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Strain aging in nickel 200

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Title:
Strain aging in nickel 200
Creator:
Cribb, Walter Raymond, 1949-
Publication Date:
Copyright Date:
1975
Language:
English
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xii, 144 leaves : ill. ; 28 cm.

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Subjects / Keywords:
Activation energy ( jstor )
Alloys ( jstor )
Atoms ( jstor )
Carbon ( jstor )
Impurities ( jstor )
Nickel ( jstor )
Precipitation hardening ( jstor )
Strain rate ( jstor )
Temperature dependence ( jstor )
Work hardening ( jstor )
Crystals -- Defects ( lcsh )
Dislocations in metals ( lcsh )
Dissertations, Academic -- Materials Science and Engineering -- UF
Materials Science and Engineering thesis Ph. D
Nickel ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 135-143.
Additional Physical Form:
Also available on World Wide Web
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Walter Raymond Cribb.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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02943131 ( OCLC )
AAT7431 ( NOTIS )

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STRAIN AGING IN NICKEL 200


By

WALTER RAYMOND CRIBB


















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA

1975








































UNIVERSITY OF FLORIDA
S1 ilI II II I II III i lH 1 I8 11I
3 1262 08552 4618

































To Mom and Dad












ACKNOWLEDGMENTS


Sincere appreciation is due many people in this department for

their help during my entire stay at the University of Florida. Most

sincere thanks are due Professor Robert E. Reed-Hill whose continued

guidance and encouragement made this dissertation possible.

Many thanks to Professor F.N. Rhines who first encouraged me and

gave me confidence to strive for a higher degree in metallurgy and

whose continued interest in my program is appreciated.

I would also like to thank the members of my committee,

Drs. Martin A. Eisenberg, Craig S. Hartley and John J. Hren for fruitful

discussions of my work.

Many thanks to my colleagues Messrs. Juan R. Donoso, R.M. Chhatre,

Francisco Boratto and to the laboratory assistants C. Barnes and

M. Brimanson who spent many hours of discussion and who cooperated in

the collection and interpretation of experimental data. The preparation

of the final manuscript by Elizabeth Seville is also greatly appreciated.

The financial support of the Army Research Office (Durham), the

International Nickel Company, and the Energy Research and Development

Administration is greatly appreciated.

Finally, I thank my wife, Kathie, whose patience, encouragement

and understanding during my course of study helped make it all possible.











TABLE OF CONTENTS

PAGE

ACKNOWLEDGMENTS ................................................... iii

LIST OF TABLES .................................................... vii

LIST OF FIGURES ............... .............................. viii

ABSTRACT .............................................. .......... xi

INTRODUCTION ...................................................... 1

CHAPTER

I PREVIOUS INVESTIGATIONS ................................... 4

1.1 Static Strain Aging .................................. 4

1.1.1 Historical Aspects ............................ 4

1.1.2 Mechanisms of Static Strain Aging in Metals
Alloys ........................................ 5

1.1.3 Summary of Important Mechanisms of Dislocation
Locking During Aging .......................... 10

1.1.4 Aspects of the Static Strain Aging Experiment.. 11

1.1.5 Static Strain Aging Stages in BCC Metals ...... 13

1.1.6 Static Strain Aging in Nickel and FCC Alloys... 17

1.2 Dynamic Strain Aging ................................. 19

1.3 Work Hardening in Metals and Alloys .................. 23

1.4 Anelastic Phenomena in Nickel and FCC Ferrous Alloys.. 27

II EXPERIMENTAL PROCEDURES ................................... 35

2.1 Materials ............................................ 35

2.2 Experimental Techniques .............................. 37








PAGE

2.2.1 Swaging ................... .................... 37

2.2.2 Annealing .................................... 37

2.2.3 Specimen Profile Measurements ................ 37

2.2.4 Tensile Testing .............................. 38

2.2.5 Static Aging Experiments ...................... 38

III EXPERIMENTAL RESULTS .................... .................. 43

3.1 The Behavior of the Lower Yield Stress Increase, Ao... 43

3.2 The LUders Extension, EL ............................. 50

3.3 The Hardening Component, AoH ....................... 50

3.4 Activation Energies .................................. 50

3.5 The Dependence of Ao and EL on Prestrain ............ 53

3.6 Comparison of Nickel 270 and Nickel 200 Static
Strain Aging ....................................... 55

3.7 The Stress-Strain Behaviors .......................... 57

3.8 The Work Hardening Behaviors of Nickel 270 and of
Nickel 200 ........................................... 69

IV DISCUSSION ..................... .......................... 80

4.1 Rationale for Static Strain Aging in Nickel 200 ...... 80

4.2 The Mechanism for Static Strain Aging Exhibited in
Nickel 200 .......................................... 82

4.2.1 The Distribution of Vacancies, Carbon Atoms
and Dislocations After Plastic Deformation .... 82

4.2.2 Vacancy Trapping by Carbon Atoms .............. 82

4.2.3 The Concentration of Carbon-Vacancy Pairs ..... 83

4.2.4 Theory of Initial Schoeck Locking by Carbon-
Vacancy Pairs ..................... ........ 85

4.2.5 The Mechanism Controlling the Increase in Ao
with Time ....................... ............. 96

4.2.6 Regarding the Behavior of Nickel 200 After
the Peak in Ao ...................... ...... 107







4.3 Summary .............................................. 'l

4.4 Comments on the Relationship Between Static Strain
Aging and Dynamic Strain Aging in Nickel 200 ......... 115

CONCLUSIONS AND OBSERVATIONS ...................................... 121

APPENDICES

A COMPUTATION OF THE INTERACTION ENERGY BETWEEN A
CARBON-VACANCY PAIR DEFECT AND SCREW DISLOCATIONS IN
FCC METALS ...................................... ... ..... 125

B DETERMINATION OF U(x), THE ENERGY OF A SCREW DISLOCATION
DISPLACED A DISTANCE x FROM THE CENTER OF ITS SNOEK
ATMOSPHERE ............................................... 132

BIBLIOGRAPHY ............................ ......................... 135

BIOGRAPHICAL SKETCH ............................................... 144












LIST OF TABLES

Table Page

1 Recognized Aspects of Strain Aging ..................... 2

2 The Diffusivity of Carbon in Nickel .................... 21

3 Estimates of the Rate of Vacancy Production During
Plastic Deformation .................................... 32

4 Alloy Compositions ..................................... 36

5 Least Squares Parameters for Static Strain Aging Data
Assuming Ao Is a Function of In t ...................... 45

6 Least Squares Parameters for Static Strain Aging Data
Assuming a Log Ao-Log t Linear Relationship ........... 49

7 The Slopes and Intercepts (at t = Is) of Ao Versus In t
Curves Calculated from Eq. 56 ......................... 108

8 Interaction Energies for Tetragonal Defects in the
FCC Structure ........................................ 128











LIST OF FIGURES

Figure Page

1 General aspects of the classical static strain
aging test .................... ...... ............... 12

2 An example of the stages of the yield return in Nb-O
alloys [8]; (a) the increase in Ao with time; (b) the
components of AM and their dependence on aging time..... 16

3 Schematic example of stage behavior in polycrystalline
fcc metals ............................................. 25

4 Schematic description of the work hardening behavior in
a metal using a log 9-log a plot ....................... 25

5 Illustrating the method used to determine the aging
parameters EL, Aa= Ly o and AcH = oExt oo. Dashed
loading line indicates the approximate loading line
which would have been observed in the absence of
misalignment of the test specimen .................... 41

6 Selected load-time curves obtained after restraining
a series of specimens 5% at 273"K, aging at 4080K for
the times indicated, and restraining at 2730K ......... 42

7 The time and temperature dependence of the return of
the lower yield stress in Nickel 200. Specimens were
prestrained to a stress level of 265 MPa. The dashed
curves are approximate corrected curves which account
for specimen heat-up in the aging baths ............... 44

8 Normalized aging curves for Nickel 200. The 3730K
curve was normalized to an assumed maximum of 28.5 MPa.
The dashed curves reflect approximate corrections for
the heat-up time of the specimens .................... 47

9 Illustrating the approximate t1/7 power law relation
governing the aging of Nickel 200 at temperatures
below 4480K. Data for the 448K (shown) and 473K
cases do not fit this relation well ................... 48

10 The dependence of the Luders strain on time and
temperature in Nickel 200 ............................ 51







Figure Page

11 The approximate behavior of the secondary hardening
component of the lower yield stress increase
(AcH = "Ext "o) ................. ...................... 52
12 The dependence of Ao and EL on prestrain. Nickel 200
specimens were prestrained at 2730K, aged for
6000 seconds at 4480K, and restrained at 2730K ........... 54

13 (a) The yield return of a Nickel 270 specimen aged for
a time to achieve a maximum in Aofor Nickel 200.
(b) Yield return for a Nickel 200 specimen aged only
one-half as long ......................................... 56

14 True stress-true plastic strain curves for Nickel 270
(- = 4.2 x 10-4 s-l) .................................... 58

15 Variation of the 0.2% yield stress and the ultimate
tensile strength with temperature of Nickel 270
(6 = 4.2 x 10-4 s-1) ............... ..................... 59

16 Variation of the uniform and total elongation with
temperature in Nickel 270 (- = 4.2 x 10-4 s-1) .......... 60

17 True stress-true plastic strain curves for Nickel 200
( = 4.2 x 10-4 s-1) ................. ... ............ 62

18 The temperature dependence of the 0.2% yield stress and
ultimate tensile strength of Nickel 200 ................. 63

19 The temperature and strain rate dependence of the 0.2%
yield stress in Nickel 200 on an expanded stress axis .... 64

20 Variation of the stresses at 5, 11, 19 and 30% plastic
strain with temperature in Nickel 200 .................. 66

21 Variation of the uniform and total elongations with
temperature in Nickel 200. Also shown are the
approximate temperature ranges over which serrations
were observed at the respective strain rates.............. 67

22 Variation of reduction in area with temperature for
Nickel 200 and Nickel 270 ................... ............. 68

23 The log O-log o curves of Nickel 270 (f = 4.2 x 10-4 s ). 70

24 The log e-log a curves of Nickel 200 (C = 4.2 x 10-4 s-1) 71

25 The variation of mil and mIII with temperature in
Nickel 270 (e = 4.2 x 10-4 -1) ....................... 73






Figure Page

26 The variation of mil and mIII with temperature in
Nickel 200 (E = 4.2 x 10-4 s-1) ......................... 74

27 The variation of the work hardening parameter (05%-c0.5%)
with temperature (C = 4.2 x 104 s )................. 76

28 The variation of e2 and e3 (the approximate strains at
which Stages II and III, respectively, begin) with
temperature for Nickel 270 and Nickel 200 deformed at
-l -
a strain rate of 4.2 x 10- s ......................... 78

29 The dependence of E3 on temperature and strain rate in
Nickel 200 .............................................. 79

30 A schematic illustration of the assumed configuration
of the carbon-vacancy pair and the three possible
independent orientations that it may assume ............. 87

31 The concentrations of dipoles in each of the three
possible orientations (B' = 0.3 eV (6900 kcal/mole),
A = 0.2 eV (4600 kcal/mole)) ............................ 91

32 Schematic illustration of the growth of a saturated
carbon-vacancy atmosphere. Rs is time dependent and the
concentration within R is assumed to be a fraction, f,
of the carbon concentration ............. ............ 99

33 The aging curves obtained from the model for strain
aging in Nickel 200 (Eq. 56); the dashed lines are the
experimental data (Figure 7) ..... ..................... 109

34 This diagram illustrates the aging stages of Nickel 200.
The solid line represents the experimental scope of
the present investigation .............................. 116

35 The interaction potential u. of a carbon-vacancy dipole
with a screw dislocation for r = b and A/b = 0.2 eV ..... 131





Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in
Partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy


STRAIN AGING IN NICKEL 200

By

Walter Raymond Cribb

August, 1975


Chairman: Robert E. Reed-Hill
Major Department: Materials Science and Engineering


Dynamic and static strain aging were observed in commercially

available Nickel 200 which contains principally 1000 ppm carbon as an

alloying impurity. Static strain aging tests were conducted on

annealed tensile specimens which were prestrained at 2730K to a stress

level of 265 MPa (approximately 0.05 strain) at a nominal strain rate

of 4.2 x 103 s-1. Under these conditions, homogeneous plastic flow

was guaranteed to occur. Specimens were aged immediately after pre-

straining for different times at 373, 408, 428, 448 or 4730K and the

time dependence of the return of the lower yield stress was observed.

The return of the yield experiments indicated that AB increased as

In t or approximately as t/7 kinetically and behaved in accordance with

an activation energy of 25 kcal/mole before the observed peak in Ao.

