Development of statistical methods to highlight differences between endurance limit of re-rolled steel sheets in an assumed homogeneous population

In this paper, we present the results of the activities carried out to estimate the axial fatigue strength of ASTM E566 Gr. 70 sheet metal. We describe the experimental setup, followed by the collected data. To highlight a possible inhomogeneity in some of the material properties, we estimate the endurance limit for the investigated material using various statistical approaches on the set of test articles. The data analysis was carried out both by considering as a whole the set of test articles and by dividing the set into two groups. As a new analysis technique, we propose a method introducing a bimodal distribution (Gaussian mixture).


Introduction
The present paper shows the results of the activities carried out in the framework of a collaboration with Toscotec S.p.A. The paper is divided into four sections. After this short introduction, section 2 is dedicated to the presentation of the investigated material and theadopted testing procedure. In section 3 the experimental results are presented, and the estimations of the material fatigue strength by the Staircase Procedure and the Maximum Likelihood Principle are reported. A new analysis technique is finally proposed, in which the population distribution is a linear combination (mixture) of two Gaussian distributions. In section 4 some concluding remarks are presented and a suggestion for the choice of the endurance limit of the material is proposed.

Materials and methods
In this section we describe some of the mechanical properties of the steel under investigation and the geometry of the test articles made of this material. We finally describe how the test campaign was carried out.

Material properties
In this work we study the fatigue strength limit of steel SA 516 -Gr. 70. The supplier has sent the material needed to manufacture the test articles in two sheets, called sheet A and sheet B, which are nominally the same material, therefore with the same fatigue strength limit. However, it could have occurred that the two sheets were obtained in two separate castings. Therefore it might be interesting to try to understand if differences can be highlighted by considering the test articles extracted from the two sheets as two different samples.

Description of test articles
The test articles have been extracted from the two sheets through thickness. The extraction positions are highlighted in figures 1 and 2. The geometric dimensions of the test articles are shown in figure 3 We obtained 20 test articles from sheet A (named A1,…, A20) and 19 test articles from sheet B (named B1,…, B19).  Figure 3. Test article design.

Experimental setup and test conditions
A campaign of high cycle number fatigue tests was carried out using a resonance testing machine. The test conditions are shown in table 2. The test campaign was conducted following the UNI 3964 standard [1] having the foresight to select test articles from sheet A in the first part of the test campaign and from sheet B in the second and final part. In each of the two parts, the selection was random. This method of conducting the test campaign allows highlighting any differences between the mechanical fatigue behaviour of the test articles extracted from sheet A compared to the ones extracted from sheet B. In fact, the number of test articles present in each of the two parts of the test allows statistically reliable values to be obtained considering separately the data obtained by analyzing the test articles extracted from each of the sheets as well. The fatigue test facility, with the resonant testing machine, is shown in figure 4. Figure 5 shows the detail of the assembly of the test article in the testing machine.

Fatigue strength estimation
It is well known that the fatigue strength is not a physical quantity measurable as such. On the contrary, we can obtain the mean value of this quantity (for a statistically homogeneous population) by carrying out a statistical analysis of a binary random variable that indicates the outcome of each of the tests in the campaign (runout/failure). In the present work, we present three methods of statistical analysis. Two of them are applied in two different ways: the test campaign is either considered as a whole or divided into two sub-campaigns.

Outline of the Staircase Procedure
We briefly recall the data processing method reported in the UNI 3964 standard [1] for determining the fatigue strength. We recall that this method was presented by Dixon and Mood [2] in a famous work in which they show, for the staircase experimental procedure, a smart approximation for the determination of the likelihood maximum. We will describe the "exact" likelihood maximization method in subsection 3.3. In table 3 we list the quantities used in the elaboration of the calculation. The method of analysis that we describe below is applicable in the hypothesis that the sample (set of test articles) is extracted from a Gaussian population distribution.
With the symbols listed in table 3, it is possible to compute the mean value ∆σ 50% of the fatigue strength S as Mean value of the fatigue strength ∆σ (50% survival probability) ∆σ 10% Fatigue strength corresponding to 10% survival probability ∆σ 90% Fatigue strength corresponding to 90% survival probability SD Standard deviation of ∆σ and the standard deviation SD of this mean value as The extremes of the 90% confidence interval for S 50% can be obtained, again assuming the validity of Gaussian population, computing

Elaboration of the staircase outcomes
In figure 6 we show the outcomes of the staircase procedure, obtained starting at an initial stress level of 235 MPa and with a staircase stress step d = 12 MPa. It should be noted that, although the knowledge of the mechanical properties of the material under examination allowed to establish a reasonable stress value for the first test, three test articles were "wasted" in determining the correct initial stress level (two consecutive different outcomes are needed to start the staircase procedure).  We note that SD is very large when compared with ∆σ 50% . Generally, in fact, the ratio SD/∆σ 50% falls in the interval [0.080, 0.16]. We also note that, when the results obtained with this procedure are inserted in the relationship ∆σ 50% − 6 · SD for the determination of the endurance limit, we obtain a negative value. This result would make this material useless in the mechanical design. Observing that all the first seven outcomes carried out using test articles from sheet B were runouts, we analyzed separately the results involving test articles extracted from different sheets. The outcomes are shown in figures 7 and 8. Note that the first six outcomes of the test articles extracted form sheet B cannot be considered in the statistical analysis. We aim to highlight a possible difference between the average values of the mean fatigue stength in the two groups.  The statistical analyses carried out following equations 1,2,3 for the computation of the fatigue strenngth limit provides the results presented in tables 5 and 6. Comparing the results reported in tables 5 and 6 with the results of table 4, we observe that the average values for the two groups are significantly different from each other and that the standard deviation of each of the two groups is about half the standard deviation obtained when the results from the test articles extracted from the two sheets are analyzed together.

