REFERENCE: Vander Voort, G. F., Examination of Some Grain ... · grain size the Jeffries method shouldbe as it is calculated as the square root of the average area thatsuggests that

REFERENCE: Vander Voort, G. F., "Examination of Some Grain Size Measurement Prob-Metauoerannv: Past, Present, and Future Anniversary Volume), ASTM STP 1165,

G. Vander J. Warmuth, S. M. and A. Szirmae, Eds., American Society forTesting and Materials, Philadelphia, 1993, pp.

ABSTRACT: The size is performed nearly always on a metallographicallyoreoarec cross section, suitably to reveal the grain structure, using methods that pertain

cross sections. These measurements are termed planar and some may be con-mathematically into estimates of the size of the three-dimensional grains. How-

of work is planar where no assumptions about grain shape aresize one- or two-dimensional terms (intercept length, diameter,

or area) on sections through the The most used measurement methodsare described in this and using the same methods are the Jeffriesplanimetric count and intercept method. Thesemethods base grain one-dimensional featuresof the microstructure, resnectivelv

Test lines can be toactual intercept length measurements may be made. Furthermore,or curved (circular test lines commonly prr',nl,r",,'-r1 \

line cause a bias in either intercenta tablet is used with a

mtersections, or arcsured. either practice causelengths compare to the mean lineal interceptintercepts intersections) unit

as the resultsber compare to that

quantitative metallography, Jeffries plammetncmetallurgical specimens,

1 Materials4662.

.haracterization, Carpenter Technology Corporation, & D Center, Reading, PA 19612-

266

VANDER VOORT ON GRAIN SIZE 267

of spatial size is very difficult recumnzint{~gnmcm (liquid metal vH'V' ••",-'H"""/

These methods are

Definition Grain Size

Different measurements have been used to assess the[metric Albert Sauveur and refinedwidely used the United States while the "'''"A''',",I-'' method, rl",'"r",I.'Yn.>rl

Heyn of was the first method usedsize was described of either

at Xl, or as the square root of theintercept u ...., ...'"... , the size was defined as mean linear

which is the of the number of intersectionslength of test or the number unit of test NL • Thesubscript,3, means that the measurement is valid in three dimensions. The use ofdto describegrain size the Jeffries method should be as it is calculated as the square root ofthe average area that suggests that are square on the cross which arenot. Calculation ofthe diameter ofa circle area circle isa slight but not much better. Use of a basic parameter, todefine the grain size is recommended.

The lack of a simple interconversion between the results of these two that is,between and or PL , has been a well-recognized weakness. is a function of the totallength of grain edges per unit volume, while or PL a smaie-nnase C'Yr"r'Y,,,"O

function ofthe grain boundary surface area per unitofgrain size are based on different characteristics of thethere is no direct relationship between Lv andthe same specimens as a function of grain size.

While the first (1930) comparison grain size chart copper and conner-oaseuscaled the grain size based on uniform linear increments of d, later charts used a power series

scaled the chart pictures based on The chart'S listed eitherNA value of grains per square inch at X or the "ASTM size .... " ........ I"or

from the following equation

n is the number of grains per square inch at XI 00 magnification. To obtainof grains per mm? at XI, n by 15.50031. is related

is measured readily by the Jeffries method, or the count method.

as theto G

PAST, PRI;;:SF:NT

Jeffries Planimetric ivrernou

This method ,."",.."",.",,,,,circular in

a known test area,is drawn on a or a

nroiectron screen or television mon-magruncation is chosen so that at least 50 are within test area. Next, the

number within the test area, and then2, are counted. To obtain an accurate count, the

bered which slows down as either a new templateor it must be cleaned between counts. Several fields must• <,T<' ,,1-,-,. .... r level of This number wouldstructure.

