This document discusses techniques for determining the spatial and size distributions of spherical inclusions in mild steel. Planar measurements taken from electropolished surfaces are compared to measurements taken from fracture surfaces. It is found that the average dimple size on fracture surfaces is always greater than the most probable first neighbor spacing from planar measurements. Additionally, the mean inclusion size is smaller on fracture surfaces compared to electropolished surfaces. These differences are believed to arise from the fracture process itself.

1. ð
ð
The background fracture studies leading up to Fractal fracture discovery
PARTICLE SIZE DISTRIBUTIONS AND
INTERPARTICLE SPACINGS IN 2.XX
DIMENSIONAL SPACE
INCLUSIONS ARE A REALITY!!
Results were used to improve welding practice
(in general) and to develop the welding practice
used on the atomic submarines
iocqCA)
w~ n IS pn.JlLr?-.
15?~"'DS-~'~ 12/pf8J

2. II 0.52 0.47 0.37
42 FRACTOGRAPHY-MICROSCOPIC CRACKING PROCESSES
PASSOJA AND HILL ON INCLUSION DISTRIBUTIONS 43
this is not the case when considering a particle sampled from the volume. Tables
5 and 6 show a comparison between planar, corrected volume, and fracture
surface measurements. The agreement is not perfect although the corrections do
tend to correct the planar measurements in the direction of the fracture surface
measurements.
TABLE 5-Comparisons between two-dimensional mean diameters, corrected
three-dimensional mean diameters [6], and fracture surface mean diameters.
and fracture surfaces for the three different cases. The plots are shown as
cumulative log normal probability plots. In all cases investigated in this study,
the frequency distributions were skewed, asymmetric, and yielded the best linear
plots when the cumulative frequency distribution was plotted on log normal
probability paper. It is felt that the distributions are, therefore, all log normal
and will be considered as such in the following discussions. Table 4 shows a
TABLE 4-Summary of particle size distribution statistics obtained from extraction replicas.
Specimen
II 0.17 0.15 0.20
A. Electropolished -Planar
Specimen XA1 (µm) XG1 (µm) UA1 (µm) 1n2 uG1
0.64 0.58 0.28 0.20
II 0.52 0.50 0.17 0.11
III 0.38 0.36 0.12 0.11
B. Fracture Surfaces-Room Temperature
XAF (µm) X"GF (µm) uAF (µm) 1n2 uGF
0.53 0.47 0.27 0.26
II 0.37 0.32 0.20 0.29
III 0.36 0.32 0.18 0.23
0.64 0.53 0.53
III 0.38 0.34 0.36
TABLE 6-Comparisons between two-dimensional arithmetic standard deviation,
corrected three-dimensional arithmetic standard deviation,
and fracture surface arithmetic mean diameter.
Specimen
0.28 0.24 0.27
III 0.12 0.11 0.18
~
It is felt that only fair agreement between the corrected values and measured
values is obtained due to the fact that the probability of observing a particle on a
fracture surface depends upon the mechanism of particle-nucleated void
formation. Both particle fracture and interface decohesion occur during the void
nucleation step. Thus, the probability of observing a particle on a fracture
surface most likely depends upon both the diameter and the surface area of the
particle.
summary of the statistical data obtained for all the distributions which were
studied. The arithmetic mean and the arithmetic standard deviation were
computed in the standard manner from the dat~. The geometric mean and
standard deviation were calculated from 3 ~
Discussion
In2uG = (::) 2
InXG = InXA -0.5 In2 uG (5)
In each case it was found that the arithmetic mean diameter of the particle
found on a fracture surface was smaller than that found on a planar surface. As
discussed by Ashby and Ebling [6], the diameter of a particle as measured on a
surface extraction replica is not the same as the true mean diameter of the
particle for the volume. The difference arises from the fact that probability of
intersecting a particle in a planar section is proportional to its diameter whereas
(4)
Spatial Distribution and Dimple Size
The combined set of measurements (that is, determination of spatial
randomness of the nearest-neighbor interparticle spacing distribution and
measurement of the mean linear dimple intercept size) constitutes a method by
3 Ashby and Ebling's [6] nomenclature is used here.

3. D. E. Passoja! and D. CHill!
Comparison of Inclusion Distributions
on Fractu re Su rfaces and in the Bu Ik
of Carbon-Manganese Weldments
REFERENCE: Passoja, D. E. and Hill, D. C., "Comparison of Inclusion Distributions
on Fracture Surfaces and in the Bulk of Carbon-Manganese Weldments," Fractog-
raphy-Microscopic Cracking Processes, ASTM STP 600, American Society for
Testing and Materials, 1976, pp. 30--46.