Fundamentals of the Maximum Likelihood Principle
To enforce the result shown above, we analyzed the same test campaign outcomes using likelihood maximization. Maximum Likelikood Estimators (MLE) must be obtained using a numerical procedure to obtain the values of the arguments at whiuch maximum is attained. In this section, we briefly outline the parameter estimation procedure based on maximization of likelihood (ML). The random variable ∆σ, assumed to be Gaussian, cannot be sampled. Given a sample made of n test articles, for the k− th test (k = 1, . . . , n), a binary response random variable e k is sampled: e k = 0 for runout and e k = 1 for failure. Let µ and S be the mean and standard deviation of the random variable ∆σ. If we introduce the cumulative distribution function ϕ (see figure 9), i.e. the probability P (for given values of µ and S) that the random variable ∆σ is less than the value of ∆σ k used in the k− th test, as we can define L(µ, S) as the logarithm of the likelihood function: We can obtain the MLEs of µ and S taking the couple of valuesμ andŜ at which L attains its maximum value.

Maximum Likelihood estimates and comparison with staircase results
In figure 10 we show the outcomes of the staircase procedure used to obtain the ML estimators. It should be noted that the three initial failure outcomes can be inserted in the calculations, increasing the value of n.
The statistical analysis carried out using maximization of 5 provides the results presented in table 7.  We note that the ML estimate of SD is smaller than the Dixon-Mood estimate. A comparison of the estimates of ∆σ 50% and SD obtained wwith different methods is reported in [3]. The bias of Dixon-Mood SD estimators has been shown in [4] and is not generally considered critical, since it tends to decrease the endurance limit. Notwithstanding this, we observe that, when these results are inserted in the relationship ∆σ 50% − 6 · SD for the determination of the endurance limit, we again obtain a negative value. Again we analyzed separately the results involving test articles extracted from different sheets. The outcomes are shown in figures 11 and 12, for which we can again use all the outcomes.    Comparing the results reported in tables 8 and 9 with the results of table 7, we observe that also for this statistical technique the average values for the two groups are significantly different from each other. This result confirm that the difference between the two mean values is not due to a limitation of the Dixon-Mood approximation, but is a consequence of a real difference between some of the mecanical properties of the two sheets, only nominally equal.

Maximum Likelihood estimation of a mixture of Gaussian distribitions
We present here an extension of the ML method using, as a distribution function, a mixture of two Gaussians. We define a linear combination ψ of two Gaussian cumulative distribution function with two differerent mean values µ 1 and µ 2 and (for simplicity) the same standard deviation S for both components as where α ∈ (0, 1) is the relative weight of the two components. In this way, a new objective function (to be maximized) L ′ (µ 1 , µ 2 ) can be defined as [e k log(ψ(∆σ k |µ 1 , µ 2 , S, α)) + (1 − e k )(1 − log(ψ(∆σ k |µ 1 , µ 2 , S, α)))] in which, for simplicity, the same standard deviation S has been kept fixed (30 MPa, a reasonable value obtained by previous analyses), and also the weights α and 1 − α have been kept fixed as well (0.42 and 0.58, values of the relative numbers of test articled from the two sheets). The numerical maximization of L ′ provides µ 1 = 191 MPa and µ 2 = 269 MPa confirming a separation of the two mean values more than twice a reasonable estimation of the standard deviation. These results do not vary much if S and α are varied within a reasonable range of values (20-40 MPa, 0.3-0.6). However, in the current state of research, it is not possible to show reasonable values of the estimate of S using a Gaussian misture using the ML method because the estimation of S is lowered until it reaches very small and meaningless values. The probability density function obtained with this method is shown in figure 13.

Conclusions
In this paper, we have shown the differences that can occur if we introduce the hypothesis that the sample is not distributed like a Gaussian. This hypothesis, which is usually assumed to be true in the analysis of fatigue tests, is very strong and conditions the interpretation of a series of fatigue tests. In staircase tests, 90% confidence intervals for the means show that ∆σ 50% seems to be different for sheets A and B nominally made of the same steel. The standard deviation S is slightly overestimated in the Staircase Procedure with respect to the Maximum Likelihood estimation. A new procedure for the estimation of means of a mixture of two Gaussians confirms the difference between the values of ∆σ 50% for the two sheets. The Staircase method tends to be conservative, and even ML estimates do not differ much. Perhaps the development of new test/analysis methodologies could provide results that improve component design. For the estimation of the endurance limit for this material (to be used in component design), we propose for ∆σ 50% the lowest of the two mean values (209 MPa) and for SD the corresponding value (31 MPa) as obtained from the analysis, carried out according to [1], of test articles extracted from sheet A only, which has the worst mechanical fatigue characteristics. For the endurance limit, we obtain therefore ∆σ 50% −6·SD = 23 MPa. Although small, it is still a value that can be used in mechanical design. In future developments of this work, we will investigate the characteristics of the Gaussian mixture method also using Monte Carlo simulations, to obtain, for instance, the confidence intervals of the estimates. It should be as well interesting a comparison of MLEs with other estimators or other statistical techniques (Fisher Information, Bayesian analysis).