It is demonstrated that the defect responsible for this anomalous increase

in Ac may be the rotation of carbon-vacancy pairs in the strain fields

of dislocations. A quantitative model is derived for the increase in

Ao before the aging peak and it is concluded that several important

stages in the aging of Nickel 200 may occur: (a) the formation of





carbon-vacancy pairs and their initial ordering, (b) the migration

of vacancies in the strain energy gradients of dislocations and the

consequent formation of more carbon-vacancy pairs near dislocations,

(c) the growth of an ordered carbon-vacancy dipole atmosphere,

(d) depletion of free vacancies in the remainder of the lattice

which decreases the flux to the ordered atmosphere and results in a

Ac maximum, (e) the migration of bound vacancies to dislocation sinks

and the resulting decrease in Ac, (f) the migration of carbon atoms in

the strain fields of dislocations and the growth of a Cottrell atmosphere,

and (g) precipitation of graphite during averaging. Items (f) and (g)

are only speculated to occur. This model is different from the Cottrell-

Bilby model and can account for the kinetics and activation energy for

strain aging observed in Nickel 200.

Tensile tests were conducted between 77 and 800oK at nominal strain

rates of 4.2 x 10-5, 10-4, 10-3, and 10-2 s-1. The results of these

experiments confirm that dynamic strain aging (DSA) in Nickel 200 is

exhibited over a temperature interval between 273 and 5750K at

4.2 x 10- s Over the DSA interval, the following phenomena were

exhibited and depended upon the strain rate: the Portevin-Le Chatelier

Effect, yield stress plateaus, ultimate stress peaks, reduction in area

minima and mild ductility minima. An analysis of work hardening indicates

that anomalous work hardening over the DSA interval is very weakly

exhibited. The mechanism for discontinuous yielding is rationalized

to be dynamic Snoek ordering of carbon-vacancy pairs during plastic

deformation and can account for the anomalously low temperature interval

(with respect to the expected mobility of carbon) over which DSA is

observed to occur.











INTRODUCTION


Currently, eight aspects of strain aging are recognized [1] as

playing a major role in the deformation of polycrystalline metals

(Table 1). The first two are static strain aging phenomena which

are obtained by restraining a set of prestrained specimens that have

been aged at an elevated temperature. The last six aspects listed in

Table 1 are characteristic of dynamic strain aging, i.e., aging which

occurs during plastic deformation. Dynamic strain aging can occur in

both substitutional and interstitial alloys. The most interesting

cases of dynamic strain aging have normally involved interstitial

solutes in transition metals.

Most research on the role of interstitial impurities in the

mechanical behavior of transition metals has been conducted using the

body-centered cubic class of metals such as Fe, Nb, Mo, Ta, 14 and V [2].

The principal interstitial impurities in these metals which are responsible

for strain aging are N, 0, C and H.

Nickel is the only metal of the commercially important Period IV

transition series of the Periodic Table that is face-centered cubic. It

is also the only fcc transition metal widely used for constructional

purposes. The other fcc transition metals Rh, Pd, Ir, and Pt are less

abundant and have not been used as major construction materials. As such,

in-depth investigations of their mechanical properties have not been

undertaken.




2




TABLE 1

Recognized Aspects of Strain Aging


1. Yield Points

2. Strengthening

3. Discontinuous Yielding

4. Strain Rate Sensitivity Minimum

5. Ductility Minimum

6. Abnormal and Rate Dependent Work Hardening

7. Yield Stress Plateaus

8. Flow Stress Transients on Changes in Strain Rate




3



The principal purpose of the present investigation was to

characterize the strain aging phenomena of commercially available

Nickel 200. This alloy contains as its principal strengthening agent

solid solution interstitial carbon (0.15 w/o maximum). To the best

knowledge of the author, a complete classical static strain aging

investigation has never been conducted using an interstitial solid

solution fcc alloy.

A prime goal in the present investigation was to develop a quanti-

tative model that could explain the kinetics and energetic of the return

of the lower yield stress. Furthermore, the tensile behavior of Nickel 200

during constant strain rate tests conducted over a wide range of

temperatures and strain rates was investigated in order to better define

the dynamic strain aging phenomena in Nickel 200.











CHAPTER I

PREVIOUS INVESTIGATIONS


1.1 Static Strain Aging


1.1.1 Historical Aspects

It has been recognized for a long time that the yield phenomenon

in iron and other bcc metals is closely related to the presence of

interstitial impurity atoms such as carbon or nitrogen. Most strain

aging investigations have centered about the iron and steel industry

since the 1930s when the phenomenon in low carbon steel first became

a major commercial nuisance.

The first metallurgical investigation of aging in mild steel was

conducted by Davenport and Bain [3] in 1935, who noted that heterogeneous

flow occurred in both annealed and deformed materials after having been

"aged" by storing before working. Subsequent work by Gensamer and

Low [4] in 1944 related the strain aging and yield point to the presence

of trace amounts of nitrogen and carbon. Since the time of these early

investigations, much interest has continued to center on iron and

other commercially significant body-centered cubic metals such as

vanadium [5], niobium [6], tantalum [7], and molybdenum [8]. At the

present time, very little research effort has been directed to the study

of static strain aging phenomena in face-centered cubic and hexagonal

metals containing interstitial impurities.




5



1.1.2 Mechanisms of Static Strain Aging in Metals and Alloys

Three main dislocation pinning mechanisms have been postulated

on the basis of experimental evidence in metals and alloys. These are

Cottrell pinning, Suzuki locking and Schoeck locking.

In.all of the previously mentioned investigations of bcc metals,

the most plausible explanation for static strain aging is due to Cottrell

and Bilby [9] who attributed the effect to the diffusion of interstitial

atoms in solution (e.g., carbon, nitrogen, oxygen or hydrogen) to

dislocations. Their concept relates the increase in flow stress and yield

point return after aging to the migration of solute atoms to the

tensile region about an edge dislocation. The effect of this segregation

is to locally lower the strain energy of the system and to consequently

stabilize the dislocation to the point where an increased flow stress

is required to remobilize the dislocation or to generate mobile dislocations.

The Cottrell mechanism is of major importance in causing the return

of the sharp yield point in steel while an increase in the steel's

ability to work harden and a reduction in ductility (in the later stages

of aging) are probably associated with precipitation of carbides and

nitrides [10]. The major contribution of the Cottrell-Bilby work was

to solve the problem related to the diffusion of an interstitial atom

in the stress field of a dislocation. The solution predicts the time-

temperature dependence of the rate of impurity migration as inferred

from internal friction measurements [11]. Cottrell and Bilby derived

the following relationship for n(t), the number of atoms arriving at the

dislocation in the time t per unit length,

n(t) = 3 ()l1/3 no(Tt)2/3 (1)








where no is the average number of solute atoms per unit volume and

the parameter A is the interaction constant which describes the

tendency for a solute atom or center of dilatation to be attracted

by an edge dislocation's hydrostatic stress field, D is the diffusivity

of the solute and k and T have their usual meanings. The principal

characteristics of Cottrell pinning as manifested in the static strain

aging experiment are (a) a t2/3 time dependence of the lower yield

stress return, and (b) an activation energy for the yield return

approximately equal to that for the migration of interstitial solute

atoms. The model assumes long range migration of solute and probably

involves about 103 atoms jumps [12] (or a net rms displacement of 30 to

50b). An empirical result is that the increase in stress necessary

to free a dislocation from its atmosphere as measured by Ao is directly

proportional to the number of atoms, n(t), which have arrived at the

dislocation. Thus, the strain energy decrease associated with long

range impurity migration is directly proportional to n(t). This model

with modifications [13,14] has survived for twenty-five years without

its concepts being significantly altered. Excellent reviews 'of the

Cottrell-Bilby theory are available in many places [15-19].

Suzuki [20] has pointed out that in face-centered cubic metals

containing extended dislocations, a completely different form of

interaction between dislocations and impurity atoms can exist. Since

the stacking fault has a locally different crystal structure from the

matrix, the solid solubility of impurities contained in the matrix can

differ appreciably within the stacking fault and outside. Consequently

a chemical potential exists across the fault, resulting in the

binding of impurity atoms to the stacking fault. Pinning is a result








of the accumulation of solute at the fault. Thus, this type of inter-

action should be characterized by an activation energy due to solute

migration. Unfortunately, while the magnitude of the locking stresses

has been calculated and applied with some success to solid solution

alloys, the kinetics of migration to the faults have not been studied [21].

Nickel has a stacking fault energy [22,23] of approximately 400 dynes/cm.

The equilibrium separation [24] between two partial dislocations is

estimated to be only 3b. The magnitude of the yield point produced by

segregation to stacking faults is related to the fault area. Hence, in

other metals such as Ag-6 w/o Al [25] where faults are estimated to

be 30b wide the effect is more important. Thus, one would not expect

Suzuki locking to be a very important pinning mechanism in nickel [26].

The third type of pinning is sometimes called short range order

locking and was proposed by Schoeck [27] and later expanded upon by

Schoeck and Seeger [28]. Schoeck and Seeger considered a bcc lattice

in which the concentration of interstitials is low enough to keep the

interaction between interstitials small. Snoek originally proposed [29] that

small atom impurities in solid solution occupy the octahedral interstices

at the center of an edge or the center of a face of the unit cell in a

bcc metal. Such sites have tetragonal symmetry since two of the six

solvent atoms surrounding the interstitial site are closer than the other

four. As a result, the octahedral sites may be classified into three

groups depending upon which one of three mutually perpendicular <100>

directions the two nearest neighbor solvent atoms are aligned along.

Thus, the three types of interstitial sites correspond to the three

directions of tetragonality and if no applied stress is acting, the

three kinds of interstitial sites will be occupied by the same fraction









of interstitials; namely, one-third will be in each of the three types

of sites. One may visualize each type of site occupied by an inter-

stitial as a dipole. The principal axis of the tetragonal distortion

gives the orientation of the dipole. If an applied (non-hydrostatic)

stress is acting, the energy of interaction between the stress and the

dipoles will in general depend on the orientation of the dipoles

(i.e., the types of sites occupied by an interstitial atom). As a

consequence, an applied stress will cause a redistribution of dipole

orientations and the population of the sites with lower energy will

increase, whereas the population of the sites with higher energy will

decrease. This process is known as the Snoek effect [29] and it gives

rise to a well established internal friction peak [30,31,32]. The

activation energy associated with stress induced ordering of interstitial

solute in bcc metals is normally that associated with diffusion of the

impurity [33]. Schoeck [27] in 1956 pointed out that a similar redis-

tribution of dipole orientations could be effected by the strain field

of a dislocation. By such a process, the energy of the system is lowered

in a period of time approximately that required for one interstitial

atom jump and, therefore, the dislocation becomes locked. Whereas the

locking due to atmosphere formation (Cottrell) requires diffusion of

interstitials over long distances, the locking due to stress induced

ordering of interstitial dipoles is accomplished merely by atomic

rearrangement between neighboring lattice sites and, therefore, takes

place in times which are orders of magnitude faster.

Schoeck and Seeger [28] examined the process in considerable detail

in 1959. Starting with the interaction energy between the interstitial

solute atoms and the dislocation and assuming the concentration of solute








was small, they showed that the line energy of a dislocation surrounded

by a Snoek ordered atmosphere is decreased by an amount U0 given by

mc A2
o = 3LkT
U= In (2)


where

c = total concentration of interstitials

A = an interaction constant

L = cut-off radius

kT = thermal energy


They next derived that the extra applied stress, AT, necessary to

pull the dislocation from the ordered atmosphere is given by

U kT
AT 2bA (3)

A more complete derivation of these results is carried out in the

discussion relative to the carbon-vacancy pair (Section 4.2.4) and is

related directly to pinning in Nickel 200.

The possible contribution to the rise in yield stress made by

ordering of solute atoms in the stress fields of dislocations has

generally been ignored probably because it occurs very quickly at the

temperatures that have usually been investigated. Snoek ordering,

however, can explain the rapid static strain aging phenomena observed

by Wilson and Russell [34] in tensile tests on a low carbon steel and

similar observations on a range of materials (for example, Carpenter

on tantalum-oxygen [12] and niobium-oxygen [35]; Owen and Roberts [36]

on martensite; Rose and Glover [37] in stainless steel). Support for
this view comes from an investigation by Nakada and Keh [38] of rapid

strain aging in iron-nitrogen alloy single crystals.









Wilson and Russell [34] verified that the rise in yield stress at

261K in iron specimens (containing 0.039 w/o carbon) prestrained 4%

was 63% complete in 100 seconds and noted that this time is in reasonable

agreement with relaxation times observed in the case of the elastic

after-effect due to the ordering of carbon in iron. Thus, the process

is complete in roughly the jump time of a carbon atom. It must be

noted that parts of their data were taken with a reduced applied load

which was generally between 80 and 90% of the load at the end of

prestrain. Aging while applying a load has been shown to influence the

size of the yield point and depends strongly upon the fraction of the

prestrain load that is used in aging [39,40,41,42,43]. A similar case

occurs in the data or Nakada and Keh [38] who used single crystal

Fe-0.l a/o C and N in their investigation.

Although Quist and Carpenter [35] did not conduct the usual

static strain aging experiments, their investigation of dislocation

pinning in Nb-0 alloys during internal friction measurements is note-

worthy. They conducted their experiments between 273 and 3130K and

attributed damping phenomena to the pinning of dislocations by Snoek

ordering of oxygen interstitial atoms in the strain fields of dislocation

line segments. They observed that pinning was effectively completed

in a period of one oxygen atom jump time.