ABSTRACf: Techniques are developed for determining the spatial and size
distributions of spherical inclusions in mild steel. The relations between these
distributions as found on electropolished surfaces and as found on fracture surfaces
are determined. It is shown that the first neighbor spatial distributions on
electropolished surfaces can be described analytically. The average dimple size on a
fracture surface is always greater than the most probable first neighbor spacing. The
inclusion size distributions both on electropolished and fracture surfaces are shown
to be log-normal. The mean inclusion size on a fracture surface is always smaller than
that on an electropolished surface. These variations are believed to arise from the
nature of the fracture process.
KEY WORDS: fractography, inclusions, crack propagation, fractures (materials),
fracture properties, carbon steels, geometric surfaces, weld metal, weldments, particle
size distribution, particle density (concentration)
Nomenclature
tv Volume fraction of inclusions
n Number of events
r First neighbor separation
r* Most probable first neighbor separation
r Average first neighbor separation
XA
Arithmetic mean
XG Geometric mean
!AI Arithmetic one-dimensional mean (after Ashby and Ebling [6])
!A2 Arithmetic two-dimensional mean (after Ashby and Ebling)
!AF Arithmetic fracture surface mean
XGF
Geometric fracture surface mean
D Dimple size (equal to 1.5 [)
Ds Average particle spacing (after Kocks [8])
K1c
Fracture toughness
I Research scientist, Central Scientific Laboratory, and research supervisor, Linde
Research Department, respectively, Union Carbide Corporation, Tarrytown, N. Y. 10591.
30
PASSOJA AND HILL ON INCLUSION DISTRIBUTIONS 31
L
Ns
Pen)
Per)
µ
a
aAI
aA2
aAF
aGF
Mean linear intercept dimple size
Average planar inclusion density
Probability as a function of n
Probability as a function of r
Number of elements in system-average number of
Standard deviation
Arithmetic one-dimensional standard deviation
Arithmetic two-dimensional standard deviation
Arithmetic standard deviation on fracture surface
Geometric standard deviation on fracture surface
There is considerable evidence to indicate that the fracture toughness of
metals is related in some Why to a characteristic feature of the microstructure.
The feature is thought to be related fundamentally to the fracture process itself,
that is, to play a determinant role in the progression of the fracture.
Considerations of the physical characteristics of the microstructure usually
precede the development of a more specific fracture model and use of such a
model to explain fracture behavior in any system.
Several investigators have recognized salient microstructural features and have
related them to fracture toughness. Using Krafft's [1] 2 tensile ligament
instability theory, Birkle, Wei, and Pellisier [2] measured the spacing between
sulfide particles on extraction replicas and were able to relate the fracture
toughness to the spacing between the particles. Hahn and Rosenfield [3]
considered several factors which influenced the toughness of aluminum alloys
which included:
I. the extent of the heavily strained region in front of the crack tip,
2. the size of the strained ligaments-which was related to the volume fraction
of cracked particles, and
3. the work required to rupture the ligaments.
They furthermore recognized that both the size, distribution, and particle type
influenced the fracture process. By plotting K1c
versus tv -! /6 ifv being the
volume fraction of cracked particles), the authors were able to develop
convincing arguments that fracture toughness in several different material.s could
be related to inclusions. In his work on high strength steels, Yoder [4] found a
somewhat coarser spacing on fracture surfaces which correlated with fracture
toughness. Furthermore, fractographs published in his work clearly indicate that
the spacing which correlated was considerably larger than the dimple size.
