Characterization of inclusions in clean steels: a review including the statistics of extremes methods

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Progress in Materials Science 48 (2003) 457–520 www.elsevier.com/locate/pmatsci

Characterization of inclusions in clean steels: a review including the statistics of extremes methods H.V. Atkinson*, G. Shi1 Department of Engineering Materials, University of Sheﬃeld, Mappin Street, Sheﬃeld, UK Received 1 April 2002; accepted 1 August 2002

Abstract The application of new secondary reﬁning techniques and non-metallic inclusion reduction techniques in steel production processes has greatly reduced the size and amount of nonmetallic inclusions remaining in molten steels and steel products. This makes the inspection of inclusions diﬃcult. Here the main methods used for the characterization of inclusions in clean steels are reviewed. The inﬂuences of inclusions on the properties of steels are discussed. Statistical methods for the prediction of the maximum inclusion size in a large volume of steel are introduced. Methods based on the statistics of extremes are described in detail. The methodology for the practical application of these methods is described and the factors aﬀecting the precision of the estimation are discussed. # 2002 Elsevier Science Ltd. All rights reserved.

Contents 1. Introduction ....................................................................................................................458 2. Characteristics of inclusions in clean steels ..................................................................... 459 2.1. Inclusions in clean steels......................................................................................... 459 2.2. Critical inclusion size in steels ................................................................................ 461

* Corresponding author. Present address: Department of Engineering, University of Leicester, University Road, Leicester LE1 7RH, UK. Tel.: +44-116-252-2525; fax: +44-116-252-2525. E-mail address: [email protected] (H.V. Atkinson). 1 Present address: TWI Limited, Cambridge CB1 6AL, UK. 0079-6425/03/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0079-6425(02)00014-2

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3. Eﬀects of inclusions on the mechanical properties of steels ............................................461 3.1. Fatigue properties...................................................................................................461 3.2. Fracture behaviour .................................................................................................465 4. Characterisation methods for assessment of inclusions in steels .....................................466 4.1. Techniques based on surface analysis by optical microscopy.................................467 4.1.1. Standard chart comparison ........................................................................ 467 4.1.2. Image analysis ............................................................................................ 467 4.2. Non-destructive testing........................................................................................... 471 4.2.1. Ultrasonic tests ........................................................................................... 471 4.2.2. Magnetism-related methods ....................................................................... 472 4.2.3. X- ray transmission method ....................................................................... 474 4.3. Inclusion concentration methods............................................................................ 474 4.3.1. Electron beam button melting (EBBM) .....................................................474 4.3.2. Cold crucible remelting (CCR)................................................................... 477 4.4. Chemical extraction of inclusions........................................................................... 481 4.5. Fracture method.....................................................................................................482 4.6. Spark-induced optical emission spectroscopy.........................................................485 4.7. Oxygen content.......................................................................................................487 4.8. Other methods ........................................................................................................487 4.8.1. Inclusion ﬁltration ...................................................................................... 487 4.8.2. Eddy current method.................................................................................. 488 4.8.3. Electric sensing zone method...................................................................... 488 4.9. Summary comparison of methods.......................................................................... 489 5. Statistical prediction of the size of the maximum inclusion in a large volume of steel ...489 5.1. Extrapolation of the log-normal function ..............................................................489 5.2. Statistics of extremes .............................................................................................. 493 5.2.1. Statistics of extreme values (SEV) method .................................................494 5.2.2. Prediction of the maximum inclusion size in a given volume of steel using the Generalized Pareto Distribution (GPD) method ........................500 5.2.3. Inﬂuence of the sampling method on the estimates of the GPD method...508 5.2.4. Comparison of the log-normal extrapolation, SEV and GPD ...................509 5.2.5. Precision of statistics of extremes methods for estimating the maximum inclusion size............................................................................................... 510 6. Summary .........................................................................................................................515 Acknowledgements...............................................................................................................516 References ............................................................................................................................516

1. Introduction Inclusions have remarkable inﬂuences on the properties of steels [1–11]. The large inclusions can be disastrous, whereas the very small inclusions are unavoidable and usually not dangerous. Of all the inclusions, the hard and brittle oxides usually have the most harmful eﬀect on the properties of steels.

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The application of new secondary reﬁning techniques [12–14] and non-metallic inclusion reduction techniques [15–17] in steel production processes has greatly reduced the size and amount of non-metallic inclusions remaining in molten steels and steel products. The inclusions in clean steels consist of a few large ones and clouds of small ones. The size, size distribution and the maximum inclusion size in a given volume of steel are particularly important for clean steels. The properties of clean steels are highly aﬀected by the few large inclusions. Prediction of the steel properties can be made from the theoretical models based on the estimation of the maximum inclusion size [18–20]. However, the progressive improvements in the cleanness of high quality steels for critical applications have led to a low inclusion content which is diﬃcult to inspect by conventional methods such as optical metallography and ultrasonic inspection. The combination of low concentration and small size requires the examination of unrealistically large areas or volumes to give a statistically signiﬁcant measure of the inclusion content. As the demand on the reliability and life of steel components continues to grow, it is necessary to develop a set of methods for clean steels capable of describing as accurately as possible, the distribution of endogenous inclusions and relate it to each process route. Methods must also be available to detect scarce exogenous inclusions and must be suﬃciently accurate and ﬂexible to be used as an aid in the development of new processes. Many methods have been developed for the characterization of inclusions in steels based on the requirements of steelmakers and steel users. In addition, the size of the maximum inclusion in a large volume of steel has to be predicted using data from a small sample of perhaps only a few grams. This requires statistics. This review introduces the main methods used for the characterization of inclusions in clean steel and highlights the recent development of statistical methods for the prediction of the maximum inclusion size in large volumes of steel. It begins with a brief introduction to the characteristics of inclusions in clean steels and the developing requirements for characterization methods. This is followed by a description of the inﬂuence of inclusions on the properties of steels, particularly fatigue and fracture, which illustrates the importance of control. The next section introduces the major inspection methods: surface analysis by optical microscopy, non-destructive testing, inclusion concentration methods, chemical extraction of inclusions, fracture methods, spark emission spectroscopy, oxygen content and other methods. In the remainder of the review, statistical methods for the prediction of the maximum inclusion size in a large volume of steel are introduced and compared including extrapolation of the lognormal distribution, the Statistics of Extreme Values (SEV) method and the Generalized Pareto Distribution (GPD) Method.

2. Characteristics of inclusions in clean steels 2.1. Inclusions in clean steels Clean steels are often regarded both by the producers and consumers as reliable, uniform products, the properties of which can be predicted from a general knowledge

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of the metallic phase. Inclusions in clean steels consist of few large particles and lots of small ones. The volume fraction of inclusions depends directly on the oxygen and sulphur content. Advances in steelmaking technology have resulted in remarkable decreases of the contents of oxygen and sulphur, and hence a signiﬁcant decrease of the inclusion content [21] as shown in Fig. 1. Because of the low incidence of inclusions and limited testing volume of most cleanness assessment methods, the measured result is very sensitive to the sampling. The assessment of inclusions essentially depends on statistical probability analysis. The main focus in characterising inclusions in clean steels is changed from point to point study of individual inclusions to statistical assessment. The clustering of inclusion particles which usually exists in traditional steels, is rarely found in clean steels. Therefore the shape, size, size distribution and the maximum size of inclusion in a large volume of steel emerge as the most important parameters. In addition, because the failure of steel components starts from larger inclusions, estimation of the maximum inclusion size in larger volumes of steel becomes an important issue for both the steelmakers and steel users. In practical situations, measurements of inclusions are usually carried out on small volumes of steels by, for example, optical metallography of polished sections. The size of the maximum inclusion is below the resolution of ultrasonic tests. The maximum inclusion size in, say, a full cast of 300 t must be obtained by predictions based on the size distribution curve with statistical analysis. The estimation of the maximum inclusion size in larger volumes of steel will be helpful for steel users needing to be aware of the potential dangers caused by the worst inclusion, and for steelmakers needing to control the inclusion size in the steelmaking process.

Fig. 1. The number of inclusions in 1 cm3 as a function of the total oxygen content and inclusion size, assuming all oxygen being in equal size Al2O3 inclusions [21].

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2.2. Critical inclusion size in steels Large oxide inclusions are dangerous for most steel properties, and much more harmful than small inclusions [22–25]. A critical inclusion size is usually deﬁned, above which inclusions are dangerous and can cause the failure of steel products [2]. This critical size is diﬀerent for diﬀerent properties of steel, for example, fatigue, welding, bend fracture strength, hot working and corrosion. For the fatigue properties, the critical size and position in relation to the steel surface have been determined [26–29]. The critical inclusion size can be estimated using fracture toughness calculations, provided that inclusions can be regarded as defects similar to cracks in the steel matrix. The critical inclusion size for fatigue failure in rotating bending of bearing steel is around 10 mm if the inclusion is just below the surface and increases to about 30 mm at about 100 mm below the surface [26–29]. There is therefore a need to estimate the maximum inclusion size in a large volume of steel and the probability of ﬁnding inclusions larger than a critical size. The critical inclusion size problem, however, relates not only to estimation, but also to assessment. It is diﬃcult to ﬁnd a safe method to determine the presence of isolated large inclusions. Sophisticated and costly quantitative microscopic methods do not solve the problem as they relate to the examination of very small volumes of steel, and the possibility of ﬁnding a large, isolated inclusion on the polished surface of a steel sample is minute. The only valid method is to extrapolate from the data on inclusions in a small volume of steel by statistical analysis. Recently, methods to do this based on the statistics of extremes have been developed and these will be described in detail in Section 5.

3. Eﬀects of inclusions on the mechanical properties of steels Inclusions are generally present as oxides or sulphides in steel, although nitrides are signiﬁcant in some products and may be associated with other inclusions. They have great eﬀects on the properties of steels depending on their size, composition and distribution. 3.1. Fatigue properties For decades, inclusions have been associated with the general problem of fatigue failure in steels. For steel applications where high fatigue strength under dynamic loading is required such as for bearings, crankshafts and gears, the main focus in production is the control of inclusion characteristics [30–32]. Non-metallic inclusions reduce the fatigue strength and endurance (Fig. 2) [33]. It is now generally accepted that the smooth specimen fatigue limit decreases with increase in the total inclusion content. The hard and brittle oxide inclusions are the most harmful and the least harmful inclusions are the MnS inclusions [34]. Duplex oxide–sulphide inclusions are less deleterious than single-phase alumina or calcium aluminates since the presence of a more deformable inclusion phase such as MnS modiﬁes the dangerous properties of the hard oxides [32].

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Fig. 2. Schematic illustration of the eﬀect of inclusions on fatigue S–N curve [33].

Stress localisation at the interface between the inclusion and matrix is the origin of fatigue failure initiation. This arises [35] from (1) the diﬀerential thermal contraction of inclusions and matrix during cooling and (2) the concentration of remote applied stresses due to elastic constant diﬀerences between the matrix and the inclusions. As a results of these stresses, cracks are initiated at the interface between the inclusion and the steel matrix during hot deformation such as rolling (Fig. 3) [36]. The eﬀect of inclusions on the fatigue crack initiation depends on the chemistry, size, density, location relative to the surface, and morphology. Brittle oxide inclusions are more detrimental to fatigue than sulphide inclusions which have a high index of deformability. Siliceous types of inclusions behave in an intermediate way [37,38]. Large exogenous inclusions of slag or refractory origin are always detrimental to fatigue properties of steels because of their large size and irregular shape. If inclusions have a low thermal expansion coeﬃcient, the residual tensile stresses generated during cooling may approach the yield strength of the matrix [39–41]. The circumferential stress ﬁeld will increase with inclusion size. Therefore large inclusions will experience larger tensile stresses in the presence of imposed cyclic loading [40], and hence be prone to earlier crack nucleation. In terms of the eﬀect of inclusion chemistry on fatigue crack initiation, the most detrimental inclusions, in decreasing order of harm, are calcium aluminates, Al2O3, and spinel. This is precisely the order in which the average thermal expansion coeﬃcients increase [32].