1.1.3 Sumnaryof Important Mechanisms of Dislocation Locking During Aging

The two most important mechanisms of dislocation locking that may

occur in metals containing dissolved interstitial impurities are related

to (a) the Cottrell-Bilby model and, (b) the Schoeck-Seeger model.







The principal features of the Cottrell-Bilby model are

(1) Solute atoms migrate toward the dislocation over long distances

under the influence of the gradient in elastic interaction energy

between the dislocation and the solute.

(2) t2/3 aging kinetics are predicted by the model and observed

experimentally.

(3) The activation energy for the yield return predicted by the model

and observed experimentally is that for the diffusion of

interstitial solute.

The principal features of the Schoeck-Seeger model are

(1) Interstitial atoms with tetragonal strain fields reorient in

the strain field of a dislocation.

(2) The aging process by Snoek ordering is completed in approximately

the time required for one atom jump.

(3) The activation energy predicted by this model is that for

diffusion of the interstitial solute.


1.1.4 Aspects of the Static Strain Aging Experiment

Figure 1 illustrates the mechanics of the classical static strain

aging test for a specimen deformed in tension (or compression) at a

constant strain rate. The initial prestrain and unloading cycle gives

the specimen a known deformation history and internal state, i.e.,

a higher "fresh" dislocation density than that in the annealed specimen.

If the material is immediately restrained after unloading, the stress-

strain curve returns to the curve which would have been attained had

the specimen not been unloaded. However, by aging under the proper

conditions (e.g., higher temperature and/or longer times) a yield point

occurs and is followed by a period of Luders flow at constant load
















a-u
G-a
Oh 7
(TO


UnSoad, age, retest


STRAIN


Figure 1. General aspects of the classical static strain
aging test.








before work hardening resumes. In fact, the process of aging results

in a gradual transition from a smooth reloading curve for very short

aging times to a curve similar to that shown in Figure 1.

The important parameters of the reloading curve are oU, oL and

OExt' the upper yield stress, the lower yield stress and the stress
increment obtained by extrapolating the post-yielding curve, respectively.

For short aging times, OExt is equal to o the value of the stress

before unloading. The extrapolated stress, Ext, is determined by

the intercept of the flow curve, i.e., that portion of the stress-strain

curve where uniform strain hardening is present, with the pre-yield

or "elastic" portion of the reloading curve. The experimental parameter,

Ao = L 'o, is the parameter that is normally associated with strain

aging as it is experimentally the easiest to determine.

Accompanying the return of the yield point is the reappearance of

heterogeneous deformation, i.e., the passage of a LUders band down the

specimen gage length, at the lower yield stress, which is also charac-

teristic of annealed metal. During the initial stages of aging, the

lower yield stress increases with aging time as does the size of the

Liders strain. The rate of change of these properties generally increases

with aging time and temperature. After aging for somewhat longer times,

depending on the metal and its history, the variation of Ao and the

LUders strain with time in bcc metals becomes much slower, in many cases

exhibiting a slight decrease with aging time.


1.1.5 Static Strain Aging Stages in BCC Metals

Five stages of aging during static strain aging have been identified

for bcc metals containing interstitials. The first stage has been









explained on the basis of observations of very rapid returns of yield

points in interstitial iron alloys [34,38,44] and internal friction

experiments in other bcc metals [12,35]. The explanation is that very

rapid pinning may be attributed to stress induced ordering of inter-

stitials in the strain fields of dislocations as previously modeled

by Schoeck and Seeger [28] (Section 1.1.2).

The last four stages have been explained [10,45,46] on the basis

of a Petch equatior of the form


o = o. + 2k d-1/2 (4)


where a is the lower yield stress, a. the lattice friction stress,

k a dislocation locking parameter and 2d is the grain size. This

equation was developed originally by Petch [47,48,49] in order to

provide a method of separating the factors contributing to the lower

yield strength of polycrystalline iron. During the LUders band pro-

pagation, it was believed that unpinned sources release many dislocations

which pile up at the grain boundaries. Thus, a feature of the model

is the grain size and the boundaries are pile-up sites which act as stress

concentrations. The pile-ups are controlled by the grain size and act

in conjunction with the applied stress to unpin nearby dislocations in

a neighboring grain. The friction stress is represented by oi and is

the stress to move an unbound or free dislocation through the lattice.

Rosenfield and Owen [50] formulated the aging phenomena in terms

of an equation of the form


Ao = AoH + 2k d-1/2








where Ao is the gain in the lower yield stress after aging, AoH the

gain in the hardening component of the lower yield stress increase,

as determined by extrapolating the load-time curve after the Luders

strain back to the reloading curve,and k and d have the usual meanings.
y
Szkopiak [6] performed static strain aging experiments on niobium-

oxygen alloys and separated the two components of the yield stress increase of

Eq. 5 as shown in Figure 2. In Figure 2a, the typical return of the

yield stress experiment on a bcc metal shows that at small aging times

the increase in lower yield stress is very rapid (depending upon

temperature of aging) and approaches a maximum. At longer aging times,

the lower yield stress increment shows a slight decrease. In Figure 2b,

the two components of A in Eq. 5 are shown separately.

The five stages of aging that have been observed in alloy systems

such as Nb-0 [6,35], Fe-C [34,45,46], and Fe-N [38,51,52] alloys and

probably occur in Ta [12,18], V [5], and Mo [8] as well are:

Stage I: This stage is observed clearly only at low temperatures since

locking occurs by stress induced ordering of interstitials in the strain

fields of dislocations and occurs within the time span of approximately

one solute atom jump. The strength of pinning agrees reasonably well

with the Schoeck-Seeger model.

Stage II: In this stage, k reaches a maximum and remains constant. The

lower yield stress reaches a maximum and the LUders strain increases very

rapidly. The rationale for this stage is that the formation of Cottrell

atmospheres takes place during aging and upon reloading dislocations become

unpinned from their atmospheres.

Stage III: Further increases in the lower yield stress are due to an

increase of the AoH parameter. In this stage, the Luders strain remains









4-
Oxygen,ppm

S600
3 .300

b //






0
S 10 100 '1000 10000

(a)







b b



0o 0
^ / p9-0-0---0-----.0---0--c
C 2 Oxyoen, ppm

o iN *0600
*0300

Fr 10 100 1000 10000
Aging Time, Minutes
(b)


Figure 2. An example of the stages of the yield return in
Hb-O alloys [8]; (a) the increase in Ac with time,
(b) the components of 0A and their dependence on time.








nearly constant as the yield stress increases and enhanced strengthening

occurs. The principal rationale for this hardening is that dislocations

have been aged to the extent that they tend to remain immobile or pinned

upon subsequent reloading. Thus, new or additional dislocations

are created and the yield stress continues to increase.

Stage IV: During this stage, solute continues to be accomodated in the

strain field of dislocations but no longer effects an increase in Ao.

Hence, Ac remains approximately constant.

Stage V: As more and more interstitial solute segragates to dislocations,

a condition of averagingg" is satisfied and precipitates may form;

hence, the loss of a coherent strain field or the robbing of solute near

dislocations and a mild decrease in the hardening component.

The above stages of static strain aging appear to hold true for

most of the body-centered cubic metals containing interstitial oxygen,

carbon or nitrogen. However, in the case of face-centered cubic metals

containing interstitial impurities no complete investigations of the

behavior of the return of the lower yield stress have been carried out.


1.1.6 Static Strain Aging in Nickel and FCC Alloys

Among the fcc commercial alloys, nickel containing carbon is

probably the most significant where an interstitial (carbon) is

deliberately added to improve mechanical properties. Other than pure

nickel, only fcc multicomponent alloys such as the austenitic stainless

steels [37,53,54] contain carbon for similar reasons and exhibit

mechanical properties similar to nickel-carbon alloys.

There exists some experimental evidence related to the static strain

aging of nickel-carbon alloys. In particular, two short notes were









published by Macherauch et al. [55] and by Macherauch and V6hringer [56]

regarding static strain aging in Ni-0.05 w/o C after restraining

slightly beyond the initial yield plateau. Their data were plotted

by this author and an approximate t/3 or t1/4 time dependence of the

lower yield stress return was exhibited. They did not speculate on

the kinetics; however, they determined an activation energy of

10.22 kcal/mole in agreement with the activation energy for diffusion

of hydrogen in nickel (see, for example, Boniszewski and Smith [57]).

The only other investigation relating to nickel-carbon alloy is

due to Sukhovarov [58] and Sukhovarov et al. [59]. Using compression

specimens deformed at room temperature and aged between 433 and 493K

for various times, they deduced with apparent difficulty (because

serrated flow occurred) that the average activation energy was

30.7 kcal/mole, somewhat lower than the carbon migration energy in

nickel. This author plotted their lower yield stress data and noted

that 6o/AOmax varied approximately as t0.3. They conclude (incorrectly,

it is believed) that the Cottrell-Bilby model explains the rise in

Ao and that probably the formation of precipitates eventually occurs;

they never observed this aspect.

In addition, since serrated flow occurred in the investigation of

Sukhovarov [58], it is probable that the data were scattered because

the lower yield stress was not as clearly defined as in the present

investigation where yield point measurements were made under conditions

precluding serrated flow. On this basis, their data should be used

judiciously.

Hydrogenated nickel exhibits strain aging behavior [57,60-65] when

deformed below room temperature. Much effort has been directed toward









understanding fracture, ductility, and other embrittlement related

phenomena attributable to hydrogen. Also, serrated flow [57, 60-65] is

exhibited in hydrogenated nickel between approximately 130 and 2250K
-4 -1
[65] at a nominal strain rate of 10- s The kinetics of static

strain aging in hydrogenated nickel were very briefly investigated by

Boniszewski and Smith [57] and they concluded that the Cottrell-Bilby

model can account for static strain aging of charged specimens. However,

they did not speculate on the exact kinetics that the experimental

data may have followed.

Marek and Hochman [66] have demonstrated the existence of static

strain aging effects in AISI 316 alloy and related it to the approximate

activation energy for diffusion of interstitial carbon in austenite.

However, the effect was most marked in the micro-yield region (0.01% proof

stress) with no effect on flow stress after yielding, UTS, or elongation,

which is indicative of a low interstitial/dislocation interaction energy.


1.2 Dynamic Strain Aging

Dynamic strain aging (DSA) is a feature exhibited in most commercial

metals and alloys [1]. In general, aspects of DSA have created very

little interest in the past, probably since in steel it exhibits its

most significant effects over a temperature range around 4500K where

steel is not normally worked. Other bcc metals such as titanium,

tantalum, niobium and vanadium exhibit DSA over a temperature range

where these metals are most needed [1].

As with steel, nickel containing carbon exhibits its effects at

relatively low temperatures (300 to 5000K, roughly). Early investigations

by Sukhovarov and Kharlova [67] confirmed that dynamic strain aging








occurs in nickel when alloyed with small amounts of carbon. In a

subsequent investigation Popov and Sukhovarov [68] indicated that the

apparent activation energies associated with the appearance and

disappearance of serrated flow are 20 and 33 kcal/mole, respectively.

No conclusion regarding the very low activation energy for the onset of

serrated flow was ventured. On the other hand, Nakada and Key [51]

have indicated that the onset activation energy in Ni-C alloys is

152 kcal/mole and that for the disappearance of serrations is 264

kcal/mole. It is a well established experimental fact that the acti-

vation energy associated with the diffusion of carbon in nickel is

approximately 35 kcal/mole (see Table 2). Generally, in interstitial

alloys the activation energy for the onset of serrated flow is

associated with the diffusion of impurity atoms and is made on the

basis that when the velocity of dislocations is approximately equal to

that of the velocity of the diffusing impurity atoms, a drag or pinning

of dislocations occurs. Thus, the pinning as observed through the

serrated flow phenomenon is assumed to be controlled by the diffusion

of impurity atoms just as in the static strain aging experiment. By

plotting log E versus 1/T, where c is the strain rate at temperature T

where serrated flow is first observed, an activation energy may be

deduced. The values of 152 and 20 kcal/mole for the onset of serrated

flow determined by the preceding authors are much too small to be related

to the diffusion of carbon in nickel. Popov and Sukhovarov [68] made

no conclusions regarding this apparent anomaly. Nakada and Keh,

however, ventured that pipe diffusion of carbon along dislocation cores

controls serrated flow in nickel.




21




TABLE 2

The Diffusivity of Carbon in Nickel


Do Q Technique
cm2/sec kcal/mole

0.048 34.8 Elastic and magnetic aftereffect [69]

0.13 34.5 Radioactive tracer [70]

0.1 33.0 Radioactive tracer [71]

-- 38.5 Thermogravimetric [72]

-- 39.7 Thenogravimetric [72]

-- 32.3 Magnetic aftereffect [73]







Regarding the activation energy associated with the disappearance

of serrations, Kinoshita et al. [74] proposed that this value may

represent the sum of the activation energies for the diffusion of solute

plus the binding energy of solute atoms to dislocations. On this basis,

Nakada and Keh [51] have deduced a binding energy of 11.0 kcal/mole

(0.5 eV) for a carbon atom to a dislocation in nickel assuming that

serrated flow in Ni-C alloys is caused by carbon directly. Popov and

Sukhovarov [68] attributed their value of 333 kcal/mole (1.4 eV) for

the disappearance of serrations to a combination of creep processes

coupled with Cottrell atmosphere formation.

