles. In general, fatigue life f N follows a lognormal distribution with a mean of f N  and standard deviation of f N  ,   ~ , f f f N N N LN   , its PDF is   2 ln 1 exp 1 2 2 f f f f N n f N f N N f N N                        (7) Since the cumulative damage relates to fatigue life as shown in Eq. (2), the relationship between the PDF of cumulative damage,   d f D , and that of fatigue life can be derived as     d n f f f D dD f N dN  (8) Eq. (8) can be graphically derived from the equal shaded areas as shown in Fig. 2, which leads to

 

   

2

   

   

D k a  

ln ln



1

 

N

f D

exp 1 2  



(9)

f

d

a



Da

2  

N

N



f

f

It’s worth noting from Eq. (9) that cumulative damage follows a similar distribution as     ~ ln , f f f N N D N LN a k a   

(10)

where the standard deviation of   f D N after transformation is

.

D N a   

f

2.3 Modeling the trend curve of damage variance

Researches shown that the variability or standard deviation of cumulative damage increases monotonically with