Chapter X - Elongation and reduction of area

Contents

1. The metallurgy of ductility

2. Data preparation

3. Data set

4. Training the model

5. Application

I The metallurgy of ductility

Ductility is a measure of the ability of a material to undergo plastic deformation without failing [1]. A material that is ductile will be protected from fracture by small plastic deformations. There are two ways of quantifying the ductility of a material:

1. elongation e_f = l_f - l_o x 100 where l_f = length of specimen after fracture;

l_o l_o = original length.

1. reduction of area q = A_o - A_f x 100 where A_f = cross sectional area of

A_o specimen after fracture;

A_o = original cross sectional area.

Elongation may be considered as having two components, due to uniform and non-uniform elongation [2]. Uniform elongation, which is dependent on the strain hardening capacity of the steel, dominates the elongation behaviour [3]. When a metal such as a steel is subjected to several cycles of loading beyond the elastic limit, it progressively increases in strength. Each successive plastic deformation increases the stress required for slip to occur in a process called strain hardening. As strain hardening increases, the steel undergoes an increasing amount of plastic deformation and hence the total elongation at fracture increases.

Reduction of area is strongly linked to the necking process which is closely linked to slip [3]. Slip is caused by the alignment of dislocations to form a plane through the specimen resulting in fracture. Necking occurs when narrow bands of slip are formed in regions of high stress concentration, for a specimen undergoing a tensile test these would be at the points of maximum load. The local deformation caused by the slipping causes the steel to elongate and the cross sectional area to reduce. In face-centred-cubic metals, such as steel at room temperature, the metal undergoes necking before it undergoes ductile fracture [4]. Therefore, reduction of area measures the amount of deformation necessary to induce fracture [3]. The stresses in the neck are very complicated and depend heavily on the geometry of the specimen. This means that the reduction of area measurement cannot be taken to be a true material property.

Steels with a high volume fraction of second phase particles, such as carbides, silicates and sulphides, have reduced ductility [5]. The inclusions act as nucleation points for dislocations, either when the inclusion cracks or when the inclusion debonds from the matrix. Inclusions are added to steel to increase strength, and so they are strong but brittle materials that are much less ductile than the matrix. The shape and orientation of the inclusions are also important since these control the shape of the stress field which in turn influences when and where cracking or debonding occurs. This sensitivity to inclusions means that measurements of ductility can be used to assess the quality, or purity, of a steel [3].

II Data Preparation

It is necessary to prepare the data before the neural network is applied since it cannot process missing or text inputs. The data kindly provided by Corus for this project were from reports dating back to 1960 and contained many missing entries which were of three sorts:

1. Missing weight percentages of alloying metals found in steel as impurities, e.g. P, S

2. Missing weight percentages of alloying metals deliberately added to steel to improve the mechanical properties, e.g. Mo, Ti

3. A text column detailing the heat treatment which needed to be expressed in a numerical format, e.g. Norm 900°C, 30 min, Ann 620°C, 12 min.

Missing inputs in the first category were set to the average value of the other entries in the column [2]. Missing inputs in the second category were set to zero. In some cases the weight percentage of an element was given as an inequality. These entries were assigned values based on the mean values of the other entries. The values used for the neural network are given in table S1.

Table S1

The original data also gave information on arsenic, cobalt, oxygen and tungsten. These data were not statistically significant, and in the case of cobalt and tungsten represent specialist steels, so they, and the few steels involved, were removed from the database.

The data for tin was distributed in two small groups: 78 inputs very close to the minimum value of 0.0020 and 4 inputs at the maximum value of 0.5200. Rather than setting missing inputs to the mean of this distribution, it would have been an improvement to remove the four high tin steels and assign the missing inputs the mean of the remaining values. This should result in the predictions for tin having large error bars.

The textual information from the heat treatment column was extracted and expressed as a mixture of binary, temperature and time columns. The binary columns were used to indicate that a given heat treatment, e.g. normalising, had been applied. The weights given by the neural network to these binary parameters should indicate which of the heat treatments are important. Table S2 gives the details of the columns used to describe the heat treatments of

Table S2

the steels. The data concerning the temperature and time of the heat treatment is contained in three sets of coupled columns. The neural network needs to recognise this. In an attempt to preserve the information given and avoid biasing the network, steels that were not described as austenitised or normalised were assigned the mean first temperature value but a zero time value. It was assumed that if there were a second or third heat treatment this would be explicitly recorded, so that if it were not it was assumed to be absent. It would have been better if a random temperature within the range had been assigned to correspond to a zero time value. In this way the model would not incorrectly determine a relationship between the mean temperature and zero time, but would determine the relationship between the two columns.