Other than the above nickel-carbon studies and the experimental

work on hydrogenated nickel [57,60-65] which shows strain aging, no other

investigations of the effect of interstitials on the stress-strain and

work hardening behaviors in pure face-centered cubic metals have been

conducted. However, some face-centered cubic ferrous alloys appear to

possess mechanical properties similar to those of nickel-carbon alloys.

In an investigation by Jenkins and Smith [54] complications due to

substitutional alloying elements such as Cr occurred. Nevertheless,

AISI 330 stainless steel (Fe-15Cr-33Ni-0.4C), exhibits similar dynamic

strain aging trends. A calculation of the energies for the onset and

disappearance of serrations revealed that 26.6 and 62.0 kcal/mole are the

onset and termination activation energies. They indicated that the

onset activation energy is very close to that for vacancy migration in

Fe-30 Ni. As the Portevin-Le Chatelier effect is absent for low carbon

content, they conclude that vacancies alone are not responsible and that

carbon-vacancy pairs account for the observed activation energy. Mention

was not made of the exact mechanism for the pinning during serrated flow.

A similar argument appears to apply in the case of Nickel 200.








Other strain aging effects occur in face-centered cubic alloys

but these arise mainly from the diffusion of substitutional solutes

and are outside the scope of this dissertation.


1.3 Work Hardening in Metals and Alloys

In studying the work hardening of metals and alloys it is desirable

to determine the mechanisms that control the rise in flow stress.

In general, this involves relating the macroscopic behavior to changes

in the microscopic structural features of the metal. For example,

observations of slip line lengths or dislocation structures can

supplement an explanation of work hardening.

From the macroscopic point of view, polycrystalline stress-strain

curves have been shown to be generally piece-wise continuous [75-81].

For example, Figure 3 shows schematically that a polycrystalline face-

centered cubic metal may deform so as to show discontinuities in its

stress-strain behavior. Zankl [75] and others [76-81] have shown that

these stages can be correlated very well with deformation processes.

According to his experimental work [75] the identifiable stages are

related to the following processes:

1. The Transition Stage. This extends from zero plastic strain

to approximately 0.1%. In this stage, multiple slip starts first

in the largest grains and then spreads into neighboring grains.

2. Stage I. This begins when all grains have begun to deform

with slip still involving multiple slip systems. Stage I in

polycrystals is thus basically different from the easy glide

Stage I of face-centered cubic crystals. It ends at approximately

E = 1.0%.
p









3. Stage II. Here slip tends to occur predominantly on a single

(primary)system but with interaction from secondary systems. The

deformation is accordingly analogous to that in Stage II of a fcc

single crystal. Large grains may break down into several regions [80]

with different primary systems. This stage extends to about

e p5.0% in pure fcc metals such as copper and nickel.

4. Stage III. As in fcc single crystal deformation, this stage

is controlled largely by dynamic recovery and has been associated

with cross slip [82].

In general, polycrystalline stress-strain curves appear to be

continuous in shape and the stages difficult to identify on such curves.

This is in marked contrast to single crystal stress-strain curves which

often exhibit well-defined stages. A sensitive empirical method [83]

for detecting polycrystalline stage behavior has been developed as a

logical projection of previous empirical analyses [84-92]. This is

based upon the assumption that each stage of stress-strain behavior can

be reasonably described by a modified Swift [92] equation:


e = Eo+com (6)


where a is the true stress, E the true plastic strain, m the work

hardening exponent and o. and c are constants.

One may solve very simply for the parameters in Eq. 6 by plotting

log 0 versus log o where 0 = a A schematic example of such a plot

is shown in Figure 4 for a typical face-centered cubic metal such as

copper or nickel. Any straight line on this type of plot has an

equation of the type:

log 0 = (1-m) log o log cm (7)




25














tI


Figure 3. Schematic example of stage
fcc metals.


behavior in polycrystalline


I







LOG 0-


Figure 4. Schematic description of the work hardening behavior
in a metal using a log C-log a plot.









Thus, the value of m typically characterizes the power law

relationship of Eq. 6. An m equal to one is a linear stress-strain

curve and a log 0 versus log o plot would show a line with a zero

slope; a parabolic stress-strain curve would show a (l-m)-value of

-1.0 (i.e., m=2) and so on. High m values correspond to curves with a

great deal of curvature, i.e., very rapidly decreasing work hardening

rate with increasing stress as in Stage III when dynamic recovery processes

reduce the work hardening rate very rapidly with continued deformation.

All other parameters held constant, a high value of m for a single

stress-strain curve would, in general, imply that the material has low

ductility, even though the material might possess a reasonably high

ultimate strength. However, it should be noted that a three or four

stage stress-strain behavior could well lead to a combination of both high

strength and high ductility depending upon the m values of the various

stages and the extent of a particular stage during deformation.

It should be noted that the stage behavior observed by using Eqs. 6

or 7 is within limits independent of the empirical power law equation

used. For example, an analysis based upon the Crussard and Jaoul

method [87-89] using a log a log E diagram shows that discontinuities

occur at the same places on the stress-strain curve as determined by

using a log 0 log a diagram.

In the current investigation, interstitial solute concentration

was the principal alloy variable known to affect stress-strain behavior.

Interstitial elements tend to significantly increase the strength of a

metal while generally decreasing its ductility. This in turn affects

the stage behavior of the parent metal.









Another factor influencing the stage behavior of metals and

alloys is the stacking fault energy. This intrinsic property of a

metal or alloy determines the separation distance of the two partial

dislocations of an extended dislocation [93].

For high SFE metals like nickel [22,23,94] or aluminum [93] the

separation distance is small and mobile dislocations may be assumed

to approach total dislocations. In low SFE metals and alloys such as

Ni-40Co and Ni-60Co [94] this distance becomes appreciable and the

partial may be separated by a wide band of stacking fault. By

effectively varying the geometry of the dislocation by lowering its

stacking fault energy, the deformation behavior might be expected to

change as well. Thus, it would appear that in a low stacking fault

energy metal a dislocation effectively loses a degree of freedom of

movement by virtue of its assuming a planar character. This should

reduce the ability of the material to undergo dynamic recovery involving

either cross-slip or climb.

One may also view a stacking fault as a building block for a

deformation twin. In general, twin boundary energies are lower in lower

stacking fault energy metals. This observation is in agreement with the

fact that twinning plays a greater role in the deformation behavior of

metals and alloys [95,96] with low stacking fault energies. Twinning

was not observed in the present investigation.


1.4 Anelastic Phenomena in Nickel and FCC Ferrous Alloys

Considerable research has been performed for many years on the

effects of interstitial solutes on the strain aging of body-centered

cubic metals and alloys [2]. In addition, much emphasis has shifted









toward examining a variety of internal friction effects (such as the'

Snoek effect, the cold-work peak and dislocation damping behavior)

that can be correlated closely to strain aging phenomena [97].

In contrast, however, there exists but a dearth of strain aging

and internal friction studies of interstitial solid solutions of face-

centered cubic metals. The reason for this lack of interest probably

stems from the fact that in terms of the mechanical behavior of these

alloys, the interstitials apparently cause a less dramatic effect on

the mechanical properties. In addition, fcc metals are not expected to

exhibit a Snoek peak because of site symmetry [30,32,98,99].

Very different specific mechanisms for the observed relaxation

peaks in fcc metals and alloys have been suggested by various authors.

Adler and Radeloff [99] reviewed the types of defects which could

account for internal friction in fcc metals and alloys:

(1) interstitial-solute clusters which have noncubic strain

fields [30,98];

(2) interstitial-substitutional solute clusters in which the

interstitial reorients preferentially under stress if it is a

near neighbor to an immobile substitutional solute [98,99];

(3) interstitial-vacancy complexes of different types, i.e., Wu

and Wang [100] suggested a defect consisting of one interstitial

occupying a vacancy with a nearest neighbor interstitial in its

normal site.

Within each category listed above are many possible specific

combinations which could in principle cause a relaxation effect.

There is evidence for relaxation due to carbon pairs in both Ni

and the fcc allotrope of Co, and for oxygen pairs in Ag [101]. The








existence of an internal friction peak associated with dissolved

C in Ni was first reported by KB and Tsien [102] and further inves-

tigated by KQ, Tsien and Misek [103]. The peak is quite small and

occurs at 5230K for a frequency of 1 Hz. By application of a saturating

magnetic field, the existence of the peak was shown to be unrelated to

the ferromagnetic nature of the sample. Further, the peak was found to

decline in strength as the carbon precipitated from solution. KB and

Tsien pointed out that unassociated C atoms, located at the body-centered

position of the fcc structure and in the equivalent positions midway

along the cube edges, could not be responsible for the relaxation, since

the symmetry of such defects is cubic. After Tsien [104] found in later

work that the relaxation strength varied essentially as the square of

the carbon content in solution, it became evident from mass-action

considerations that carbon pairs were the responsible defect.

Diamond and Uert [69] also investigated the diffusion of carbon

in nickel utilizing elastic aftereffect measurements above and below

the Curie temperature and noted no discontinuity in carbon diffusivity.

Hence, magnetic transformation effects do not affect carbon diffusivity.

In addition, they concluded that the elastic aftereffect is due

principally to interstitial diffusion of C-C pairs in agreement with

the results of the previously mentioned group of Chinese investigators

[102-105]. The simplest interstitial pair configuration in their view

consists of two atoms occupying nearest neighbor octahedral sites which

are the largest for fcc metals, e.g., the two sites 00 and 00. The

stress induced reorientation of such pairs (designated 110 pairs) which

gives rise to anelastic effects, is the result of one of the atoms

jumping into an unoccupied nearest neighbor interstitial site. A summary









of Ni-C diffusion data is presented in Table 2. It should be noted

that all methods agree reasonably well with an activation energy of

35 kcal/mole for the migration energy of carbon in nickel. No

distinction is made in the last four references cited in the table

concerning the nature of the mobile species, e.g., dicarbon complexes.

C.T. Tsien [104] considered the effect of impurities on an internal

friction peak in a carburized 18.5 w/o manganese steel. The principal

experimental observation was that the internal friction peak height

varies linearly with carbon content. It was proposed that the addition

of Mn to Fe-C alloys may reduce the opportunity of forming carbon-

carbon atom pairs and there may be greater probability of forming

Mn-C pairs instead. Thus, in high manganese steels the internal

friction peak is not attributed to rotation of the carbon atom pairs but

due to that of carbon-manganese pairs. As a result, the height of the

internal friction peak was observed to be directly proportional to the

carbon content.

This brings us to the point of the possible mechanism involved in

Nickel 200 which has 0.18 w/o Mn and smaller amounts of iron and copper.

Presuming a pair mechanism (in order to obtain a tetragonal defect which

would account for an internal friction peak) one might presume a possi-

bility of C-C atom pairs or C-Mn atom pairs causing the static strain

aging effects. However, in the investigations previously sited, not one

was conducted using a deformed metal and this factor may be an important

consideration. Hence, a third and possibly fourth speciesmay be involved,

namely, the vacancy and the self-interstitial. It has been demonstrated

by Seitz [106] that in fcc metals the predominant defect produced

during plastic deformation is the vacancy. Table 3 shows the data of








a number of authors [107] who have described the vacancy concentration

during plastic deformation for a variety of fcc metals and alloys as


Cy = Ken (8)


where cv = vacancy concentration (atom fraction)

e = true strain

K = proportionality constant

n=

The principal experimental technique used to determine c involves

monitoring resistivity changes during deformation and subsequent

annealing of the specimens [106,108]. The identification of the

defects annealing out during each recovery stage has been the subject

of extensive previous work in nickel. Studies have been made of the

resistivity recovery spectrum following neutron irradiation, electron

irradiation and quenching from high temperatures [109,110]. In addition,

changes in magnetic properties have also been studied in conjunction

with the resistivity recovery process [73,111,112].

On this basis [108] cold rolled nickel behaves as


c = 2.1 x 10-'4 (9)


In addition, it has been estimated that approximately eight times as

many vacancies as self-interstitials are generated.

Point defects are generally believed to be generated during plastic

deformation by two basic mechanisms [31,113]: (a) nonconservative

motion of jogs on screw (or mixed character) dislocations and

(b) recombination of dislocations containing edge type components.

The second mechanism is due to annihilation of edge dislocations and may









TABLE 3*

Estimates of the Rate of Vacancy Production
During Plastic Deformation


Material K Type of Deformation

Cu 1.9 x 10-4 tensile elongation

Ni 2.1 x 10-4 cold rolling
-4
Ni 12 x 10- shock forming

NaC1 1 x 10- compression

70-30 Brass 5.9 x 10-4 tensile elongation

Au 2.9 x 10-4 tensile elongation

Al 0.2 x 10-4 tensile elongation


* From Ulitchny and Gibala [107]








not be important relative to the first mechanism in the production

of vacancies until Stage III deformation occurs. The relation between

vacancy concentration and strain is also probably only valid to strains

of the order of 10%. It has been noted that the vacancy concentration

tends to approach a saturation value at large strains [108]. Thus,

vacancies may play an important role in deformed and aged nickel.