In order to determine more definitively the relationship between features
found on fracture surfaces, and those of the microstructure, and to quantify the
existence and nature of such differences, we performed experiments to compare
the features of fracture surfaces with those of plane-polished surfaces in three
ferrous materials. In light of the findings just ;eferenced, we chose to use a
2 The italic· numbers in brackets refer to the list of references appended to this paper.

4. 46 FRACTOGRAPHY-MICROSCOPIC CRACKING PROCESSES
The second moment can be written as
(II)
So that
(I2)
References
[1]
[2]
[3]
Krafft, J. M.,Applied Materials Research, VoL 3, 1964, p. 88.
Birkle, A. J., Wei, R. P., and Pellissier, G. E., Transactions, American Society for
Metals, VoL 59, 1966, p. 98L
Hahn, G. T. and Rosenfield, A. R., 5th Spring Meeting of the Metallurgical Society
of the American Institute of Mining, Metallurgical, and Petroleum Engineer's, 29
May to 1 June 1973.
Yoder, G. R., Metallurgical Transactions, Vol. 3, 1972, p. 185 L
Hill, D. C. and Passoja, D. E., Welding Journal, VoL 53, 1974, p. 481-s.
Ashby, M. F. and Ebeling, R., Transactions, Metallurgical Society of the American
Institute of Mining, Metallurgical, and Petroleum Engineers, VoL 236, 1966, p.
l396.
Hilliard, J. E., Metal Progress, Vol. 85, 1964, p. 99.
Kocks, U. F., Philosophical Magazine, VoL 13, 1966, p. 541.
[4]
[5J
[6]
[7]
[8]
(

5. (V)
M
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6. 34 FRACTOGRAPHY-MICROSCOPIC CRACKING PROCESSES
•
0{]
0
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'" ". '"
:;: Sl ~
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..,
.:.-~
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PASSOJA AND HILL ON INCLUSION DISTRIBUTIONS 35
'.
A
>-
• <" ...• . . ,
.
.
:
. .
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"
...
.
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FIG. 2-111ustrative example of concentric circle measuring technique. (A) shows a tracing
of the unetched, polished metallographic section shown in (B). Five concentric circles are
shown superimposed on a particle marked as A on the micrograph. The magnitude of the
vector R is shown in (A); original magnification, x]OOO.
Appendix II. The agreement between the computed values and measured values
is fairly good considering that only 200 distances were measured on each sample.
It is important to note, however, that the computed curve shape is similar to the
measured values in each case, and that the trends (that is, decreasing values) are
the same for both the computed and measured values. It would appear the
discrepancies between the computed and measured values arise from either a loss

7. PASSOJA AND HILL ON INCLUSION DISTRIBUTIONS 37 36 FRACTOGRAPHY-MICROSCOPIC CRACKING PROCESSES
.0.1
;
Comparison of Computed
and
Measured lnferpor lie Ie
r--
Distribution Spac i n 9 s
)
ill
r-- Opticol- EJectropolished
-
r-!!...
• •;
'--
• •)
•
•;
• r-
•i--
Il· .
.0.2.0
r.oj
I
PI,'
C.'
Comparison of Computed
r- and
Measured interparticle
5 Distribution Spacings
I--
I
• • Optical Electropolished
• •1--....-
.0
fT - •
..... ~.;
• --, -
•• ~
IT
• • .Ial_ .. - - .d
1..0 2.0 3.0 4.0
r(lIm) ..
5..0 6.0 7..0 B.C
.0.25
0.21
rPltl
1..0 2.0 3[)
r(pm) - )'
4..0 S.C 6[) 7..0 B.C
.0.05
.0..0
FIG. 3-Frequency distribution for planar first nearest-neighbor interparticle spacing, Steel 1.
t
Comparison of Computed
and
r---- Measured Interparticle
Distribution Spacing
,
IT..-
• ITr.- Optical - Electropalished
I-a-'---
)
• •
... •it--
•
• •~.I
•
0.20
G_ 5-Frequency distribution for planar first nearest-neighbor interparticle spacing, Steel
measured_ It is well known that, during ductile fracture, voids usually
leate on inclusions, grow, and link-up to form the fracture surface_
lsurements of the dimple size should, therefore, be related to the interparticle
~ing, in some manner, but should be complicated by the fracture surface
metry_ We attempted to demonstrate this in the following manner.
