WT.OF T ECHNolo
OCT 2 1961
DUCTILE
FRACTURE
BY
THE
GROWTH
OF
PORES
by
WARREN J. RHINES
B.M.E.,
The Cooper Union
(1960)
SUBMITTED
IN
PARTIAL
THE REQUIREMENTS
MASTER
FOR
OF
MECHANICAL
FULFIILLMENT
DEGREE
THE
SCIENCE
OF
OF
IN
ENGINEERING
at the
INSTITUTE
OF
TECHNOLOGY
July, 1961
Signature Redacted
Signature of the Author '................)
Department '6f Mechanical
Enaineerinx, July 5, 1961
Signature Redacted
.........
Certified by
T,,xnesis Advisor
Signature Redacted
,
MASSACHUSETTS
Accepted by
....0
Chairman, Departmental Committee on
Graduate Students
-2-
DUCTILE FRACTURE BY THE GROWTH OF PORES
by
Warren J. Rhines
Submitted to the Department of Mechanical Engineering on July 5, 1961
in partial fulfillment of the requirements for the degree of Master
of Science in Mechanical Engineering.
ABSTRACT
Ductile fracture by the growth and coalescence of holes
formed at inclusions has been studied with the aid of plasticine
models.
Commonly observed fracture phenomena in metals such as the
decreased ductility of smooth and notched specimens with increased
inclusion size and concentration and increased specimen size,
anisotropy, the effect of pre-torsion on tensile fracture, and
the decreased ductility in the presence of triaxial stress and
strain have been obtained in the plasticine models containing
inclusions.
The cup-cone fracture was not achieved in these models.
An approximate solution is given to the critical spacing
at which holes in an ideal plastic material, under simple uniaxial
tension, will coalesce.
Thesis Advisor:
Frank A. McClintock, Ph.D.
Title
Professor of Mechanical Engineering
-
-3
ACKNOWLEDGENT
I wish to express my sincere thanks to
"Bill" Henry for his experienced advice in the preparation of specimens for this investigation.
I am indebted to Hughes Aircraft Company
for the generous financial support they provided me.
To Professor Frank A. McClintock I owe a
debt of gratitude for the advice, guidance, and
generous donation of his time which gave shape and
objective to this investigation.
-4-
TABLE OF CONTENTS
ABSTRACT
*.
ACKNOWLEDGMENT
..
..
..
-.
-
.
..
--
TABLE OF CONTENTS
INTRODUCTION
--
..
..
..
.
2
-*
-
..
-
3
..
..
..
..
.
4
--
-.
--
--
.
5
*.
..
.
8
.
GROWTH AND COALESCENCE OF HOLES
PIASTICINE MODELS
..
..
..
SPECIMENS AND INCLUSION CONTENTS
..
..
..
..
.
13
..
..
.
16
..
..
.
26
33
11
UNIAXIAL DUCTILITY TESTS
..
NOTCH SENSITIVITY TESTS
..
HISTORY EFFECTS
..
..
..
..
..
.
*
--
-.
..
.
a 39
CONCLUSIONS
APPENDIX:
so
..
*.
INVESTIGATION OF THE GROWTH
AND COALESCENCE OF HOLES
USING LIMIT ANALYSIS
REFERENCES
00
00
00
*0
..
-
.41
45
INTRODUCTION
Knowledge of ductile fracture is,
at present, very limited
and there is no quantitative criterion for it.
Whereas rupture
is the separation of material by plastic flow until the cross section
vanishes, ductile fracture is the separation of material under plastic
flow which occurs at lower extensions than would be expected in the
case of rupture.
Several investigators have put forward the view that this
fracture is caused by the growth and coalescence of holes which are
formed at inclusions. Puttick (1959), by sectioning through the neck
of a copper tensile specimen, has observed a large central cavity
surrounded by numerous small holes.
This is shown in FIG.l.
These
holes were found to originate at inclusions either by drawing away
of the metal or by fracture of the inclusion itself.
The cavities,
he suggests, are opened out by the triaxial stresses and strains
accompanying formation of the neck and eventually link up to form the
large fissure which determines final fracture.
well illustrated in FIGS. 1 and 2.
This coalescence is
Cottrell (1959) proposes that we
think of fracture as a plastic cavity or an "internal neck", rather
than a "crack", and consider it
to be growing outward, without any
fracture, to meet the "external neck" which is growing inward.
He
explains the formation of a thin lens-shaped cavity by many small
cavities becoming nucleated simultaneously,
all
each growing outward in
directions, becoming elongated in the direction of the tensile
axis, and coalescing with neighboring cavities to become part of a
-r6o-
large thin cavity.
This process is shown in FIGS. 1 and 2.
He
presents experimental evidence that the plastic cavities are nucleated at foreign particles, and that, if such particles were not
present, the specimen would pull apart entirely by the inward growth
of the external neck, giving nearly 100% reduction of area, i.e., the
specimen would fail by rupture.