The sum and is U....H"p"''''~

with the 79.8-mm-diameter circle to obtainvalue is obtained from

j=

2.is obtained from the reciprocal

where M is the linear magnmcation. If a circle ofa different diameter is ......u,<.,''-'u, substitute itsarea for 5000 in

The average area,

where A is in mrrr', or

A= (3b)

where A is in square micrometresbut use of d should not be encouragedcalculated from

oiameter d, is the square rootcross sections are not square. G can befrom

G = (3.222 - 2.954

or, from Table 2 of ASTM 12-88.

rUl,IP_,lt""ll,U'iT Count Method

This utilizes Euler's law obtain NA • A circleof known size is or a such a circle is over aground glass screen or television monitor. no statistical studies thismethod have been should be chosen so that at least 40 to 50 triplepoints are present. Too a may make and difficult.Again, a number of such counts should be made and the must be marked offasthey are counted to obtain an accurate count. If a IS whichis rare, it is a count of two. NA is determined from

(5)

VANDER VOORT ON GRAIN SIZE 269

where P is the number of triple points (junction of three grains on the of polish)\"V'.UH...,U, AT is the test area in mrrr' at XI, and is the number per mrrr' at XI. Gis calculated from 4 or from Table 2 of ASTM E 112-88.

nf"'...."""nt Method

Unlike the Jeffries method and the count the methoddoes not off of the or intersections to obtain an accurate count.Hence, this method is faster than the other two, and the same of canobtained in less time. The method is useful for deformed structures, and asure of their elongation can be obtained while G is determined. Interest in themethod in the United States was stimulated the work of John Hilliard [8] and HalleAbrams [9].

Lines of known are drawn on a templatenrorection screen, or TV monitor. For a deformed structure, one or moreparallel straight test lines are over the grain structure with orientationsIJ\""U"~UA<"U to the deformation (longitudinal direction. For an struc­ture, ASTM E 112-88 recommends circular test lines concentric circles with a total cir­cumference of 500 Circular test lines are also used structuresto average out the amsotronv

For a structure, a count can be made ofeither the number inter-cepted the test or the number intersections with the testP. Both approaches produce equivalent results. With test it may be easier tomake N counts where the grains partly intercepted the ends of the lines the end lies

on a grain boundary, which seldomly) are counted as halfvalues. For circulartest grids, there are no line ends to deal with and either P or N counts can be made.raters to make P counts (grain boundary intersections). The number of N or P countsisdivided by the true test line length (mm at XI) to obtain or The true test lineis found by dividing the total test line length by the magnification. For a single-phasestructure, Sv is twice PL or

The mean lineal (or linear) intercept length, is the of NL or (for aphase grain structure). For a deformed grain structure, N L or PL will be higher using test linesperpendicular to the deformation axis than for test lines to the deformation axis.Because is the reciprocal of NL or the intercept lengths are inversely n s-r,nr,rl'An"i

ifstraight test lines are placed parallel (/I) and perpendicular (-l) to the deformation axis, ananisotropy index, AI, can be defined as

AI =

Toobtain the grain size of a deformed grain structure, we can average the above andvalues, use randomly placed straight test lines, or use circular test lines.

The determination of the ASTM grain size number, from mean lineal valueshas been a weak part of this method. The first published relationship between 13 and G was inASTM Method for Estimating the Average Ferrite Grain Size of Low-Carbon Steels 89),introduced in 1950 but discontinued in 1961 after ASTM E 112 was adopted. This standardsuggested making intercept counts on two sets of mutually perpendicular test lines to obtainNu and N L11at magnification M. The following equation was used to calculate the number ofgrains per square inch at X 100, n

changed to

over thefor the three specrmens.

In theare not measured. Forbased uponthen ':nT~>r'l'"Jpri

ofcirclesmeasurements

Semiautomatic diginzrngand

- 3.288

measure individual"'A"'A'J'~nH this is tedious and

VANDER GRAIN

TABLE I-Grain size measurements ofhot-work tool steel specimens.

JEFFRIES PLANIMETRIC

Numberof Fields

4

3

40.7529.7525.67

30.25

f

134.95133.2485.26

Specimen

ABRAMS

Number of Fields

a L T is the total testb Straight test lines

4.92 100.2 10.0 10.06.19 83.0 12.17.41 67.8 14.8 8.9

G

9.99.68.7

INDIVIDUAL t"vr''COr''CY)T

Number of Intercepts

500432379

81012

1925°F1950°F1975°F

Spectmen Code

been 1'Y'lP'C>'''"Y',o.rt

eter, ECD. ~llrthp""Yl(.,..rp

for a .....,'nr,,,,,,,,,,,,,

After a suitable number of measurements have beenclassescan be £'A''Y> ....,''+<>rl

for 't ... t,or,.-.,o ....t

The oigmzmg

was number fraction or as a linearpercentage per class dividedtimes puts greater on the data in the class intervals. 2sent these data with the number at the top and the lineal fraction at the bottom.

