Estimating the size of the maximum inclusion in a large sample area of steel

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1 Estimating the size of the maximum inclusion in a large sample area of steel Nuša Pukšič 1,2, Monika Jenko 1 1 Institute of Metals and Technology, Ljubljana, Slovenia 2 Jožef Stefan International Postgraduate School, Ljubljana, Slovenia nusa.puksic@imt.si Abstract. Non-metallic inclusions influence the properties of steel and finished steel products. The methods relating to statistics of extremes are effective for the purpose of predicting the size of the maximum inclusion, which is an important parameter in the quality control and lifetime estimation. A steel sample with two types of inclusions is used to demonstrate the use of the extreme value theory in practice. The results of the general extreme value method are compared to the results of the mixture and the competing risk models. The competing risk model gives the best fit to data, and predicts the largest inclusions. Keywords: steel; inclusions; statistics of extremes; GEV; mixture; competing risk; model 1 Introduction Non-metallic inclusions, formed during the steel production process, have a great impact on the properties of steel and finished steel products. The detection and estimation of the size of the largest inclusions is an important consideration in quality control and lifetime estimation of steel and steel products. The size distribution of inclusions in steels is found to have a log-normal form [1,2,3]. The standard method of fitting the log-normal distribution requires a quantitative measurement of inclusion sizes right across the size range to obtain a good fit. Measurements of small inclusions with sizes smaller than 3 µm are unreliable. On the other hand, the statistic of large inclusions is affected by their low occurrence rate.

2 When using prediction methods based on the extreme value theory, only measurements of the maximum inclusions in randomly chosen areas are needed. The general extreme value (GEV) statistics method can be used to estimate the maximum size of inclusions in a large amount of steel. The estimation of the sizes of extreme inclusions is affected by the presence of multiple types of inclusions in a single steel grade. When the presence of multiple types of inclusions is obvious and the types can be distinguished by their shapes, it is good practice to apply the method to each type of inclusions separately. This will also enable one to consider each set of inclusions in connection with its harmfulness [4]. Unfortunately, this approach prolongs the manual analysis and it is difficult to implement it in an automatic image analysis. The mixture and the competing risk models were suggested, where the diversity of the inclusions is taken into account statistically [4,5]. In this paper, an overview of the statistical approach is given, followed by the results of the analyses of the data obtained by the automatic image analysis from a spring steel sample. 2 Overview of statistical methods 2.1 The general extreme value (GEV) method For distributions decreasing exponentially at upper tails, the distribution of the largest values can be described by Gumbel distribution. If the distribution decreases following a power law, the distribution of the largest values is either Fréchet- or Weibull-like. The GEV distribution groups the three types: 1/ x P ( x) exp 1, (1) where P(x) is the cumulative probability, λ and α are the location and scale parameters, and ξ is the tail index. The tail index determines the type of the distribution: a Fréchet distribution for ξ > 0, a Weibull distribution for ξ < 0, and a Gumbel distribution for ξ 0.

3 A standard inspection area S 0 is defined. The area of the maximum inclusion in S 0 is measured in N such areas. Then, the square root of the area of each measured inclusion is calculated, z area max largest measured inclusion size z i can be calculated by: where z i is the i-th in the series of of ln lng( z ) i. The cumulative probability G(z i ) of the i-th zi zi G( zi ) exp exp, (2) N 1 area max, i ordered by size. The probability plot versus z i can then be used for the basic diagnostic [1]. For the estimation of the extreme inclusions size in a large examined area of steel S, the return period is defined as T S / S0. The characteristic size of the maximum inclusion, denoted by z S, expected to be exceeded exactly once in an area S, can be defined by solving the equation G( z S ) 11/ T to give: 1 z S 1 ln1. (3) T 2.2 The mixture model and the competing risk model The mixture model assumes multiple types of inclusions and the Gumbel distribution for the maximum inclusions of each type. When we have two types of inclusions, the areas are then also of two kinds: containing inclusions of the type 1 and of the type 2, the proportion of the second kind being p. The observation process is such that the kind of area being measured remains unknown (as in automatic image analysis, where sizes of inclusions are recorded, but not types) [4]. The distribution function is then of the form: F mix ( 1 2 x x) (1 p) G ( x) pg ( ), (4) where G i are Gumbel distribution functions for i = 1, 2 and 0<p<1. The more natural assumption is that inclusions of both types are present throughout the material and the measuring process detects the inclusion that happens to be the largest in a given area. The competing risks model assumes, that the sizes of the largest inclusions of different types follow independent Gumbel distributions G 1 and G 2 [4]. The distribution function is then of the form:

4 F risk x) G ( x) G ( ). (5) ( 1 2 x 3 Results and discussion To obtain the data, 544 sample areas, each of 0.27 mm 2, from a single steel slab were investigated. The area of each inclusion larger than 3 µm 2 was measured using automatic image analysis. The results of the GEV analysis are given first, followed by the results of the mixture and the competing risk models. The fit of the GEV model to the data gives the estimates for the parameters of the distribution, Table 1. With the estimated parameters, the size of the largest inclusions can be calculated (Eq. 3) as a function of the number of sample areas S 0 to be investigated. Results are shown in Figure 1. Manually inspecting the samples, we see two types of inclusions contributing to the set of maximum inclusions. The parameter values obtained by fitting both models to the data are gathered in Table 1. The fit of both models to data and estimated inclusion sizes are shown in Figure 2. Predictions of the three models show appreciable discrepancies. The competing risk model, which seems to best capture the underlying features and also gives a good fit to the data, predicts the largest inclusions. Table 1: Estimated parameter values for the GEV model, the mixture model and the competing risk model. GEV Mixture model Competing risk model Parameter Value Parameter Value Parameter Value α 1.96 α α λ 6.37 λ λ ξ α α λ λ p 0.501

5 Figure 1: The maximum inclusion size estimated from the parameters of the GEV model with 95% confidence intervals. Figure 2: The Gumbel probability plot with a comparison of the mixture model (M) and the competing risks model (CR) fits to the data, left. The estimated inclusion size for the largest inclusions in each model, right. References: [1] C. W. Anderson, G. Shi, H.V. Atkinson, and C.M. Sellars. The precision of methods using the statistics of extremes for the estimation of the maximum size of inclusions in clean steels. Acta Materialia, 48: , [2] H.V. Atkinson, and G. Shi. Characterization of inclusions in clean steels: a review including the statistics of extremes methods. Progress in Materials Science, 48: , [3] Y. Murakami. Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions. Elsevier, [4] S. Beretta, C. Anderson, and Y. Murakami. Extreme value models for the assessment of steels containing multiple types of inclusion. Acta Materialia, 54: , 2006 [5] C.W. Anderson, G. Shi, H.V. Atkinson, C.M. Sellars, and J.R. Yates. Interrelationship between statistical methods for estimating the size of the maximum inclusion in clean steels. Acta Materialia, 51: , 2003.

6 For wider interest The properties of steel and finished steel products are affected by non-metallic inclusions, formed during the steel production process. The detection and estimation of the size of the largest inclusions can be an important parameter in the quality control and lifetime estimation of steel and steel products. Statistical methods can be helpful, since they can provide an additional insight not necessarily apparent from the raw data. There are a few options that allow us to estimate the size of the largest inclusion to be expected. Unfortunately, there can be great discrepancies in the predictions from different models. Care should be taken when choosing a method and a model to investigate and analyse your product. Any of the models presented in this paper, on the other hand, can be used as a means of comparing different grades of steels or to define bounds, within which the quality of a given grade of steel is still acceptable.