Additional experimental evidence to
support these views has been given by Tipper (1949), and Rosi and
Abrahams (1960), and Crussard et al (1959).
This mode of ductile fracture by the growth and coalescence of holes will be investigated in an attempt to understand
some of the following fracture phenomena:
1)
The effect of inclusion size and concentration
in smooth and notched specimens.
2)
The effect of specimen size in smooth and
notched specimens.
3)
The effect of stress and strain history.
4)
The effect of triaxiality of stress and strain.
5)
The cup-cone fracture.
-
- 7
FIG. 1 - SECTION THROUGH NECK OF A COPPER
TENSILE SPECIMEN FRO PUTTICK (1959)-
FIG. 2 - CAVITIES COALESCING IN CENTRAL
REGION OF FIG. 1 FRC1! PUTTICK (1959).
- 8-
GROWTH AND COALESCENCE OF HOLES
The growth and coalescence of several holes formed at
inclusions in copper is shown in FIG.2 from Puttick (1959).
The
critical spacing at which holes will grow and coalesce rather than
grow independently can be partially investigated using the theorems
of limit analysis.
See APPENDIX
for the details of this analysis.
Deformation modes are postulated to obtain upper bounds.
These are shown in FIG.3.
In one mode (dotted lines), the hypothet-
ical deformation consists of shearing on planes passing through the
holes at 45 0 angles and the holes will grow independently and elongate in the direction of the tensile axis.
In the second mode (solid
lines), deformation is localized to the region of the logarithmic
spiral slip lines between the holes and they will grow and coalesce
with a marked decrease in the required overall elongation of the
specimen.
It has not been shown that a lower bound exists for
either of these modes.
This would require an analysis as described
by Bishop (1953) in which equilibrium is demonstrated in the region
which is not yielding as well as the region in which the deformation
is taking place.
Our analysis has shown that the critical spacing at which
the hypothetical deformation mode and upper bound changes occurs at
a ratio of inclusion spacing to inclusion diameter of s/d = e = 2.72.
-
- 9
This should give an order of magnitude of the actual hole spacing
at which fracture becomes localized in the plane of the holes.
10
-
-
P
'SP
FIG. 3 - HYPOTHETICAL
DEFORMATION MODES FOR
THE GROWTH AND COALESCENCE OF HOLES
IN A TENSILE SPECIMEN
11
-
-
PLASTICINE MODELS
The difficulty of an exact solution, especially in the
three-dimensional case, makes a simple model with which to study
this
possible mechanism of ductile fracture by the growth and
coalescence of holes formed at inclusions very desirable.
The
recent and illuminating results of using models of plasticine to
simulate the plastic flow of metals has led the author's attention
in that direction.
For these applications of plasticine models to
problems of plastic flow in metals see Green (1951a, 1951b, 1954,
1955), Mortimer (1952) and Landberg (1958).
Green (1951a) studied the properties and behavior of
plasticine and compared them with those of metals. He attributes the
marked similarity under conditions of plane strain to the following
factors: any element of either material deforms in shear under an
approximately constant stress;
both are virtually incompressible
provided the air bubbles are removed by working the plasticine;
and their principal axes of stress and strain-increment coincide.
He also presents a stress-strain curve for plasticine under uniaxial
compression and the similarity of its shape to that of the stressstrain curve of metals is immediately seen. Both Green (1951a) and
Mortimer (1952) comment on the obvious advantages of using plasticine
models.
These are its inexpensiveness, the ease of fabricating models
with it, and the low forces required to work it.
To sum up:
12
-
-
the evidence that ductile fracture often
results from plastic flow around inclusions, and the fact that
plastic flow of metals can be simulated with plasticine models,
has led the author, in this research, to an investigation of the
fracture of plasticine specimens containing inclusions.
-
-13
SPECIMFNS AND INCLUSION CONTENTS
Harbutt's stone-colored plasticine was used for all
specimens.
Polystyrene spheres were used as inclusions in our
experiments because of their availability and low adhesion to plasticine.
The two sizes which were available for our use, courtesy of
the Plastics Division of the Koppers Company, Inc., were .006"
diameter spheres and .010" diameter spheres.
These are several
orders of magnitude larger than the particles found in pure plasticine.
They were kneaded into our plasticine to produce the desired
inclusion contents and a homogeneous distribution was assumed when
successive tests on a given specimen gave consistent results.
The results of preliminary testing with various concen-
trations of inclusions dictated the inclusion densities used in our
work.
TABLE 1 describes the ranges of inclusion sizes, specimen
sizes, and inclusion concentrations used and contains data on the
inclusion contents of some typical metals for comparison.
The
inclusion spacing, s, for the plasticine specimens was calcuated
on the basis of a simple cubic packing, that is, as the cube root
of the reciprocal of the number of inclusions per unit volume of the
specimen.
The plasticine specimens are smaller relative to their
inclusions than are the typical metal specimens, having a lower
ratio of specimen diameter to inclusion diameter,
do/d.
