Determining true pearlite lamellar spacings from observed apparent spacings

173

METALLOGRAPHY 23:173-188 (1989)

Determining True Pearlite Lamellar Spacings from Observed Apparent Spacings

H. S. FONG

Department of Mechanical Engineering, National University of Singapore, Kent Ridge, Republic of Singapore

The method of Roosz, Gacsi, and Baan for determining true pearlite lamellar spacings from the apparent spacing frequencies observed on a metallograpnic surface is examined. It is shown that the method as it stands does not give very accurate results. Modifications to the method are proposed, and these are shown to lead to large improvements in the accuracy of the method.

Introduction The mechanical properties of a pearlitic steel are associated with the lamellar spacings of the pearlite, which, in turn, depend on the pearlite transformation temperature, steel composition, and other factors. The ability to measure the pearlite spacings is vital in any attempt to correlate these spacings with the conditions of formation of the pearlite. Such correlations will allow for control and manipulation of the pearlite structure and, therefore, the properties of the steel. Vander Voort and Roosz [1] have surveyed the various methods of measuring pearlite spacings as well as examined their accuracies. One important technique involves the initial determination of the frequency of occurrences of pearlite colonies of various lamellar spacings S as observed on a metallographic surface. These are referred to as apparent spacings and are related to the true spacings L of the pearlite by the relationship of Belaiew [2] which may be written in the following form: L = Ssin0

(1)

where 0 is the angle at which the pearlite lamellae intersect the metallographic surface. From this frequency distribution of apparent spacings S, it is possible to deduce the frequency distribution of the true spacings © Elsevier Science Publishing Co., Inc., 1989 655 Avenue of the Americas, New York, NY 10010

0026-0800/89/$3.50

H. S. Fong

174

L. This has been done by Pellissier et al. [3]; but these authors did not fully describe how they did it. Roosz et al. [4] have, however, presented a simple way of doing this, the essence of which will now be described. To understand this method (referred to as the "RGB method" consider the following. Shown in Fig. 1 is the probability density function, P(L), of true pearlite spacings, L, ranging from Lmin to Lmax in a specimen. Also shown is the cumulative probability function, C(S), of apparent spacings, S, as seen on a metallographic section of the specimen. The value of S will range from Smi, (= Lmin) to S . . . . Ideally Smax = 0% but in practice Smax will be some finite value beyond which the probability of occurrence of spacings S is so low that these spacings are not detectable. There is also a limitation to the value of S brought about by the finite size of the pearlite colony. This is because, beyond a certain degree of obliqueness of cut of the metallographic plane into the pearlite colony, this plane passes through only a few lamellae, so that the lamellae become highly irregular in appearance, and the spacings are not measurable. Pellissier et al. [3] have shown that for a single or unique true pearlite spacing, L, the cumulative probability, C(S), of finding apparent spacings

0-

,

Lmin Smin

Lmax

SPACING

,

,

,

L , S

F~o. 1. True spacing frequency distribution P(L) and its associated apparent spacing cumulative frequency distribution C(S). The various notations are explained in the text.

175

True P e a r l i t e L a m e l l a r S p a c i n g s

with values up to S, for the case where the pearlite colonies are randomly oriented in the specimen, is C ( S ) = cos(sin-I(L/S)) = X/I -

(2)

(L/S) 2

Thus, in the case of a continuous distribution of true spacings, L, as in Fig. 1, the cumulative probability, C(S), of finding apparent spacings with values up to S is C(S)

=

__ ,=1

Ai

l

-

,

a + k

(3)

where the range of L values between Lmin and Lmax have been divided into an integral number of segments of very small width D so that Lmax - Lmin = ND for some positive integer N; Stain =

Lmin

=

a D for some positive value of a,

t4) (5)

which need not be an integer; S = (a + k ) D for some positive integer k; /max =

N or k whichever is smaller, and

(6) (71

(a + i ) D

Ai

= J(a-

i--1)D

P ( L ) dL,

(8)

where Ai is the probability of having a true spacing between (a + i 1)D and (a + i)D. The expression A,~I - [(a + i - ½)/(a + k)]2} 1/2 represents the contribution of true spacings in the minute range (a + i 1)D to (a + i)D (which are on the average of spacing (a + i - ½)D) to the probability C ( S ) of apparent spacings being in the range Smi. to S. Such a segment of spacings, considered to be (a + i - ½)D in average spacing magnitude, will be said to be "centrally weighted." In the RGB method, the spacing interval D is chosen such that the range Smm to Smax is divided into 10 segments so that -