tereo photomicrographs were taken randomly from ten different areas on
ldard Charpy specimens fractured at room temperature_ In each instance, a
Ipled fracture surface was present. three lO-cm circles were located on each"
eomicrograph pair. When viewed in the stereo viewer the circles merged and
y three circles appeared to be located over the fracture surface. The
rsections between the circles and the dimples were marked off and the
[section rules outlined by Hillard [7] for measuring grain sizes were followed
lustrated in Fig. 6. Table 2 shows a summary of the measured values .
.s in the nearest-neighbor planar interparticle measurements, Table 1, the
Ids exhibited in Table 2 are similar, but in every case the mean-linear-dimple
rcept size is greater than the nearest-neighbor planar interparticle spacing.
differences between the measured distances can be explained in the
owing manner: dimples are created during the void link-up stage upon final
lration of the fracture surface. The fracture surface is created as a result of
I nucleation and growth in a volume of material near the crack tip. For this
on the spacings between the dimple forming voids do not necessarily
'espond to the most probable value of a single nearest-neighbor interparticle
;ing, but rather it depends upon a number of factors such as the local stress
0.15
P(r)
0.1
0.0
1.0 2.0 ~O 4~
r0m) •
11.0 6.0 7.0 8.0
FIG. 4-Frequency distribution for planar first nearest-neighbor interparticle spacing, Steel
II.
of particles during the extraction step or arise from counting only 200 (a limited
number of) nearest neighbor distances with the concentric circle method.
Dimple Size Measurements
In order to compare the planar nearest-neighbor interparticle spacing with
some meaningful fracture surface feature, the mean linear dimple intercept size

8. PASSOJA AND HILL ON INCLUSION DISTRIBUTIONS 39
38 FRACTOGRAPHY-MICROSCOPIC CRACKING PROCESSES
"0
<J)
';
I ~ ": «:0-
E
0
~IU
~
0
"0
~
I
<J)
....
I ~i5l q ~'t;
'"
N
'" <J)
>:>.
~<00
~
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"-
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"-
~ "0
,S <J)
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I ~ M
"!0 0-
M
"'" E M N
..:::: 0
,~
~ U
"2 E
<00 ~
~ 1"-
%l "0
~ <J)
"
::;
I 0 .-. '<:t
..:::: '" N N....
'"<J)
"-
~,0.....,
<00
".2!
'""'1:l
" "0
"- <J)
::;
';<00
I ~ '"' q%l 0-
E E N N N
'1:l
0
~ U~
~'"'1:l
~
"::; "-
>:>. "0
E "....
0 i5l
I N 0 "!
" '"
N N
~ <J)
" ~
"
'--- I t
""'"~0
FIG. 6-( A) shows polished, etched section of the typical microstructure observed in ,'="
'""-
'" ,~
teels I through III; original magnification, xl 000. (B) shows one photo of a stereo pair >:>.
'0..
'ith two circles used to measure the dimple size, Intersections between the dimples and the E .......
" '" '" t"-
O '" J::t:: 0 0 0
I ,..., ,..., ,...,'rcles are marked on the micrograph; original magnification, x3250, I E <:: x x x,..., u 0
~ ~ N
'"' 00
w u 0 ,..., ,...,
....l <: ;'! .,., r..:
Ild the particle size distribution. Dimples are thus some irregular shape, such as
I
a:l ;;:<C
f-< :.:.J
11 n sided polygon, when viewed as a planar projection in the SEM.
Kocks [8] has shown that in some instances the average distance between a
article and its two or three neighbors is more a meaningful distance than the
c
earest-neighbor distance. This distance can be calculated by <J)
E
'u I - = -D = 1 18 N -1/2
<J)
(3) 0-
s . s u:>

9. 40 FRACTOGRAPHY-MICROSCOPIC CRACKING PROCESSES
PASSOJA AND HILL ON INCLUSION DISTRIBUTIONS 41
TABLE 2-Summary of mean linear dimple intercept measurements.