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Fig. 3. Schematic diagram of formation of cracks and voids around inclusions during rolling [36].

Fig. 4. Relationship between the average oxide inclusion size on fractured Rotating Bending Fatigue (RBF) specimens and RBF fatigue limit [42].

The fatigue properties are signiﬁcantly aﬀected by the size of inclusions. It is now generally accepted that the rotating bending fatigue limit of bearing steels decreases with the increase of inclusion size (Fig. 4) [42]. In Fig. 4, moving up the curve to the left represents improvements in steelmaking practice and cleaner steels. There is a critical minimum inclusion size for fatigue crack initiation. Inclusions close to or on

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the surface are more dangerous than those in the interior since the surface alters the principal stress distribution about a buried inclusion subject to uniaxial loading [32]. Spherical oxide inclusions larger than 20 mm and embedded in or close to the steel surface are potential nucleation sites for the main crack [37]. The shape of inclusions has a great inﬂuence. The maximum principal stress at equivalent locations on an inclusion boundary due to an applied uniaxial load can diﬀer by a factor of more than 100% for spherical and elliptical inclusions having the same modulus. Inclusion orientation changes relative to the applied stress ﬁeld can produce stress alterations with the same magnitude [32,37]. Therefore, the critical minimum size is decreased if the oxide inclusions are angular or if the stringer type is aligned perpendicular to the stress axis. Small inclusions are unimportant for crack nucleation but may contribute to fatigue crack propagation. Fracture is frequently initiated by large inclusions, followed by void formation and growth around small inclusions [43] as illustrated in Fig. 5. Kiessling [37] related the crack propagation rate to the stress intensity factor and suggested an alternative dimple-rupture and cleavage-rupture mechanism for fatigue crack propagation, observing that the dimples are always associated with inclusions smaller than 0.5 mm in diameter. Barbangelo [41] found that the inﬂuence of inclusions on the fatigue crack growth rate was a function of the local stress intensity factor range at which fracture propagates. For low local stress intensity factor values, the crack growth rate in the steel with high inclusion content was lower than in the steel with lower inclusion content. As the local stress intensity factor increased, an inversion in the diﬀerence between the two rates occurred. The propagation rate decreased with increasing steel cleanness but was independent of the inclusion composition. The inclusions are more dangerous at lower tempering temperature than at higher because the matrix then has higher strength [38]. For the improvement of the fatigue properties of steel, the number of the large sized inclusion should be reduced and since the control of individual inclusions is diﬃcult the

Fig. 5. Schematic diagram of microvoid nucleation by incoherent inclusions [43].

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most practical way to achieve this is to lower the oxygen content. However, as it is impossible to produce steels without inclusions, control over the size distribution becomes more and more important since it largely determines the probability of the presence of a large inclusion at a critical location. 3.2. Fracture behaviour Ductile fracture in steels is caused by the nucleation, growth, and coalescence of voids which are nucleated at hard particles such as inclusions, pearlite nodules, and carbides [1,3]. Inclusions are harder than the surrounding matrix at room temperature. This leads to the stress and strain concentration during matrix deformation which can produce voids by matrix-inclusion decohesion or fracture of inclusion particles. Voids nucleate more easily if the inclusion particle is rigid, has a low cohesion with the matrix, or has a low internal fracture strength. Basic studies of the inﬂuence of inclusions on the mechanics of ductile fracture have shown [3,44] that for the hard and brittle oxide inclusions, the formation of the void occurred by decohesion at the inclusion/matrix interface at negligibly small strains. For the elongated sulphides, voids were nucleated by breaking up of the particles into segments. The elongated sulphides have great eﬀects on the fracture properties of steels. During the fracture process, voids are ﬁrst formed at MnS inclusions, which are usually the largest, then at smaller oxide inclusions and ﬁnally at small carbides [45]. For a given matrix and inclusion type there is a minimum particle size below which voids will not form. Reducing the volume fraction of inclusions and control of the inclusion shape will improve the mechanical properties associated with high strain levels, i.e. reduction of area, Charpy ductile shelf energy and fracture toughness. It was found by Biswas et al. [46] that sulphide parameters such as volume fraction and aspect ratio increase linearly with the increase in sulphur content whereas the projected length varies non-linearly. The number of inclusions per unit area increases with increase in the sulphur content. Through-thickness ductility in plate materials shows improvements with reduced sulphur content and shape control leading to globular rather than elongated inclusions, thus enhancing the resistance to lamellar tearing. Experimental studies [47] and pioneering calculations [48] suggested that the upper shelf fracture toughness of ultra-high strength steels is sensitive to the inclusion spacing and inclusion volume fraction. Reducing the content of inclusions increases the resistance to void nucleation and is eﬀective in improving the mechanical properties of steels [49]. The reduction of sulphur content may be a means of achieving reasonable fracture toughness. Knott and co-workers [50,51] have also found that crack opening displacement was signiﬁcantly increased by decreasing the volume fraction of sulphide inclusions containing manganese. Ray and Paul [52] found that Charpy shelf energy and impact transition temperature were signiﬁcantly aﬀected by inclusion morphology. The modiﬁcation of stringer inclusions to tiny granular ones resulted in considerably higher shelf energies, lowering of impact transition temperatures, and minimal anisotropy of impact properties. Similar results were obtained by Tomita [53,54].

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Pacyna [55] found that in hot working tool steel, the inﬂuence of inclusions on the toughness is related to the inclusion distribution and the hardness level of the steel. Uniform arrangement of inclusions in the steel matrix can be considered as harmless for the fracture toughness. At the high hardness level, the inclusions, because of their action with a very small plastic strain zone, can be treated as natural obstacles to crack propagation. At the low hardness level, the role of inclusions in the process of crack formation is limited by the carbide precipitation from martensite. In summary, for the fatigue related situations, the failure originates from large oxides rather than sulphides. Sulphide inclusions are more important for the anisotropic behaviour of steels e.g. fracture toughness. For most of the high strength steels such as bearing steels, it is the oxide inclusions rather than sulphide inclusions which cause most of the failure.

4. Characterisation methods for assessment of inclusions in steels The parameters to be investigated can be grouped as follows:

quantitative parameters (amount, size, etc.) shape distribution chemical composition speciﬁc properties (physical, corrosion, electrical etc.) eﬀects on the properties of steels.

Determination of the composition of inclusions and tracing their origins is routine in cleanness assessment today. The properties of inclusion phases, as well as their inﬂuences on the properties of steels, have been well recognised. Therefore, the main interest here is focused on the quantitative parameters, such as the size, amount and distribution of inclusions. Methods for the characterization of inclusions can be grouped as follows: 1. techniques based on surface analysis by optical microscopy 2. non-destructive testing ultrasonic tests magnetism-related techniques X- ray transmission 3. inclusion concentration methods 4. chemical analysis 5. fracture methods 6. oxygen determination 7. spark emission 8. statistical prediction

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4.1. Techniques based on surface analysis by optical microscopy 4.1.1. Standard chart comparison The comparison chart method has been the traditional approach for rating inclusion content of steels. The common method is the Jernkontoret (J-K) picture chart [56], based on comparison of the steel sample with a series of images in chartform, most commonly by the ASTM Practice for Determining the Inclusion Content of Steel (E45–85) and by German Standard Microscopical Examination of Special Steels for Non-Metallic Inclusions Using Standard Micrograph Charts (VDEh 1570–71). Some approaches have also been made to improve the measurement reliability using comparison charts. For example, worst ﬁeld measurement results can be better described by producing frequency distributions of the worst ﬁeld ratings [57]. Chart ratings can also be supplemented by data on the length and thickness of inclusions [58]. The chart ratings may be combined to produce a single index that reﬂects the overall inclusion content [59]. This was further developed by combining the traditional comparative charts and optical image analysis [60–62], which can give quantitative information on inclusions detected. The standard chart comparison method is simple and quick. The main problems are that it gives less quantitative information about the size and morphology of inclusions than image analysis, and many ﬁelds of view are needed for steels with low inclusion incidence. The traditional comparative charts have no pictures for diﬀerentiating between a clean and a very clean steel. This situation results from the out of date standard charts used in the assessment. The standard charts were made according to the quality of steels and industry needs in the 1960s. In addition, nitride inclusions are not considered in the J-K standard charts but, in steels with the addition of Ti and other nitride forming elements, the properties of steel can be aﬀected by the nitride inclusions [63]. 4.1.2. Image analysis The application of computer controlled automatic image analysing systems makes it possible to obtain the quantitative features of non-metallic inclusions. Several alternative methods based on statistical analysis have been developed to characterise the features of inclusions in steel by the combination of image analysis and statistical treatment. The accuracy can be very high in the high resolution image processor. With image analysis, not only the amount and size of inclusions can be measured, but also the distribution and clustering of inclusions can be determined quantitatively. Many techniques have been developed based on diﬀerent statistical analyses. 4.1.2.1. Number density distribution. In this method, the number of inclusion particles per unit area NA is counted in successive locations or ﬁelds on the polished section of the steel sample [64–68]. A quantitative measure for the degree of inhomogeneity or clustering of the inclusion distribution in the steel sample is the standard deviation, , deﬁned as: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 12 X ¼ NA i NA n i

ð1Þ

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where NAi is the observed number of the inclusions per unit area in the ith location (ﬁeld of view) and NA is the average number of inclusions per unit area in n ﬁelds viewed on the sample. Maximum uniformity is characterised by a minimum standard deviation, and the degree of uniformity decreases as the standard deviation increases. In order to compare the relative homogeneity of inclusion dispersions in samples having diﬀerent number densities, the standard deviation has to be normalised by the value of the mean, i.e. the coeﬃcient of variation, CV, deﬁned as: CV ¼ ð2Þ NA The measured variation may depend on the area of the measured ﬁeld relative to the area or the size of the clusters. For example, if the ﬁeld area is so large that it contains a number of clusters, one may anticipate that the standard deviation will be very small. 4.1.2.2. Area fraction measurement. This method involves measuring the area fraction of the particles with a small test area. The area fraction AA, standard deviation of the ﬁeld values of the area fraction, and the coeﬃcients of variation are determined [69,70]. The more homogenous the distribution, the lower the standard deviation and the coeﬃcient of variation. Another method for assessing particle distributions based on area fraction measurements [71,72] is based on the area fraction measured using square ﬁelds of small size arranged contiguously is a square grid, for example, 20 ﬁelds in the X direction, repeated 20 times in the Y direction for a total of 400 ﬁelds. A work sheet is prepared with the data arranged in rows and columns in the same arrangement as taken on the specimen. In the basic method, a one is placed in the position corresponding to the ﬁeld location (that is, in the proper row and column) if the area fraction measured for that ﬁeld is greater than the mean area fraction for all the ﬁelds. If the area fraction for a ﬁeld is less than or equal to the mean, a zero is placed in the proper row/column location. The resultant table is called a null–one matrix. Next, the numbers of ones in each row and each column are counted and listed to the right of the rows and beneath the columns. The standard deviation of the total number of ones in each row, Sr, and each column, Sc, is then determined. The ratio of phase arrangement anisotropy, , is calculated as Sr/Sc, that is, the ratio of standard deviation of the sums of ones in rows to that in columns. Then, the frequency of local segregation occurrence, , is calculated by dividing the total number of ones in the null-one matrix by the number of matrix terms, i.e. the number of ﬁelds. Then, a generated quantitative index of particle segregation, , is calculated as /. The smaller the value of , the more uniform the spatial distribution of inclusions. The above basic method can be modiﬁed by listing the actual area fraction measurements in the matrix table rather than ones or zeros. 4.1.2.3. Nearest neighbour spacing distribution. This is a commonly used method to determine the nearest neighbour spacing distribution of the particles and compare