D = (S .... - Sm~,)/10

(9)

Actually, Roosz et al. [4] assign, somewhat arbitrarily, a slightly smaller value to Smin than observed experimentally. The segment size D obtained by their approach is quite practical for observing a reasonable number of spacing occurrences within each segment. But it is too large so that the expression on the right hand side of Eq. (3) now describes C ( S ) only approximately. Furthermore, in effect they took the average spacing of the ith segment (shown shaded) in Fig. 1 to be (a + i - l)D (i.e.. the

176

H. S. F o n g

segments are "left edge w e i g h t e d " ) , so that Eq. (3) for t h e m has the form:

irnax / C(S) = .__~Ai

( I -

,

)2 a + i -

1.

(10)

a+k

T h e y , in fact, viewed the true spacings to consist of discrete values [a + (i - 1)]D, i = 1, 2 . . . . . 10, with a probability of o c c u r r e n c e A;. Cumulative probabilities or frequencies C~ = C(S) m a y be obtained experimentally for S = Smin q- kD for k = 1, 2 . . . . . 10. W h e n this is done, R o o s z et al. [4] derived the probabilities or frequencies A,. of true spacings in Eq. (8) from the following set of equations, which follow from Eq. (10). A1 = Cl/~v/1 - [a/(a + 1)] 2 A2 = {C2 - A1~/1 + [a/(a + 2)]2}/~v/1 - [(a + 1)/(a + 2)] 2 A~ =

{

Ck -

i=1 Airy/1 -

~v/'l - [(a + k -

[(a + i -

(11)

}/

1)/(a + k)] 2

1)/(a + k)] 2

k=3,4

.....

10

R o o s z and Gacsi [5] have subsequently p r o d u c e d a table showing the numerical values of the inverse matrix elements that will allow the probabilities AI, A2 . . . . . Alo a b o v e to be calculated m o r e expeditiously. H o w e v e r , with the wide availability of c o m p u t e r s n o w a d a y s , these m a y be easily determined directly from Eq. (11) without need of the inverse matrix elements. The p u r p o s e of this article is to point out the limitations of the R G B m e t h o d and to describe how it could be modified to give a m o r e accurate f r e q u e n c y distribution of the true spacing L.

L I M I T A T I O N S OF T H E R G B M E T H O D T h e main limitations of the RGB m e t h o d are: 1. The true spacings in the segment (a + i - 1)D to (a + i)D are t a k e n to be of average spacing (a + i - 1)D (i.e., left edge weighting applied) rather than the average spacing of (a + i - ½)D (i.e., centrally weighted). 2. The segment width D is rather large.

True Pearlite Lamellar Spacings

177

T o a p p r e c i a t e l i m i t a t i o n (i), let us t a k e a h y p o t h e t i c a l d i s t r i b u t i o n o f p e a r l ite l a m e l l a r s p a c i n g s L w i t h t h e p r o b a b i l i t y d e n s i t y

P(L) = 0,

L < 0.1, L > 0 . 4

20 sin2 [2w(L 3

(12)

0.1)/0.6],

0.1 -< L -< 0.4

w h e r e L is in m i c r o m e t e r s . T h e f u n c t i o n P(L) has the s h a p e s h o w n in Fig. 1. T h e c u m u l a t i v e f r e q u e n c y c u r v e f o r this d i s t r i b u t i o n is s h o w n as t h e c u r v e in Fig. 2. B y u s i n g E q . (3), the c o r r e s p o n d i n g c u m u l a t i v e freq u e n c y c u r v e o f a p p a r e n t s p a c i n g s S, C(S), h a s b e e n c o m p u t e d (using a s i m p l e c o m p u t e r p r o g r a m ) a n d is p l o t t e d as t h e c u r v e in Fig. 3. A D v a l u e o f 0.0001 p.m w a s u s e d , a n d this was f o u n d to g i v e C(S) v a l u e s w i t h an a c c u r a c y o f b e t t e r t h a n 0.01%. A l s o p l o t t e d (as c i r c u l a r p o i n t s ) in Fig. 3

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FIG. 2. Comparison of true spacing cumulative frequencies obtained from frequencies of occurrence of apparent spacings grouped into 10 segments by the RGB method, RGB method with central weighting, and RGB method with central weighting and interpolation. The spacing distribution is given by P(L) = -~ sin2 [2~(L - 0.1)/0.6], 0.1 -< L -< 0.4. It is seen that the original RGB method gave results somewhat deviated from the correct values. Considerable improvement was achieved with central weighting, while the interpolation technique gave very satisfactory results.