As concerns the manganese impurity in Nickel 200, a qualitative

argument may be made against the existence of Mn-C complexes as opposed

to C-V complexes. The gram atomic volume of Mn [114] is 7.39 cm while

that for Ni [115] is 6.59 cm3. Thus, a Mn atom is only approximately

4% oversize on a nickel lattice site. In an interstitial octahedral

lattice site, a carbon atom is about 14% oversize. Relative to binding

to a vacancy, the carbon atom should provide a stronger compressional

center of dilatation than the manganese atom even though in the former

case an octahedral interstitial site is occupied and in the latter a

normal lattice site is occupied. Hence, the binding energy of a vacancy

to a carbon atom should be higher than that of a manganese atom to a

carbon atom or a vacancy. In addition, Nickel 200 contains approximately

three times less manganese than carbon on an atom fraction basis.

The value for the diffusion energy of a vacancy through a lattice

is very sensitive to the presence of impurities which tend to slow down

a freely migrating vacancy because of binding to impurity atoms [116,117].

In pure nickel, the vacancy migration energy is between 0.8 and 0.9 eV

and for impure nickel [117,118] (99.9% pure plus an unspecified amount of

carbon) is approximately 1.1 eV. Thus, one may deduce that the approximate

binding energy of carbon to vacancies is between 0.2 and 0.3 eV. In

short, the carbon-vacancy interaction could cause the formation of a








defect complex which might cause strain aging in Nickel 200. It is

not clear whether or not dicarbon-vacancy defects may be completely

ruled out as a possible complex causing strain aging.

An excellent discussion concerning internal friction and strain

aging to carburized ferrous austenite by Ulitchny and Gibala [107]

suggests that the relaxation phenomena in these alloys are attributable

to the rotation of carbon-vacancy pairs. Their conclusion is based

upon experiments in which vacancies were produced in stainless steels

by (1) quenching, (2) deformation and (3) irradiation. These processes

have in common that they (a) increase the vacancy concentration in

the metals, (b) increase the observed peak heights of the bound pair

peak and (c) increase the peak heights in proportion to the relative

numbers of vacancies they are anticipated to produce. These alloys

possess mechanical properties quite similar to nickel-carbon alloys.

In addition, the diffusivities of both carbon atoms and vacancies in

the austenitic stainless steels are similar to those in nickel-carbon

alloys [107] suggesting that a similar mechanism in the present investi-

gation of Nickel 200 should not be ruled out.













CHAPTER II

EXPERIMENTAL PROCEDURES


2.1 Materials

Nickel 200 bars of 0.75 inch diameter were obtained through a

local supplier and from the International Nickel Company. These two

heats had slightly different compositions (Table 4); the mechanical

properties as a result were somewhat altered, albeit small. All static

strain aging experiments were conducted utilizing Nickel 200b. The

standard tensile tests were conducted using Nickel 200a. This procedure

was followed in order to minimize possible scatter of the data,

particularly in the static strain aging experiments. In addition,

Nickel 270 was also purchased; its composition also appears in Table 4.

Nickel 200 contains approximately 0.18 w/o Mn and 0.10 w/o C.

The highest equilibrium solubility limit [119]for the nickel-carbon

system is 0.27 w/o carbon. The room temperature solubility limit of

carbon in nickel is only 0.02 w/o [120]. Also, nickel carbides are

unstable in Ni-C alloys [119,121]. The development of visible graphite

in nickel during cooling is generally agreed to occur very slowly.

The present specimens, which were furnace cooled from the annealing

temperature (10730K) at a rate of approximately 2.0/s, are believed

to have retained all of the carbon in excess of the equilibrium solubility

in solution since graphite was not observed either by optical or

transmission electron microscopy of the annealed specimens.




36




TABLE 4

Alloy Compositions


Ni 270

0.01

<0.001

0.002

<0.001

<0.001


Cu <0.001


All other impurities less than 100 ppm each.


Ni 200a

0.08

0.27

0.05

0.005

0.06

0.01


Ni 200b


0.10

0.18

0.01

0.005

0.04

0.01









2.2 Experimental Techniques


2.2.1 Swaging

The 0.75 inch (19.1mm) bar stock of all the materials were cold

swaged in a Model 3F Fenn rotary swaging machine to a diameter of

0.25 inch (6.4mm). Intermittent annealing was not necessary. The

resulting swaged bars were machined into threaded-end specimens with a

nominal reduced section of 0.8 inch (20mm) and a gage diameter of

0.15 inch (3.8mn).


2.2.2 Annealing

Annealing was accomplished in a Vacuum Industries Minivac furnace

assembly. This unit utilizes a resistance heated tantalum element.
-3
Pressures as low as 10 millitorr can be maintained. No cold trap

was used. All Nickel 200 specimens were annealed for 30 minutes at

800C (10730K); Nickel 270 specimens were annealed for 32 minutes at

5950C (868K). These treatments resulted in specimens with a mean grain

intercept of approximately 22pM. Annealing twin boundary intercepts

were not counted in obtaining this result.


2.2.3 Specimen Profile Measurements

A Jones and Lamson Optical Comparator capable of measuring to

0.0001 inch (2.5pM) in the vertical and 0.001 inch (25pM) in the

horizontal directions was used to measure the profile of as-annealed

specimens. For all tests a specimen gage length was assumed to be

identical to its reduced section and was determined to within 0.005 inch

(0.13mm) with experience.








2.2.4 Tensile Testing

All tensile testing was performed on two Instron machines (Model TT-C

and Model FDL of 10,000 and 20,000 pound capacities, respectively). The

standard crosshead speed was 0.02 inch/minute resulting in a nominal

specimen strain rate of 4.2 x 10-4 s Three additional strain rates

were also employed with the Nickel 200 specimens. All tests were conducted

between 77 and 9000K. Above 2970K tests were carried out in a capsule

using commercial purity argon gas. At no time was oxidation visible on the

specimen surfaces. Below ambient temperature, liquid nitrogen (770K),

dry ice-acetone (1960K), or ice-water baths were employed.

Load-time curves were processed to yield true stress versus true

plastic strain curves as well as the slope of these curves as a function

of stress or strain. The stage behavior of the specimens was analyzed [83].

In brief, the procedure involves the plotting of log E versus log o

and identifying portions of the curves through which straight lines may

be passed. Each linear interval is assumed to represent a stage.

This assumption was tested against results of Zankl [75], Schwink and

Vorbrugg [77] and others [76,78-81] and a good correlation was obtained

between stages determined in this manner and the method of Zankl and

others using different plotting and metallographic procedures. To each

linear region a different set of parameters (m,c, o) may be deduced

corresponding to the Swift [92] equation

E'Eo+Cm (10)


2.2.5 Static Aging Experiments

Annealed tensile specimens were prestrained approximately 5% to

a stress level of 38.5 ksi (265 MPa) at 2730K and immediately unloaded,








removed from the testing jig and immersed in a silicon oil bath at

473, 448, 428 and 4080K or in a boiling water (distilled) bath for

times as long as 2 x 106 seconds (approximately two weeks). Control

of the constant temperature baths was held to within approximately

one-half degree. Upon completion of the aging treatment, the specimen

was removed and quickly quenched into cold water and tested immediately.

The reloading temperature for the static strain aging tests was 273K

as in the prestrain. It was recognized early in the investigation that

at room temperature and a strain rate of 4.2 x 104 s- (the rate

corresponding to a crosshead speed of 0.02 inch/minute, standard at

this laboratory), discontinuous flow occurred in Nickel 200. To

alleviate this feature it was decided to conduct the prestrains in

ice-water baths at a rate of 4.2 x 10-3 s This had the additional

benefit of producing a stable and reproducible lower yield stress

plateau and, in addition, the ice-water bath assured that specimens

were at the test temperature after removal from the high temperature

aging baths.

Quite apparent from the start was the fact that specimen alignment

offered a problem. Upon removal and replacement of a specimen in the

tensile jig, it was evident that exact repositioning was difficult and

uncertain. Therefore, on restraining a specimen after aging it, a

small bending moment normally develops which may cause yielding to

occur nonunifonnly across the gage section of the specimen. The result

is that the upper yield point as observed on the machine chart was

usually absent. As a consequence all data reported are lower yield

stresses. Compilation of the data included interpreting the Luders

extension as the chart displacement which occurred at constant load









(Figure 5). This method proved to provide the most consistent set of

data and tended to alleviate apparent alignment or reloading displacements.

Occasionally (perhaps 10% of the time), a yield point was observed and

the data fell consistently in line with other LUders extension data

recorded by the former method.

The latent hardening achieved during long term and high temperature

aging treatments was computed by linearly extrapolating the post-

yielding curve back to the reloading line as demonstrated in Figure 5.

This method also proved consistent. However, in most cases of short

aging times or low temperature aging, the extrapolated stress fell

below the prestrain value. This is physically unreasonable and is

attributed to alignment effects which generally eliminate the yield

point as mentioned previously and may cause the extrapolations to come

back to the reloading curve somewhat low. However, the resulting data,

again, proved to be consistent.

As an example of typical experimental results, Figure 6 shows a

series of load-time curves which were obtained after restraining a

series of specimens at 2730K to 5%, aging at various times at 408K,

and restraining at 2730K. Figures 6b and 6d show examples of partial

yield points that were obtained in a few cases during the present inves-

tigation.











Aged 2.92 x 104s/4480




/I


Ext.- -157 N







O 3s



Figure 5. Illustrating the method used to determine the aging
parameters EL, AC= oLY o and AOH = OExt o .
The dashed loading line indicates the approximate
loading line which would have been observed in the
absence of misalignment of the test specimen.










^ --157 N


(a)(b)

/ s/ 960 s
60s




157 N

(d)
(c)


1.54 x 104s 1.74 x I's




157 N

(e)

5.78 x 10 s


CI


Figure 6. Selected load-time curves obtained after restraining
a series of specimens 5% at 2730K, aging at 4080K
for the times indicated, and restraining at 2730K.














CHAPTER III

EXPERIMENTAL RESULTS


3.1 The Behavior of the Lower Yield Stress Increase, At

Figure 7 illustrates the dependence of the lower yield stress

increment, Ao, on time and temperature over five decades of time and

at five temperatures. The principal features of Figure 7 are

(1) The curves appear to approach a common value at small times.

It would seem that the data obtained at the higher aging temperatures

and for very short aging times are influenced by the time required to

heat the specimen to temperature. This was confirmed by assuming a

fixed heat-up time and displacing the curve (at each temperature) to

shorter aging times. Thus, a heating time of approximately 40 to 60

seconds straightens out the start of the higher temperature curves to

approximately the same linear dependence as exhibited by the 3730K curve.

In addition, a simple heat transfer calculation indicated that a time

constant of approximately 40 to 60 seconds should describe the specimen

heat-up time. The dashed lines in Figure 7 show the approximate

corrections necessary to account for heating the specimens in the baths.

(2) Each Ao curve shows a roughly linear increase with log t for

times before reaching the maximum in Ac.

(3) All curves show a well-defined peak whose height increases

slightly, the lower the aging temperature.




44









30-



25 -


473\
20 _/448














0
5-



I 2 3 4 5 6
10 10 10 10 10 10
.Time (sec)


Figure 7. The time and temperature dependence of the return of
the lower yield stress in Nickel 200. Specimens were
prestrained to a stress level of 265 MPa. The dashed
curves are approximate corrected curves which account
for specimen heat-up in the aging baths.




45





TABLE 5

Least Squares Parameters for Static Strain Aging Data
Assuming Ao Is a Function of In t


t (sec)


480 t 1.43 x 106

240 t 6.62 x 104

60 t 2.16 x 104

30 t 3.4 x 103

not fitted


Slope Intercept r
(MPA/In sec)

2.088 -4.922 0.996

2.647 -3.356 0.998

2.602 1.618 0.995

2.318 6.689 0.998


Temperature









(4) Ao decreases significantly after max is passed. Just

after the peak, the decrease in Ao is almost linear with log t.

However, at 473"K, the highest temperature investigated, the data at

very long aging times show that the lower yield stress decreases to

a constant value of approximately 17 MPa.

Table 5 lists values obtained by the method of least squares for

the slopes and intercepts (at t=1s) of the aging curves in Figure 7.

A Ao versus In t relationship is assumed to hold for times before

Ac reaches a maximum.

In addition, Figure 8 shows a set of Am curves which were normalized

to their respective maximum peak heights, Ammax. An approximate value

of 28.5 MPa for Aomax was assumed for the 3730K curve since, for the

aging times investigated, a peak was not attained. This diagram has

the effect of making the aging curves more nearly parallel.