The
inclusions in plasticine are more closely spaced relative to the
inclusion diameters than are the inclusions in typical metals, the
-r
-
-14
plasticine specimens having a lower ratio of inclusion spacing to
inclusion diameter, s/d.
The inclusion sizes available for our use
were such that very large specimens would be required to make our
specimen diameter ratios comparable to those of the metals.
With
present equipment it was not feasible to do this and so it was
also necessary to use lower inclusion spacing ratios than in the
metals to obtain significant changes in the fracture behavior of
our specimens from that of pure plasticine.
The round tensile specimens were formed by rolling the
plasticine between flat glass plates.
Rollers were used between
the plates to give the desired diameter and insure cylindrical specimens.
Large ends were left on the specimens to provide for gripping.
The notched specimens were carved from blocks of plasticine
with a fine wire.
-
- 15
TABLE 1 - INCLUSION CONTENTS
SPECIMEN
SOURCE
INCLUSION SPACING RATIOS
SPECIMEN
DIAMETER
RATIO
d
within
rows
s
0
=
average
S2
s
d
d
d
taking
a0
between
rows
.50"
unless
otherwise
specified
I
SAE (1959)
4
4
4
Steel - plate#i
500 - 1000
1
Steel - plate#5
500 - 1000
2
Steel - plate#8
330 - 1000
1
8 - 20
7075 Aluminum
2000
7 - 20
2024 Aluminum
2000
5 - 10
Plasticine Specimens:
.010" spheres densely packed
50
1.15
.010" spheres lightly packed
50
1.87
.006" spheres densely packed
83
1.34
.006" spheres lightly packed
83
.006" spheres lightly packed
(d 0 = .90")
150
1.84
.006" spheres lightly packed
(d0 = 1.50")
250
1.84
MacGregor
and
Grossman
(1952)
L.
-16-
UNIAXIAL DUCTILITY TESTS
As inclusions are added to plasticine the mode of failure
changes from rupture, with a reduction of area of 100%, to ductile
fracture, with a reduction of area which decreases as more inclusions
are added.
The first group of specimens in TABLE 2, also shown
in FIGS. 4 through 8, have specimen diameter ratios of 50 and
reductions of area down to 36%.
The second group of specimens in
TABLE 2 illustrates the same effect with an inclusion diameter giving
a specimen diameter ratio of 83.
Kneading the inclusions into the
plasticine to achieve greater inclusion concentrations than shown
in the table was prohibitively difficult.
creased ductility with increased
This phenomenon of de-
inclusion content has been shown
in metals several times, for example by Cottrell (1959).
The specimens with the greatest inclusion concentrations,
seen in FIGS. 7 and 8, necked very little
prior to fracture.
The
values of fractional elongation presented in TABLE 2 show that a
large amount of overall plastic deformation takes place prior to
fracture so that considerable reductions of area can be obtained
with negligible necking.
To account for the large changes in ductility with what
appears to be a small change in inclusion concentration it
must be
noted that, although the difference in inclusion spacing ratio from
17
-
-
the least to the most densly packed specimen varies by not
more than a factor of 1.7, the actual space between inclusions, which
is approximately equal to the mean distance between inclusion centers,
s, minus the inclusion diameter, d, does vary by a factor several
times as great.
Specimens 6, 9, and 10 in TABLE 2 show the decrease
in ductility with increasing specimen diameter for a given inclusion
content.
This "size effect" is a commonly observed phenomenon in
the testing of metals.
Specimen 11 has the same inclusion content as specimen
10 but fractured with a considerably larger reduction of area because
of the presence of the small axial hole in the center of it.
effect is seen by comparing FIGS. 9 and 10.
The
The axial hole
suppressed the high triaxial tension which is usually present at
the axis in the neck of a tensile specimen and a far greater
ductility was achieved.
Similar results were obtained in metals
by Uzhik (1948) by deliberately drilling a small axial hole in the
center of the tensile specimen.
The cup-cone fracture was not obtained in our plasticine
models.
The ductile fractures which have been obtained are best
illustrated in FIGS. 9, 11, 12 and 13, all of specimen 10.
fracture surfaces are coarse and grainy.
Both
One half does resemble the
familiar cup-cone fracture but the other half is relatively flat.
18
-
-
From FIGS. 12 and 13 it is concluded that fracture started at the
center of the specimen and proceeded perpendicular to the tension
axis.
Considerable plastic deformation takes place as the specimen
is further elongated and this gives rise to the lip around the
edge of one fracture surface.
The inclusions are visible on the fracture surface, under
low magnification, giving proof that the bond between the plasticine
and inclusions does break and voids are opened up at the inclusions.
An unexpected result was that the ductility of a specimen
of plasticine containing a fixed inclusion content markedly increased
as the specimen aged as shown in specimens 3, 6 and 8 of TABLE 2.