H. S. Fong

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FIG. 3. Apparent spacing cumulative frequencies derived from a true spacing frequency distribution of the form P(L) = ~ sin2[2"rr(L - 0.1)/0.6], 0.1 -< L -< 0.4, seen to be approximately similar to the pearlite apparent spacing cumulative frequency distribution in a normalized DIN 39Cr4 steel observed by Roosz et al. [4]. are the a p p a r e n t pearlite spacing cumulative frequency data for a normalized D I N 39 Cr 4 steel reported in the p a p e r by R o o s z et al. [4]. It is seen that the hypothetical pearlite spacing f r e q u e n c y distribution considered here is quite c o m p a r a b l e to the normalized D I N 39 Cr 4 pearlite spacing distribution and, therefore, will serve as a good a p p r o x i m a t e model of pearlite spacing frequency distributions. The cumulative frequencies for values of S f r o m 0.1 to 1.1 i~m, comp u t e d f r o m the hypothetical distribution in steps of 0.1 ~m, have been tabulated in column (b) of Table 1. I f we had a pearlite spacing distribution of the f o r m given by Eq. (12), then we would o b s e r v e a cumulative apparent spacing frequency as in column (c) in Table 1, where we have a s s u m e d that S values greater than 1.1 i~m are not o b s e r v e d , b e c a u s e of their low probability of o c c u r r e n c e or e x t r e m e obliquity of intersection of the metallographic plane with the pearlite colony. If we regard column (c) of Table 1 to represent a set of o b s e r v e d apparent spacing cumulative f r e q u e n c y values with the range S m i n = O. 1 txm to S m a x = 1.1 I~m divided

179

True Pearlite Lamellar Spacings TABLE 1 Apparent Spacing Cumulative Frequency Distribution from a True Spacing Frequency Distribution Given by Eq. (12)

(c) (a) S (~m)

(b) Cumulative frequency (%)

Normalized cumulative frequency (= (b) × 100/97.2) (c/c)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

0.0 8.9 47.5 75.9 85.7 90.3 93.0 94.7 95.8 96.6 97.2

0.0 9.2 48.9 78.1 88.1 92.9 95.7 97.4 98.6 99.4 100.0

into 10 equal segments of width D = 0.1 ~xm, then we have the setting for applying the RGB method of obtaining the true spacing frequency distribution. This is done here by applying Eq. (11) to the values in column (c) in Table 1. The cumulative frequency values for the true spacings, C ( L ) , thus obtained, have been plotted in Fig. 2 as the dark circular points. The deviation from the actual distribution given by the curve is seen to be quite considerable. If we apply central weighting, however, Eq. (11) become: A1 = C1/X/1 -

{

Ak =

[(a + ~)/(a

k--I i=1

+

l)] 2

)/

Ck -- ~ , A i X / 1

- [(a + i - ½)/(a + k)] 2

(13)

X/1 - [(a + k - ½)/(a + k)] 2 k=2,3

.....

10

If one uses these equations, the cumulative frequency values of true spacings L work out to be the open circular points plotted in Fig. 2. These values are clearly much better than those obtained by the RGB method with left edge weighting. Table 2, (b) columns, show the true spacing cumulative frequency values, C ( L ) , obtained above for left edge and central weighting. The cu-

H. S. Fong

180

TABLE 2 True Spacing Frequency Distribution Obtained from Apparent Spacing Frequency Data in Column (c), Table 1 RGB method (Left edge weighting) Cumulative frequency (%)

(a) L (Ixm) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

(c) (b)

Normalized

0.0 10.6 62.8 97.0 101.2 102.1 102.4 102.6 102.7 102.8 102.8

0.0 10.3 61.1 94.4 98.5 99.4 99.7 99.9 99.9 100.0 100.0

RGB method (Central weighting) Cumulative frequency (%)