Data quality

The data used for this project have been gathered from Corus reports written between 1964 and 1975. This data may be inaccurate and imprecise. The accuracy of the instruments used would have increased over the period within which the reports were written so the quality of the data is dependent on when it was produced. The attitude of the researchers towards accuracy may also have improved. Conversations with researchers working on steel in the 1960s suggest that many of the composition values were very difficult to measure and the results were not accurate and were often educated guesses, particularly at the beginning of the period. This uncertainty in the data may have increased the perceived noise in the models.

III The Data Sets

Models to study elongation and reduction of area were developed. Table S3 shows the distribution of the data in the prepared database.

Table S3

A neural network is used to find information about the many-dimensional information space. When the network encounters a region of information space that is densely populated, the predictions produced are more accurate than where the data is sparse. The information space itself is very difficult to visualise but plots of individual variables against the target variable can highlight sparsely populated regions (figures S1a and S1b). The problem with the tin variable is revealed by the graphs of its information space. Filling in the missing entries with the mean value in some many cases has had a clear effect on the information space that may have had adverse affects on the model.

The frequency distribution of reduction of area and elongation are compared in Fig. S2. There are 1418 reduction of area data points and 1984 for elongation.

Fig. S2 (Fig. S1a&b on next page)

The spread of reduction of area points for low values is probably due to the dependence of reduction of area on the geometry of the specimen. Elongation is more reproducible property than reduction of area and hence the frequency data for elongation follows a Gaussian distribution. The strong influence of stress fields on reduction of area could explain why the frequency distribution appears to be composed of a uniform distribution with a narrow Gaussian superimposed.

IV Training the Models

Both of the models were trained using 20 hidden units and 5 variables. The resulting set of 100 models was compared and a committee was selected. For the reduction of area model, the log predictive error (LPE) was relatively high (Fig. S3a) and the test error (TE) was

relatively low (Fig. S3b). This, combined with the approximately uniform distributions of LPE and TE meant that the choice of committee was relatively straight forward. As shown in Fig. S3d, the combined test error of the proposed committees had a minimum value of eleven models. Hence, the models were ranked by decreasing LPE and the top eleven models were chosen to form a committee. The perceived noise level, sigma-nu, shows a minima at nine hidden units as shown in Fig S3c and follows the expected trend. The models selected for the committee mostly had between six and ten hidden units, thus minimising the noise in the final committee model (see Table S4).

Table S4

For the elongation model, the log predictive error (LPE) was very unevenly distributed (Fig. S4a). The test error (TE) was also somewhat erratic (Fig. S4b). These probably mean that a better model is possible than the one reported here, however, results were adequate as shown later.

As shown in Fig. S4d, the combined test error of the proposed committees had a minimum value of 21 models. Hence, the models were ranked by decreasing LPE and the top 21 models with a TE of less than 3.5 were chosen to form a committee. The perceived noise level, sigma-nu, appears to be monotonically decreasing (Fig. S4c). This could mean that a model with more than 20 hidden units would be more accurate. The models selected for the committee mostly had a wide variety of hidden units (see Table S5). These figures would all suggest that the committee chosen could have been improved upon had more time been available.

Table S5

V Application

Elongation and reduction in area are approximately proportional properties as can be seen in Fig. S5. The group of points in the box represent the steels that had been cold rolled. Rolling reduces the cross sectional area of a steel ingot and produces a cross section of the desired shape [6]. The deformation caused by the rollers will create stresses within the steel, these will be much greater for cold rolling. These stress concentrations reduce the amount of necking that occurs and so the reduction of area reduces. The elongation is not influenced as much by stress concentrations and so the elongation is less affected by cold rolling.

Fig. S5

Given the relationship shown in Fig. S5 it is expected that both elongation and reduction of area will have similar trends when the model is applied.