Figure 9 is a plot of the increase in lower yield stress versus

t /7 using data from the 373, 408, 428 and 4480K aging curves of

Figure 7. Least squares analysis of log Ao versus log t curves indicated

that Ao varies approximately as the 0.14 and 0.15 power of time. See

Table 6 for the complete results. This represents an approximate time

dependence of t/7

It is important to note that this author prepared a specimen which

had been aged for approximately 6 hours at 5250K (specimen taken from

the undeformed threaded end of a deformed tensile specimen) and observed

it carefully in a transmission electron microscope. No evidence of

precipitates or free graphite in the grains, at dislocations or at grain

boundaries was observed. Thus, it is highly unlikely that precipitation

of carbon occurs during aging between 373 and 4730K in Nickel 200.
























































Figure 8. Normalized aging curves for Nickel 200. The 3730K
curve was normalized to an assumed maximum of
28.5 MPa. The dashed curves reflect approximate
corrections for the heat-up time of the specimens.












o I 2 3 4 5 6



0


0

0- 0



5-


0-
o 4480K
// 4280K
5- 4080K
3730K


o I 2 3
t (min7)



Figure 9. Illustrating the approximate t1/7 power law relation
governing the aging of Nickel 200 at temperatures below
4480K. Data for the 4480K (shown) and 473K cases
do not fit this relation well.









TABLE 6

Least Squares Parameters for Static Strain Aging Data
Assuming a Log Ao-Log t Linear Relationship


Temperature t (min Slope Intercept r

373 1.0 < 23,752 0.145 0.804 0.993

408 1
428 2 t 710 0.137 1.109 0.997

448a 2t 100 0.099 1.232 0.998

473b 10< t 5 30 0.055 1.307 1.000




a Three points

bTwo points








3.2 The Liders Extension, EL

Figure 10 illustrates the dependence of the LUders extension on

log t. The behavior is similar to that exhibited in Figure 7 for

curves of Ao versus log t. However, note that the peak in the Luders

extension occurs (at a given temperature) at an earlier time than the

corresponding peak in Ao. This may be indicative of the onset of

hardening and is consistent with results obtained in bcc metals. Note

also that at long times the extensions tend to return to their short

time values; that is, they do not remain constant after reaching their

maximum values. An interesting point is that the highest temperature

curve (4730K) reaches an apparent minimum at approximately 1500 minutes.


3.3 The Hardening Component, AoH

Figure 11 is a plot of AoH, the hardening component of the increase

in lower yield stress. This parameter was deduced as noted in Figure 5

by using the equation AoH = OExt oo. Figure 11 shows that this

hardening component appears only at discrete times. Note also that

AOH peaks at approximately the same time as Ao and decreases to a value

higher than that observed at very short aging times. It should be noted

that the values of AoH cannot be taken as exact due to alignment problems

which affected the choice of OExt (see Figure 5).


3.4 Activation Energies

In order to establish a mechanism for static strain aging, the

apparent activation energies associated with particular time dependent

aging events were deduced. The activation energy for the return of the

lower yield stress where in the interval an approximate logarithm of






I- 1Iq -I I I I I ,lll -- I I ;11H I I I 1111 I i I 1 111 I] -
1.0-
4284
448 .' --4 3730





0.8 --
473*






")
3 0.2


0.0 I I
10 102 103 104 10 106
Time (sec)


Figure 10. The dependence of the LUders strain on time and temperature in Nickel 200.





I I I11 1 1 1 M4 I III1 I I IllIr I'll I I I-


Nickel 200
3730K
o 408
o 428
A 448
O 473


0


0 0
. . ..... I ,


. . .. I


.....I


105 106


101 102 103 104
Time (sec)


Figure 11. The approximate behavior of the secondary hardening component of the
lower yield stress increase (AoH= OExt ao).


I I I i 1 M I1 I I I'll I IiP r I . .









time behavior is exhibited was calculated on the basis of the respective

times to achieve a stress increase of 15.0 and 20.0 MPa. On this basis

the activation energy for the return of the lower yield stress in Nickel

200 is 25.2 3.2 and 26.4 2.7 kcal/mole.

The activation energy for the development of a 0.6% LUders

extension is similar, 24.3 kcal/mole. In addition, the shift of the

peaks of Ao versus log t is consistent with an activation energy of

approximately 22 kcal/mole. The downward trend of the aging curves behaves

in a manner corresponding to an activation energy of 29.0 kcal/mole on

the basis of the method of cuts at Ao= 23 MPa.


3.5 The Dependence of Ao and EL on Prestrain

To more fully characterize static strain aging, a series of

specimens were prestrained various amounts and then aged at 4480K for

a fixed time of 6000 seconds, the approximate time required to achieve

the maximum Ao at this temperature when the prestrain was 5% (see

Figure 7).

Figure 12 illustrates the dependence of Ao and EL on the amount

of deformation at a constant aging time and temperature. The Ac curve

shows that this parameter increases with prestrain, exhibits a broad

maximum and then decreases slowly. It is interesting that Stage III

of the work hardening behavior (see Section 3.8), as determined from

log O versus log o plots, begins at approximately 18% true strain at

2730K. Stage III is normally associated with dynamic recovery and in

view of the broad peak and subsequent decrease in Ao, it is possible

that Ao is reflecting the dynamic recovery.












1 30- 6

b 1.2

25 -

2 .8


20-

.4

15 IIIIII
0 5 10 15 20 25 30
Engr. Strain (%)


Figure 12. The dependence of Ao and EL on prestrain. Nickel 200 specimens were
prestrained at 2730K, aged for 6000 seconds at 4480K, and restrained
at 273'K.









The Liders extension increases continuously with prestrain as

indicated in Figure 12. The LUders strain is determined not only

by the size of the lower yield stress but also by the magnitude of

the work hardening rate. The latter decreases continuously with strain

and tends to make EL increase with strain. The fact that EL continues

to increase with E in Figure 12 is probably due to this cause. This

is similar to the case of Type A Luders bands which exhibit an increase

in LUders strain during plastic deformation [122].


3.6 Comparison of Nickel 270 and Nickel 200 Static Strain Aging

Several experimental observations indicate that the higher purity

Nickel 270 does not contain sufficient carbon to give rise to measurable

dynamic strain aging phenomena. Specifically, the Portevin-Le Chatelier

effect was not observed in this metal. Also, even at the highest

temperatures investigated, yield points or yield plateaus were not

observed in annealed material. Thus, Nickel 270 may be nearly repre-

sentative of pure nickel in terms of its mechanical properties.

To test if static strain aging is weakly exhibited in Nickel 270

a specimen was prestrained 5% at 273K and aged for 1200 seconds at

4730K. The resulting curve shown in Figure 13a appears to indicate some

aging since a short yield plateau is exhibited. However, the lower

yield stress increase for this specimen was only 4.76 MPa which is small

compared to the 23.00 MPa value for the commercial purity Nickel 200

specimen (Figure 13b) which was achieved in only one-half this aging time.

It is also possible that a significant portion of the 4.76 MPa yield

effect that has been observed in face-centered cubic metals [17,123].

Thus, one may be reasonably assured that strain aging phenomena in

Nickel 270 are generally weak.









Nickel 270


1200 s/4730


3s


Nickel 200


-127 N 600s/473.


P

t


5% Pre-strain


Figure 13. (a) The yield return of a Nickel 270 specimen aged for a time to achieve a
maximum in Ao for Nickel 200. (b) Yield return for a Nickel 200 specimen
aged only one-half as long.








3.7 The Stress-Strain Behaviors

The details of the basic mechanical behavior of the high purity

nickel, Nickel 270, are shown in Figures 14-16. Those for commercial

purity Nickel 200 are shown in Figures 17-22. Nickel 200 unlike

Nickel 270 exhibits serrations and yield points.

As indicated in Figure 21, the Portevin-Le Chatelier effect was

observed over four orders of magnitude of strain rate in Nickel 200.

The figure also shows the approximate temperature intervals over which

Types A, B and C serrated flow were observed. Type C serrations were

sudden load drops appearing at regular intervals on the load-time

curve. The serrated flow intervals correspond closely to those observed

by Nakada and Keh [51] in nickel-carbon alloys indicating that the

presence of the manganese in Nickel 200 does not appreciably affect the

dynamic aging effects. The data of the present investigation are not

extensive enough to calculate the apparent activation energy for the

onset of serrations in Nickel 200 with accuracy. However, the data

appear to be consistent with the apparent activation energy 152 kcal/mole

calculated by Nakada and Keh for the somewhat purer alloys [51]. In

addition, the apparent activation energy for the disappearance of

serrations calculated by Nakada and Keh [51], 264 kcal/mole, also

appears to be reasonable for Nickel 200.

Figure 14 shows representative stress-strain curves for Nickel 270

obtained at several temperatures. The 0.2% offset flow stress and the

ultimate tensile strengths of Nickel 270 are plotted in Figure 15 as

functions of the temperature. Note that the ultimate stress decreases

monotonically with temperature without any undue irregularity. This

type of stress-temperature variation is characteristic of a metal which

















E 12

S10-
N
0 77*K
8-


R.T.

700"K


0 0.1 0.2 0.3 0.4 0.5


Figure 14. True stress-true plastic strain curves for Nickel 270 4.2 x
Figure 14. True stress-true plastic strain curves for Nickel 270 (e = 4.2 x I0-4 s1).
















































0 100 200 300 400 500 600 700 800 900
T (K)


Figure 15. Variation of the 0.2% yield stress and the ultimate
tensile strength with temperature in Nickel 270
(E = 4.2 x 10-4 s-l.





















NICKEL 270
00-
Total
o Uniform


60 0


50 -


40 -

0
30-


20-


10-



0 100 200 300 400 500 600
T('K)


C
0
0


700 800 900


Figure 16. Variation of the uniform and total elongation with
temperature in Nickel 270 (E = 4.2 x 10-4 s1).








does not exhibit pronounced dynamic strain aging. The total and

uniform elongations are illustrated in Figure 16 and show no anomalies.

A minimum in ductility was observed at 8350K and surface cracking was

noted on the specimens. Cracking also appeared on the specimen tested

at 8500K. The Nickel 270 specimens were annealed at 8680K and this

temperature was the upper testing limit. The elongations are reasonably

constant over a wide range of temperature (approximately 200 to 6500K).

A representative sample of Nickel 200 stress-strain curves are

shown in Figure 17. It should be noted that the curves at 300 and 5250K

were serrated. Only average stress-strain behavior can be shown in

these cases. Note that at 525K the curve shows an anomalously high

ultimate strength. The enhanced strengthening during work hardening

is best illustrated in Figure 18 which shows dependence of the 0.2% flow

stress and the ultimate stress on the temperature.

Because of the scale of the drawing in Figure 18 the 0.2% offset

stress appears to decrease monotonically with temperature. However,

a plot of the 0.2% offset stress for Nickel 200 at two strain rates with

the stress axis expanded as in Figure 19 shows that there is a small

yield stress plateau between approximately 300 and 4750K. This is

generally characteristic of dynamic strain aging in bcc and hcp metals.

The stresses were not normalized with respect to the elastic modulus, as

is customary, because nickel exhibits a large magnetostriction [124,125] and the

choice of modulus is uncertain below 626K, the Curie point [126].

This plateau is weakly exhibited compared to that of titanium [127-129],

for example. In comparison to the 0.2% stresses observed in Nickel 270,

Nickel 200 exhibits a larger temperature dependence (compare Figures 15

and 18).












NM 12-

S 10- 77. K
5250K
S 8 2K 192"K



S----------- 700"K
4-
850*K



0 0.1 0.2 0.3 0.4
Ep



Figure 17. True stress-true plastic strain curves for Nickel 200 (c = 4.2 x 10-4 s-1).















Nickel 200


* 4.2 x 10-5 sec-1
-4 -1
* 4.2 x 104 sec1
0 4.2 x 10-3 sec-
S4.2 x 10-2 sec-


0-^






es \ \


400 -


0 0.2 I
I. 0-
* I I I I. -


200 400


600 800


Figure 18. The temperature dependence of the 0.2% yield stress
and ultimate tensile strength of Nickel 200.


12001-


1000-


True
Ultimate


_ I I 1 I I 1










200




150-




100-




50





0


I I *
Nickel 200
4.2 x 10-4 s"
o 4.2 x 10'3 s"















S 1 I I


200 400
T


~eo ~800


Figure 19. The temperature and strain rate dependence of the
0.2% yield stress in Nickel 200 on an expanded
stress axis.


I









An important feature to note in Figure 18 is that anomalous

strengthening is exhibited between 300 and 6000K. In Figure 20 the

strain dependence of the flow stress is shown for 5, 11, 19, and 31%

plastic strain. This figure demonstrates that only above approximately

11 to 19% plastic strain does the anomalous strengthening become

significant. That is, the strengthening effect is not due to anomalous

work hardening as Sukhovarov and Kharlova [67] previously suggested.

Note that unlike the behavior of the ultimate tensile strengths plotted

in Figure 18, the stress levels attained at 31% strain exhibit a rate

dependent shift in the peaks. These peaks have a rate dependence

corresponding to an activation energy of approximately 38 kcal/mole.

Figure 21 illustrates the temperature dependence of the uniform

and total elongations in Nickel 200 at several strain rates. The total

elongation at a strain rate of 4.2 x 10- s-l shows a mild ductility

minimum (blue-brittle effect) between approximately 300 and 450K. This

minimum is not well-defined. However, a well-defined but small reduction

in area minimum does occur in Nickel 200 that is strain rate dependent

as shown in Figure 22. This minimum was noted to have shifted in

accordance with an apparent activation energy of approximately 25 kcal/mole.