(b)

(c) Normalized

0.0 13.9 80.6 107.7 102.5 103.1 102.9 102.9 102.9 102.9 102.9

0.0 13.5 78.3 104.8 99.6 100.3 100.0 100.0 100.0 100.0 100.0

mulative frequencies flatten out at about 102.8%, which is above the maximum possible value of 100%. This is because the maximum apparent spacing, S . . . . was taken to be finite, i.e., 1.1 Ixm, when actually it should be infinite. The actual apparent spacing cumulative frequency at S = 1.1 p.m is 97.24%. Because it was taken to be higher, i.e., 100%, the frequency values obtained for the true spacings are higher by a factor of 100/97.24. The appropriate frequency values for the true spacings given in columns (b) of Table 2 should therefore be normalized before use to the values in columns (c) of the table. The values plotted in Fig. 2 are these normalized values. This is a point apparently not considered by Roosz et al. [4], as can be seen from Table 3, which shows in column (b) their cumulative frequency results for the true spacings of the pearlite of the normalized DIN 39 Cr 4 steel, as best as can be seen from their paper. (The frequency values have been ascribed here to the whole spacing segment rather than to individual spacings at the left edge of the segments, as was done by Roosz et al.) It is seen that the cumulative frequency flattens out at 106.6% and should therefore be properly normalized, as shown in column (c). Analysis redone here of the apparent spacing data in the paper by Roosz et al., both by left-edge and central weighting, produced the results given in the other columns in Table 3. The values by left-edge weighting, column (d), are quite consistent with those of Roosz et al., with small deviations being accountable to possible errors in reading the data in their paper.

True Pearlite Lamellar Spacings

181

TABLE 3 True Spacing F r e q u e n c i e s from the A p p a r e n t Pearlite Spacing Observations of Roosz, Gacsi, and Baan [4] Cumulative frequency (%)

F r o m R o o s z et al. [4]

By left-edge weighting this paper

By central weighting this paper

L (~m) (a)

(b)

(c) Normalized

(d)

(e) Normalized

(f)

(g) Normalized

0.125 0.213 0.301 0.389 0.477 0.565 0.653 0.741 0.829 0.917 1.005

0.0 6.0 56.3 86.6 94.8 98.8 103.8 103.8 106.6 106.6 106.6

0.0 5.6 52.8 81.2 88.9 92.7 97.4 97.4 100.0 100.0 100.0

0.0 5.6 56.4 86.7 94.0 99.7 103.0 102.2 104.7 105.3 105.3

0.0 5.3 53.6 82.3 89.3 94.7 97.8 97.1 99.4 100.0 100.0

0.0 7.4 74.4 96.1 96.7 102.4 104.5 102.2 105.9 105.5 105.5

0.0 7.0 70.5 91.1 91.6 97.1 99.0 96.9 100.4 100.0 100.0

TABLE 4 Frequencies of Occurrence (Normalized) of True Pearlite Spacings in 0.088 ~ m Intervals Deduced from the A p p a r e n t Spacing Observations of Roosz, Gacsi, and Baan [4] Normalized frequency (%)

L (~m) (a) 0.125-0.213 0.213-0.301 0.301-0.389 0.389-0.447 0.447-0.565 0.565-0.653 0.653-0.741 0.741-0.829 0.829-0.917 0.917-1.005

(b) From Roosz et al. [4]

(c) By left edge weighting this paper

(d) by central weighting this paper

5.6 47.2 28.4 7.7 3.8 4.7 0.0 2.6 0.0 0.0

5.3 48.3 28.7 7.0 5.4 3.1 - 0.7 2.3 0.6 0.0

7.0 63.5 20.6 0.5 5.4 2.0 - 2.2 3.6 -0.4 - 0.1

182

H. S. Fong

This confirms that the application of Eq. (1 I) with left-edge weighting, as done here, is equivalent to the RGB method. The central weighting approach gives the results in columns (f) and (g). In view of the above discussion, these may be expected to be closer to the actual true spacing frequency distribution. The corresponding normalized values of frequency of occurrence of true spacings within each spacing segment of 0.88 txm width between 0.125 and 1.005 p~m is given in Table 4, which raises another point worth commenting on. It is noted that in some segments spurious negative frequency values apply. This is due to the fact that the observed apparent spacing frequency values for these segments are too low, or the values for the preceding segments are too high. These effects are the natural outcome of limitations in experimental measurements, experimental scatter, statistical variation, and even the lack of true randomness of the pearlite orientations. So long as the negative values are small, they represent acceptable natural experimental limitations and error. The limitation brought about by too large a segment width D is discussed in the next section.