The relative significance, or weight, assigned by the models to each input are shown in Fig. S6. The elongation model shows that for elongation the most important variables are

1. Silicon

2. MLI

3. ASTM

4. Manganese

5. Nickel

6. Niobium

7. Vanadium

8. Annealing

It is something of a surprise that the greatest importance should be attached to silicon content. At concentrations below 0.3, the silicon is able to completely dissolve into the ferrite, thus strengthening the steel without causing any loss in ductility [7]. Because the silicon atoms are small and the iron atoms are large, the silicon can sit in the interstices of the iron lattice. This strengthens the lattice and is called solid solution strengthening.

Fig. S6

At concentrations above 0.3, the silicon that cannot dissolve into the ferrite forms silicates. These inclusions are brittle and therefore decrease the ductility of the steel. The predictions for elongation when silicon was varied support this trend (fig. S7)

Fig. S7 - Si content (x-axis) vs % Elongation (y-axis)

The increasing error bars are a reflection of the fitting error in the model. There were few entries in the original database for steel with high silicon concentration. This uncertainty is reflected by the increasing magnitude of the error bars. However, given the initial tightness of the error bars and the population of the information space, it is likely that the decreasing trend continues as expected.

MLI and ASTM are both measures of the grain size. If grains are large then the plastic deformations they can undergo before they fracture is larger than if the grains are small. It is well known that small grain structures increase the strength of a steel, this is why so much effort is invested in heat treated the steel to achieve this desirable property. It is therefore somewhat surprising that the trends indicated by the database show that elongation decreases as the grain size decreases. This is an observation the author is unable to explain.

Fig. S8 - grain size vs % elongation

Manganese has many complicated effects on a steel. It solid solution strengthens the ferrite in the same way as silicon. Manganese combines with sulphur to form the deformable MnS inclusion [5]. This has the effect of increasing the ductility of the steel since it negates the embrittlement caused by sulphur, which is always present as an impurity. However, despite the MnS inclusions being deformable, they can still act as nucleation sites for voids, which would cause the amount of elongation to decrease. The trends found by the database indicate that this second effect dominates and that the manganese reduces the ductility. This trend is one of the results that the author is most confident with, since the information space for manganese was well populated and evenly distributed. (see fig. S9) Manganese is an austenite stabiliser, that is the presence of manganese expands the temperature range at which austenite is stable, in some cases to include room temperature. Increasing manganese content in steel should therefore increase the content of soft and ductile austenite. This does not seem to be the case in practice.

Nickel is also an austenite stabiliser and so should increase the ductility of the steel. However, a significant number of the steels containing nickel in the database also contained chromium, and so these steels fall into the category of nickel based superalloys. In a superalloy, a metastable secondary phase is formed giving the material high strength, particularly at high temperatures. The inclusions of this secondary phase will to some extent encourage the formation of voids, leading to ductile fracture. The effect of the gamma primed phase dominates and indeed such a phase would not be possible unless the austenite had been stabilised. The trends indicated by the model suggest that increasing the nickel content does reduce the elongation (Fig. S10)

Fig. S9 - Mn vs %Elongation

Fig. S10 - Ni vs %Elongation

Vanadium forms a particularly hard carbide that is completely soluble within the ferrite lattice. The effect is massive solid solution strengthening of the steel. The addition of vanadium to steels was found have a profound effect on the elongation trends; it caused marked reduction in elongation. The vanadium is so effective at strengthening the steel, and therefore reducing its ductility, because the vanadium carbides are so evenly distributed throughout the ferrite matrix. The vanadium carbides are brittle and so crack easily, creating voids leading to fracture. Fig. S11.

Fig. S11 - V vs % Elongation

Again the increasing error bars reflect the distribution of data within the information space.

The process of annealing involves heating a steel is heated to a given temperature, holding it there for a given time and cooled it at a given rate. Annealing a material has many beneficial effects, including increasing the ductility of a material by reducing the number of brittle inclusions. During the annealing there may be enough time for the carbides to decompose, thus reducing the number of inclusions and homogenizing the composition. As the annealing temperature is increased, the ductility is increased. This is because as the temperature increases the solid state diffusion rate increases, allowing more alloying elements to escape from the carbides at a faster rate. This prediction has been supported by the results of the model. (see Fig. S12)

From Fig. S6, the variables with the most significant effect on the reduction of area are

1. Manganese

2. Grain size

3. Carbon

The remaining inputs are of a similar weight. This is not too surprising, since the reduction of area is not a material property and is not consistently reproducible.