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the observed mean and variance with the expected mean and variance for a random Poisson distribution [73–78]. To determine the nearest neighbour spacing(nns), the distance between each particle and all others in the ﬁeld of view is determined. First, all x and y co-ordinates of all particle centroids in the dispersion are obtained, and then distances between all pairs of particles are calculated. In a dispersion of N particles, the total number of paired spacings is N(N1)/2. However, only one spacing for each particle is the smallest spacing corresponding to its nearest neighbour. Therefore, there are N nearest-neighbour spacings in a dispersion of N particles. These nearest-neighbour spacings can be measured easily by automatic image analysis, and a frequency distribution can be determined readily. To describe the distribution as being ordered, random or clustered, comparisons should be made between the mean nearest neighbour spacing and its variance with the expected values obtained for ordered, random and clustered dispersions of points. For an ordered distribution, a hexagonal arrangement of points can be assumed. The expected nearest neighbour spacing for this ordered distribution is given by: 1 EðXÞ ¼ pﬃﬃﬃﬃﬃﬃﬃ NA

ð3Þ

where NA is the average number of points per unit area. The expected variance E(S2)=0. For a random dispersion of points, a Poisson distribution can be assumed. The probability distribution function of nearest neighbour spacing X is given by : FðXÞ ¼ 2NA Xexp NA X 2

ð4Þ

The expected mean for nearest neighbour spacing is: 1 EðXÞ ¼ pﬃﬃﬃﬃﬃﬃﬃ 2 NA and expected variance 4 1 E S2 ¼ 4 NA

ð5Þ

ð6Þ

For a dispersion consisting only of clusters, the expected mean for nearest neighbour spacing is: 1 EðXÞ << pﬃﬃﬃﬃﬃﬃﬃ 2 NA

ð7Þ

and the expected variance is: 4 1 E S 2 << 4 NA

ð8Þ

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Therefore, it is possible to characterise the dispersion of inclusions by comparison of the mean nearest neighbour spacing and its variance with the expected values obtained for a random distribution. Based on the ratio Q of the observed to the expected (for random) mean nearest neighbour spacing and the ratio R of the observed to the expected variance of the nearest neighbour spacing the comparisons can be made as follows: for: random dispersion Q=1, R=1 ordered dispersion 2> Q > 1, 0< R < 1 clustered dispersion 0 < Q < 1, 0< R < 1 The nearest neighbour spacing technique can be very useful in describing the observed inclusion distribution as being ordered, random, clustered, or composed of clusters superimposed on a random inclusion distribution. It is the last distribution that is observed most frequently. In this case, the nearest neighbour spacing technique can give some indication of the extent of clustering, but it is not a very sensitive technique since it is based on the nearest neighbour only. Better results are obtained if the distributions of second and third nearest neighbours are determined. However, this makes the analysis more diﬃcult. Edge corrections have been shown to be valuable in such work. Considerable computer time is required for this analysis, especially as a number of ﬁelds must be evaluated. Again, this method gives no direct information concerning the number of clusters, the percentage of particles in clusters, the number of particles per cluster, or the size of spacing between clusters. 4.1.2.4. Dirichlet tessellation. The Dirichlet tessellation is a technique for subdividing a region containing a dispersion of a secondary phase into subregions to analyse the spatial distribution or clustering of inclusions. It takes account of the association of an inclusion with more than just its nearest neighbour. It is a geometrical construction of a network of polygons around the points in a point distribution. The result of the tessellation process is the division of the plane into convex polygons each surrounding a single point. Each polygon is uniquely deﬁned by the positions of all near neighbours around the point. The polygons are constructed by the normal bisectors of all the lines joining the particular point to all neighbour points as shown in Fig. 6. To construct the Dirichlet network for an inclusion dispersion, the coordinates of the centroids of inclusions and the area of each inclusion are recorded using image analysis. A computer program is then used to construct normal bisectors and assign a unique area to each inclusion in the dispersion. The near neighbour distances, the number of near neighbours around each particle, the cell area and cell shape, the number of sides per cell and the ratio of each particle to cell area [79–82] are then measured automatically. Comparisons are made between observed values and those for a random distribution of non-metallic inclusions to see whether the size distribution of inclusions is random or not. This method produces an excellent description of clustering and discriminates the change of the size distribution of inclusions during cold and hot working processes.

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Fig. 6. Construction of Dirichlet cells [64].

Methods based on surface analysis by optical microscopy are time-consuming and the results are aﬀected by the preparation of samples. When the oxygen content is low, the frequency of oxide inclusions is also very low and sample preparation defects will have a proportionately greater inﬂuence on the result compared with a case when the amount of inclusions is higher. In automated optical image analysis, the sample preparation defects and other artefacts cannot be separated from inclusions. This problem can only be avoided by very careful specimen preparation [83,84]. Using ﬁner diamond paste during the ﬁnal polishing, the number of ﬁelds with oxide and oxisulphide inclusions increases and the number of ﬁelds without inclusions decreases at diﬀerent magniﬁcations [84] as given in Table 1. 4.2. Non-destructive testing 4.2.1. Ultrasonic tests Ultrasonic tests of steel products are well-established. The detection of inclusions by ultrasonic testing is based on the diﬀerence in acoustic properties between the steel matrix and defects [85]. Ultrasonic testing is now routinely used for the

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Table 1 Eﬀect of magniﬁcation and ﬁneness of ﬁnal polish on number of inclusions observed by optical microscopy [84]

Oxygen determined by chemical analysis [O]cha [ppm] Registered number of inclusions Number of inclusions per unit area, mm2 Examined area, mm2 Total surface of inclusions, mm2 Oxygen in inclusions [O]incl [ppm] ([O]incl/[O]cha) (%) a

Magniﬁcation 400

Magniﬁcation 1024

1 mm diamond

0.25 mm diamond

1 mm diamond

54

54

54

54

142 15

165 18

30 (589)a 63 (63)a

35 (687)a 74 (74)a

9.296

9.296

0.001328 33.02 61.15

0.001475 36.73 68.02

0.4736 (9.296)a 0.00009263 45.89 84.98

0.4736 (9.296)a 0.00009583 47.31 87.61

1/4 mm diamond

Normalized values to the area of 9.296 mm2.

inspection of defects whose size is greater than 200 mm. The major advantage is that a large volume can be assessed, decreasing the chance of missing large harmful exogenous inclusions compared with conventional techniques [86,87]. Conventional ultrasonic testing systems have probe frequencies lower than 10 MHz and do not therefore reveal smaller defects such as clusters of inclusions of 50– 100 mm long. The development of a high frequency ultrasonic testing using 30–100 MHz probes is able to detect defects (holes or inclusions) smaller than 100 mm in diameter [88]. However, as the frequency is increased in order to detect smaller inclusions, then the ultrasonic signals do not penetrate so deeply into the materials. The main advantages of ultrasonic tests are the possibility of on-line testing, the non-destructive nature and the large tested volume of steel. However, the tests are severely dependent on surface quality and structural homogeneity. Typical in-line ultrasonic inspection devices at the billet stage therefore are limited in capability to detecting high contrast defects such as pipe, ﬂakes and large pores. For further worked and area reduced products, the structural homogeneity is improved and the surface quality is enhanced, therefore the inspection capability increases. However, the increase in the amount of working reduction is directly associated with closure of pores and break up of hard and brittle inclusions. This means that the better the detectability, the less there is to detect. Inclusions are of course, not removed by metal working. Even if they become harder to detect as the size distribution changes, they remain as potentially harmful particles in the steel. 4.2.2. Magnetism-related methods Magnetic methods have also been used for the internal inspection of metals and alloys. One of the commonly used magnetism related methods is the Magnetic Leakage Flux. This method is mainly used for the internal inspection of coil or samples from slabs, as well as on those from hot rolled strip [89–92]. The basic

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Fig. 7. Schematic illustration of the procedure for magnetic leakage ﬂux method [91].

principle is that when a discontinuity such as a defect or inclusion is present near the surface of a ferromagnetic material, a leakage ﬂux develops. The magnetic resistance in that position increases to a great extent compared with the rest of the material as illustrated in Fig. 7. If the ferromagnetic material is magnetised close to saturation, the magnetic ﬂux is perturbed by the discontinuity and deviates into the space over the defective part as leakage ﬂux, which can be detected by a magnetic sensor. A quality assurance system which can inspect tinplate steel strip on both surfaces along the entire length for internal defects measuring 5104 mm3 or over without hindrance to line productivity has been developed [89–91]. This method is only sensitive to elongated inclusions. Inclusions that are too short are easily lost in the random background noise. In addition, when inclusions are elongated beyond an optimum value, they become too thin to have any noticeable eﬀect on the magnetic ﬁeld.

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4.2.3. X- ray transmission method X-ray microradiography is a another commonly used non-destructive testing method for detecting inclusions in metals. A barely enlarged shadow picture of the object can be obtained using standard equipment. When using X- ray tubes with microfocus, a 5–10 times enlarged image can be produced [93,94]. An alternative is image enhanced ﬂuoroscopy, which avoids the use of radiographic ﬁlms, giving real time images. However, it is diﬃcult to separate holes and inclusions with this method. Moderate rolling might be necessary to close the holes. It is conceivable to perform the detection on slab pieces in the as-cast condition. On radiographs the holes appear blacker than inclusions of same size, as the former have a higher transmission coefﬁcient. This method is an option of interest for inclusion detection. The investment is large and the operational costs are high. 4.3. Inclusion concentration methods The low incidence of inclusions in clean steels means that the number of detectable inclusions on the polished surface of samples is limited. A large number of samples have to be used in order to get statistically meaningful results. Inclusion concentration methods were developed to concentrate the non-metallic inclusions of a bulk sample into a small area of the surface so that the inclusions could be more easily examined. Two methods have been developed in recent years, i.e. electron beam button remelting (EBBM) and cold crucible remelting (CCR). 4.3.1. Electron beam button melting (EBBM) This method has attracted much attention in recent years. The steel sample (up to 1 kg) is melted by an electron beam in vacuum. Drops of the liquid metal are collected in a water cooled copper mould and a molten pool formed. The inclusion particles concentrate in a small area (‘raft’) on the surface of the button as illustrated in Fig. 8. The collected inclusions can subsequently be assessed by a variety of techniques ranging from simple visual comparison to detailed chemical classiﬁcation and sizing in the scanning electron microscope [95–98]. This technique was originally developed for assessing nickel based alloys, and has since been successful for the collection and subsequent analysis of inclusions in a variety of metals and alloys [99]. The conditions to obtain high collection eﬃciencies in diﬀerent steels are diﬀerent. They have been found to depend on several factors [95] including: 1. the wettability of the inclusion in the melt which depends on the composition and structure of the inclusions; 2. the liquid metal ﬂow directions which depends on the purity of the melt, in particular the sulphur contents. The technique has been used for the cleanness assessment in relation to a wide range of problems in clean steels such as [95–99]:

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Fig. 8. Schematic illustration of the basic concept for evaluation of non-metallic inclusions by electron beam button remelting [95].