7-Comparison of log normal cumulative probability plot of particle diameters for
nd fracture surface for Steels f through fl!.
InclU$lon [)oomeler
Cl.rnulolive Probob,hly
I
([>lrocllonRephcosJ
,
.,
..,
.6
.,
••
•• O~ 0
• 0
• 0
• 0
• 0
eEleclropOI,shed
o• DRoam Temperoture
Chorpy
Cumulohve ProbabIlIty
'~r'------------------------,
.,
6
,
Inclusion Dio'7'lller
Cumulative PrObability
IT
(E.'roction Replicas)
,.,.8
.'.6
• C
• 0
• 0
• 0
o
o
o
o eElecl'opollshod
o Room Temperalur,
Chorpy
.I' 0 I
0.01 10 20 JC) 40 50 60 10 00 90 95 99 999
Cumuloli". Probablhty
Cumuklilve Probablhty
m
(hlroct_Repllcas)
! '(5.9
~~~, 6
~ 5
,.
..
:JAoom Tlln'lpe.olure
Cho'P)'
.11 ' , I
01.1 10 20 30 405060 70 80 90 95 99 ~99
C.."....,lal,ve ProbabIliTy
L(µm)Specimen a(µm)
3.3 0.65
2.6II 0.48
2.5III 0.19
Table 3 shows a comparison between measured D (equal to 1.5 L) and Ds as
calculated by Kocks equation. Thus, it can be seen that the number of nearest
neighbors is important when considering the mean linear intercept dimple size
measurement. There is only fair agreement between Ds and the measured values
due to a number of contributory factors such as:
I. the statistical scatter due to the topographical features (non-planarity),
2. the physical aspects of the fracture process resulting in a physically
meaningful standard deviation, and
3. local differences in the number of nearest neighbors-the most common
number projected in a plane appears to be five.
TABLE 3-Comparisons between measured dimple sizes and computed spacings.
Specimen D (µm)
4.95.0
3.9 4.4II
III 3.7 3.4
Particle Size Distribution
Direct carbon extraction replicas were made of fracture surfaces and
electropolished surfaces taken ~0.5 cm away and parallel td' the fracture surfase.
One thousand particles which could uniquely be associated with dimples were
measured on the fracture surface replicas from each specimen. One thousand
particles were measured on the planar cut and electropolished surfaces from each
specimen. Mechanical polishing was tried, but gave inconsistent results with too
few particles and unreaJistically large mean sizes. Following Ashby and Ebling's
[6] particle extraction efficiency arguments, our assumptions were that the
particle extraction efficienoy was not 100 percent but there was equal
probability of extracting a particle over the entire size range (that is, particle
extraction was not selective).
Figure 7 shows cumulative probability plots of particles extracted from planar

10. PASSOJA AND HILL ON INCLUSION DISTRIBUTIONS 45 44 FRACTOGRAPHY-MICROSCOPIC CRACKING PROCESSES
(7)
(8)
which planar surface features can be compared in a meaningful manner with
fracture surface features. Successful application of these methods to our
problem required that two criteria be met.
1. The fracture mode must be related to particle nucleated microvoids.
2. The center-to-center spatial distribution must be random. (This can be and
was checked by means of the Poisson test.)
Differences between planar surface features and fracture surface features can
best be understood by considering the topographical differences which exist
between a fracture surface and a planar surface. The mean linear dimple
intercept, for example, is larger than the most probable planar nearest-neighbor
interparticle spacing due to the fact that voids grow around inclusions and link
up in a volume of material in front of the crack tip. A single planar,
nearest-neighbor interparticle spacing, therefore, does not describe the dimple
formation process since the volume distribution of nearest neighbors is a more
relevant description of the fracture process. For this reason Kocks' correction
appears to bring agreement between planar and fracture surface features.
~PPENDIX I
¥eldment Preparation
Three weldments were made in ASIS baseplate using an E70S-3 electrode4 by
he GMA process according to American Welding Society Standard AS. 18-69.