Fig. S12 - Annealing temperature vs elongation

The effect of manganese has been discussed elsewhere in this report and so only the graph shall be reproduced here. Fig. S13.

Fig. S13 - Mn vs reduction of area

Grain size and shape has a string influence on the stress fields within a specimen undergoing a tensile test. The larger a grain is, the large the stress field will be. This is relevant to the necking process. Although it is clear that grain size affects the necking process, which in turn affects the reduction of area, it is not clear exactly what the relationship is. The predictions

Fig. S14 - Grain size (ASTM) vs reduction of area

of the model show trends of two types: trends similar to that in fig. S14 or trends similar to that in fig. S15. It is not therefore possible to draw any conclusions about the influence of the grain size from this model.

The presence of carbides increases the amount of void nucleation in a clean steel. It is therefore claimed by Bhadeshia [8] that ductility will decrease with increasing carbon content in bainitic steels. The neural network would support this theory for both elongation and reduction of area. The graphs produced do have significant error bars, although these represent the error in fitting and do not include the perceived noise, sigma-nu. The amplitude of the error bars increases as the carbon content increases, to such an extent that although a trend is suggested, it is by no means certain.

Fig. S15 - grain size vs reduction of area

The presence of carbides increases the amount of void nucleation in a clean steel. It is therefore claimed by Bhadeshia [8] that ductility will decrease with increasing carbon content in bainitic steels. The neural network would support this theory for both elongation and reduction of area. The predictions have large error bars and so this trend can only be supported, but not validated. It should be noted that most of the steels in the database were not bainitic.

It is well known that the addition of sulphur decreases ductility through the precipitation of MnS and FeS. The hard inclusions form nucleation sites for voids and so the elongation and reduction of area reduce as the number of inclusions increases. This trend could only be seen in some of the steels studied, such as the steel plotted in fig. S16. The remaining steels displayed such wide error bars that no trend could be inferred from the predictions at all. This may be due to the restricted range of sulphur content.

Fig. S16 - S vs reduction of area

V Contradictory results

Despite the clear positive correlation between reduction of area and elongation as shown in Fig. S5, it was found that for some steel the elongation would decrease and the reduction of area increase, when the same input variable was varied. The steels that exhibited this property had a wide range of compositions, although all had only one heat treatment, either austenitising or normalising. For an example set of graphs, compare Fig. S16 with Fig. S17. This is clearly a result that requires further research to investigate fully, however, I shall outline a working hypothesis below.

Both elongation and reduction of area can be thought of as having uniform and non-uniform components of strain. In elongation, as discussed above, the uniform strain dominates. In reduction of area, the non-uniform strain and the stress field are the important quantities. It is therefore possible that for some steels, some factor is affecting the elongation and another factor is affecting the reduction of area. These factors may be having opposite affects on the behaviour of the steel.

The necking process determining the reduction of area is initiates at a stress concentration. It is conceivable that the stress concentration is increased as length increases, thus creating a smaller reduction of area.

Fig. S17 - S vs elongation

References

[1] Encyclopaedia of Materials Science and Engineering, M.B. Bever (Ed), Peragmon Press, 1986

[2] S.H. Lalam, H.K.D.H. Bhadeshia, D.J.C. MacKay, Estimation of mechanical properties of ferritic steel welds Part 2: elongation and Charpy toughness, Welding and toughness of materials, Part II, Science and technology of welding and joining, 2000, volume 5, 3, 149-160

[3] Mechanical Metallurgy, G.E. Dieter, Mcgraw-Hill, 1998

[4] The Plastic Deformation of Metals (2nd Edition), R.W.K. Honeycombe, Edward Arnold, 1984

[5] Steels: Microstructure and Properties (2nd edition), R.W.K. Honeycombe & H.K.D.H. Bhadeshia, Edward Arnold, 1995

[6] The Manufacture of Iron and Steel, volume 4: mechanical treatment of steel, G.R. Bashforth, Chapman & Hall, 1962

[7] Ferrous Physical Metallurgy, A.K. Sinha, Butterworths, 1989

[8] Bainite in Steels, H.D.K.H. Bhadeshia, Institute of Materials, 1992