PSI - Issue 4

Shun-Peng Zhu et al. / Procedia Structural Integrity 4 (2017) 3–10 S.P. Zhu et al. / Structural Integrity Procedia 00 (2017) 000 – 000

5 3

2.1 Modeling the mean value of cumulative fatigue damage

Consider a railway axle subjected to constant amplitude loading. Assuming that cumulative fatigue damage   D n at loading cycles n is initially equal to 0 D , and then accumulates monotonically. While ignoring the environmental and frequency - based effects on   D n , the damage accumulation rate generally depends on 0 D , the actual state of damage, and the loading stress amplitude S . For the real engineering components, this dependency on the actual damage state can be characterized by the number of loading cycles n . Note that a nonlinear description is more appropriate for damage accumulation modeling according to the nature of fatigue. Based on the above definitions, the general form of cumulative fatigue damage curve in Fig. 1 can be expressed as     0 0 , a D n D f S D n   (1) where   0 , f S D describes the rate of damage accumulation, a is a damage exponent. Based on the boundary conditions and failure criterion, assuming that failure occurs when the cumulative damage   D n equals the critical damage threshold C D , which leads to

a

 N           0 0 C f n

  D n D D D

(2)

f N and stress level S as

The S - N curve describes the relationship between fatigue life

m f N S C 

(3)

where C is a fatigue strength constant and m is the slope of the S - N curve. Substituting Eq. (3) into Eq. (2) leads to

 C           0 0 m a C S

  D n D D D

a

n

(4)

Fig. 1 illustrates possible damage accumulation curves as a function of loading cycles described in Eq. (4). Damage accumulation curve begins at the initial value 0 D and passes through the location under failure criterion. In addition, Eq. (4) is a general damage model under constant amplitude loading. Similarly, assuming that no initial damage and damage evolves nonlinearly in a power law by using i a at i S and failure occurs when 1 C D  , Eq. (4) under multi - level stress condition is

1      i i S   C  i m a j

j

  i   D n

 

a

D n

n

(5)



i

i

i

1



Therefore, Eq. (5) represents a general damage model with nonlinear evolution. Using Eq. (4) and Eq. (5), the mean value of cumulative fatigue damage at any given loading cycles can be estimated under VAL. However, fatigue failure is stochastic in nature. Thus, it is extremely important to model fatigue damage accumulation as a random process and estimate the distribution of damage accumulation.

2.2 Distribution of cumulative fatigue damage

Until now, many models have been developed to describe the average fatigue damage accumulation behavior. However, the individual fatigue damage accumulation path may diverge significantly from its mean. Thereby, the distribution of cumulative fatigue damage depicted in Fig. 1 needs to be modeled. By treating fatigue life as random which follows a certain distribution, then the distribution of damage accumulation can be derived through the one to - one PDF transformation [22]. A probabilistic scheme of general damage accumulation curve is depicted in Fig. 2.