Using a reverse life plot for estimating fatigue endurance/limit

1 Introduction

Traditionally, fatigue limit σ_FL is usually determined conducting a constant amplitude cyclic loading with zero mean stress, σ_m = (σ_min + σ_max)/2 = 0, or load ratio, R = σ_min/σ_max = −1, till failure, where σ_min and σ_max are the minimum and maximum applied stresses, respectively (ASTM 1997; Morrow 1968). Due to inherent scatter in the fatigue data approaching fatigue limit, a ‘staircase’ method is often utilized, which allowed for statistical analysis but needed relatively large number of specimens (around 15–30) to be tested to give a spread in the data for statistical analysis (Dixon and Mood 1948). On the other hand, one can use the existing S-N data to estimate a fatigue endurance/limit, σ_FL by extrapolating them to a predetermine endurance life, N_e. Figure 1a illustrates such extrapolating method to estimate fatigue limit at endurance life N_e (say 10⁷ cycles) using the S-N curve on a log-log scale. Since, the S-N curve corresponds to a 50 % failure reliability, the estimated values of σ_FL by this method is similarly 50 % reliable. On the other hand, an alternative approach to estimate fatigue limit σ_FL is presented in Figure 1b, using a linear plot of stress amplitude σ_a versus 1/N_f and extrapolating the data to 1/N_f = 10⁻⁷ (where 1/N_f = 0 represents infinite life). This method is also 50 % reliable but can use existing high-cycle fatigue (HCF) S-N data. Usually, 3–5 data points are sufficient to estimate the fatigue/endurance limit at a given endurance life, N_e, e.g., of 10⁷ or 10⁶ cycles. By choosing 1/N_f equal 10⁻⁷ or 10⁻⁶ the obtained σ_FL value is linked to the endurance life of N_e = 10⁷ or 10⁶ cycles, respectively.

Figure 1:

An illustration of the fatigue limit σ_FL estimation using (a) extrapolation of the log-log S-N curve at endurance life N_e = 10⁷ cycles, (b) the same data in terms of the stress amplitude σ_a versus 1/N_f extrapolated at 1/N_f = 0, which corresponds to N_f→∞.

In this paper we propose and discuss the use of stress amplitude σ_a versus reverse life 1/N_f plot, to estimate a fatigue endurance/limit, σ_FL, for a given endurance life, N_e, using HCF data alone.

2 Procedure for fatigue limit estimation using 1/N_f method

It is customary to plot the S-N curve where an abscissa (the x-axis) corresponds to the number of cycles to failure N_f. However, N_f is a dependent variable, which should be plotted as an ordinate (the y-axis) when using the best fit analysis. This fact is reflected in the procedure for the proposed 1/N_f method specified below.

1. Select fatigue data corresponding to the lowest three stress levels from the S-N data.

2. If there are multiple lives data at a given stress level (multiple samples) use all of them or calculate an average N_f at each stress level.

3. Plot 1/N_f as an ordinate (the y-axis) versus stress amplitude σ_a as an abscissa (the x-axis) for the chosen three stress levels from the S-N data. Note that σ_a is the independent variable whereas 1/N_f is the dependent variable.

4. Perform the best fit linear regression using Excel: 1/Nf=mσa+c and extrapolate the best fit line to 1/N_e to estimate fatigue/endurance limit, as an endurance life, N_e.

3 Background on staircase method

The staircase method proposed by Dixon and Mood (1948) is often used for fatigue limit estimation. Several techniques exist for evaluating staircase tests showing various accuracies (Little 1972; Mackay and Byczynski 2021; Müller et al. 2017; Pollak et al. 2005).

The Dixon-Mood method (Dixon and Mood 1948) provides formulas to estimate the mean and standard deviation of the endurance limit. The staircase method is notably accurate in quantifying the mean fatigue endurance strength, σ_FL, but is not so accurate in estimating the standard deviation. This is inherently associated with the staircase procedure, in particular, when the number of samples used is relatively small (Little 1972; Pollak et al. 2005). In the staircase method the applied stresses oscillate up-and-down, near the mean fatigue limit, therefore is more difficult to obtain an accurate estimation of the standard deviation. In the past, a number of approaches have been proposed to improve the accuracy of the estimated standard deviation (e.g., Rabb 2003; Svensson et al. 2000; Wallin 2011, and others).

In the typical staircase method, the first specimen is subjected to a stress amplitude corresponding to the expected fatigue limit at a predefined endurance life, N_e of 10⁷ or 10⁶ cycles. If the specimen survives the predefined endurance life in terms of number of cycles, N_e, the test is stopped, and the next specimen is tested at a stress amplitude that is one increment above the previous. When a specimen fails prior to reaching the predefined endurance life, N_e, the number of cycles to failure is recorded, and the next specimen is subjected to a stress that is one stress increment below. Such a sequence of loading is continued during the staircase procedure depending on if a specimen survives N_e, or fails before reaching, N_e cycles. This staircase procedure is often referred to as the up-and-down method where the sum of failures and run outs is equal to the number of specimens tested. It requires a relatively large number of specimens between 15 and 30 with an average being usually about 25. Dixon and Mood (1948) and Dixon (1965) suggested that the selected stress increment should be about the logarithmic standard deviation to obtain optimal results. However, this leads to the obvious problem that the logarithmic standard deviation is usually unknown before the tests are conducted. It is apparent that engineering judgment and experience is important in such instances.