IMPROVEMENTS TO THE RGB METHOD As shown in the preceding discussion, one improvement to the RGB method is to use central weighting. The second improvement is to reduce the segment width D. When the segment width D is too large, then the assumption that each spacing in the segment may be taken to be (a + i 1)D or (a + i - ½)D (see Fig. l) for the case of left-edge and central weighting, respectively, becomes rather tenuous, and the right-hand side of Eq. (3) becomes a poor representation of the cumulative apparent spacing frequency C(S). Indeed, it was found that in the case of the hypothetical spacing frequency distribution given by Eq. (12), the cumulative apparent spacing frequencies should be known accurately at 0.01 txm intervals to deduce the true spacing distribution of Eq. (12) to an accuracy of better than 0.2% in the frequency values. Opposed to this is the requirement that D be large enough to avoid excessive statistical scatter in the observed frequencies. It is now suggested that one way around this problem is to obtain cumulative apparent spacing frequency values at several other points within each segment by a suitable interpolation process. Thus, in the case of a pearlite spacing frequency distribution of the form given by Eq. (12), although practical considerations may dictate that C(S) values be obtained at 0.1 i~m intervals as exemplified in column (c) of Table 1, it is possible -

True Pearlite Lamellar Spacings

183

to interpolate for cumulative frequency values at 0.01 i~m subintervals within each of the 0.1 p~m segments so that, consequently, C(S) values at intervals of 0.01 p,m are known, and the range Smin--Sma× may be considered divided into 100 segments each of much smaller width D (= 0.01 ~m) than before. This degree of subdivisioning was mentioned earlier to be about the extent required to achieve a reasonable degree of accuracy in the true spacing frequency distribution. Such an interpolating process was carried out for the apparent spacing cumulative frequencies shown in column (c) of Table 1. Details of the interpolating process are given in the Appendix. Then, with the use of appropriate equations in the form of Eq. (13), the Ai values giving the frequency of occurrences of true spacings in 0.0! p.m intervals between 0.1 to 1. I ~m were obtained. From these the cumulative frequencies for true spacings were worked out at 0.02 ~m intervals and plotted as triangular points in Fig. 2. It is seen that these points are much closer to the actual cumulative frequency values (given by the curve) than for the case when no interpolation was done, and when the D value was 0.1 ~tm. The direct way of getting better values of the true spacing frequencies is to increase the number of segments in the range Smin--Sm~x without introducing excessive statistical scatter in the segmental frequencies. This may be possible if the number of apparent spacing observations is drastically increased, e.g., by doubling or tripling. The effect of increasing the number of segments beyond 10 has been examined for the case of the hypothetical pearlite frequency distribution of the form given by Eq. (12). In the first instance, C(S) values between 0.1 and 1.1 gm at intervals of 0.0625 Ixm were computed using Eq. (3) and then treated as '~observed'" apparent spacing cumulative frequency values from which the true spacing frequency values were to be determined. Such observations constitute a divisioning of the Smin--Smax range into 16 segments each of width D = 0.0625 ~m. When the true spacing frequencies were obtained using Eq. (13), with k now having values of 1, 2 . . . . . 16, the cumulative true spacing frequency values given by the dark triangular points in Fig. 4 resulted. It is seen that the points are very close to the solid curve that delineates the actual cumulative frequency values, in fact, the points are nearer to the solid curve than even the case of interpolation with 10 segments. Next, the case of 20 segments was explored. The true spacing cumulative frequency values obtained now are shown as open triangular points in Fig. 4, and are visually not noticeably better than for the 16-segment case. In other words, the observed apparent spacing range needs to be divided into no more than about 16 segments to achieve almost the maximum improvement in the true spacing results that can be achieved by