1. comparison of the steel cleanness from diﬀerent production processes, i.e. vacuum induction melting(VIM) and VIM-vacuum arc remelting(VAR), etc; 2. assessment of eﬀects of diﬀerent refractories on VIM bar stock; 3. comparison of eﬃciency of ﬁlters for inclusion ﬁltration; 4. ranking of cleanness of diﬀerent alloy batches; 5. identifying the heterogeneous distribution of inclusions in cast bar stock; 6. behaviour of inclusions in the tundish during continuous casting; 7. pickup of inclusions during the investment casting process; 8. cleanness evaluation of super clean steels; 9. determination of the size distribution of inclusions; and 10. prediction of sheet surface defects. The main advantages of this method are [95]: (1) High ﬂoating eﬃciency of inclusions Sixty–eighty per cent of inclusions (in terms of the total oxygen) will ﬂoat out under suitable irradiation energy. This allows the inspection of the larger inclusions in a sample. The size distribution of inclusions can be obtained by measuring the size of inclusions collected. (2) Close correlation with the optical microscopy method The results of regression calculations correlating the EBBM method to the optical microscopy method, chemical dissolution method, and oxygen method showed that the EBBM method has a close correlation to the optical microscopy method. The EBBM method was found to be applicable to evaluate the cleanness of steels from diﬀerent processes as shown in Fig. 9. (3) Over ﬁve times higher measurement eﬃciency than that with the optical microscopy method.

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Fig. 9. Comparison of speciﬁc oxide area determined by EBBM method (open points) and number of inclusions determined by the optical microscope analysis (closed points) [95].

The EBBM method has over ﬁve times higher measurement capacity than with the optical microscopy method, drastically reducing time and observation labour as shown in Table 2. (4) Measurement area equivalent to over 350,000 visual ﬁelds by the optical microscopy method Suppose a sample with an area of 25 mm2 and a weight of 1 g is observed at a magniﬁcation of 400. When the observation of the area is completed, the surface is polished by 10 mm, to produce and observe a new surface. By repeating this operation, the number of visual ﬁelds needed to complete sample observation is over 350,000. Since the EBBM method provides a wide visual ﬁeld, it excels over other methods in representation, providing more opportunity to measure large-sized inclusions. (5) Higher accuracy Since inclusions can be gathered in one place, the analysis accuracy for their particle size, shape and composition can be improved. (6) Less contamination The steel is molten in a water cooled copper hearth and hence it is possible to prevent contamination with e.g. refractories, oxygen and nitrogen. When adequate melting conditions are used, it is possible to probe the actual state of the primary deoxidisation products. (7) Larger sample size

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477

Table 2 Comparison in the required measuring time of inclusion analysis between the EBBM method and optical microscopy analysis (for 30 samples) [95] Evaluation methods

EBBM method Microscopy

Required time (min)

Total

A

B

C

D

Sample preparation

EB melting (sample setting, vacuum evacuation, melting, cooling, sample take-out)

Sample embedding and polishing

Measurement of inclusions

90(25) 330(17.4)

140(38.9) 0

0 990(52.4)

130(36.1) 570(30.2)

360 1806

Sample preparation: started on a pin-shaped sample with the EBBM method, on a concave sample with the optical microscope method. The amount and size of inclusions are measured by a Quantitative Television Microscope. The ﬁgure in ( ) shows the percentage to the total measurement time required.

The large sample ensures that statistically signiﬁcant values can be obtained for inclusion concentration and size in super-clean materials [100,101]. The major disadvantages of EBBM are [95]: 1. Carbon containing steel undergoes a carbon boil problem which results in the reduction of oxide inclusions; 2. Inclusions with low melting points tend to sinter during the process, which can lead to diﬃculties in evaluating inclusion size distribution; 3. The apparatus is relatively expensive.

4.3.2. Cold crucible remelting (CCR) In this method, a cold crucible is used to melt the sample. It utilises a copper crucible constructed in segments, each of which is water cooled, as illustrated in Fig. 10. A high frequency current is applied to a coil surrounding the crucible. The slits in the crucible ensure that the eddy currents induced in the sample have the same sign as those induced in the crucible and this results in a repulsive force between the sample and crucible. The repulsive force causes the sample to levitate and consequently there is no contact between the alloy and the crucible [102,103]. Previous work by Barnard and co-workers [102] has found that inclusion particles within the sample will ﬂoat to the surface of the remelted button during the melting process. Fig. 11 shows the appearance of a cold crucible remelted button of a steel. Several veins, which start from the bottom of the button, are apparent on the surface of the remelted button. The inclusion particles are not collected in one area like the ‘raft’ in electron beam button remelting, rather they are congregated on the surface of the button along the veins (opposite to the slits in the crucible) and on the surface areas between veins. The size and number of the areas where inclusion particles congregate are diﬀerent

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Fig. 10. Schematic diagram of the cold crucible levitator [102].

for diﬀerent steels [104], as shown in Fig. 12. The collected inclusion particles are exposed on the surface so that the diameter of inclusions can be measured by observation under a SEM. Most of the inclusions collected are oxide particles. Both large and small inclusion particles are collected on the surface of the remelted buttons. Many inclusions are less than 3 mm in diameter and hence would have been diﬃcult to observe under optical image analysis. The size distribution of the inclusion can be determined by measuring the size of collected inclusions. The size distribution is of a log-normal form as shown in Fig. 13. The main advantages of this method are as follows: 1. It can handle samples of about 100 g, and thus produce statistically signiﬁcant results for alloy cleanness. The melting process can be carried out very quickly.

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Fig. 11. Appearance of the cold crucible remelted button of a clean steel.

Fig. 12. (a and b) Inclusions collected on the surface of cold crucible remelted buttons; (c and d) inclusion particles collected on the surface of cold crucible remelted buttons.

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Fig. 13. Size distribution of inclusions collected on the surface of a cold crucible remelted button.

2. It can be operated at 1 bar pressure of argon (or Ar–H2), thereby avoiding problems with carbon boil. 3. The collection eﬃciency is similar to that obtained in EBBM. 4. The method is relatively inexpensive, and it would appear to have potential for routine determination of product cleanness in super-clean materials. The collected inclusion particles are spread over a large area on the surface of the remelted button, which makes it diﬃcult to view and count them all under the SEM. Furthermore, the curvature of the surface of the remelted button introduces problems in focusing on all the inclusion particles in the ﬁeld at high magniﬁcations. It is impractical to inspect all the inclusions particles on the surface of the button. Consequently, the following measures have been taken [102] in an attempt to concentrate the inclusions into one region: 1. The molten metal was cooled at diﬀerent rates. 2. The liquid metal was quenched by ﬂooding the chamber with hydrogen. 3. The position of the crucible relative to that of the coil was altered by adjusting the height of the support arm; lowering the crucible resulted in a solid base and molten cap and raising the crucible in a molten base with a solid cap.

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Barnard and co-workers [102] found that none of these measures resulted in the concentration of inclusions into a single area. The ﬂuid ﬂow in the molten button was responsible for the collection of the inclusions along the veins on the surface of the remelted button. It was suggested that the deposition of inclusion particles along the veins is due to the formation of vortices in the liquid area between any two slits as a result of localised increases of magnetic ﬂux density. Butler and co-workers found that not all inclusions are trapped on the surface of the button. Most alumina-based particles occurred within 100 mm of the surface of the button [105], as illustrated in Fig. 14. Silicon rich inclusions were trapped inside the buttons and most of the oxides which have been brought close to the surface are aluminium or calcium rich. To overcome this problem, a simple electrolytic method was developed by Butler and co-workers to extract the inclusions from the surface of the button and those close to the surface. The extracted inclusions can then be analysed using a Laser Diﬀraction Particle Size Analyser. The combination of cold crucible remelting, electrolytic extraction of particles and laser sizing is signiﬁcantly faster than conventional extraction routes and samples a larger volume than conventional metallography. It is estimated that a full distribution of sizes including sampling, sample preparation and analysis of a comparatively large mass of material could be achieved within 4 h which compares with about 2–4 days for electrolytic extraction for a similar volume or about 8 h for a mass a hundred times smaller by metallographic analysis [105]. 4.4. Chemical extraction of inclusions The metal matrix (100–200 g)can be dissolved by either chemical [106–109] or electrochemical dissolution [106,110,111], in a speciﬁc organic extraction liquid.

Fig. 14. Map of inclusions in section of a cold crucible remelted button [105].

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After recovering the insoluble non-metallic inclusions, the morphology, size distribution and structure can be analysed by SEM, laser particle size analysis and X-ray diﬀraction. The amount of inclusions can then be calculated [112]. The collected particles can also be dissolved in a strong acid and analysed using atomic absorption spectroscopy to determine the amount of inclusion forming elements. Attention must be paid to contamination during dissolution and some chemical treatments must be performed under a dust controlled atmosphere. The chemical etchants are systematically ﬁltered before the treatment. The heat treatment of the materials prior to the dissolution has a great inﬂuence on the dissolution. Without a speciﬁed heat treatment, the dissolution front is not ﬂat, and big particles of metal could be removed from the massive sample and could thus disturb the subsequent ﬁltration. This method seems to be well adapted for identifying the process route. Investigations performed on diﬀerent melts show a certain diﬀerence between the nonmetallic inclusion populations. It can also be used to identify exogenous inclusions originating from refractories. Indeed, it has allowed the validation of several hypotheses concerning the origin of these rare but detrimental inclusions. The main advantage of this method comes from the large volume of the investigated sample and the available quantitative information obtained (in term of number of particles and of chemical analysis). The drawbacks are that it is time consuming (2 weeks between the beginning of the dissolution and the end of the energy dispersive spectrometer (EDS) analysis and interpretation) and it is necessary to be very careful when using chemical treatment (contamination, dust, etc.). The question of whether the solvent distinguishes completely between the matrix and the inclusions also exists, particularly if the inclusions contain transition metals. 4.5. Fracture method The suitability of many steels for their application is judged by the mechanical properties attainable, hence fracture tests have also been used as inclusion assessment methods. One possible approach is in the use of short transverse reduction of area (STRA) tests for lamellar tear resistance. The STRA method is a good inclusion assessment method for ductile structural steels, and can be augmented by fractographic examination to determine the nature of the non-metallic inclusions [3,93,113]. Other methods [3] currently in use include the through-thickness tensile testing on hot rolled slabs from concast stock; transverse tensile testing on hot rolled billet and the blue brittle test [114]. For the blue brittle test, a notch is preformed on the surface of samples which are then heat treated to a high ductility and fractured. The presence of large inclusions or inclusion clusters will induce failure and the worst inclusions will be exposed. For the blue brittle test, the fresh fracture surface is blue treated. The inclusion clusters in the tested volume are exposed on the fracture surface and appear as bright trails on the blue background. They can be easily observed under an optical microscope. A typical inclusion cluster band on a fracture surface is shown in Fig. 15.