.11 welding was done in the flat position using automatic equipment. The
IOminal welding conditions were: current, 200 A; voltage, 27 V; and travel
peed, 30 em/min. The shielding gas composition was varied to provide different
)xidizing potentials and different levels of inclusions in the weldments. The
.hielding gases used and weld identifications are: CO2, I; Ar-25% CO2, II; and
r-2% O2, III. Weld compositions are given in Table 7.
TABLE 7 - Weldmetal compositions.
ield Designation c Mn Si o
0.08 0.45 0.12 0.099
II 0.09 0.78 0.32 0.044
III 0.08 1.00 0.46 0.042 Particle Size Distributions
Inclusions which are present in the bulk are the same set of inclusions which
cooperatively participate in fracture and form the fracture surface. The
differences which are observed between inclusion sizes found on a planar surface
and on a fracture surface are believed to result both from the geometrical
differences involved in planar versus volume sampling and from the fracture
process.
.PPENDIX II
ipatial Distribution Function
Equation 2a represents the incremental number of inclusions surrounding a
ingle inclusion in a thin strip of width /::;.r:
/::;.Ns = rrrNs exp (-r2 Ns) /::;.r (6)
lhe exponential term represents the probability that the distance between any
wo inclusions lies between rand r + /::;.r. The distribution can be normalized by
:hanging Eq 6 into a continuous distribution, equating the integrand to I, and
olving for the normalization constant, k:
Conclusions
I = kJ~rrNs r exp (- Nsr2) dr
k= ~
rr
Techniques have been developed for quantifying and comparing both the
spatial and size distributions of inclusions on fracture surfaces and in the bulk. A
nearest-neighbor interparticle spacing distribution function of the form
(2b)
4 All terms in this section refer to the American Welding Society designations.
apparently describes a random planar spatial distribution of particles for the
three ferrous materials used in this study.
A correction must be made for two or more nearest neighbors in order to
bring agreement between dimple size measurements performed on a fracture
surface and particle density measurements performed on planar surfaces.
Particle size measurements indicate that the particle sizes are log normally
distributed both on planar surfaces and fracture surfaces. Differences between
the two distributions can be rationalized by considering the details of the
particle sampling process during fracture.
,0 that
¢ (r) = 2Ns r exp (-Nsr2) (9)
~quation 9 is the continuous distribution function from which various moments
)f r can be calculated. For instance, the average value of r is
'----
r=.r: r¢(r)dr= 1/2 J 1T (10)
Ns

11. I = kJ~ 7TNs r exp (- Nsr2) dr
k = .2
7T
(7)
(8)
PASSOJA AND HILL ON INCLUSION DISTRIBUTIONS 45
APPENDIX I
Weldment Preparation
Three weldments were made in AS IS baseplate using an E70S-3 electrode4 by
the GMA process according to American Welding Society Standard AS. 18-69.
All welding was done in the flat position using automatic equipment. The
nominal welding conditions were: current, 200 A; voltage, 27 V; and travel
speed, 30 cm/min. The shielding gas composition was varied to provide different
oxidizing potentials and different levels of inclusions in the weldments. The
shielding gases used and weld identifications are: CO2, I; Ar-2S% CO2, II; and
Ar-2% O2, III. Weld compositions are given in Table 7.
TABLE 7-Weldmetal compositions.
Weld Designation c Mn Si o
0.08 0.45 0.12 0.099
II 0.09 0.78 0.32 0.044
III 0.08 1.00 0.46 0.042
APPENDIX II
Spatial Distribution Function
Equation 2a represents the incremental number of inclusions surrounding a
single inclusion in a thin strip of width I::lr:
I::lNS=7TrNS exp (-r2 Ns)l::lr (6)
The exponential term represents the probability that the distance between any
two inclusions lies between rand r + I::lr. The distribution can be normalized by
changing Eq 6 into a continuous distribution, equating the integrand to I, and
solving for the normalization constant, k:
So that
¢(r)=2Nsrexp(-Nsr2) (9)
Equation 9 is the continuous distribution function from which various moments
of r can be calculated. For instance, the average value of r is
(l0)
4 All terms in this section refer to the American Welding Society designations.