A reduction in area minimum is not always manifested in dynamic strain

aging [130]. It is interesting to note, however, that 25 kcal/mole is

approximately the activation energy for vacancy migration in nickel [109,118].

Not only is the loss in reduction in area not marked, but the lowest

value recorded is still above 75%. Adjunct observations of the fracture

surfaces under a low-power microscope did not reveal any striking

difference in fracture mode at the reduction in area minimum.












700

\E19%
600- N.-- .


S500 -


400--


300 5 %


200 -

4.2 x o10 s'
10 o 4. x 103 s-
4.2 x 10 s

0 100 200 300 400 500 600 700 800 900
T


Figure 20. Variation of the stresses at 5, 11, 19, and 30% plastic strain with
temperature in Nickel 200.
















Nickel 200
4.2 x 10 sec'
4.2 x I0 sec' -
o 4.2 x 10-3 sec'-
s80 4.2 x 10-2 sec-


: ./ ,V/ Total

G e- '- o
\Q N. A'--- ,
G '. "
40

B .C. \" Uniform -

20 L A iABC
A A+B
A
......................
1 --- -------1 --- --I---'-

0 200 400 600 800
T

Figure 21. Variation of the uniform and total elongations with
temperature in Nickel 200. Also shown are the
approximate temperature ranges over which serrations
were observed at the respective strain rates.















00-

98

96- Ni 270

94 ^ 4.2 x 1064s-

92 A

90

s88 Ni 200

8s .. ..... .. 0. / 42 x 1O- .s-

\. *\ 4.2 x lO s-1

\'\o I\
80 -



78 \ -

76

740 20 40 -O
0 200 400 500 SOO !000


Figure 22. Variation of the reduction in area
for Nickel 200 and Nickel 270.


with temperature








Note that the higher purity Nickel 270 shows more ductility at

all temperatures than does Nickel 200. Although a mild reduction in

area minimum does occur in Nickel 270, it is spread over a wide

interval between approximately 425 and 750K and is not as pronounced

as that in Nickel 200. Even the lowest reduction in area observed in

this investigation is considerably higher than that in most commercially

available bcc metals and alloys.

The strain rate sensitivity in Nickel 200 and Nickel 270 was not

investigated because the temperature interval of serrated flow in

Nickel 200 was some 3000K wide and it was believed that strain rate

changes conducted during discontinuous plastic flow would prove in-

conclusive. It was noted, however, that at room temperature during

moderately heterogeneous plastic flow, changes in rate resulted in the

appearance of flow stress transients in Nickel 200 and a steady state

strain rate sensitivity very close to zero was observed.

In summary, whereas the higher purity Nickel 270 shows no anomalies

in ultimate strength and elongation with increasing temperature, Nickel 200

between 300 and 6000K shows anomalous strengthening, serrated flow, a

small yield stress plateau and a mild elongation minimum.


3.8 The Work Hardening Behavior of Nickel 270 and of Nickel 200

Figures 23 and 24 show a cross-section of the log 0 versus log o

curves for Nickel 270 and Nickel 200, respectively, deformed at a

nominal strain rate of 4.2 x 10-4 s.1 These curves satisfactorily

represent the general trend of work hardening at all strain rates

investigated. No attempt has been made to draw in the straight lines

representing the stage behavior in order to reduce the complexity of

the figures.

















0'0
o o
.. Oo

%


95








CD 9.0

(9
0
-J


850K 800*K


S (Stress in N/n)


7.6 7-8 8 0 8.2
LOG 0-


8.4 8.6 8.8
8.4 8.6 8.8


Figure 23. The log 0-log a curves of Nickel 270 (e = 4.2 x 10-4 s- ).


NICKEL 270


4 ,
* 4

0 4

0a *192-K
S R.T
S 550'K


.700 K


* 0.


8.5-


77*K

*~











NICKEL 200
= 4.2x i04 s"7


\ee sooo" t **. e o
A,

0 o0
0 0o


(Stress in MPa)


0o 0 0 o
o.

7Ooo





525AK .
A On








o '650*K


o750*K


* 850*K


7.8 8.0 82 8-4 8-6
LOG (o


8.8 9.0


Figure 24. The log 0-log a curves of Nickel 200 (- = 4.2 x 10-4 s ).


IIIIIIIIII








The work hardening behavior of Nickel 270 and of Nickel 200 are

similar to those found earlier in pure face-centered cubic metals. The

stages appear in the same manner as Zankl [75], Schwink and Vorbrugg [77],

and others [76,78-81] have shown to be the case for pure face-centered

cubic metals. It should be noted that hexagonal close packed metals

such as zirconium and titanium [131] as well as body-centered cubic

metals such as iron and niobium [132] show much higher m values (order

of 7 to 40) than the fcc alloys presently being investigated. That is,

all log o versus log a plots for these metals show much steeper slopes.

Also, these metals tend to show only one or two stages of deformation

behavior indicating that the deformation in these metals is possibly

controlled by a different set of deformation phenomena.

Figures 25 and 26 show the m values obtained for Nickel 270 and

Nickel 200 by measuring the slopes of log 0 versus log a plots at

different temperatures. The error of each particular mil value is

approximately 0.1 as measured by the plausible maximum and minimum

slopes that might conceivably characterize a particular work hardening

stage. It should also be noted that Stages I and III are difficult

to characterize in many cases. The parameter minI is plotted in the

figures only to show trends in the third stage as a function of temperature.

They are not accurately defined since in Stage III log O-log o plots

are not linear but curved. However, Stage II is generally uniquely

defined by a straight line on log O-log a curves.

Nickel 270 and Nickel 200 possess mil values which are close to 1.5

as shown in Figures 25 and 26. Note that in Nickel 200 mlI remains

constant over a wide range of temperatures and is strain rate independent.









NICKEL 270
8 ml











E 4 m







1MA




0 100 200 300 400 500 600 700 800 90S
T (


Figure 25. The variation of mi[ and milj with temperature in Nickel 270 (E = 4.2 x 10-4 s-1.







7- "' 4.2 x 10-5-'1
7 4.2 X 10-48-1
Nickel 200 o 4.2 x Io-33



5
0 f0
0 -
0 0 -- o
4o _- o-- o m0


2 *n




---" ^- ^





0 200 400 600 800 1000

T


Figure 26. The variation of m11 and mlII with temperature in Nickel 200
(E = 4.2 x 10-4 s-1).








It is important to emphasize that the apparent work hardening peak

observed in Nickel 200 (Figures18 and 20) is not a true work hardening

peak in the sense that Nickel 200 stress-strain curves show anomalous

increases in work hardening over a uniquely defined temperature interval.

The peak is associated with an increase in the uniform elongation

around the temperature of the ultimate stress peak. Figure 27, a plot

of (o5%-0.5%) versus temperature, readily shows this behavior. This

parameter (where o5% and o0.5% are the true stresses at 5 and 0.5%

true plastic strain, respectively) confirms that there is little

temperature dependence of the work hardening. The constancy of mil in

Figure 26 also substantiates this observation. Figures 28 and 29

show very clearly that the onset of Stage III is delayed over exactly

the same temperature interval as develops the peak in ultimate stress.

These facts imply that there is no prominent work hardening peak in

nickel as earlier authors have indicated [67]. There is, however, a

significant delay in the onset of Stage III or of dynamic recovery.

In Stage III deformation in both a polycrystalline or single

crystal fcc metal or alloy, the stress-strain curve shows a high

curvature, i.e., work hardening decreases very rapidly with continued

deformation. Thus, Stage III m values tend to be high. Figures 25 and

26 show that mIII increases with increasing temperature. This is

consistent with the concept that dynamic recovery becomes more important

with increasing temperature.

Now consider the extent or length of Stage II which is not illus-

trated well in the log 0-log o diagrams. In purer metals such as

Nickel 270 it is observed that Stage II, which is governed primarily by

slip on single slip systems within each grain [75], is lengthened by
























0 100 200 300 400 500 600 700 800 900
T (K)

Figure 27. The variation of the work hardening parameter (5%-' 0.5%) with temperature
(E = 4.2 x 10-4 s-).








the presence of interstitial carbon as in Nickel 200. Figure 28

compares the approximate strain at which Stage II deformation begins,

E2, and that at which Stage III begins (and Stage II ends), E3, for

Nickel 200 and Nickel 270. The principal effect of interstitial

carbon in the DSA interval in this respect is to postpone Stage III

dynamic recovery processes to a later time in the deformation history.

Also the general level of the Nickel 200 curve is higher than that

for Nickel 270. It is not clear at this time how the carbon accounts

for the general delay in Stage III.

Nickel 200 exhibits an e3 maximum over the dynamic strain aging

interval. This maximum is closely related to the maximum exhibited

in the uniform elongation (Figure 21) and accounts for the strengthening

observed over this temperature interval. This maximum in e3 is strain

rate dependent as shown in Figure 29. Not only do the peaks shift to

higher temperatures with increasing strain rate but the peak heights

decrease as well. This is similar to the behavior of the flow stresses

described in Figures 19 and 21. The shift in peaks follows an approximate

activation energy of 37 kcal/mole, similar to that obtained for the

flow stress peaks.











50- 3 NICKEL 270
o C2

1 3 NICKEL 200

(All strains ore true plastic strains)
40 -


201-


I s? ,. .= Z z--= ..... __ _o" o ....
0 100 200 300 400 500 600 700 800 900
T(oK)


Figure 28. The variation of E2 and E3 (the approximate strains
at which Stages II and III, respectively, begin) with
temperature for Nickel 270 and Nickel 200 deformed at
a strain rate of 4.2 x 10-4 s-1.







I I i I I- I I I I


Nicke
* 42 x
* 4.2 x
o 4.2 x
> 4.2 x


401-


1200
id5 s'

102cs,
lo-Vf


30-


20-


F F I I F


0 200 400
T


600 800


Figure 29. The dependence of e3 on temperature and strain rate
in Nickel 200.













CHAPTER IV

DISCUSSION


4.1 Rationale for Static Strain Aging in Nickel 200

The static strain aging data for Nickel 200 aged between 373 and

4730K after a 5% prestrain indicate:

1. The activation energy for the return of the lower yield stress

for times before the maximum in Ao is approximately 253 kcal/mole.

2. The kinetics of the rise in the lower yield stress can be

described by a In t time law.

According to the previous discussion three principal causes for

strain aging in other metals have been identified: (a) Cottrell pinning,

(b) Suzuki locking and (c) Snoek pinning.

It would appear on the outset that aging in Nickel 200 is such that

Cottrell pinning must be ruled out as a candidate mechanism. First,

the kinetics for aging as described by Cottrell and Bilby [9] are t2/3

In Nickel 200 the rise in Ao with time is much slower, approximately t /7

this is very different from the Cottrell-Bilby time law. Secondly, the

activation energy deduced to govern the early stages of pinning in

Nickel 200 is not consistent with the migration energy of carbon, the

principal impurity in Nickel 200, as the Cottrell-Bilby model would

suggest. The migration energy of carbon in nickel is approximately

35 kcal/mole (see Table 2) and thus significantly different from the

observed activation energy. Accordingly, the random migration distance








for carbon at all temperatures investigated during the time required to

cause maximum strengthening is only about 8b which is a very short

diffusion distance compared to distances involved in the formation of

a Cottrell atmosphere.

Suzuki pinning is not believed to be responsible for the observed

strain aging behavior in Nickel 200 primarily because the stacking fault

energy of pure nickel is very high. Hence, the probability of forming

stable faults in nickel is very low. It is believed that carbon

impurities do not lower the stacking fault energy enough to have a

significant effect with regard to this mechanism.

Snoek pinning by itself cannot account for the rise in Ao unless

anisotropic defects exist in the material in sufficient numbers and

have a sufficiently high interaction energy with dislocations. As will

be demonstrated the carbon-vacancy pair can satisfy these requirements.

However, Snoek pinning should occur in times, very short at the temperatures

investigated, compared to the times observed for the rise in Ao.

The formation of ordered carbon-vacancy pair atmospheres near

dislocations can account for the observed static strain aging behavior.

The very long times required to reach the maximum in Ao can be rationalized

in terms of a stress assisted migration of vacancies toward dislocations

where enhanced trapping by carbon occurs. The result is an accumulation

of ordered carbon-vacancy pairs near the dislocation and the concommitant

growth of an ordered carbon-vacancy pair atmosphere. In short, the

proposed model utilizes aspects of both the Cottrell-Bilby and the

Schoeck-Seeger theories of static strain aging.

In summary the following stages are envisioned: (a) At the end of

deformation vacancies diffuse to and become trapped in large measure








by adjacent carbon atoms. (b) The carbon-vacancy pairs thus formed

undergo Snoek ordering within the stress fields of the dislocations.

(c) Then a slow migration of vacancies toward the dislocation develops

an atmosphere of carbon-vacancy pairs near the dislocation.

(d) Eventually the vacancy concentration decreases because of losses

to sinks. This reduces the pair concentration and causes an eventual

loss of strengthening. These stages will now be considered in detail.