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The cleanness levels of diﬀerent steels can be evaluated by comparing the number, length and width of inclusion clusters on the fracture surface. For the remelted super clean steels, a clean fracture surface is found (Fig. 16) whereas for air melted clean steels clusters of oxide particles and large sulphides are visible (Fig. 17).

Fig. 15. Banded cluster of small inclusion particles on the fracture surface of a clean steel.

Fig. 16. Fracture surface of a super clean steel.

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Fig. 17. Inclusions on the fracture surface of a clean steel.

The number of inclusion particles exposed on the fracture surface is much larger than that on a comparable polished surface. This is because the possibility of sectioning inclusions by a polished surface is much smaller than that by a fracture surface where the advancing crack tends to ‘‘seek out’’inclusions. On polished samples, the number of inclusions detected per unit area can be approximately estimated using [115]: NA ¼ NV D

ð9Þ

where D is the average size of inclusion particles. The situation in the blue brittle test is diﬀerent from that in image analysis as illustrated in Fig. 18. As steels are heat treated to a high ductility, the fracture is usually initiated from, and propagates [3] through, the worst inclusions in the tested sample, such as clusters of small inclusion particles or large individual particles. Therefore the chance of missing the worst inclusions is decreased. The number of inclusions which could possibly be detected per unit area can be approximately estimated using: NA ¼ NV L3 13 1 L3 ¼ 0:554 NV

ð10Þ ð11Þ

L3 is the average distance between particles, which is much bigger than D for the same steel. Therefore the number of inclusions observed on the fracture surface is much larger than that on the polished surface. The main advantage of the fracture method is that it allows characterisation of the size, type and distribution of the worst inclusions in a sample volume considerably larger than can be utilised economically for metallographic assessment. However, the fracture is localised to the preformed notch in the slice, and the samples have to heated treated before and after the fracture. The volume tested is a small part of a slice and the statistical evaluation of the results is diﬃcult. It is also not eﬀective for super clean steels. For example, very few oxide inclusion particles are found on the fracture surfaces of the VAR and ESR steels.

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Fig. 18. Schematic illustration of the exposure of inclusions on a polished surface and a fracture surface.

4.6. Spark-induced optical emission spectroscopy OES-PDA (Optical Emission Spectroscopy: Pulse Discrimination Analysis) is a physical technique of analysis using an optical emission spectrometer. The sample surface is sparked with a ﬂow of spectrometric argon at a certain frequency. One spark of 100 Hz is performed for 20 s, consisting of 2000 elemental sparks of 10 ms each [88,116,117]. The volume analysed by an elemental spark is about 5–8 mm in diameter and 10–50 mm in depth [ < 0.5 mm3 (< 5 mg)]. This method, which requires only rough surface grinding, is extremely rapid and has classically been used to determine the nominal chemistry of metals and alloys. Large areas of sample surface can be analysed by the continuously overlapping sparking. The analysis of several elements can be carried out simultaneously. The elemental signal of the eight elements is then stored in a database and subsequently treated. The low intensity pulses correspond to the elements dissolved in the metal, whereas the high intensity pulses are related to the elements contained in the inclusions. Multi-elemental

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Fig. 19. Frequency histogram of the size in 2 mm of oxides detected by image analyser and OES pulse intensity for two heats (52100 grade-Al-killed steels) [88].

correlation on high intensity pulses allows the determination of the proportion of diﬀerent kinds of oxides contained in the sample, with respect to their composition [88]. Considerable progress has been made in this technique, which is regularly used by steelmakers [118] to monitor melting and reﬁning processes. The qualitative and quantitative evaluation of the inclusions in a steel now takes only a relatively short time. The size distribution of oxide inclusions can be determined from spark emission spectrometry. The minimum diameter of inclusions which can detected by this method can be as small as 1.5 mm [118]. There is a good correlation between the height of pulse intensity and the size of the oxide inclusions determined by optical metallography coupled with image analysis [88], as shown in Fig. 19. This method is well suited for description of the population of deoxidisation, reoxidation and solidiﬁcation inclusions in steels. It is also well suited to following the steel cleanliness during steelmaking. A comparison of the total oxygen content in the liquid steel at diﬀerent stages of the process route with the number of oxide inclusions identiﬁed by OES shows a very good correlation (Fig. 20) [88]. However, taking into account the volume sampled, it is diﬃcult to detect detrimental exogenous inclusions [88].

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Fig. 20. Comparison of the oxygen content of the liquid steel at successive reﬁning stages and OES results in terms of number of oxide inclusions (arbitrary time scale) [88].

4.7. Oxygen content Chemical analysis of oxygen is usually taken as a measure of the bulk inclusion content in steel and used as a criterion of steel cleanness in the steelmaking process control [119–121]. Since the solubility of oxygen in the metallic matrix is extremely low, the bulk concentration of oxide inclusions can be calculated, based on the oxygen analysis provided that the chemical composition of inclusions is known or can be satisfactorily anticipated. The chemical analysis results can be converted to area fractions of inclusions by: AA ðoxidesÞ ¼ OWt% ðdmatrix =dinclusion Þ ðMoxide =Mmatrix Þ

ð12Þ

where AA is the area fraction of inclusions on a polished surface, dmatrix and dinclusion are densities of the steel matrix and inclusions respectively, and Moxide and Mmatrix are the molecular weight of the oxide inclusions and steel. However, the oxygen content does not give any information on the size, morphology and distribution of inclusions. Steels with the same oxygen content might have diﬀerent size distributions. 4.8. Other methods 4.8.1. Inclusion ﬁltration The size and number of inclusions can be reduced substantially by passing liquid metal through ceramic ﬁlters. Filtration is routine for Al and certain superalloys [122], but application to steels has been limited by the lack of ﬁlter materials suitable for high liquid metal temperatures and problems of priming the ﬁlters. Interest in this technique has grown considerably with understanding of the ﬁltration mechanism and improvement of the ceramic ﬁlter quality. There has been

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considerable work on the use of ﬁlters for cast iron, stainless steels and a range of superalloys [123–128]. Honeycomb or foam ﬁlters, made from Al2O3, ZrO2, SiC or mixtures of these ceramics have been used. They are mounted either in the tundish or in the runnerware or trumpet. Removal of inclusions from the liquid metals is by both trapping of particles too large to pass through the pores and impingement of smaller particles on the ﬁlter. Thus the eﬃciency of a ﬁlter is governed by several parameters, of which the most important in addition to pore size are: melt viscosity; ﬁlter morphology; and materials of construction, together with the initial size range of inclusion particles in the liquid metal [129]. In general, the principal application currently remains with the superalloys, for which the ﬁltration provides a signiﬁcant improvement in cleanness over the product of conventional vacuum induction melting. So far, at least where mass produced steel is concerned, a successful application has not been reported. This is because the ceramic ﬁlters used for a steel melt are exposed to severe conditions compared with a non-ferrous melt, that is, for example, high temperature, high density and high interfacial tension. They are required to withstand thermal and mechanical stresses, and erosion by the steel melt. At the same time, they must oﬀer an adequate ﬁltration eﬃciency without any serious resistance to the ﬂow of molten steel. A ceramic loop ﬁlter has been developed [130] by piling up ceramic strings with a uniform diameter to a required thickness, which shows stronger resistance to the stresses imposed by the steel melt. However, major development is required before it is applied in the steel industry. 4.8.2. Eddy current method An eddy current instrument operates like an electrical transformer [93]. A primary winding is connected to an ac power source, the core is the metal which is being examined, and the secondary windings are small eddy current loops induced in the metal. This method is surface sensitive, hence the surface condition of samples may dominate over the eﬀects of deeply located inclusions. Since the eﬀects of inclusions and holes are essentially the same, the sample has to be rolled and should be of very good surface quality. 4.8.3. Electric sensing zone method This method has been developed to directly measure the number density and the size distribution of inclusions in liquid metal systems [131]. It is similar to that of a Coulter Counter where non-conductive particles pass through an electrically insulated oriﬁce in the presence of an electric current for counting and sizing. The principle of this method is that when small particles, suspended in a measuring medium, pass through an electrically insulated oriﬁce, the electric resistance of the ﬂuid ﬂowing through this sensing zone increases in direct proportion to the volume of a particle. In the presence of an electric current, a resistive voltage pulse is generated when a particle is in the oriﬁce region, residence times typically being of the order of a millisecond. The quantitative information about the inclusion particles can be obtained according to the resistive voltage pulse. However, this method gives no information on the morphology of inclusions.

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4.9. Summary comparison of methods A comparison of the main methods in Section 4 in terms of the investigated volume, the type of information obtained, the duration of the analysis and the advantages and drawbacks is given in Table 3 and Fig. 21. It can be seen that no single technique is capable of describing the whole inclusion distribution and giving all the formation needed. An appropriate combination of methods is necessary to reach the speciﬁc goals. The inspection time can vary from a few minutes to 100 h, for the analysis of 1 mg to several hundreds of grammes. Statistical analysis is needed to assess the results obtained.

5. Statistical prediction of the size of the maximum inclusion in a large volume of steel 5.1. Extrapolation of the log-normal function The distribution of the size of most phases in metals and alloys has a log-normal form, such as the grain size in metals and alloys [132,133]. The size of inclusions in steels is also found to have a log-normal form [134–136]. The cleanness of diﬀerent steels can be discriminated by comparing the mean and standard deviation of the log-normal distribution. In Fig. 22, Steel A is an air melted clean steel and steel B is a remelted super-clean steel. The data was obtained using cold crucible remelted samples. The size distributions of the inclusions in the two steels have a log-normal form but with diﬀerent parameters. The probability of ﬁnding larger inclusions in steel A is larger than for steel B. One method of predicting the size of the maximum inclusion in a large volume of steel is by extrapolating the log-normal distribution. A detailed procedure has been given by the authors [138]. Suppose the size distribution of inclusions has a log-normal form. The probability density function for inclusions with size x can then be calculated by [138]: 1 1 y

2 fðyÞ ¼ pﬃﬃﬃﬃﬃﬃ exp 1
ð13Þ

where y=Ln(x) and and are the standard deviation and mean of y. The cumulative proportion P(x) of inclusions no greater than size x is therefore: 1 PðyÞ ¼ pﬃﬃﬃﬃﬃﬃ 2

ð LnðxÞ 1

1 y

2 exp dy 2

ð14Þ

Consider now a volume V of steel, and deﬁne the characteristic size of the maximum inclusion (CSMI) in V as the size of inclusion, xV say, which it is expected will be exceeded exactly once in V. If NV is the mean number of inclusions per unit

490

Table 3 Comparison of methods for the characterisation of non-metallic inclusions in steels Time for Time for sample analysis preparation

Volume tested

Volume Size of Location of Number of Morphology Chemical Advantages fraction inclusions inclusions inclusions information information per unit area or volume

Drawbacks

J-K assessment

2–3 h

30–60 min

10–100 mg

No

No

No

No

No

No

Simple; Quick

Less quantitative; Many view ﬁelds

Image analysis

2–3 h

4–10 h

10–100 mg

Yes

Yes

No

Yes

No

No

Large quanitity of information Quantitative

Time consuming; Not suitable for very small inclusions

Ultrasonic tests

2–3 h

1–2 h

Large volume

No

Yes

Yes

Yes

No

Yes if +EDX

On-line examination Quantitative

Time consuming; Not sensitive to small inclusions

Blue brittle test

10–20 h

0.5–1 h

Large volume

No

Yes

Yes

No

Yes if +EDX

Yes if +EDX

Exposure of the Time consuming; worst inclusions Statistical diﬃculties

Cold crucible remelting

1–2 h

0.5–1 h

100 g

No

Yes

No

No

Yes if +EDX

Yes if +EDX

Concentration of inclusions in the surface of the remelted sample

Diﬃcult to quantify

Sparking emission

1 to 2 h

5–10 min

1–10 mg (single sparking)

No

No

Yes

No

No

Yes

Information about the distribution of diﬀerent inclusions

Expensive; Diﬃcult to quantify

Electron beam button remelting

1–2 h

20–30 mins 1 kg

No

Yes

No

No

Yes if +EDX

Yes if +EDX

Concentration of inclusions and large quantity of metal

Expensive; Non-stable inclusions

100–250 g Yes

Yes

No

Yes

Yes

Yes

Large quantity of metal and quantitative information

Time consuming

Electrolytic dissolution 1–2 h +SEM/EDX

2 weeks

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Methods

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491

Fig. 21. Schematic representation of the diﬀerent methods for describing inclusion distributions in steels (BFT: blue fracture test, US: ultrasonic testing, OES-PDA: optical emission spectroscopy with pulse discrimination analysis) [119].