The scaled demonstrate that the data arenature. The lineal fraction percent, bottom of 1, better reveals the emerging rl" .....l"'",+"

the size distribution than the number fraction percent. It is also apparent that theof the distribution increases with This has been observed in ..... 1'Y'l,o,rn".., n,..,,,,r cr .. rL

ies In contrast, the revealformly curves a distribution of the data.data have also concluded that size distributions are someresearchers have that certain other more esoteric distributions may exhibit some­what better fits to the data. In any case, grain size distribution data are not normal

it is not correct to compute an arithmetic standard deviation for individualUAC",UUUUVU0. UAU.AV~'''''U this is done due to the of such calculations.

272

Lineal I= ...,,.tinn40 -r-----------------------------,

o

30 + .Y.;/"'_~ I

20 -1- // / ~'\'"'"

15.610 -+- II /......... "-" ,,,............................... ···························1

err­boundary intersec-

to and theis an arc distance. On the

circular test patterns and the chord distance

SIZE 273

Abrams three-circle method wasof the E4 members claimed that

claim was on a tneoreticatments. 'J'~j,,,,".uH."H

since the 1982 r""",c,rw'l

12.2 statesneghgrblc when Nom mended that the testso that N or exceeds six

VANDERVOORT ON GRAIN SIZE 275

In their first made intersection measurementsmagruncanon, and number of piacemems

A total of 606 intersections were counted an average-"U.Jl-"UAU.<AVAA, the total test line would have been 4.7 mm. intersection

counts were made six circles with true diameters circle diameters not16 to 67.6 Jim. Two of the circles had diameters less than the mean ' e-d-."' ...,.."'.....+

than a constant number of counts with each which would haveincreased the number of intersection counts as the circle diameter

decreased. Not the 95% confidence limits imnrovednumber of counts increased. The ,",,, ....... 1,,,,,,-r-

calculated. Table 2 presents their results for the circular patterns forrange of intersection data). The bottom ofTable shows the calculated number of circlements, the total circumference line and the number of ""1",,..,,,,..,,..,,,,",+,, "''''''',,,,,c,,,rl

obtain about 500 intersection counts each circle.Note that for the P was 482 and this was obtained with 18 ptacements

"fields" and the total test line was 4.7 mm. For the smallest P was 1274625 and a test line of 10. should be

obvious to any thatmeasurements with the smallest circle. For each ptacement.

....... , ,C''''''.<:'ror,,,,,,,, retocused, the count made and Even if a constantof the test 245 than fields must be rated with

if the data are to be evaluated in a manual eachcalculator or PC that takes much more time for 625

The Willis and Lake mean lineal values showtest circles that suggests is no bias between use of test lines andcurved test lines for the case ofintersection counts. However, these authors state thatfor individual measurements, there is a bias between arc and chord measure-ments. if very small test can be used obtain

the effort is far greater than for test with

TABLE 2-Data a/Willis and Lake [10].

Circle Diameter, Number of Average Number of L 3, 95%,urn Intersections Intersections ,urn

67.6 482 26.8 7.90 0.4630.6 554 12.3 7.8 0.4319.0 707 7.86 7.59 0.3813.4 822 5.41 7.78 0.356.66 854 2.74 7.64 0.355.16 1274 2.04 7.94 0.28

INFORMAnON CALCULATED FROM THEIR OATA

For CountsCircle Diameter, Number Total Length, PL ,

,urn of Fields" mm mm- I 'BPi Number of Fields"

67.6 18 4.703 128.9 509 1930.6 45 3.823 126.1 504 4119.0 90 5.372 131.6 50313.4 152 6.399 128.5 498 926.66 312 6.528 130.8 501 1835.16 625 10.132 125.8 500 245

a N umber of times the test circles are applied.

276 METALLOGRAPHY:PAST,PRESENT,ANDFUTURE

o

1000

No. of1400 ,-------.:......-------------:=-----:-----~

1200

FIG. 3-Plot showing the increased number riP intersections as a/unction ofthe numbermentfields for circular test lines used by Willis and Lake [10] versus conditions/or constant intersecuoncounts.

Additionally, with the use of small test circles, the test data will not be Gaussian and the usualarithmetic statistical approaches are not valid.

