Issue 38

C. Xianmin et alii, Frattura ed Integrità Strutturale, 38 (2016) 319-330; DOI: 10.3221/IGF-ESIS.38.42 320 Several probabilistic approaches have been established for damage accumulation evaluation. Many issues related to statistical characteristics of fatigue have been involved, such as the combined randomness of loading process and fatigue resistance of materials [6], probabilistic density function of failure cycles [7], complexity of loading conditions [8], random critical damage [9], frequency domain approach [10] and time domain approach [11]. However, few of them focused both on the load sequence effect and the probabilistic nature of fatigue damage. Based on Morrow’s nonlinear plastic work interaction damage rule, W.-F.Wu and T.-H. Huang [8] proposed an approach to predict the fatigue damage and fatigue life under random loading considering the loading history and the statistical feature. In this paper, a new model is proposed to predict fatigue damage and fatigue lives of specimens under spectrum loading. The effect of high load retardation on damage accumulation is taken into account by introducing an exponential function exp(-fj) in the model. The independent variable fj is related to the stress ratio of high and low stress levels. The statistically self-consistency between the probabilistic distributions of fatigue damage and failure cycles is also achieved by introducing a consistent index b (greater than one) and a random disturbance Δ. In the damage model, the fatigue damage and fatigue life are predicted using the probabilistic sampling method supported by test data. As for the critical damage Dc, i.e. the damage value at failure, there are mainly two points of view: (1) Dc is deterministic, equal to or smaller than unit; (2) Dc is a random variable, with mean value equal to unit. In the present paper, the second view point is adopted. The fatigue life N is assumed to follow the two-parameter Weibull distribution, W(α,β) [12]. The shape parameter α is set as 4 [13] for aluminum alloy according to a lot of fatigue test data. In order to verify the reliability and feasibility of this model, the predicted fatigue life is compared with that of the fatigue test results. A STATISTICALLY SELF - CONSISTENT MODEL FOR FATIGUE DAMAGE AND FATIGUE LIFE PREDICTION or constant amplitude loading, Palmgren-Miner’s linear damage accumulation rule, which is widely used in engineering, is expressed as 1 1 1 n f c i i D N     (1) where N i is the fatigue life under the i th stress level, which actually obeys some distribution; n f is the cycle number to failure; and the critical damage value D c is assumed to be one. Obviously, when N is taken as a constant, Eq. (1) can be simplified to the form of n f / N =1, which is the most common form of Miner’s damage criterion. However, in actual situations, N has probability characteristics, thus n f is also correspondingly statistical rather than deterministic. With a given distribution of N , the equation can be called statistically consistent if n f obtained from Eq. (1) with the Monte Carlo sampling method is approximate to N with respect to probability distribution. In practice, fatigue life N is exactly reflected in the number of load cycles to failure, so it is necessary that the damage criterion be statistically consistent. However, it can be verified that Eq. (1) is statistically inconsistent as follows. Through numerical simulation with the Monte Carlo sampling method [14], it can be found that the cycle number n f obtained from Eq. (1) is not identical to the original fatigue life N considering probability distribution. Assuming N ～ W ( α , β ), with α =2.1, β =1129, the mean value μ =1000, and the standard deviation σ =500, sufficient random sampling(2,000 times) of Eq. (1) is performed. As a result, 2000 different n f s can be obtained and then statistics of n f are calculated as: μ =670, σ =41. It should be noted that both the mean value and standard deviation are smaller than those of the original N , especially for standard derivation. Thus it can be concluded that Eq. (1) is statistically inconsistent. In order to enlarge the mean value and the standard deviation of n f , a consistent index b (greater than one) and a random disturbance Δ with mean value equal to zero are introduced to the left and right sides of Eq. (1) respectively, which makes the linear fatigue damage prediction model statistically consistent, given by     1 1 1 , b n f i i i N E N N E N               (2) where, N i is a random fatigue life from certain distribution, which corresponds to the i th load cycle, and E(N) is the mean value of the life distribution. F