Fig. 22. Size distribution of inclusions in two diﬀerent steels.

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volume in the tested volume, then, since the probability of an inclusion larger than x is 1P(x), it follows that: NV Vð1 PðxV ÞÞ ¼ 1

ð15Þ

The solution to Eq. (15) is given by: xV ¼ exp þ F1 1 ðNV VÞ1

ð16Þ

where F is the standard Normal distribution function [138]. The parameters and can be estimated from the measured sample by the maximum likelihood method (which takes Q the estimates of and to be the values which maximize the likelihood L= ni¼1 f ðyÞ based on the observations y1. . .yn ). The likelihood method yields also approximate proﬁle likelihood conﬁdence intervals for the parameters and for xV [139–141]. The estimated characteristic size of the maximum inclusion increases with the increase of the logarithm of the volume of steel and there is no upper limit for the estimated inclusion size [137] as shown in Fig. 23. The estimated sizes are diﬀerent for steel with diﬀerent cleanness levels. The width of the conﬁdence intervals increases with the increase of the volume of steels for the extrapolation, as shown in Fig. 24. The standard method of ﬁtting the log-normal distribution requires quantitative measurement of inclusion sizes right across the size range to obtain a good ﬁt, a laborious procedure for the many small inclusions for two reasons. Firstly, the resolution in optical microscopy systems makes it diﬃcult to detect the smallest inclusions in the size distribution curve. Secondly, the low number of the large

Fig. 23. Characteristic size of the maximum inclusion in large volumes of steel estimated by extrapolating a log-normal distribution in two diﬀerent steels [137].

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493

Fig. 24. Characteristic size and conﬁdence intervals of the maximum inclusion in large volumes of steel estimated by extrapolating a log-normal distribution (Steel A in Fig. 23).

inclusions aﬀects that part of distribution statistically. Usually, only inclusion particles in the central part of the distribution can be easily measured. There are ways of ﬁtting which dispense with the requirement to measure the full size range, but nevertheless, the method relies on the assumption that a log-normal distributionform for the middle range of inclusions implies that the same distribution will describe the unobserved large values, something which is neither logically inevitable or testable [137]. 5.2. Statistics of extremes The methods relating to statistics of extremes are eﬀective for this purpose because extreme value theory is intrinsically about extrapolation. In its simplest form the problem is, given a set of independent data from an unknown distribution, to estimate accurately the tail of the distribution. The problems of measuring small inclusions can be avoided by using prediction methods based on the extreme theory where only measurements of the size of inclusions larger than a threshold value or measurements of the maximum inclusions in randomly chosen areas or volumes are needed. Two methods have been developed for practical applications based on the diﬀerent branches of the theory of statistics of extremes. The ﬁrst is termed the statistics of extreme values (SEV) method. It is based on measuring the maximum size of inclusions in randomly chosen areas or volumes and has been standardised by Murakami and co-workers [142–150] for discrimination between super-clean steels and the estimation of the maximum size of inclusions in a large volume of steel. The second method is the Generalized Pareto Distribution (GPD) method, developed by the authors [137,151], where the size of all inclusions

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larger than a chosen threshold value is measured. Both SEV and GPD methods allow data on inclusion sizes in small samples to be used to predict the maximum inclusion size in a large volume of steel. 5.2.1. Statistics of extreme values (SEV) method The statistics of extreme values method (SEV) has been used in many ﬁelds relating to metals, such as (1) the estimation of the pit depth of the localised corrosion of metals, (2) the maximum segregation of impurities in the solidiﬁed microstructure of alloys [152], (3) estimation of grain size during the recrystalization of alloys [153], and (4) estimation of the service life arising from stress corrosion [152,154]. Murakami and co-workers [155–158] were the ﬁrst to apply this method to estimate the size of the maximum inclusion in a large volume or area of steel from data from the polished surface. The basic concept of extreme value theory is that when a ﬁxed number of data points following a basic distribution are collected, the maximum and minimum of each of these sets also follow a distribution, which is diﬀerent from the basic distributions such as normal, exponential, log-normal etc. The distribution function was given by Gumbel [159] as follows: GðzÞ ¼ expðexpððz lÞ=ÞÞ

ð17Þ

where G(z) is the probability that the largest inclusion is no larger than size z, and and l are the scale and location parameters. If the reduced variate, y: y ¼ ðz lÞ=

ð18Þ

is introduced, then from Eq. (17) its distribution function is HðyÞ ¼ expðexpðyÞÞ y ¼ ðz lÞ= ¼ LnðLnðHðyÞÞÞ

ð19Þ

In the practical measurement by the method of Murakami and co-workers, a standard inspection area S0 (mm2) is deﬁned. The area of the maximum inclusion in pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ S0 is measured. Then, the square root of the area areamax of the maximum inclusion is calculated. This is repeated for N areas S0 as illustrated in Fig. 25. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ The values of areamax;i are classiﬁed, starting from the smallest, and ranked with i=1,2. . .N. Then: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ areamax;1 4 areamax;2 4 . . . 4 areamax;N The cumulative probability of inclusion sizezi can be calculated simply by: Hðyi Þ ¼ i=ðN þ 1Þ

ð20Þ

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495

Fig. 25. Illustration of the procedure for the SEV method [171].

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where i is the ith of the ordered areamax;i . If y=Ln(Ln(i/(N+1))) is plotted against inclusions size zi, a straight line of slope 1/ and intercept on the vertical axis (l/) is obtained, i.e. pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð21Þ areamax ¼ y þ l An illustrative example is shown in Fig. 19. The values of and l can be estimated by least squares [155,157], moment [160,161] and the maximum likelihood method [151,162–165]. The maximum likelihood method has been found to be the most eﬃcient estimator [166] because its estimators of the extreme defects have the lowest error. The likelihood is given by the product of the probability density functions corresponding to Eq. (9): zi l ð z i lÞ exp þ exp L¼ i¼1 N Y 1

ð22Þ

and the estimates are obtained numerically as the values of , l and which maximize this. The maximum likelihood method gives the best use of the observed data and a more precise estimation than linear regression.

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For the prediction of inclusion size in a large volume of steel V, the return period, T, is deﬁned as [155,157]: ð23Þ T ¼ V=V0 The characteristic size of the maximum inclusion (CSMI) in volume V ( i.e. the size expected to be exceeded exactly once in volumeV) can be deﬁned by solving Eqs. (21) and (23) with G(z)=1–1/T to give: y ¼ LnðLnððT 1Þ=TÞÞ

ð24Þ

Vo is the standard inspection volume and is deﬁned by Murakami and co-workers [155–157] as: V0 ¼ h S0

ð25Þ

whereh is the mean value of the P pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ areamax;i h¼ N

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ areamax;i as illustrated in Fig. 26. ð26Þ

Hence, the maximum inclusion in the large volume V, zV can be predicted with Eq. (21). In eﬀect, the y value corresponding to the large volume V has been calculated using Eqs. (24) and (25). Then the intercept between thisy value and the pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ straight line plot ofln(ln(H(y)) versus areamax has been found and the corresponding maximum inclusion size identiﬁed as illustrated in Fig. 27. The theoretically speciﬁed error of the estimation, known as the Cramer–Rao lower bound, is [166]: SECR ½xv ¼ pﬃﬃﬃ 0:60793y2 þ 0:51404y þ 1:8066 ð27Þ n where y is the reduced variate, y=ln(ln(1–1/T)). The error of the estimates of xV obtained using a given method such as the proﬁle likelihood method is larger or at least equal to the theoretical speciﬁed deviation.

Fig. 26. Deﬁnition of standard unit volume V0 [155].

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Fig. 27. Schematic illustration of the estimation of the maximum inclusion size in a large volume of steel by the SEV method.

In practice, estimates of the parameters , l and xV are based upon limited data and so are subject to uncertainty. Moreover, the greater the degree of extrapolation, the greater will be this uncertainty. The analysis of the 95% conﬁdence intervals by the proﬁle likelihood is based on calculation of the maximum of the likelihood L corresponding to diﬀerent ﬁxed values of the parameter of interest [137]. The proﬁle likelihood interval is then the range of values of the parameter for which L diﬀers no more than would be expected by chance from its maximum value as illustrated in Fig. 28. What this means in practice is that the proﬁle maximum likelihood value L is plotted against possible CSMI values. The peak in the curve corresponds to the CSMI prediction. Dropping down by 1.92 on either side gives the 95% conﬁdence limits on that predicted value. Thus, the interval consists of values of the parameter judged to be consistent with the observed data, this consistency being assessed on the basis of the assumed distributional model. A further source of uncertainty is the validity of the distributional model beyond the largest inclusion observed—something which the observations do not allow one to check. The conﬁdence interval should therefore be regarded as indicating lower bounds of uncertainty only: the true uncertainty being greater. For the predictions from the SEV method, the characteristic size of the maximum inclusion in a given volume increases linearly with the increase of the logarithm of weight of steel. There is no upper limit for the estimated size [151]. The precision of the estimation depends on the number of sample areas used for the measurement

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Fig. 28. Schematic illustration of the estimation of conﬁdence limit interval by the Proﬁle Likelihood Method [137].

and the volume of steel used for the extrapolation [167]. The estimated size is little aﬀected by the number of samples but the width of the conﬁdence intervals decreases with the increase of the number of sample areas. The larger the volume of steel for the extrapolation, the larger the width of the conﬁdence intervals [167]. Similar results have been obtained by Beretta and Murakami [166] with computer simulated data. Overall the number of samples needed for the estimation with the SEV method depends on the precision and the volume of steel needed. For a given precision, the minimum number of samples tends to increase with return period T (T=V/V0) and the shape ratio /l as shown in Fig. 29. Major work has been done on this method by Murakami and co-workers [142– 150,155–158]. The SEV method has been used to predict the lower limit of fatigue strength of high strength steels with inclusions at diﬀerent locations [155,157], as shown in Fig. 30. For example, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1=6 areamax pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1=6 for subsurface inclusions: w ¼ 1:41ðHV þ 120Þ= areamax pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1=6 for interior inclusions: w ¼ 1:56ðHV þ 120Þ= areamax for surface inclusions: w ¼ 1:43ðHV þ 120Þ=

where w is the lower limit of the fatigue strength, HV is the hardness of the steel and pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ areamax is the size of the maximum inclusion in the volume of steel tested.