Circular Intercept Experiment

Because the Willis and Lake experiment did not use a constant count approach and the datadistribution was not reported; and, because the arc versus chord individual interceptmeasurements were reported to be biased, but not the circular versus straight line intersectioncount data, the writer decided to repeat their experiment. A specimen of AISI 304 stainlesssteel with an apparent ASTM austenitic grain size of about 5 was selected for the measure­ments. The specimen was etched electrolytically using 60% aqueous nitric acid thatreveals the grain boundaries but not the twin boundaries, 4. The 101.6 by 127 mm (45 in.) negative was enlarged and 203 by 254 mm (8 by 10 in.) prints were made at a maznm­cation of X207.3.

First, the ASTM three-circle method was employed to measure the mean lineal mterceot.The grid was dropped onto the print thirteen times until about 500 intersection counts wereobtained. The results are given here (triple points intersections were counted as tangenthits as

ASTM Three-Circle Intercept Method

Number of fields = 13Number of intersections (P) = 527

Total test line length = 31.356 mm (6500 mm on the photographs)PL = 16.807 per mmL 3 = 59.5 ± 4.6 ~m

ASTM grain size, G = 4.85

VANDER VOORT ON GRAIN SIZE 277

FIG. 4-Grain structureofAfSf 304 stainlesssteel specimen revealedby electrolytic etchingwithaque­ous60% HN03 (6 V de, 2 min, Ptcathode, 20°C). Note that the grainstructureappears to be slightlyduplex.

Next, a series of parallel diagonal lines were drawn across the photographs, with bothonal orientations (45° to longitudinal direction), and the lines spaced about 10 mm apart. Indi-vidual lengths were measured with the following

Individual Intercept Measurements (Diagonal Lines)

Number of intercepts = 463Total line = 28.041 mm (5812.7 mm on photographs)

= 60.4 ± 3.7 Jlm

ASTM G = 4.80

The distribution ofthese intercept lengths was examined, Fig. 5, but was notIt was right skewed = 1.51) and the kurtosis ({32) was 5.178, a bit more than for anormal distribution. The logarithmic distribution appears to be a bit more symmetrical.

a set of lS0-mm-long straight, parallel test lines were applied using and per-pendicular orientation. Forty-five placements in each orientation were made to obtain about500 intercepts (N) per orientation with the results.

Straight Lines-Parallel to 254-mm (lO-in.) Side

Number of lines = 45Number of intercepts, N = 505 (466 ± 78/2)

Total line length = 32.562 mm (6750 mm on photograph)= 15.509 per mm

L 3 = 64.5 ± 3.1 Jlm

ASTM grain G = 4.61

50...,.------------------------------,

20+··················································· /

o -I ;/ /.. +.... . \ \........ I

= 45mtercents. N = 529

= 32.562= 16.246 per

Side

3 alsoand the percentwith ,nr''''''''''''''-''''

the standard deviation rises

+280 0

0 Meanl

I

II

oo

L-----------------------

70 -,-------------------------------,

65

++ + +

+ + ++ +

----------------------- -28

+

FIG. 6-Plotswhen withofthe data except for that from the s-mm-aiameter circte

all

s(P IL) or

VANDER VOORT ON SIZE 281

o 50 200

FIG.7-Plot the ofthe standard deviation the %RA ofPL measurements madeusing 8- to l Stl-mm-diameter test circles and the AISI 304 specimen.

times the mean lineal The 95% confidence limits are calculated from thestandard the Student-t value and the number of fields. the 95%CL is divided the mean value and as a percentage to determine the % RA.Note that these except for the 8-mm-diameter test are constant with avery decrease as the test circle increased. with the confidencelimits Willis and Lake where 95% CL decreased consioerabtvtest circle diameter increased the effort as the test circle diameters

no intersections for the 8­intersections for the

be onfor

placement for the 8-,8, the number of

on the x-axis. The x-axis is whole unitshalfrather than whole

piacements with 0.5

test data due toH.nHH.JU. Note that in each

hits. For the 24­9, all

ptacements fordata sobe normal.

trmre-nomr count measurementstotal

GRAIN SIZE 283

FIG. 10-Graph the vmrtnt irvn

and the triple-point count method usingstructure.

",-""",·tnrl were held as constant as desired.ruacement was used because to

otantmetnc method304 grain

284

steel specimen.

Circle Diameter, Number NA , ASTMmm of Fields mm ? G

79.0 8 49.25 0.1141 224.6 4.999.0 6 72.17 0.1791 207.0 4.719.5 4 108.75 0.2610 212.2 4.8

140.0 2 150.0 0.3582 212.2 4.8160.0 2 196.5 0.4679 212. 4.8180.0 248.0 0.5922 21 4.8

Percent60

050

40

30

20

10

20 30 40Grain sq.

