Issue 38

C. Xianmin et alii, Frattura ed Integrità Strutturale, 38 (2016) 319-330; DOI: 10.3221/IGF-ESIS.38.42 321 The self-consistent index b is related to statistical parameters α and β of fatigue life distribution. It can be solved by comparing the mean values of n f with N through numerical simulation. The right-hand side term (1+Δ) indicates that the critical damage is a random variable. Assuming N ～ W(α, β), (1+Δ) obeys W(α, β/μ) distribution, with its mean value equal to unit and standard derivation equal to σ/μ , respectively. Here, σ is the standard derivation of fatigue life N . Assuming N ～ W ( α , β ), α =2.1, β =1129 and accordingly μ =1000, σ =500, through large number of Monte Carlo sampling tests, it is found that if the consistent index b is taken as 1.065, the mean values of n f and N can be comparable. Repeating Monte Carlo sampling of Eq. (2) for 2000 times, the statistics of n f are calculated as: α =2.1, β =1139, μ =1009, σ =513. Comparing with the original fatigue life, it can be noted that n f is close to N with respect to the mean value and standard deviation, as well as the maximum likelihood estimations of parameters α and β . Furthermore, in the goodness of fit test with the original Weibull distribution, n f has passed K-S test and W 2 test quite well. Therefore, the new cumulative damage criterion expressed in Eq. (2) can be considered statistically consistent. Through some appropriate transformation of Eq. (2), a statistically consistent fatigue damage model that can quantitatively calculate the statistical properties of damage under spectrum loading conditions can be achieved. For simplicity, the case of constant amplitude loading is first presented. It should be mentioned that in practice, for the case of constant amplitude loading, the concept of loading block generally does not exist. Here, it is used for the generalization to the case of variable amplitude loading. Assuming there are n load cycles in one loading block, the equation to calculate the damage D B caused by one loading block can be established by moving the random disturbance Δ to the left-hand side of Eq. (2), written as     1 1 b n i B i i N E N D N E N            (3) Then, the fatigue life can be predicted by 1 f B N n D   (4) Only if the predicted fatigue life N f obtained by Eq. (4) is approximate to the original N , can the fatigue damage model be called statistically consistent. A coefficient relative to the cycle number n must be added in front of Δ to make D B satisfy the boundary conditions: (1) damage value is zero when the cycle number is zero; (2) the mean value of damage is approximate to 1 when the cycle number equals to the mean value of fatigue life. In addition, the coefficient should increase with the number of cycles. Through the large number of Monte Carlo sampling tests, it is found that when the coefficient in front of Δ is defined as n / N i , N f can be close to N for the probabilistic properties. Thus, the damage caused by one loading block D B can be given by     1 1 b n i B i i i N E N n D N N E N            (5) Eq. (5) can be extended to the case of spectrum loading naturally, written as     1 1 1 1 1 b j n j m j j j B ji j j i j m f j B j N E N n D N N E N N n D                            (6) where D B is the damage introduced by one loading block with m stress levels; m is the number of stress levels in a spectrum loading; n j is the cycle number under the j th stress level; a j is the self-consistent exponent dependent on the fatigue life distribution under j th load level , which makes the mean value of n f (Eq.(2 )) approximate to the original N ; N j