Ductile fracture by the growth of pores

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 FULFIILLMENT

THE REQUIREMENTS FOR THE DEGREE

OF OF MASTER OF SCIENCE IN

MECHANICAL ENGINEERING at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

July, 1961

Signature Redacted

Signature of the Author '...)

Department '6f Mechanical Enaineerinx, July 5, 1961

Signature Redacted

Signature Redacted

,

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.

3

-ACKNOWLEDGENT

I wish to express my sincere thanks to

"Bill" Henry for his experienced advice in the pre-paration 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.

3

4

5

GROWTH AND COALESCENCE OF HOLES *. .. .

8

PIASTICINE MODELS .. .. .. .. .. 11 SPECIMENS AND INCLUSION CONTENTS .. .. . 13

UNIAXIAL DUCTILITY TESTS .. .. .. .. .

16

HISTORY EFFECTS .. .. .. .. .. .

33

CONCLUSIONS so * -- -. .. .

a

39

APPENDIX:

REFERENCES

INVESTIGATION OF THE GROWTH AND COALESCENCE OF HOLES

USING LIMIT ANALYSIS .. .41

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. This coalescence is well illustrated in FIGS. 1 and 2. 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, each growing outward in all 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

nuc-leated 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 coa-lescence 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

elong-ate 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

-'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 stress-strain 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.

- 12

-To sum up: 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

-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

plast-icine. The two sizes which were available for our use, courtesy of

the Plastics Division of the Koppers Company, Inc., were

.006"

orders of magnitude larger than the particles found in pure plasti-cine. 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 spec-imens. Large ends were left on the specimens to provide for gripping.

The notched specimens were carved from blocks of plasticine

- 15

-TABLE 1 - INCLUSION CONTENTS SOURCE

SAE (1959)

a

INCLUSION SPACING RATIOS within rows s d between rows S2 d average s d I 4 4 4 1 Steel - plate#i Steel - plate#5 Steel - plate#8 7075 Aluminum 2024 Aluminum Plasticine Specimens:

.010" spheres densely packed .010" spheres lightly packed .006" spheres densely packed .006" spheres lightly packed

.006" spheres

.006" spheres

(d

=

.90")

(d

=

1.50")

500

-

1000

500

-

1000

330

-

1000

50

50

83

83

250

7

1.15

1.87

1.34

1.84

1.84

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. This phenomenon of de-creased ductility with inde-creased 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,

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. The effect is seen by comparing FIGS. 9 and 10. 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. Both

fracture surfaces are coarse and grainy. 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 FRESH SPECIMEN ELONGATION, FRACTIONAL REDUCTION OF AREA, FRACTIONAL SPECIMEN ONE DAY OLD ELONGATION, FRACTIONAL REDUCTION OF AREA, FRACTIONAL SPECIMEN TEN DAYS OLD ELONGATION, FRACTIONAL REDUCTION OF AREA, FRACTIONAL 1~t t t I 4 4 L plasticine

1.87

1.34

1.19

1.15

1.84

1.50

1.34

1.84

1.84

1.84

.80

.6o

.50

.30

.30

.65

.50

.40

.70

.85

.95

.84

.64

.36

.36

.93

.73

.53

.87

.70

.75

1

3

4

5

6

7

8

9

.80

.60

.80

.50

(g-)

(2)d

50

50

50

50

83

83

83

150

250

250

50

50

50

50

.50

.50

.50

.50

.90

1.50

1.50

.90

with axial hole)

.84

j

.88

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

FIG. 9 - FRACTURE OF TENSILE SPECIMEN 10

WITH d

/d

1.84;

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

- 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. The deformation factor,

FD ,

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

inter-mediate, 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 singly-grooved 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

-

28

-Except for the large sizes of specimen 3 in which the

incon-sistency 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

re-vealed 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

3

and 4

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

29

-TABLE 3 - NOTCH CONFIGURATIONS

P

t

as

Radius,

_P

TP

2as

20n

w

Root

P

Radius,

P

P

SINGLY- GROOVED

Small

Specimen

as

an

t

p

W

.40

.20

1.00

,001

300

SPECIMENS

Large

Specimen

1.20

.60

3.00

.001

30*

Standard Deformation, D =

an

DOUBLY-GROOVED SPECIMENS

as

an

t

P

w

Standard

Small

Specimen

.26

.04

1.00

.001

300

Large

Specimen

.78

.12

3.00

.001

30*

Deformation, D

=

an

ASYMMETRICALLY-GROOVED SPECIMENS

as

an

t

P

P

Root

RadiuslP

IP

Small

Specimen

.20

.04

1.20

.001

300

600

Large

Specimen

.60

.12

3.60

.001

300

60*

Standard Deformation, D

=an

TABiE 4 - NOTCH SENSITIVITY DATA

When a range of values is indicated, three specimens were tested.