H. S. Fong

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TRUE SPACING L (lJm) Fro. 4. True spacing cumulative frequency values obtained by the central weighted RGB method with the range of observed apparent spacings divided into 16 and 20 segments. The spacing frequency distribution is that ot P(L) = 3 sin2[2~r(L - 0.1)/0.6], 0.1 <- L -< 0.4. The obtained cumulative frequency values are very close to correct values for both cases. TABLE 5 True Spacing Frequencies Deduced from the Cumulative Apparent Spacing Frequencies of Column (c), Table I, by Various Methods Frequency (%) 10 Segments

16 Segments

20 Segments

L (Ixm)

Ideal

LEW

CW

CWI

CW

CWI

CW

CWI

0.1-0.2 0.2-0.3 0.3-0.4 0.4-0.5 0.5-0.6 0.6-0.7 0.7-0.8 0.8-0.9 0.9-1.0

19.5 61.1 19.5 0.0 0.0 0.0 0.0 0.0 0.0

10.3 50.8 33.3 4.1 0.9 0.3 0.2 0.1 0.1

13.5 64.9 26.4 -5.1 0.7 -0.2 0.0 0.0 0.0

16.9 62.3 24.2 -4.3 0.6 0.1 0.0 0.0 0.0

17.3 62.1 22.1 -1.4 0.0 0.0 0.0 0.0 0.0

19.9 59.5 20.9 -0.2 -0.1 0.0 0.0 0.0 0.0

17.2 63.1 20.5 -0.7 -0.1 0.0 0.0 0.0 0.0

19.0 61.5 19.8 -0.4 0.0 0.0 0.0 0.0 0.0

Abbreviations: LEW, left-edge weighting; CW, central weighting; CWI, central weighting and with interpolation.

True Pearlite Lamellar Spacings

185

an increase in the divisioning of the apparent spacing range beyond the 10 proposed by Roosz et al. [4]. Further divisioning brings about only slight further improvements, which may not be worthwhile, in view of the much larger number of observations this further divisioning requires to counter statistical scatter. The results with 16 and 20 segments can be further improved by interpolation. This can be seen in Table 5, which gives the frequencies of true spacings occurring over intervals of 0.1 ~m determined by the various approaches as well as the actual (or ideal) frequencies for the case of the spacing frequency distribution given by Eq. (12). The interpolation in the 16- and 20-segment case has been done in intervals of 0.01 ~m and, as can be seen in Table 5, has brought the true spacing frequency values very close to actual. AN APPLICATION OF THE CENTRALLY WEIGHTED INTERPOLATION METHOD By way of immediate application, the centrally weighted interpolation version of the RGB method described above was applied to the 183 apparent pearlite spacing observations for an as-rolled AISI 1040 steel which Vander Voort and Roosz [1] obtained by transmission electron microscopy measurements on replicas of etched samples of the steel. The apparent spacings ranged from 0.124 to 1.130 ~m, but it may be taken here, with hardly any difference, that Smax = 1. 134 txm, thus allowing interpolation and division of the apparent spacing range into 100 segments of 0.0101 ~m width. The data forming the basis for the interpolation are the apparent spacing cumulative frequency data of Vander Voort and Roosz given in steps of 0.1006 ~m starting at 0.2031 ~xm and ending at 1.2078 ~m. The C(L) values thus obtained are plotted in Fig. 5. Also plotted are the normalized C(L) values as determined by Vander Voort and Roosz [1] using the RGB method. Their values deviate by a fair amount from those of the centrally weighted interpolation method and are therefore very likely to be incorrect, in view of what has been discussed earlier about the superior accuracy of the centrally weighted interpolation method. The average true interlamellar pearlite spacing determined by Vander Voort and Roosz was 0.236 ~m. This is in error because of their approach of ascribing the spacing frequency value for a segment to the true spacing at the left edge of the segment. This frequency value should more correctly be assigned to the spacing at the center of the segment when one is computing the mean spacing value. Their average true spacing should therefore correctly be 0.236 txm + 1/2 (segment width) = 0.236 + 0.0503 = 0.286 ~xm. The average true spacing from the results of the centrally

186

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1.2

FIG. 5. Cumulative frequencies of pearlite lamellar spacings in an as-rolled AISI 1040 steel derived from the apparent spacing measurements of Vander Voort and Roosz [1] according to their method and by the central weighted interpolation technique of this paper.

weighted interpolation method works out to be 0.259 ~m, which is very close to the value of 0.254 and 0.255 txm obtained by Vander Voort and Roosz, respectively, for the same steel sample by TEM measurements on replicas by the random spacing method and on thin foils by direct measurement of true spacings between pearlite lamellae inclined perpendicular to the foil surface, respectively, and to the values of 0.250 and 0.255 ~m they obtained by SEM measurements by direct and random spacing techniques, respectively. This further supports the view that the centrally weighted interpolation technique gives results close to actual.