In ten days specimen 6 was as ductile as pure plasticine and specimen
8 had an increase in reduction of area of 31%.
The observation that
plasticine sticks better to glass after is remains in contact with
it
for a period of time makes it
seem
likely that the adhesion of
the plasticine to the polystyrene inclusions improves with aging,
perhaps by the diffusion of the grease lubricant into the spheres.
TABLE 2 - UNIAXIAL DUCTILITY DATA
NO.
SPEC.
DIAM.
d
GAGE SPEC. INCLUSION
FRESH
SPECIMEN ONE
LENGTH DIAM.
SPACING
SPECIMEN
DAY OLD
RATIO
RATIO
ELONGATION,
REDUCTION
ELONGATION,
REDUCTION
10
0
d
FRACTIONAL
OF
FRACTIONAL
OF AREA,
AREA,
(in.) (im.)
FRACTIONAL
FRACTIONAL
(g-)
(2)d
SPECIMEN TEN
DAYS OLD
ELONGATION, REDUCTION
FRACTIONAL
OF AREA,
"
SPEC.
1~t
t
t
1
50
2.0
2
50
2.0
50
3
50
2.0
4
50
5
I
4
.80
1.00
1.87
.6o
.84
50
1.34
.50
.64
2.0
50
1.19
.30
.36
.50
2.0
50
1.15
.30
.36
6
.50
2.0
83
1.84
.65
.93
7
.50
2.0
83
1.50
.50
.73
8
.50
2.0
83
1.34
.40
.53
9
.90
2.0
150
1.84
.70
.87
10
1.50
2.0
250
1.84
.85
.84
11
1.50
2.0
250
1.84
.95
.96
pure plasticine
FRACTIONAL
4
L
.80
1.00
.60
.88
H
.80
.70
.50
.75
.90
j
(large specimen with axial hole)
1.00
-
- 20
FIG . 4 - RUPTURE OF PURE PLASTICINE TENSILE SPECIMEN 1,
REDUCTION OF AREA = 100O
FIG. 5 - FRACTURE OF TENSILE SPECIMEN 2 WITH
do/d = 50; s/d = 1..87; R.A. = 84%
-
- 21
FIG. 6 - FRACTURE OF TENSILE SPECIMEN 3 WITH
d0/d = 50; s/d = 1.34; R.A. = 64%
FIG. 7 - FRACTURE OF TENSIIE SPECIMEN 4 WITh
dO/d = 50; s/d = 1.19; R.A. = 3qo
-
- 22
FIG. 8 - FRACTURE OF TENSIIE SPECIMEN 5 WITH
d /d = 50; s/d = 1.15; R.A. = 36%
LA~
FIG. 9 - FRACTURE OF TENSILE SPECIMEN 10
WITH d /d = 250; s/d = 1.84; R.A. = 84%
FIG. 10 - FRACTURE OF TENSILE SPECIMEN 11
WITH d /d = 250; s/d = 1.84; R.A. = 96%.
NOTE TfE AXIAL HOLE IN THE CENTER OF THE
SPECIMEN.
24
-
-
FIG. 11 - TOTAL VIEW OF TENSILE SPECIMEN 10.
FIG. 12 - VIEW OF TENSIIE SPECIMEN 10 SHOWING CUPPED
FRACTURE SURFACE ON RIGHT. SURFACE ON LEFT IS
RELATIVELY FLAT.
25
-
-
FIG. 13 - SECTION THROUGH TENSIIE SPECIMEN 10
SHOWING LIP AROUND EDGE OF FRACTURE SURFACE ON
LEFT. FRACTURE, IT IS CONCUDED, STARTED AT THE
CENTER OF THE SPECIMEN.
-
26-
NOTCH SENSITIVITY TESTS
Tests were made on singly-grooved, doubly-grooved, and
asymmetrically-grooved specimens of two sizes as shown in TABLE 3.
Requirements on shoulder widths and specimen depths as presented
by McClintock (1961) have been complied vith.
FD
,
The deformation factor,
is defined as the ratio of the actual deformation to the standard
deformation for ideally ductile specimens as given by McClintock (1961).
The deformation factors for the various specimens are presented in
TABLE 4, first on the basis of the elongation to initial cracking,
and, second on the basis of elongation to separation at the center
of the specimen.
In general, our deformation factors are high compared to
those observed in notched aluminum specimens by McClintock (1961).
This may be caused by the creep and strain-hardening of plasticine
as observed by Green (1951a).
The data shows that, based on deformation factors at
initial cracking, the singly-grooved specimens are most notch
sensitive, the asymmetrically-grooved specimens are intermediate,
and the doubly-grooved specimens are least notch sensitive.
Based
on deformation factors at separation, the singly-grooved specimens are
again the most notch sensitive, the doubly-grooved specimens are
now intermediate, and the asymmetrically-grooved specimens are least
notch sensitive.
