The bootstrap method is mostly used to estimate statistical characteristics of small sample data. However, the limitations of the bootstrap method itself lead to a reduction in the reliability of small-sample estimates. In this article, an improved bootstrap method is developed to address this problem. In the statistically significant error range (the sample average error and the limit error of sampling) of the original single sample data, expanding the virtual test data that obey two distributions to overcome the limitations of the bootstrap method itself. This article compares and analyses these two methods through the case; the result indicates that the improved bootstrap method can enhance the reliability of the estimation results without changing its probability distribution. We also discussed how to reduce the fluctuation of the improved bootstrap method. And the effectiveness and feasibility of this improved method are discussed in the analysis of fatigue life test data.

In order to determine the effect of different loads on fatigue damage, a strength degradation model is proposed according to the law of residual strength degradation of metal materials. The model is verified with the strength degradation test data, and the results show that the model can describe the strength degradation process of general metal materials well. Combined with the strength degradation model, an improved equivalent damage model for different loading sequences is proposed. On this basis, a nonlinear fatigue cumulative damage model based on strength degradation is derived. The cumulative damage model is applied to the estimation of fatigue residual life under two-, three-, and four-stage loads to investigate the effects of different loading sequence on fatigue damage under various loading conditions. Combining with experimental data, it is verified that the cumulative damage model can accurately estimate the fatigue life under two-, three-, and four-stage loads.