Area Fraction,

4030

specimen shown with linear class

20

5 "'11····· j cr--.~~~. . /...........,....··················1

15 -+ I + \ .......j

25 -+ i··::;;"'····· ...'.., , =~•.••..= r·····..····- L' I

20 -+ II ..········ ..··.. \ .

30 -t , .

10 -+1· ·· /· ..····· ..· v.\.............................. I

35...,-----------------------------,

FIG. ll-Distributionsize intervals and with

VANDER VOORT ON GRAIN SIZE 285

Percent50 --,-----------------------------,

N 1

10 ..+ ~ -l ".................................................................................................. I

10000010000010000

sq.1000100

O-t--e"""-..,.....,...,-rrr-..,.....-.-.-rrrrrr-----.,,--,r-+,.-,-,rrrT--.,.---,-.,.-,-''l-r-r-rT''''-€Ir-,.....,...,...,..rrr'

10

Area Fraction, %

1000001000001000

50 -A

40

30

20

10

010 100 10000

sq.FIG. 12-Distribution ofgrain areas for the AlSl 304 stainless steel specimen shown with logarithmic

classsize intervals and with number percent (top) and areafraction percent (bottom)frequencies.

linearclass intervals, Fig. 11. However, if a logarithmic scale is employed, Fig. 12, and ofthe grain areas are evaluated, the skew and kurtosis values fit a lognormal distribution.

From the grain area values, the Jeffries diameters were calculated and plotted in the sameway, Figs. 13 and 14. The linear distribution and data analysis indicate a slightly nonnormaldistribution, while the logarithmic plots and data analysis reveal a lognormal distribution. Themean d value was 55 Mm, somewhat less than L 3• The standard deviation was 32.68 Mm.

SIZE

288

D

Percent

m

300

areas/or the AlSf 304 stain­a/the size classes.

the data. Note that the reveals a more normal appearanceand the skew and kurtosis values of the Ioganthrmc values indicate conformance to a lognor­mal distribution.

The final consisted individual chord and arc adja-cent grain intersections test circles diameter and a digitizing tablet.Because it is far easier to measure chord than arc more chord measurements

VANDER VOORT ON GRAIN SIZE 289

Percent30,---------------------------------,

N

250 300

lineal Percent30,-------------------------------,

N 428

300250

lIB 5.41

60.9 p,m

.. 1.60

15010050

20 f- /··1····················· \ .

5

10 1-··············1··························· + .

15 1- ·················1············· / \.... I

251-············································· +··=·············c·········

200

mFIG. 16-Distribution ofintercept length measurements made with a digitizing tablet and straight test

lines with the AISI 304 grain structure plotted using linear size classes and number percent (top) and linealpercent (bottom) frequencies.

were made. The writer made chord length measurements with each of the test circles and arclength measurements with the 16- and 79-mm-diameter circles. Then, two different metal­lurgy students at Lehigh University (Operators 2 and 3) measured both chords and arcs usingfour of the test circles. The students were unaware of any of the other test data in this paper,and they were not aware of each others results. Figure 18 shows an of the test data listed in

17-Distributionlines withlineal percent

VANDER ON GRAIN

a linear size class thewith the 79-mm-diameter test

This

Arc data is OV"-"'O"-'" al'" uu"\-u.n

and moredecreases.

19 illustrates the distribution curves,ber percent, for measured arc and chordcircle. The two curves are shown astrates the very close agreement that can be obtaineda test circle much greater in diameter than is

plottmg distribution curves, one must decide howclasses """1"",,1"1,,1''''' a loss of if is ofsizeclasses can be used if the number ofmeasurements is verythe mode the percentages per class will be rather erratic. In b~,..~.,~A,

should be used If the number of measurements is somewhat use 8 to 10If N is very above a number of classes may be

Conclusions

of the same revealed excellent correlation un,v"s,

produced by the Jeffries the trmie-nomtAbrams or by individual ,...1-,"'''1',,,,.-.1-

straight or circular test lines. In general, values

METALLOGRAPHY: PRESENT,

TABLE 6-Intercept lengths using arcs and chords circles ofvarying diameter.