The reason for this inversion is that, although the
-27-
asymmetrically-grooved specimens start cracking at lower elongations
than the doubly-grooved specimens, they separate with a slight
shear lip and therefore require greater elongations to separation
than do the doubly-grooved specimens which crack straight across
between the notch roots.
In the aluminum specimens tested by McClintock (1961),
based on deformation factors, the asymmetrically-grooved specimens
are most notch sensitive, the singly-grooved specimens are intermediate, and the doubly-grooved specimens are least notch sensitive.
The greater notch sensitivity of the singly-grooved specimens in our
tests is &ttributed to the fact that the ligament width in the singlygrooved specimens is more than twice as large as the ligament width
in the doubly-grooved and asymmetrically-grooved specimens.
The
size effect, described by McClintock (1961) in aluminum specimens and
exhibited by our data of deformation factors at cracking in the small
and large sizes of specimen 3, predicts this increased notch sensitivity
with increased size.
The deformation factors at separation in the small and large
sizes of specimen 3 are not consistent with this size effect.
The
large specimens were carved from a large block of plasticine which
was very difficult to work.
The fracture surfaces are predominated
by tearing and delamination as shown in FIG 14.
the elongations to separation.
and this effected
This tearing and delamination is
further investigated under HISTORY EFFECTS.
28
-
-
Except for the large sizes of specimen 3 in which the inconsistency is again attributed to the tearing and delamination described
in the previous paragraph, the data for the doubly-grooved specimens
shows that cracking and separation occur at practically the same
elongation.
This is consistent with the observations made by Neimark
(1959) of very rapid crack growth starting in the center of aluminum
specimens.
The data for the asymmetrically-grooved specimens shows
that cracking started in the small notch root where it was first
observed in our tests.
This is caused by the high amount of
localized strain at this point as described by McClintock (1961).
The decrease in ductility of the notched specimens with
increasing inclusion content at a fixed inclusion diameter is revealed by comparing corresponding notch configurations of specimen 3
and specimen 4.
This is analogous to the decreased reduction of area
with increased inclusion content observed in all the uniaxial
ductility tests and, as previously stated, has been observed in metals.
Specimens 1, 3 and 4 were tested when fresh and were again
tested when they were ten days old.
As in the uniaxial ductility tests,
the specimens, except for pure plasticine which remains unchanged,
increase in ductility as they age.
Specimen 3, very notch sensitive
when fresh, became as ductile as pure plasticine when it was ten days
old and the notched specimens made of it failed by rupture rather than
fracture as shown in FIGS. 15 and 16.
-
- 29
TABLE 3 - NOTCH CONFIGURATIONS
SINGLY- GROOVED SPECIMENS
P
Small
Specimen
t
as
Large
Specimen
as
an
.40
1.20
.20
.60
t
1.00
3.00
p
,001
.001
W
300
30*
Standard Deformation, D = an
Radius, P
TP
DOUBLY-GROOVED SPECIMENS
Small
Large
Specimen
Specimen
as
.26
.04
.78
.12
an
t
1.00
3.00
.001
.001
P
30*
300
w
Standard Deformation, D = an
2as
20n
w
Root
P
Radius, P
P
ASYMMETRICALLY-GROOVED SPECIMENS
Small
P
Specimen
.60
.12
an
3.60
t
.001
.001
P
300
300
WI
60*
600
W2
Standard Deformation, D =an
as
W
W2
Root
RadiuslP
IP
.20
.04
1.20
Large
Specimen
TABiE 4 - NOTCH SENSITIVITY DATA
DEFORMATION FACTOR,
FD
The first entry is based on the elongation to initial
cracking and the second entry is based on the elongation to separation at the center of the specimen.
When a range of values is indicated,
three specimens were tested.
SPECIMEN NUMBER
1
2
INCLUSION DIAMETER, d (in.)
INCLUSION SPACING RATIO,
SINGLY
GROOVED
DOUBLY
GROOVED
ASYMMETRICALLY
GROOVED
Small specimen,
width
ligament
a
.20"
n
1.84
.20
.65
1.34
.10
.30
Specimen ten
days old
rupture
1.50 - 1.60
.25 .30
-95 - 1.00
Specimen
fresh
rupture
2.50 - 3.00
2 an
ligament width,
=08"
Specimen ten
days old
rupture
2.50 - 3.00
Large specimen,
2a = .24"
Specimen
fresh
n
.006
1.34
Small specimen,
Large specimen,
2a = .24"
.006
plasticine
rupture
1.50 - 1.6o
Specimen
fresh
I
1.00 - 1.25
1.00 - 1.25
1.00
rupture
2.50 - 3.00
.75 - 1.00
1.00 - 1.25
.33-
rupture
.75
3.00 - 3.25
2.00
t
4
rupture
3.00 - 3.25
.50 -
.42
1.25
.25
.1
rupture
1.25 - 1.50
I
.75
4
.75
0
1.25
1.75
1.00
I
Specimen
fresh
.25 .30
.95 - 1.00
rupture
1.50 - 1-55
.08 - .10
.83 - 1.50
Specimen
fresh
Specimen ten
days old
4
.010
Specimen
fresh
Large specimen,
an = .60"
Small specimen,
ligament width,
2a = .08"
n
3
3.00 - 3.25
I
I
.25 - .33
1.25 - 1.50
.75
.50 - .50
1.25 - 1.50
-
- 31
IN THE LARGE
,
FIG. 14 - TEARING AND DELAMINATIO1
FRESH, SINGLY-GROOVED SPECIlEN 3 WITH a
s/d = 1.84.