Conclusion Thus, it is seen that there is a significant degree of inaccuracy inherent in the RGB method of obtaining true pearlite lamellar spacing frequency distributions from the apparent spacing frequencies. Central weighting of the apparent spacing segments brings about a marked improvement in the accuracy of the results. Further marked improvements are brought about

True Pearlite Lamellar Spacings

187

by using smaller spacing segment widths either by interpolating between observed data, or by obtaining apparent spacing frequencies over smaller segments, or by a combination of both smaller segment observations and interpolation. The tasks of interpolation and the solution of 100 equations or so of the form in Eq. (13) are readily performed by computer. The computer programs needed are relatively simple and short. It is recommended that efforts be made to divide the pearlite apparent spacing range S m i n - S . . . . into about 16 segments, which, from the preceding discussions, is an adequate degree of divisioning to ensure a good level of accuracy in the true spacing results. It is judged that about 500 pearlite apparent spacing observations taken at random would be sufficient to ensure that the scatter effect in the apparent spacing frequency distribution over these 16 segments would be minimal. Then the RGB procedure may be applied with confidence using central weighting. The results would be further enhanced in accuracy if the interpolation process is additionally adopted. This is to be encouraged, as it involves no further experimental effort but only additional mathematical processing readily achieved with a computer.

Appendix The interpolation method used above to obtain values of cumulative apparent spacing frequency values, C(S), for values of apparent spacings S intermediate between experimental values is the standard Lagrange method described in various mathematical texts, e.g., that by Williams [6]. If the S values at which C(S) values have been experimentally determined a r e Smin, Smin~ 1, Smin+2 . . . . . . Smax 2, S m a x - I, and S . . . . then interpolation for C(S) values for S between S m i n + n - I and Sm~. . . . for some integer n, was done using the experimental values of C(S) at S = Smin+n 2, Smin-n-1, Smi . . . . and Smi. . . . 1, except for S > Sm~x 2 or S < Smin+ I. That is, interpolation of cubic order was generally used. Higher orders may tend to produce an oscillating curve for C(S). For S > Sma×- 2, Lagrangian interpolation of quadratic order was done instead based on the experimental values of C(S) at S = Smax-2, Smax 1, and S . . . . This is because the C(S) curve tended to become almost linear for these values of S. For S < Stain+ l, the following quadratic order interpolation was done instead:

C(S)

=

Sm~n+1 ~ Smi

C(S~i, + i)

(14)

188

H. S. Fong

This expression gives a quadratic curve of C(S) versus S, which passes through the experimental points (Smin,C(Smin)) and (Stain+ ~,C(Smin+l)), and is, at the same time, of zero gradient at S = Stain. This interpolation expression was preferred for S < Stain+ ~ since the Lagrangian interpolation (of order higher than 1) tended to produce negative values for C(S) for S near Smin. The Lagrangian interpolation of first order based on two points is a linear interpolation and was felt too approximate. References 1. G. F. Vander Voort and A. Roosz, Measurement of the interlamellar sPacing of pearlite, Metallography 17:1-17 (1984). 2. N. T. Belaiew, The inner structure of the pearlite grain, J. Iron Steel Inst. (London) 105:201-239 (1922). 3. G. E. Pellissier, M. F. Hawkes, W. A. Johnson, and R. F. Mehl, The interlamellar spacing of pearlite, Trans. ASM 30:1049-1086 (1942). 4. A. Roosz, Z. Gacsi, and M. K. Baan, A simple method for determining the true interlamellar spacing, Metallography 13:299-306 (1980). 5. A. Roosz and Z. Gacsi. The correlation of the average true interlamellar spacing of pearlite and the transformation temperature in DIN 39Cr4 steel, Metallography 14; 129-139 (1981). 6. P.W. Williams, Numerical Computation, English Language Book Society, Southampton, U K (1978), pp. 120-122.

Received March 1988; accepted November 1989.