The traditional staircase testing can be time-consuming and a relatively large number of specimens are needed. However, when a limited number of specimens are available, then the modified staircase method can be used. In the modified staircase method, the first specimen is subjected to a stress level that is well below the expected fatigue limit. If the specimen survives the predefined number of cycles, N_e, at its initial stress amplitude, then it is subjected to a stress amplitude level increased by one increment above the previous. This is continued with the same specimen until failure. Then, the number of cycles is recorded at the last stress level at which specimen has failed. The next specimen is subjected to a stress amplitude that is at least two increments below the level where the previous specimen failed. In this modified approach the number of run-offs is usually greater than failures which are equal to the number of specimens tested. The results from the modified staircase method should be used with caution since each specimen before failure, was trained at lower stress level for N_e cycles. This may result in an exaggerated fatigue limit for materials that exhibit high cyclic strain hardening. In the modified staircase method, the minimum number of specimens can be much smaller than 15.

In the traditional and the modified staircase methods, an analysis uses only the less frequent occurrence in the test results to determine the fatigue limit, i.e., if there are more runouts than failures, then the number of failures is used, and vice versa.

4 Comparison between staircase method versus 1/N_f method

It is worth mentioning that in the open literature there is a very limited number of staircases data results available, in particular, for a corrosion environment. This is probably due to the high confidentiality of such data.

Figure 2a lists the stress levels and corresponding number of cycles for the staircase testing method of 7000 Al alloy in lab air whereas Figure 2b shows applied stresses at corrosion chamber with NaCl 0.5 % solution sprayed for 60 min and dried for 40 min. The corresponding values of the estimated fatigue limits, σ_FL, with probability of 50 % are 146.6 and 63.6 MPa, respectively. For each staircase test 15 specimens were used.

Figure 2:

Staircase fatigue limit in (a) air and in (b) corrosion (x means failure, o means run out). Material Al7000: σ_0.2 = 514 MPa, σ_u = 571 MPa, RA = 12.5 %, f = 120 Hz.

Figure 3 depicts the specimen geometry with dimensions in mm used in the staircase testing and to obtain the S-N data in HCF region.

Figure 3:

Geometry of the specimen (dimensions are in mm).

Figure 4 shows the bilinear S-N curve for 7000 Al alloy tested in air and 0.5 % NaCl solution. The horizontal part of the S-N curves corresponds to the fatigue limit, σ_FL, estimated from the staircase method (see Figure 2). Open symbols in Figure 4 represent recovered samples that didn’t fail at the previous lover stress level. It can be seen from Figure 4 that these recovered samples’ lives are similar or somewhat longer than that for regular samples.

Figure 4:

Bi-linear fatigue HCF data for Al 7000 in air and corrosion of 0.5 % NaCl solution with the estimated fatigue limit σ_FL from the staircase method shows in Figure 2.

Figure 5a and b shows the estimated fatigue limits for 7000 Al at air and 0.5 % NaCl from the average lives of the three lowest stress level of the S-N data above σ_FL. An examination of Figures 4 and 5 regarding the estimated σ_FL demonstrates a very close agreement between these two methods.

Figure 5:

Fatigue limit estimation of Al 7000 alloy using 1/N_f method: (a) in air and (b) in 0.5 % NaCl environment.

An addition example shown in Figure 6 illustrates how this method estimates endurance limits for 4140 steels tested in three environments: dry air, air with 93 % RH, and aerated 3 % NaCl solution (Lee and Uhlig 1972). The close examination of the original S-N data plotted in semi-log plot didn’t show clearly how to estimate the σ_FL for each environment. On the other hand, Figure 6b show the bilinear S-N cures together with the estimated σ_FL for each environment. It is commonly noted that in a chemical environment, defined fatigue limit is difficult to obtain, as in Figure 6a. Hence, the bi-linear method applied to chemical S-N fatigue allows one to estimate reliably the fatigue limit.

Figure 6:

Data for 4140 steel. (a) Semi-log original fatigue S-N data in three environments, and (b) bilinear S-N curves with fatigue limits at 10⁷ cycles estimated using 1/N_f method.

5 Summary

1. This short review paper proposes and describes a simple reverse fatigue life method for estimation of a fatigue limit at a desire predetermine endurance life, N_e, utilizing only the existing HCF data.

2. The utilization of the existing HCF data allows us to estimate the fatigue limit without a need to run time consuming staircase testing method.

3. The method is applicable for different alloy-environment systems and demonstrates a good reliable agreement with the traditional staircase method.