/d = 100;
YO~
FIG. 15 - FRACTURE IN THE SMALL, FRESH,
ASYMMETRICALLY-GROOVED SPECIMEN 3 WITE
2a /d = 13; s/d = 1.84.
FIG. 16 - RUPLURE IN TEE SMALL, TEN DAY OLD,
ASYMMETRICALLY-GROOVED SPECIMEN 3 WITE
2a /d = 13; s/l = 1.84.
-
- 33
HISTORY EFFECTS
In the study of ductile fracture caused by the growth and
coalescence of holes, the history of the specimen prior to testing
will have a major effect because the size, shape, and orientation of
the holes prior to a given test is determined largely by the amount
and kind of prestraining.
The effect of simple tension on the
growth of originally circular holes in a bar of plasticine is shown
in FIG. 17.
When this is compared with FIG. 2 of the growth and
coalescence of holes in copper from Puttick (1959), a marked similarity
is noted.
Another history effect is the prior plastic working of the
bar or rod from which the test specimen is taken.
There was consider-
able difficulty in properly working the center of the large specimens
because of the great amount of material used.
FIGS. 18 and 19 show
these large specimens and the orientations of the small, .50"
diameter, tensile specimens which were used to test the anisotropy of
the material which resulted from this poor working.
The axial and radial specimens from the large round rod
are shown in FIG. 20.
The axial specimen shows the same fracture and
ductility as the same specimen from the uniaxial ductility tests
with a reduction of area of about 95% whereas the radial specimen shows
a fracture with considerable delamination and tearing with a reduction
of area of about 65%.
The cause of this phenomenon is that cracks
which could not be worked out of the center of the large specimens were
-34-
oriented parallel to the axis of the large specimen and, therefore,
cut across the face of the radial specimen.
Similar results were
obtained for the longitudinal, transverse, and lateral specimens
which were cut from the large block of material and they are shown in
A great deal of tearing and delamination results in the
FIG. 21.
transverse and lateral specimens because of the direction of forming
and this explains the considerable delaminations observed in the notch
sensitivity tests with large specimens.
Anisotropy
is a common phen-
omenon in the testing of metals and is described to some extent by
Honeycombe (1959).
Backofen, Shaler, and Hundy (1954) performed
tensile tests
on copper specimens with a more complicated prestraining.
The
specimens were twisted to certain values of surface shear strain and
then pulled to fracture in a uniaxial tensile test and the fractures
they obtained are shown in FIG. 22.
Numbers beneath the specimens
indicate the amount of surface shear strain and the last specimen
shows the tensile fracture after twisting and then completely untwisting before tension was applied.
The
specimens obtained from
the same tests on plasticine models with specimen diameter ratios of
50 and inclusion spacing ratios of 1.34 are shown in FIG. 23.
fractures obtained with copper and with plasticine are almost
identical.
The
-
- 35
1
I
I I I I -1
S
I
hi
1III I
I
I
6
FIG. 17 - GROWTH AND COALESCENCE OF ORIGINALLY
CIRCULAR HOLES IN A BAR OF PURE PLASTICINE
UNDER SIMPLE TENSION IN THE VERTICAL DIRECTION.
COMPARE WITH FIG. 2 OF HOLES GROWING IN COPPER.
V
-
- 36
4.0"
2.0"
Radial
FIG. 18 - ORIENTATION OF TENSILE SPECIMENS FOR
ANISOTROPY TESTS ON A ROUND ROD
4.0"
2.0"
Lat eral
Longitudinal
Transverse
3.0
FIG. 19 - ORIENTATION OF TENSILE SPECIMENS FOR
ANISOTROPY TESTS ON A RECTANGULAR BLOCK
-
- 37
AXIAL SPECIMEN
R.A. = 95/
FIG. 20
-
RADIAL SPECIMEN
R.A. = 650/
ANISOTROPY IN A LARGE ROLLED ROD.
6-.
IIIlly,"-w
LONGITUDINAL
TRANSVERSE
SPECIMEN
SPECIMEN
FIG. 21 - ANISOTROPY IN A LARGE BLOCK.
I NMI
LATERAL
SPECIMEN
-
- 38
0
2
I
4
3
FIG. 22 - TENSILE FRACTURES OBTAINED BY BACKOFEN, SHALER, AND HUNDY
(1954) IN COPPER AFTER PRESTRAINING BY TWISTING TO THE VALUE OF
SURFACE SHEAR STRAIN INDICATED BENEATH THE SPECIMEN.