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499

Fig. 29. Maps of iso-T curves that show the minimum number of defects needed to obtain a stability of bias: (a) gain of 0.1% on 10 sampled defects; (b) gain of 0.1% on 20 sampled defects [166].

Fig. 30. Various locations of inclusions causing fatigue fracture and the fatigue limit prediction equation [157].

The merits of the SEV method, as compared with conventional methods, are (1) it distinctly discriminates the cleanliness of super-clean steels, and (2) it predicts the size of large inclusions contained in a domain larger than the inspection domain. The predicted size presented in the SEV method contains more information than the traditional methods. First, from the point of view of metallurgy, the predicted result of the inclusion size can be used as a database for the reduction of inclusion size in the steelmaking processes. Moreover, the method proposes a prediction of the lower band of the fatigue limit. The prediction can be used for diﬀerent types of specimens and applied to the design of components made from high strength steels. This method is also useful for the quality control of steels and for improvement of the steelmaking processes. However, for the estimates from the SEV method, there is no upper limit for the estimated inclusion size. The estimated inclusion size increases linearly with the volume of steel, which is not expected in practical steelmaking.

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5.2.2. Prediction of the maximum inclusion size in a given volume of steel using the Generalized Pareto Distribution (GPD) method 5.2.2.1. Procedure of the GPD method. The Generalized Pareto Distribution (GPD) is the standard family of statistical distributions used as a basis for modelling data which arise as exceedances over some thresholds. For a set of data collected on the basis of all values, both the number, n, and their values xi are random variables. For suﬃciently high thresholds, and for a wide variety of initial distributions, the number of exceedances over the threshold u, is reasonably modelled by a Poisson distribution and their values, x, by a Generalized Pareto Distribution with distribution function [137]: FðxÞ ¼ 1 ð1 þ ðx uÞ= 0 Þ1=

ð28Þ

0

where > 0 is a scale parameter and ð1 < < 1Þ is a shape parameter. The range of (xu) is 0 < xu < 1 if 50 and 0 < xu < 0 / if < 0. When x=0, F(x) is interpreted as the limit as tends to zero, which is F(x)=1exp((xu)/ 0 )), an exponential distribution. If x is the size of inclusions, F(x) is the conditional probability that an inclusion is no greater than x, given that it is at least u. The parameters 0 and in Eq. (28) can be estimated by the maximum likelihood method from those observations, x1, . . ., xk say, larger than u. The likelihood is given by the product of the probability density functions corresponding to Eq. (28): k Y 1

ðxi uÞ ð1= Þ1 L¼ 1 þ ð29Þ 0 0 i¼1 and the estimates are obtained numerically as the values of 0 and which maximize this. In practice, estimates of the parameters 0 and , and hence of the CSMI xV and upper end point u( 0 / ), are based upon limited data and so are subject to uncertainty. The conﬁdence intervals for the parameters can be estimated by the proﬁle likelihood technique. The data needed for the GPD method is the number and size of inclusions larger than a certain size as illustrated in Fig. 31, which is diﬀerent from that for the SEV method (see Fig. 25). When the GPD is used to model the sizes of inclusions which exceed a threshold u, the characteristic size of the maximum inclusion (CSMI) xV in a volume V can be found as follows [137]. The expected number of inclusions in volume V exceeding a size x is equal to the product of the expected number in V exceeding u and the probability, given that an inclusion is larger than u, that it is larger than x. This last probability is given immediately by Eq. (29). Thus, if NV(u) denotes the expected number of exceedances of u in unit volume, the size xV which is expected to be exceeded exactly once in volume V satisﬁes: NV ðuÞVð1 FðxV ÞÞ ¼ 1

ð30Þ

With F as in (20), this gives xV ¼ u

0 1 ðNV ðuÞVÞ

ð31Þ

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501

Fig. 31. Illustration of the procedure for the GPD method [171].

NV(u) may be estimated from a Woodhead Analysis based on the planar sizes of inclusions intersected by a polished surface [137,151]. The probability of ﬁnding inclusions larger than a given size, x say, greater than the threshold u may be found from the distribution function (29). The data needed for the GPD approach are the numbers of inclusions NV(u) of size larger than a threshold u in a test volume and their sizes, as illustrated in NV(u). Therefore, the practical diﬃculty in measuring the many small inclusions less than the threshold can be avoided. The Generalized Pareto Distribution method can be used for data conforming to various distributions in their central parts, e.g. lognormal, exponential, normal, Weibull, without the assumption of a particular form for the upper tail [137,151]. If < 0, then (NV(u)V) < < < 1 when V is very large, so from Eq. (31): xV ¼ u ð 0 = Þ

ð32Þ

Thus, u( 0 / ) will be the upper limit for the inclusion size. The probability of ﬁnding inclusions 5u( 0 / ) is zero. The choice of the threshold u is a major practical issue in this method, partly because there is no point in collecting data on inclusion sizes much below the threshold since these data will then be discarded in the analysis, hence saving labour. A preliminary choice of the threshold u can be obtained by plotting the mean observed excess of the inclusion size over the threshold u against u, giving what is

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Fig. 32. Illustrative mean excess plot for the GPD method [137].

called a mean excess plot. If observations aboveu indeed come from a Generalized Pareto Distribution, then the mean excess plot beyondu will be approximately linear with an intercept on the vertical axis 0 /(1 ) and slope /(1 ), as illustrated in Fig. 32. It is sometimes diﬃcult to judge whether the chosen threshold is right. Actually, it has been found by the authors that the estimation of the GPD is relatively insensitive to the selection of the threshold. The present authors explored the use of quantile– quantile plots to judge the ﬁtting of the threshold to the GPD model [137]. Inspection of a sequence of such plots for a range of values of u can aid choice of the critical value of the threshold on which further analysis can be based. In general, the critical threshold should be high enough to ensure a good ﬁt to the GPD whilst retaining enough data above it to allow optimum estimation of the parameters of the GPD. The procedure of the GPD method can be summarized as [137]: 1. Choose a size above which the inclusions are to be measured. This might be 3 mm if the analysis is with optical metallography and image analysis on polished cross sections. It might be smaller if the quantitative measurement is of inclusions viewed in the SEM after e.g. cold crucible remelting. If it is larger than 3 mm, say, then a larger area will need to be viewed in order to gather suﬃcient data for the conﬁdence limits to be reasonable

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503

2. Once the inclusions have been measured, plot the mean excess versus the threshold. The mean excess for a particular threshold is found by taking the diﬀerence between the size of each particle and the threshold and averaging all these diﬀerences. This is repeated for a whole series of potential thresholds and the results summarised on the mean excess plot. An illustrative example is shown in Fig. 32. 3. The plot is inspected to ﬁnd where it becomes reasonably linear, rather than displaying the steep curve that tends to occur at the left hand end. The point beyond which it becomes reasonably linear is termed the critical threshold. For graphical estimation of s0 and , the line is drawn: the intercept on the vertical axis gives s0 /(1 ) and the slope is /(1 ). These two equations can then be solved for rough estimates of the s0 and parameters. For a more objective procedure, the maximum likelihood method is applied by computer, using all the data points above the critical threshold (including those from the right hand end of the plot which embody the information about the biggest inclusions actually observed) to ﬁnd the best values of s0 and . 4. NV(u) is then found from the measured data. 5. All the parameters for Eq. (31) are then known and the characteristic size of the maximum inclusion (CSMI) xV can be calculated for the chosen large volume V. The conﬁdence limits can also be predicted, along with the upper limit for the inclusion size from Eq. (32) if < 0 and the probability that no inclusion is larger than a certain size from the distribution function F(x) in Eq. (28). 5.2.2.2. Estimation of the upper limit for the inclusion size in clean steels. One of the main characteristics for the GPD method is that there is an upper limit for the estimated inclusion size when the shape parameter is negative. The estimated inclusion size is below the upper limit whatever the volume of the steel as is shown in Fig. 33. Estimates of have been found to be generally negative in diﬀerent steels [137,151]. The probability of ﬁnding inclusions larger than the upper limit is zero. This accords with the expected situation for inclusions in steels. In the steelmaking process, the ﬁnal size and distribution of oxide inclusions depends on the formation and extent of removal of the deoxidation products from the melt before the steel solidiﬁes. The ﬂoating out of inclusions from a steel melt can be approximately described by Stokes law [168]: Ut ¼

2gr2 ðm i Þ 9m

ð33Þ

where Ut is the terminal velocity of the inclusion, r is the radius of inclusions, m and i the densities of the liquid steel and inclusion respectively, m the viscosity of the liquid steel, and g the acceleration due to gravity. Provided the steel is allowed to stand for suﬃcient time, inclusions greater than a certain size will ﬂoat out, however many casts of steel are made by a certain processing practice. Previous modelling work has found that spherical particles in the tundish of 50–100 mm in size are likely to ﬂoat out, leading to a physically based maximum size [169]. If the same concept is applied to electroslag remelting, a terminal velocity less than the withdrawal velocity leads to a much smaller maximum size.

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Fig. 33. Illustration of the maximum inclusion size and the upper limit of Inclusion size estimated by the GPD method in a clean steel.

With the estimation of the upper limit of the inclusion size, it is possible for steelmakers and steel users to know the probability of ﬁnding inclusions larger than a critical size in steels and to predict the potential dangers caused by inclusions in steel products. This is particularly important for those steels used in critical situations where the inclusion size must be below a certain size. For example, for high performance bearing steels, the failure is usually by fatigue caused by the large hard and brittle oxide inclusions. It has been proved that the critical inclusion size for fatigue failure in rotating bending of bearing steel is about 30 mm for the subsurface inclusions [2]. The GPD method can be used to estimate the probability of ﬁnding inclusions larger than the critical size. 5.2.2.3. The effects of the GPD parameters on the estimated result. There are three parameters in the GPD method, threshold u, the shape parameter and scale parameter 0 . The parameters and 0 are determined based on the chosen threshold. Previous work from the authors showed that the characteristic size of the maximum inclusion in a large volume of steel estimated by the GPD method [137,167,170,171] is relatively insensitive to the selection of the threshold. However, the precision of the estimation from the GPD decreases at higher thresholds because of the smaller number of inclusions used for the estimation at higher thresholds. Therefore more data should be collected for estimates of GPD with higher thresholds. The parameter has a stronger eﬀect on the estimation in the GPD method than 0 . The characteristic size of the maximum inclusion increases with the increase of 0

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505

Fig. 34. Relationship between the characteristic size of the maximum inclusion in 1 kg steel (NV(u)=1, number of data=100) and the GPD parameters and 0 [172].