DEFORMATION FACTOR, FD

The first entry is based on the elongation to initial cracking and the second entry is based on the elong-ation to separelong-ation at the center of the specimen.

SPECIMEN NUMBER 1 2 3 4

INCLUSION DIAMETER, d (in.) .010 .006 .006

INCLUSION SPACING RATIO, plasticine 1.34 1.84 1.34

Specimen rupture .20 .10

Small specimen, fresh 1.50 - 1.6o .65 .30

ligament width

SINGLY a .20" Specimen ten rupture .25 - .30 rupture .25 - .30

GROOVED n days old 1.50 - 1.60 -95 - 1.00 1.50 - 1-55

.95

Large specimen, Specimen .08 - .10

an = .60" fresh _{.83 - 1.50}

Specimen rupture _{1.25 .75}

Small specimen, fresh 2.50 - 3.00 1.75 1.00

DOUBLY ligament width, Specimen ten rupture 1.00 - 1.25 rupture .75 - 1.00 GROOVED 2an =08" days old 2.50 - 3.00 1.00 - 1.25 2.50 - 3.00 1.00 - 1.25

Large specimen, Specimen _.33- .42

2a = .24" fresh _{1.00 - 1.25} ASYMMETRICALLY GROOVED Small specimen, ligament width, 2a = .08" n Large 2a = n specimen, .24" Specimen fresh Specimen days old ten rupture

3.00

-

3.25

.75

3.00

-

3.25

.50

-

.75

1.25

-

1.50

3.00

-

3.25

_1.25

.25 -

_-

_1.50

.33

.75

.50 -

.50

1.25

-

1.50

31

-FIG. 14 - TEARING AND DELAMINATIO1 _{IN THE LARGE ,} FRESH, SINGLY-GROOVED SPECIlEN 3 WITH a

/d

FIG. 15 - _{FRACTURE IN THE SMALL, FRESH,}

ASYMMETRICALLY-GROOVED SPECIMEN 3 WITE

2a /d

=

13; s/d

=

1.84.

FIG.

16

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 FIG. 21. A great deal of tearing and delamination results in the

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 un-twisting 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. The fractures obtained with copper and with plasticine are almost identical.

35

-1

I

S

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

4.0"

2.0"

Radial

FIG. 18 - ORIENTATION OF TENSILE SPECIMENS FOR

ANISOTROPY TESTS ON A ROUND ROD

Lat

4.0"

eral

3.0

Longitudinal

Transverse

2.0"

FIG. 19 - ORIENTATION OF TENSILE SPECIMENS FOR

ANISOTROPY TESTS ON A RECTANGULAR BLOCK

37

-AXIAL SPECIMEN RADIAL SPECIMEN R.A. =

95/

650/

FIG. 20 - ANISOTROPY IN A LARGE ROLLED ROD.

LONGITUDINAL

SPECIMEN

TRANSVERSE

SPECIMEN

FIG. 21 - ANISOTROPY IN A LARGE BLOCK.

LATERAL

SPECIMEN 6-.

-

38

-2

0 I

3 4

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

INDICATED BENEATH THE SPECIMEN. COMPARE WITH FIG. 22 FROM COPPER

-

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

36%

2) The ductility of smooth and notched specimens has decreased with increasing specimen size. In one case, the reduction of area decreased from

93%

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.

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.

-- 1

l

-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

- 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

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 spirals

r

= 2k [,+ln(l+ ] where

d r is the radius of the root = ,

x is the distance measured from the root, and 2k = Y

Substituting this expression for OT , integrating, and simplifying yields:

ppr= Yt c-d-s+s.ln(S)

P2,upper d)

Comparing the two upper bound yield point loads we have:

P2,upper Plupper yt

Esds.1n(jiI

and P2 = lupper when ln( .) = 1 ;

,. ,upper d

-

43

-Zero

P

Stress

Uniform

Stress

1111111

d

-c

b

d

s=b+d

.*a

I I

t

FIG. 24 -

A SIMPLE LOWER BOUND SATISFYING

EQUILIBRIUM

111114

age-- 44

-P

Shear

Surface

0

FP

FIG. 25 - DEFORMATION

K

MODE NO.

P

Logarithmic

Spirals

N

Shear

Surface

It

p

Step I

4;

+

''II

1~4ipo~

Step 2

FIG. 26 - DEFORMATION MODE NO. 2

9

0

Backofen, W.A., Shaler, A. J., and Hundy, B.B. Bishop, J.F.W. Cottrell, A.H. REFERENCES

1954

1953

1959

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Green, A.P. 1951b

Green, A.P. 1954

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1959

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1950, page 248.

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₁₉₅₉

1959

1949

1948

1953

"Ductile Fracture in Metals" ,

Philosophical Magazine, Ser.8,

Vol.

4,

1959, pages 964-969.

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