Issue 38

C. Xianmin et alii, Frattura ed Integrità Strutturale, 38 (2016) 319-330; DOI: 10.3221/IGF-ESIS.38.42 327 accumulation if the difference between load levels is large enough. Usually the retarding effect is more remarkable than the acceleration effect and easier to be defined quantitatively. Therefore, the retardation effect of high stress level is taken into account only in this work. The retardation effect is due to the high stress level in the loading spectrum. The fatigue damage accumulation rate of specimens under a low stress level slows down since a high stress level may leads to micro-plasticity of material and compressive residual stress. The retardation effect is apparently related to the disparity between high and low load levels. In a variable amplitude stress condition, according to Morrow’s plastic work interaction damage rule [19], the fatigue damage caused by the stress amplitude S k can be written as: max d k k k k n S D N S        (9) where S max is the maximum stress amplitude in the stress history; n k is the cycle number of stress peak at level S k ; N k is the number of stress peak to cause failure if the constant amplitude S k is considered; and d is Morrow’s plastic work interaction exponent which can be considered as the stress sequence effect on the fatigue damage. Using the similar principle as Morrow’s plastic work interaction damage rule, an exponential function exp(- f j ) is introduced to modify the fatigue damage caused by the low stress level, seen in Eq. (10). As the independent variable , f j , in this exponential function is larger than one and related to the stress ratio of high and low load level , i.e. 1 max j j S S  , where S j is the stress peak of the j th stress level and 1 max j S  is the maximum stress peak among the stress levels from S 1 to S j-1 . In addition, a constant coefficient v is defined to accommodate the model to the test data, which can be fitted by comparing the calculated fatigue life through large number of Monte Carlo sampling with the test data. In order to reduce the damage accumulation rate of the lower stress levels, negative one is set as a multiplier to the independent variable f j . The modified fatigue damage model with load retardation effect under multilevel spectral loading is given as:       1 1 1 1 max 1 1 max 1 1 1 max max 1 exp 1 0, 1 , max ~ , a j n j m j j j B j ji j j j i m f j B j j j j j j j j j j N E N n D f N N N N n D j or S S f and S S S S v S S S                                                         (10) where v is a constant depending on the material itself and the assumption of fatigue life distribution. The load retardation effect index has no effect on the fatigue damage produced by the first stress level or the stress level higher than the previous maximum stresses , since exp (-f j ) =1 in Eq. (10). However, the fatigue damage is affected by the load retardation effect index when the current stress level is lower than the previous maximum stresses , since exp (-f j ) <1. Simulation work (detailed steps presented in section 4.2) has been done by using this modified model to determine the value of v . It is found that the fatigue life distribution of the model matches that of the test well when v =1. The statistical parameters estimated by the modified model are given in Tab. 10. Load level ˆ  ˆ  (×10 6 ) E (×10 6 ) R E  (×10 5 ) R  N 95/95 (×10 5 ) R N 95 N 1 Fall into ( E 1 , E 2 ) Fall into (  1 ,  2 ) 5 3.643 1.689 1.523 0.882 4.647 0.960 7.397 0.924 0 yes yes 3 3.888 0.589 0.533 1.148 1.533 1.177 2.716 1.262 0 yes yes Table 10 : Comparisons of parameters based on the modified statistical fatigue damage model with those of the fatigue tests.