and . Larger 0 and less negative will give larger characteristic sizes of the maximum inclusion [172] as shown in Fig. 34. A similar relationship exists between the GPD parameters and the upper limit of inclusion size [172]. The upper limit of the estimated size increases steeply as becomes less negative. For the estimation of the characteristic size of the maximum inclusion in diﬀerent volumes of steel, the more negative the value of , the more quickly the estimated size approaches the upper limit [172] as shown in Fig. 35. However, the inﬂuence of 0 on the characteristic size of the maximum inclusion is diﬀerent. The upper limit of inclusion size increases gradually with the increase of 0 . For a ﬁxed value, the characteristic size approaches the upper limit when larger volumes of steel are considered [172] as shown in Fig. 34. The diﬀerences between the characteristic sizes of the maximum inclusion in 106 kg steel and the upper limits estimated by the GPD with diﬀerent 0 in Fig. 36 are much smaller than in Fig. 35. The width of the conﬁdence intervals increases with the increase of 0 and as approaches zero. There is a steep increase between =0.1 and =0 [172], as shown in Fig. 37. This indicates that , the shape parameter, has a more pronounced eﬀect on the estimation than . For a given number of inclusion particles, a small negative (particularly between 0.1 and 0) will give bigger estimated sizes and wider conﬁdence

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Fig. 35. Relationship between upper limit, characteristic size, weight of steel used for the estimation and for a ﬁxed 0 =4.0 [172].

intervals than a more negative . The value of is related to the size distribution of inclusions in diﬀerent steels. The more negative the value of , the smaller the probability of ﬁnding larger inclusions [172] as shown in Fig. 38. As the value of becomes less negative, the size distribution extends toward larger sizes. For practical application of the GPD method, the value is an indication of the size distribution of the inclusion size in steels. If more negative values are obtained in a steel, the probability of ﬁnding larger inclusions will be smaller; the characteristic size and upper limit of the maximum inclusion estimated by the GPD method will be smaller and the characteristic size of the maximum inclusion will approach the upper limit more quickly. 5.2.2.4. Relationship between the number of inclusions and confidence intervals of estimates of the characteristic size of the maximum inclusion in a large volume of steel. The precision of estimation of the GPD depends on the number of data. The number of sample areas has less eﬀect on the estimated size. However, the width of the conﬁdence intervals decreases with the increase of the number of inclusions [168]. Once the number of inclusions is greater than a certain value decline is slow, as shown in Fig. 39. The uncertainty also increases as the prediction is made for larger and larger volumes of steel. Similar results have been obtained with computer simulated data by the Monte Carlo method [172] as shown in Fig. 40. The relationships

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507

Fig. 36. Relationship between upper limit, characteristic size, weight of steel used for prediction and 0 for ﬁxed =0.1 [172].

Fig. 37. Relationship between average width of 95% conﬁdence intervals for the characteristic size of the maximum inclusion and GPD parameters 0 , (1 kg steel, NV(u)=1, number of inclusions=100) [172].

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Fig. 38. Comparison of probability density functions of GPD with diﬀerent values ( 0 =4.0) [172].

between the number of inclusions and width of the conﬁdence intervals are similar for estimation with diﬀerent and 0 . For practical application of the GPD method, the number of inclusion data needed is dependent on the precision of the estimation required and the volume of steel for the extrapolation. For the same conﬁdence, more data should be collected if the estimation is to be made in a larger volume of steel. If a particular volume of steel is considered, more data should be collected if a higher precision is needed. For a given set of data, the error of the extrapolation can be estimated. 5.2.3. Inﬂuence of the sampling method on the estimates of the GPD method One of the features of the GPD method is that only the size and number of inclusions larger than a certain starting size need be measured. Estimates of the GPD method have been compared by the authors [173] using data on inclusions from optical microscopy of polished samples (measuring inclusions larger than 3 mm) and SEM of the surface of the cold crucible remelted samples (which allows the measurement of inclusions down to 1 mm) The results from the two diﬀerent data sources are consistent as shown in Fig. 41; an important conclusion for the practical application of the GPD method. This means that there is no disadvantage in using a method such as optical metallography of polished sections and starting to measure inclusion size at 3 mm, 5 mm or even larger, provided suﬃcient numbers of inclusions

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509

Fig. 39. Inﬂuence of the number of sample areas on the characteristic size and conﬁdence intervals of the maximum inclusion in 100 t of steel for GPD method predicted by the maximum likelihood method [167].

are measured. This saves the cost of remelting and avoids diﬃculties in measuring inclusions on the surface of cold crucible remelted buttons. 5.2.4. Comparison of the log-normal extrapolation, SEV and GPD A summary of the comparison between the log-normal extrapolation, and SEV and GPD methods is given in Table 4. Fig. 42 shows the predictions from log-normal extrapolation and GPD. The results are similar for a small volume of steel but for the log-normal extrapolation, an approximately linear relationship exists between the characteristic size of the maximum inclusion and log10 (weight of steel). The larger the volume of steel, the bigger the maximum inclusion [137]. In contrast, with the GPD method there is an upper limit for the size of the maximum inclusion, no matter how large the volume of steel, provided is negative. The conﬁdence limits for the estimated CSMI values for the log-normal are wider than for the GPD [137]. The results for the SEV and GPD methods are similar in a small volume of steel but diverge for a large volume of steel [170, 71], as shown in Fig. 43. The estimated size from SEV increases approximately linearly with the increase of the logarithm of the weight of steel but there is an upper limit for the GPD. The existence of the upper limit makes the GPD a powerful tool in the design of components for fatigue

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Fig. 40. Computer simulated relationship between the average width of 95% conﬁdence intervals for the characteristic size of the maximum inclusion, number of inclusions and weight of steel used for the estimation ( 0 =0.5, =0.01, NV(u)=1) [172].

resistence [174]. Both the SEV and GPD methods are eﬀective in discriminating the cleanness of diﬀerent steels [170,171]. 5.2.5. Precision of statistics of extremes methods for estimating the maximum inclusion size The conﬁdence intervals of the estimation, which are crucial for design engineers, are dependent on the number of inclusions measured and the variations of the parameters for both the SEV and GPD methods. A certain number of inclusions must be measured to obtain a certain precision. There are fundamental diﬀerences between the two methods, with SEV predicting ever-increasing values, with relatively narrow conﬁdence intervals, and GPD converging on a maximum value, but with widening conﬁdence intervals as shown in Fig. 44 [167]. An important question is: why do the SEV and GPD methods lead to diﬀerent estimates and conﬁdence intervals? The answer depends on the relationship between the two ﬁtted distributions as discussed in [167], and is crucial for the interpretation and practical application of the two methods. Fig. 46 compares SEV, EXPGPD, GPD and GEV. EXPGPD is the GPD approach with ﬁxed to zero. The predictions are similar to SEV but the conﬁdence intervals are narrower [Fig. 45(b)]. The diﬀerence between the SEV and the EXPGPD is that the SEV uses only the sizes of the largest inclusions in the separate

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Fig. 41. Characteristic sizes and conﬁdence intervals of the maximum inclusions in a large volume of steel estimated by the GPD method comparing results from remelted button samples and polished samples [173].

Table 4 Comparison of methods for estimating the size of maximum inclusion in large volumes of steel Log-normal

Statistics of extremes

GPD

Measure all detectable inclusions Assumed distribution form

Measure the maximum inclusion in S0 Assumed distribution form

Have to measure more inclusions No upper limit for inclusion size

Avoids the diﬃculty of measuring small inclusions No upper limit for inclusion size

Measure inclusions above a given size in random areas No assumed distribution form and works whatever the distribution form of inclusion size such as log-normal, exponential and normal, etc. Avoids the diﬃculty of measuring the small inclusions Upper limit of inclusion size—in line with practical expectation of steelmakers More credible estimation based on more data and fewer assumptions

Only uses the maximum and ignores other large inclusions

Each method allows you to calculate the probability of ﬁnding an inclusion larger than a critical size.

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Fig. 42. Comparison of predictions and conﬁdence limits for the characteristic size of the maximum inclusion in a large volume of steel estimated by the GPD and by log-normal with the same data from cold crucible remelting. AM=Air Melted steel. ESR=Electro-slag Remelted steel.

Fig. 43. Comparison of the characteristic sizes of the maximum inclusion estimated by the SEV and GPD methods in two diﬀerent steels [171].

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Fig. 44. Comparison of the characteristic size of the maximum inclusion and maximum likelihood conﬁdence intervals estimated by the SEV and GPD methods in 40 sample areas [167].

sampled areas whereas the EXPGPD utilises all measured sizes greater that the threshold. Hence the EXPGPD is able to use more information causing the conﬁdence intervals to be narrower. The SEV is a special case of the Generalised Extreme Value (GEV) distribution with =0. GPD and GEV are compared in Fig. 45(b). The conﬁdence intervals in GEV are much wider than those in GPD. The GPD and GEV are related by the fact that under the GPD assumption about excess sizes, the distribution of the size of the largest inclusion is GEV with the same shape parameter as in the GPD. Thus the apparent diﬀerence in precision of the SEV and GPD methods results from an assumption about (that =0 in SEV) not derived from the immediate data. If there is no external evidence to suggest a zero value of then the conﬁdence intervals calculated under this assumption in the SEV method will give an overoptimistic assessment of the precision of estimation. If cannot be assumed to be zero, then the general GPD method is recommended as making the best use of available information. The estimated inclusion size of GPD with a negative is below an upper limit. The probability of ﬁnding inclusions larger than the upper limit is zero. This accords with the expected situation for inclusions in steels for two reasons. In the steelmaking process, the ﬁnal size and distribution of oxide inclusions depends on the formation and extent of removal of the deoxidation products from the melt before the steel solidiﬁes. This suggests that

in GPD should be taken to be negative to give an upper limit for inclusion sizes, rather than set to zero as in SEV method. Preliminary modelling of the ﬂotation of inclusions in steel-making process further suggests that the negative value of may

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Fig. 45. Comparison of the characteristic sizes and conﬁdence intervals of the maximum inclusion in a large volume of steel estimated by the statistics of extremes method in steel A. (a) SEV and EXPGPD, (b) GEV and GPD.

be as low as 0.75. If were known to have a speciﬁc negative value then the GPDbased conﬁdence intervals would become much narrower [167], as illustrated in Fig. 46 for =0.1. In practice it is unrealistic to expect ever to be able to specify exactly, but more reﬁned modelling of ﬂow and ﬂotation dynamics in the steelmaking process may limit the range of possible values. In applying these concepts, it must be recognized that a cluster of inclusions, which could form on a refractory and be dislodged into the liquid late in the casting process, is considered to be exogenous

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Fig. 46. Characteristic size and conﬁdence interval of the maximum inclusion in a large volume of steel estimated by the GPD method with ﬁxed =0.1 [167].

and not part of the measured distribution. Further, in applying the results of inclusion size measurements it must be assumed that such exogenous inclusions are detected by NDT procedures and result in rejection of the volume of steel in which they occur.

6. Summary There is a continuing drive for increasing steel cleanness. This makes the inspection of inclusions diﬃcult. Here the main methods have been reviewed including: techniques based on surface analysis by optical microscopy; non-destructive techniques such as ultrasonics, magnetism related methods and X-ray transmission; inclusion concentration methods; chemical analysis; fracture methods; oxygen determination; and spark emission. Recently there has been a move to use the statistics of extremes to predict the maximum inclusion in a large volume of steel based on observations in small samples. Two approaches have been developed: Statistics of Extreme Values (SEV) and Generalized Pareto Distribution (GPD). Here these approaches have been compared and contrasted. If the parameter cannot be assumed to be zero, then the general GPD is recommended as making the best use of available information. In some circumstances, the GPD allows an upper limit for the inclusion size to be estimated. This is of importance to design engineers assessing the potential dangers caused by the inclusions in a steel component.

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Characterization of inclusions in clean steels: a review including the statistics of extremes methods

Characterization of inclusions in clean steels: a review including the statistics of extremes methods

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