I
I
JI
1
1
1
1
9
(1.6)
(2.6)
(3-T)
(4.2)
(
3.1)
FIG. 23 - TENSILE FRACTURES IN PLASTICINE WITH d /d = 50; s/d = 1.34
AFTER PRESTRAINING BY TWISTING TO THE VALUE OF SURfACE SHEAR STRAIN
INDICATED BENEATH THE SPECIMEN. COMPARE WITH FIG. 22 FROM COPPER
SPECIMENS.
39
-
-
CONCLUSIONS
Two holes in an ideal plastic material under simple
tension will coalesce when the ratio of inclusion spacing to
inclusion diameter, s/d, is within an order of magnitude of
e(2.72).
As regards fracture phenomena:
1)
The ductility of smooth and notched specimens of
plasticine has decreased as the inclusion concentration is increased.
Reductions of area of
2)
36% have been achieved.
The ductility of smooth and notched specimens has
decreased with increasing specimen size.
of area decreased from
In one case, the reduction
93% to 84% when the specimen diameter increased
from .50" to 1.50".
3)
Anisotropy is revealed in large rolled rods and large
rectangular blocks of plasticine.
This has been attributed to the
difficulty of adequately working the center of these specimens.
The
effect of pre-torsion on tensile fracture of the plasticine specimens
shows a marked similarity to those obtained with copper specimens.
4)
Relieving the triaxial stress at the axis in the neck
of a plasticine tensile specimen has resulted in considerable increases
in ductility.
5)
The cup-cone fracture was not achieved with plasticine
models but one fracture surface, in many tests, did contain a shear
lip whereas the other surface was relatively flat.
Evidence is given
that the bond between the plasticine and the polystyrene spheres
does break and that voids are opened up at these inclusions.
-
- 4o
All of these fracture phenomena have been observed in
metals.
The fact that plasticine is a non-metallic, non-crystalline
material strongly suggests that ductile fracture is more a
question of mechanics than of metal physics and that more intense
study of problems, such as the coalescence of holes in a plastically
deforming body, is urgently needed for its understanding.
1l
-
-
APPENDIX: INVESTIGATION OF THE GROWTE AND COALESCENCE OF HOLES USING
LIMIT ANALYSIS
A complete description of the technique to be used here is
contained in
Prager (1959) and the specimen is shown with dimensions
in FIG. 24.
A simple lower bound satisfying equilibrium is seen from
FIG. 24 to be:
Plower = Y(c-2d)t
Two deformation modes are postulated to obtain upper bounds.
In Deformation Mode #1 (see FIG. 25) the holes grow independently by
successive shearing through them.
The upper bound is obtained by
equating external work to internal dissipation:
external work = internal dissipation
P
= . AV- (c -d) 4-2 - t
P2
lupper = Y(c-d)t
In Deformation Mode
#2
(see FIG. 26) the holes grow and
coalesce by the plastic flow of the material out of the space between
the holes.
This phenomenon was also studied in notched bars with
semicircular roots by Wang (1953) and by Garr, Lee, and Wang (1956).
The deformation is pictured in two steps as also shown in FIG. 26.
In step 1, the center ligament is assumed to be parted and as the
specimen elongates under the external load the wedge shaped sections
move in toward the center as shown by the arrows.
External work is
done by the force P and energy is dissipated on the shear surfaces.
-
- 42
In step 2, the stress distribution a
does work to close the gap in
the center ligament and this is equal to the energy dissipated in the
center ligament, the region of the logarithmic spiral slip lines.
Again the upper bound is obtained by equating the external work to the
energy dissipated:
external work = energy dissipated
P
S
=
+
F2.2a2-t
energy dissipated in
ligament between holes
Step 2 of the deformation shows that:
energy dissipated in ligament between holes
t
=
(V)dx
Cabining these gives:
PE
=-
2-2a2-t
+
t
(-)dx
Hill (1950) has shown that in the region of the logarithmic
is given by:
=
2k [,+ln(l+
where
]
r is the radius of the root =
d
,
r
spirals
x is the distance measured from the root, and
2k = Y
Substituting this expression for
OT
,
integrating, and
simplifying yields:
ppr=
P2,upper
Yt
c-d-s+s.ln(S)
d)
Comparing the two upper bound yield point loads we have:
P2,upper
and
P2
,.
Plupper
= lupper
,upper
yt
Esds.1n(jiI
when
ln( .) = 1 ;
d
T=
e
=2.72
43
-
-
Zero
Stress
Uniform
Stress
P
111114 1111111
d -c
b
d
s=b+d
.*a
age-
t
c
I I
-
FIG. 24
A SIMPLE LOWER BOUND SATISFYING
EQUILIBRIUM
44
-
-
P
Shear
Surface
0
0
9
K
FP
FIG. 25 - DEFORMATION
MODE NO. I
P
Shear
Surface
Logarithmic
Spirals
+
N
It
p
I I I
I
''II
1~4ipo~
Step I
FIG. 26 - DEFORMATION
4;
MODE NO. 2
Step 2
- 45--
REFERENCES
Backofen, W.A.,
1954
"Mechanical Anisotropy in
Copper", Transactions of the
American Society for Metals,
Vol. 46, 1954, pages 655 - 675.