Circle Diameter, True Diameter, Number of ASTMmm .urn Chords s, .urn G

16 78.3 90 43.0 19.4 5.81

24 17.5 80 57.0 29.3 5.032 156.6 79 60.6 30.1 4.840 195.8 130 57.5 35.4 4.950 244.7 64.4 36.9 4.679 386.7 157 61.6 37.7 4.879 386.7 15 58.1 37.6 4.999 484.6 157 61.6 38.5 4.8

119.5 582.5 158 64.5 38.3 4.6140 680.4 159 60.0 38.3 4.8160 783.2 97 70.4 39.2 4.4180 88 1 199 63.7 40.0 4.750 244.7 300 63.0 36.3 4.72

119.5 582.5 300 40.97 4.7140 680.4 300 59.6 35.25 4.8160 783.2 300 64.8 39.00 4.650 244.7 565 64.4 41.86 4.63

119.5 582.5 45 63.2 43.12 4.7140 680.4 64 63.6 34.00 4.7160 783.2 114 63.2 37.73 4.7

Circle Diameter, True Diameter, Number of Arc ASTMmm .urn Arcs .urn s, .urn G

16 78.3 169 64.2 42.8 4.61

79 386.7 157 62.1 39.1 4.750 244.7 300 63.6 39.54 4.72

119.5 582.5 300 62.9 39.97 4.7140 680.4 300 62.2 38.19 4.7160 783.2 300 65.7 40.24 4.650 244.7 231 61.0 41.45 4.83

119.5 582.5 57 60.4 36.35 4.8140 680.4 64 62.7 34.10 4.7160 783.2 14 62.9 37.41 4.7

1,2,3 Pertain to data developed by Operators 1, 2, and 3 (students were Operators 2 and 3).

sistently slightly smaller than this difference was so small thatthe ASTM G values were unaffected.

Use oftest circles with true diameters less than three times the mean lineal interceptis not recommended. With single test circles, the effort to obtain the same degree ofprecision increases as the circumference decreases. For efficient measurements,the diameter of the circle circles) should be greater than six times A total line of500 mm, as recommended ASTM with three concentric circles to minimize the sizeof the test about 500 intercepts or intersections with 13 placements(for this grain size and magnification, a lower would have been pre-ferred for best The 160-mm-diameter test circle also about 500 intersec-tions with 13 placements,

Test circles smaller than the largest grains should never be First, they are highlyinefficient. Second, some will produce no intersections. For such work, if P = 0and PL = 0, L 3 = co, which is absurd. in such cases, P or data must be averagedand the mean values used to calculate the individual field data cannot be converted to L 3

VANDER VOORT ON GRAIN SIZE

Percent50..,...----------------------------,

o N =

30025020015010050

40+·· ·..··········································· I····················································· i

Percent50 -,--------------------

oN =

30025020015010050

10 +,1················································1·· "-: .

40+··················································· \ j

20 + I······································ j '\ ············································1

30

FIG. 19-Distributions ofarc (top) and chord (bottom) intercept lengths made simultaneouslythe digitizing tablet and the 79-mm-diameter test circle with the AfSf 304 grain structure showinglinear size class intervals and number percent frequencies.

valuesand then an average value calculated. Statistical analysis must be performed on thedata rather than L 3 data.

ASTM E 112 is a totally manual method and only describes the Jeffries planimetric methodand several types of intercept count methods-but not individual arc or chord measurementson test circles. There are statements in ASTM E 112 concerning bias introduced using smalldiameter test circles. This information will be revised.

294 METALLOGRAPHY: PAST, PRESENT, AND FUTURE

Percent

numbers oflinear size classes on the shape ofthe histogram

Measurement of individual a rI",i·h7H''''

test lines. Measurements with test either chord or arcmade small diameter test circles. Such measurements are tedious. All arc

measurements are very tedious. Chord diameter test circles> 6) appear to be free of bias. circles little ",rb"", ...,t",,",p

over lines except for the fact that can be out. With testmay be several different orientations to even out structural anisot-

can be and to the deformation axis to assess theamsotronv with the data combined to obtain an average.

References

.....HHU.'V', E. B., Transactions, American Institute of Mining, Metal-Engineers, 1916, pp. 594-607.

Vol. 1916, pp. 40-56.

Vol. 35,1982, pp. 595-600.Applications of Topology," Metal Interfaces,

IVIPTnll1JrO'/f'n Vol. 21, No.2, 1987, pp. 1733-1736.