﻿ 拟合威布尔随机寿命S-N曲线的新算法 New Approach for S-N Curve after Fitting Random Fatigue Life Data to Weibull Distribution

Statistics and Application
Vol.06 No.02(2017), Article ID:21179,9 pages
10.12677/SA.2017.62024

New Approach for S-N Curve after Fitting Random Fatigue Life Data to Weibull Distribution

Guijin Wang

Alloy Steels Department, Central Iron & Steel Research Institute, Beijing

Received: Jun. 9th, 2017; accepted: Jun. 24th, 2017; published: Jun. 28th, 2017

ABSTRACT

This paper presents a new approach to build up S-N curve by first fitting fatigue datasets to three parameter Weibull distribution under the scheme of adjusting location parameter and maximum fatigue life to match the shape parameter of skewness and kurtosis from sample fatigue data with that of fitted Weibull distribution. Thereafter, the dimensional parameter of the Weibull distribution under various cyclic stresses can be used to calculate S-N curve, its expected standard deviation and accumulated failure life rate L10, L63, L90. This new approach has been justified due to available experimental data of Zhao etc.

Keywords:S-N Curve, Weibull Distribution, Skewness and Kurtosis, Fitting Indicator

1. 引言

(1-1)

(1-2)

(1-3)

2. 两参数威布尔分布的拟合

2.1. 极大似然法

(2)

(3)

2.2. 威布尔分布的斜度和峭度

(4)

(5)

2.3. 拟合威布尔形状参数κ的具体步骤

(6)

1) 先由小到大排列试样疲劳寿命值，用极大似然法算出两参数威布尔分布的形状参数，与此同时根据试样数组的内禀斜度，峭度计算它们对应的形状参数最后得到拟合指数。如果偏离期望值1较远，则；

2) 令位置参数为非零值，调整重复步骤1，逐步使靠近1；

3) 如果仍然偏离1较远，可调整和名义全失效寿命计算直到它接近最佳值1。这就是拟合的形状参数

3. G20CrNi2Mo轴承钢的S-N曲线

3.1. G20CrNi2Mo轴承钢疲劳寿命的威布尔分布

Table 1. Fatigue life of G20CrNi2Mo steel under four levels of cyclic stress

*指经to和tf修正得到最佳拟合指数后归一化的无量纲寿命：

1) 原始数据的拟合

(7)

2) 数据经过to和tf修正后按斜度和峭度的拟合

Table 2. Fitting four original datasets of fatigue life to two parameter Weibull distribution

Table 3. After correction of and, fitting four datasets of fatigue life to three parameter Weibull distribution

3.2. S-N曲线

(8)

1) 原始数据的S-N曲线

Table 4. After correction of and, fitting normalized four datasets of fatigue life to two parameter Weibull distribution

Figure 1. The S-N curve from Weibull distribution of fatigue life data before correction of skewness and kurtosis

(9)

2) 按斜度和峭度拟合的S-N曲线

(10)

3.3. 标准差

(11)

Figure 2. The S-N curve from Weibull distribution of fatigue life data after correction of skewness and kurtosis

3.4. L10，L63和L90的计算

4. 结果和讨论

1) 通过调整位置参数to以及名义全失效时间tf，G20CrNi2Mo轴承钢在四组定幅交变应力下的疲劳寿命都服从三参数威布尔分布，它们的拟合指数都很接近1，因此得到的形状参数，尺寸参数和位置参数to都是比较合理的。由于样本寿命在威布尔拟合中采用了三次矩和四次矩，它比现有的拟合方法对样本寿命的随机性要求更严格。不仅保证这两者之间具有最小均方偏差，而且分布密度曲线的不对称性和峰型也比较一致。赵永翔等人的原始数据已经接近要求，只需对to和tf作不大的调整就取得令人满意的拟合指数。相信其他研究者只要严格控制试样的制作以及试验参数，也能做到。实际上调整to和tf上只是把寿命数据组内禀的正确统计量揭示出来。如果经过多次调整to和tf还达不到良好的拟合指数，那可能是样本数量不足，存在系统误差或者疲劳数据服从其他分布所致。

2) 经过调整位置参数to和名义全失效时间tf，四种应力幅值的威布尔寿命形状参数的分散度比未经调整的小得多。可以参考McCool提出的判据。当然越接近1越好。在各拟合指数接近1的场合下，用威布尔参数可以得到可靠的S-N曲线，及其标准差和各种累计失效率，供可靠性工程参考。

3) 本文通过取得在不同应力幅值下寿命数据组的尺寸参数，能够对应力的影响做出唯象的定量推断。同样也可根据不同环境条件(诸如温度，湿度，酸碱度等等)下尺寸参数的变化，对它们的影响做定量研究。因此本方法有可能在实验数据基础上为工程应用建立S-N曲线数据库。

Table 5. The standard deviation of S-N curve from Weibull distribution and that of fatigue data after correction of skewness and kurtosis

Table 6. The L10, L63 and L90 of S-N curve from Weibull distribution after correction of skewness and kurtosis

Figure 3. The L10, L63 and L90 of S-N curve from Weibull distribution

4) 本文结论只适用于所述应力区间内极大值与极小值比R为−1的场合。其他场合，例如定幅应变，随机载荷，温度变化，以及腐蚀环境等等须另行研究。至于疲劳裂纹的萌生及扩展的微观机制研究，都不在本文范围，有兴趣的读者可参阅文献 [1] 及 [9] [10] [11] 。

New Approach for S-N Curve after Fitting Random Fatigue Life Data to Weibull Distribution[J]. 统计学与应用, 2017, 06(02): 210-218. http://dx.doi.org/10.12677/SA.2017.62024

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