1953
"On the Complete Solution to
Problems of Deformation of a
Shaler, A. J., and
Hundy, B.B.
Bishop, J.F.W.
Plastic-Rigid Material", Journal
of the Mechanics and Physics
of Solids, Vol. 2, 1953-54,
pages 43-53.
Cottrell, A.H.
1959
"Theoretical Aspects of Fracture"
Fracture - Proceedings of an
international conference on the
atomic mechanisms of fracture
held in Swampscott, Mass., April
12-16, 1959, pages 20-44.
1959
Plateau, J., Tainhankax R.,
Henry, G., and Lajeunesse, D.
"A Comparison of Ductile and
Fatigue Fractures", Fracture
Proceedings of an international
conference on the atomic
mechanisms of fracture held in
Swampscott, Mass., April 12-16,
-
Crussard, C.,
1959, pages 524-558.
Garr, L., Lee, E.H. and
1956
Wang, A.J,
"The Pattern of Plastic
Deformation in a Deeply Notched
Bar with Semicircular Roots",
Journal of Applied Mechanics,
Vol. 23, 1956, pages 56-58.
Green, A.P.
1951a
"The Use of Plasticine Models to
Simulate the Plastic Flow of
Metals", Philosophical Magazine,
Ser. 7, Vol. 42, 1951, pages
365-373.
Green, A.P.
1951b
"A Theoretical Investigation of
the Compression of a Ductile
Material between Smoota Fiab uies",
Philosophical Magazine, Ser. 7,
Vol. 42, 1951, pages 900-918.
Green, A.P.
1954
"The Plastic Yielding of etal
Junctions due to Combined Shear
and Pressure", Journal of the
Mechanics and Physics of Solids,
Vol. 2, 1953-54, pages 197-211.
-46-
Green, A.P.
1955
"On Unsymmetrical Extrusion in
Plane Strain", Journal of the
Mechanics and Physics of Solids,
Vol. 3, 1954-55, pages 189-196.
Hill, R.
1950
The Mathematical Theory of
Plasticity,
The Clarendon Press, Oxford,
1950, page 248.
Honeycombe, R.W.K.
1959
"Sumkary of Current Status and
Needs for future Research: Session
on Fatigue and Ductile Fracture",
Fracture - Proceedings of an
international conference on the
atomic mechanisms of fracture held
in Swampscott, Mass., April 12-16,
1959, pages 9-14.
Landberg, P.
1958
"Impact Extrusion", Microtechnic,
Vol. 12, No. 6, 1958, pages 301-312.
MacGregor, C.W.
and Grossman, N.
1952
"The Effects of Cyclic Loading
on the Mechanical Behavior of
24S-T4 and 755-T6 Aluminum Alloys
and SAE 4130 Steel", National
Advisory Committee for Aeronautics, Technical Note 2812,
Washington, October 1952.
McClintock F.
1961
"On Notch Sensitivity",
Welding Journal Research Supp-
lement, Vol. 26, No- 5, May 1961,
pages 202S - 209S.
Mortimer, F.
1952
"Model Forging Experiments: The
Uses of Plasticine", Iron and
Steel, Vol. 25, 1952, pages
433-436.
Neimark, J.E.
1959
"The Initiation of Ductile
Fracture in Tension", ScD
Thesis, Mechanical Engineering
Department, Massachusetts Institute of Technology, 1959.
Prager, W.
1959
An Introduction to Plasticity,
Addison-Wesley Publishing Company,
Inc., Massachusetts, 1959, Chap.3,
pages 35 - 44.
-
- 47
"Ductile Fracture in Metals"
Philosophical Magazine, Ser. 8
,
1959
,
Puttick, K.E.
Vol. 4, 1959, pages 964-969.
Rosi, F.D. and
1960
Abrahams, M.S.
"Porosity in Plastically
Deformed Single Crystals",
Letter to the Editor, Acta
etallurgica, Vol. 8, No. 11,
Nov. 1960, pages 807-808.
Society of Automotive
Engineers
1959
Society of Automotive Engineers Handbook, 1959, page 122.
Tipper, C.F.
1949
"The Fracture of etals",
etallurgia, Vol. 39, 1948-49,
pages 133-137.
Uzhik, G.V.
1948
"Resistaice to Brittle Rupture
and Strength of Metals", U.S.S.R.
Wang, A.J.
1953
"Plastic Flow in a Deeply
Notched Bar with a Semicircular
Root", Quarterly of Applied
Mathematics, Vol. 11, 1953-54,
pages 427-438.