4.2 The Mechanism of Static Strain Aging Exhibited in Nickel 200


4.2.1 The Distribution of Vacancies, Carbon Atoms and Dislocations
After Plastic Deformation

After the prestrain of 5%, the vacancy concentration in nickel

is approximately 10-5 atomic fraction as given by Eq. 9. The

particular heat of Nickel 200 used in this investigation contains

approximately 0.1 w/o or 0.5 a/o carbon. Thus, at 5% strain the carbon

to vacancy ratio is approximately 500 to 1. If the carbon and the

vacancies are distributed at random throughout the nickel lattice, then

the mean carbon atom spacing will be approximately 6b; the mean vacancy

spacing will be approximately 50b.

After plastic deformation to 5% strain, the dislocation density [133,134]

is estimated to be between 3 and 8 x 109 cm-2. Assuming an equidistant

array of straight dislocation lines as a first approximation to the

dislocation configuration, then their mean spacing is approximately

450 to 730b.


4.2.2 Vacancy Trapping by Carbon Atoms

In the absence of other defects such as grain boundaries or

dislocations, vacancies in a metal are attracted to oversized impurity








atoms by a hydrostatic pressure gradient. The strain energy released

when a vacancy is moved from an infinite distance away from an impurity

to the impurity is known as the binding energy of the vacancy-impurity

pair. This phenomenon is commonly called trapping and it is generally

accepted that impurity atoms can trap vacancies in metals. For example,

it has been shown that the presence of carbon in austenitic stainless

steel forestalls radiation induced swelling in certain critical nuclear

reactor parts. This indicates that void formation is hindered by the

presence of carbon due to trapping of vacancies [135]. Trapping of

carbon by vacancies in irradiated mild steel has been demonstrated as

well [136].

4.2.3 The Concentration of Carbon-Vacancy Pairs

After deformation in Nickel 200 and before aging of a specimen,

it is reasonable to assume that vacancies become trapped by carbon

atoms initially. This probably occurs very rapidly even at 3730K, the

lowest aging temperature used, since the vacancies do not have to travel

far to become trapped by carbon atoms. Vacancies in nickel are very

mobile as evidenced by their approximate 0.8 to 0.9 eV migration energy

and assuming that an equilibrium is established between the carbon-vacancy

pairs and free vacancies, their concentrations may be calculated on

the basis that the binding energy of the carbon-vacancy pair can be

estimated to be 0.2 to 0.3 eV per pair from diffusion data (Section 1.4).

For the moment, it is assumed that the dislocations in the metal do not

affect the concentrations. The concentrations of single (free) vacancies

(c ) and bound vacancies in carbon-vacancy pairs (c ) vary with carbon

concentration (Cc), temperature T, and carbon-vacancy binding energy (B)








according to the equations [116]

ccv = Z cf eB/kT (11)

and

cv cf + (12)
v= CC V

where cv is the vacancy concentration generated during plastic defor-

mation, cc is the free carbon concentration and Z is 6, the nearest

neighbor coordination number for a carbon atom in an octahedral site

of the fcc lattice.

Substituting Eq. 12 into Eq. 11 gives

cc = Z(cc-Ccv)(cvcv )eB/kT (13)

For c = 5 x 103 and c = 105 one may write on the basis that
c v
cc>>Cv,


Ccv = Zcc(c -ccv)eB/kT (14)


Substituting a value for B equal to 0.3 eV and a temperature of

408K into Eq. 14 gives the carbon-vacancy pair concentration of

9.93 x 10-6, very close to the total vacancy concentration of 10-5.
-R
The free vacancy concentration from Eq. 12 is then 6.92 x 108 atom

fraction. Thus, almost all (99.3%) vacancies are bound to carbon atoms

based upon the choice of B.

In the absence of carbon atoms the concentration of vacancies at

thermal equilibrium is given by

-Qf/RT
cv e (15)








where Qf is the vacancy formation energy. At 4080K and a Qf of

approximately 39 kcal/mole (1.7 eV) [118] the vacancy concentration

is approximately 1.8 x 1021 atom fraction. This is many orders of

magnitude smaller than the free vacancy concentration calculated above

and suggests that the deformed metal is not in its lowest energy state

with regard to numbers of vacancies. Thus, there exists a distinct

tendency for vacancies to migrate to sinks. Because of the high binding

to carbon atoms, vacancies spend a large portion of their migration time

bound to carbon atoms and, hence, the process of annealing to dislocations

and grain boundaries is sluggish and takes a considerable time.

Consider next the influence of a dislocation on the carbon-vacancy

pair concentration. It has been demonstrated that the carbon-vacancy

pair may be visualized as an elastic dipole which can reorient in the

strain field of a dislocation. In Eqs. 11-14, the binding energy, B,

is completely general. That is, in the specific case which includes

Snoek ordering, B becomes position dependent. Specifically,


B = B' + u(r) (16)


where B' is the binding energy of a carbon-vacancy pair and u(r) is

the position dependent interaction energy of a carbon-vacancy pair with

a dislocation. Thus, the effect of u(r), which varies as r- is to

enhance the binding of vacancies to carbon atoms near the dislocation.

This will be explained in detail presently.


4.2.4 Theory of Schoeck Locking by Carbon-Vacancy Pairs

In the case of nickel-carbon alloy, the carbon-vacancy pair acts

as a dipole (Section 1.4). Appendix A presents a calculation of the








approximate interaction energy between an assumed carbon-vacancy

dipole and a screw dislocation in a fcc lattice. The interaction

calculation related to an edge dislocation is more complicated [137].

However, as indicated first by Nabarro [138] and later expanded upon

by Cochardt et al. [137], the strength of interaction between a dis-

location and impurity atoms may be assumed to be very similar whether

the dislocation is in the screw or edge orientation.

Let us consider the case of locking of a screw dislocation by an

ordered atmosphere of dipoles in the fcc lattice. The reasoning is

analogous to that for the calculation made by Schoeck and Seeger [28]

for carbon (or nitrogen) in alpha iron. However, the interaction

energy for a dipole in a fcc lattice as shown in Figure 35 in Appendix A

is somewhat different. The carbon atoms in nickel are assumed to occupy

octahedral interstitial sites, e.g., the body-centered position of the

fcc unit cell. The six nearest neighbors are face-centered atoms. A

dipole is formed when a vacancy is situated on a face-centered position

as schematically shown in Figure 30. Thus, the carbon-vacancy dipole

may be oriented in one of three possible 100 type directions. These

positions are denoted by an orientation number 1, 2 or 3. The dislocation

line is assumed to lie along one of four 101 type directions.

It is shown in Appendix A that for the specific case of a screw

dislocation, each of the three possible orientations of the carbon-

vacancy dipole interacts differently with the dislocation. This gives

rise to three possible interaction energies and one of the three

orientations is most stable. That is, a carbon-vacancy dipole in,

say, the 1-orientation can flip to another orientation (say, 2) and

the result is that the interaction energy between that particular dipole
















































Figure 30. A schematic illustration of the assumed configuration
of the carbon-vacancy pair and the three possible
independent orientations that it may assume.




Full Text

PAGE 1

STRAIN AGING IN NICKEL 200 By WALTER RAYMOND CRIBB A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1975

PAGE 2

/ERSITY OF FLORIDA 3IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 3 1262 08552 4618

PAGE 3

To Mom and Dad

PAGE 4

ACKNOWLEDGMENTS Sincere appreciation is due many people in this department for their help during my entire stay at the University of Florida. Most sincere thanks are due Professor Robert E. Reed-Hill whose continued guidance and encouragement made this dissertation possible. Many thanks to Professor F.N. Rhines who first encouraged me and gave me confidence to strive for a higher degree in metallurgy and whose continued interest in my program is appreciated. I would also like to thank the members of my committee, Drs. Martin A. Eisenberg, Craig S. Hartley and John J. Hren for fruitful discussions of my work. Many thanks to my colleagues Messrs. Juan R. Donoso, R.M. Chhatre, Francisco Boratto and to the laboratory assistants C. Barnes and M. Brimanson who spent many hours of discussion and who cooperated in the collection and interpretation of experimental data. The preparation of the final manuscript by Elizabeth Beville is also greatly appreciated. The financial support of the Army Research Office (Durham), the International Nickel Company, and the Energy Research and Development Administration is greatly appreciated. Finally, I thank my wife, Kathie, whose patience, encouragement and understanding during my course of study helped make it all possible. iii

PAGE 5

TABLE OF CONTENTS PAGE ACKNOWLEDGMENTS i i i LIST OF TABLES vii LIST OF FIGURES viii ABSTRACT xi INTRODUCTION 1 CHAPTER I PREVIOUS INVESTIGATIONS 4 1.1 Static Strain Aging 4 1.1.1 Historical Aspects 4 1.1.2 Mechanisms of Static Strain Aging in Metals Alloys 5 1.1.3 Summary of Important Mechanisms of Dislocation Locking During Aging 10 1.1.4 Aspects of the Static Strain Aging Experiment.. 11 1.1.5 Static Strain Aging Stages in BCC Metals 13 1.1.6 Static Strain Aging in Nickel and FCC Alloys... 17 1.2 Dynamic Strain Aging 19 1.3 Work Hardening in Metals and Alloys 23 1.4 Anelastic Phenomena in Nickel and FCC Ferrous Alloys.. 27 1 1 EXPERIMENTAL PROCEDURES 35 2.1 Materials 35 2.2 Experimental Techniques 37

PAGE 6

PAGE 2.2.1 Swaging 37 2.2.2 Annealing 37 2.2.3 Specimen Profile Measurements 37 2.2.4 Tensile Testing 38 2.2.5 Static Aging Experiments 38 III EXPERIMENTAL RESULTS 43 3.1 The Behavior of the Lower Yield Stress Increase, Ao... 43 3.2 The LLiders Extension, e. 50 3.3 The Hardening Component, Aa„ 50 3.4 Activation Energies 50 3.5 The Dependence of Aa and e^^ on Prestrain 53 3.6 Comparison of Nickel 270 and Nickel 200 Static Strain Aging 55 3.7 The Stress-Strain Behaviors 57 3.8 The Work Hardening Behaviors of Nickel 270 and of Nickel 200 69 IV DISCUSSION 80 4.1 Rationale for Static Strain Aging in Nickel 200 80 4.2 The Mechanism for Static Strain Aging Exhibited in Nickel 200 82 4.2.1 The Distribution of Vacancies, Carbon Atoms and Dislocations After Plastic Deformation 82 4.2.2 Vacancy Trapping by Carbon Atoms 82 4.2.3 The Concentration of Carbon-Vacancy Pairs 83 4.2.4 Theory of Initial Schoeck Locking by CarbonVacancy Pai rs 85 4.2.5 The Mechanism Controlling the Increase in Aa with Time 96 4.2.6 Regarding the Behavior of Nickel 200 After the Peak in Aa 107

PAGE 7

PAGE 4.3 Summary 11^ 4.4 Comments on the Relationship Between Static Strain Aging and Dynamic Strain Aging in Nickel 200 115 121 CONCLUSIONS AND OBSERVATIONS APPENDICES A COMPUTATION OF THE INTERACTION ENERGY BETWEEN A CARBON-VACANCY PAIR DEFECT AND SCREW DISLOCATIONS IN FCC METALS "'25 B DETERMINATION OF U(x), THE ENERGY OF A SCREW DISLOCATION DISPLACED A DISTANCE x FROM THE CENTER OF ITS 5N0EK ATMOSPHERE 1 32 BIBLIOGRAPHY 135 BIOGRAPHICAL SKETCH ""^^

PAGE 8

LIST OF TABLES Table Page 1 Recognized Aspects of Strain Aging 2 2 The Diffusivity of Carbon in Nickel 21 3 Estimates of the Rate of Vacancy Production During Plastic Deformation 32 4 Alloy Compositions 36 5 Least Squares Parameters for Static Strain Aging Data Assuming Aa Is a Function of In t 45 6 Least Squares Parameters for Static Strain Aging Data Assuming a Log Aa-Log t Linear Relationship 49 7 The Slopes and Intercepts (at t = Is) of Aa Versus In t Curves Calculated from Eq. 56 108 8 Interaction Energies for Tetragonal Defects in the FCC Structure 1 28

PAGE 9

LIST OF FIGURES 1 General aspects of the classical static strain aging test '^ 2 An example of the stages of the yield return in Nb-0 alloys [8]; (a) the increase in Aa with time; (b) the components of Aa and their dependence on aging time 16 3 Schematic example of stage behavior in polycrystal line fee metals '^^ 4 Schematic description of the work hardening behavior in a metal using a log G-log a plot 25 5 Illustrating the method used to determine the aging parameters e,_, Ao^a^y " ^o ^"^ ^^H " '^Ext " ^o" ^^^^^^ loading line indicates the approximate loading line which would have been observed in the absence of misalignment of the test specimen 41 6 Selected load-time curves obtained after prestraining a series of specimens 5% at 273°K, aging at 408°K for the times indicated, and restraining at 273°K 42 7 The time and temperature dependence of the return of the lower yield stress in Nickel 200. Specimens were prestrained to a stress level of 265 MPa. The dashed curves are approximate corrected curves which account for specimen heat-up in the aging baths 44 8 Normalized aging curves for Nickel 200. The 373°K curve was normalized to an assumed maximum of 28.5 MPa. The dashed curves reflect approximate corrections for the heat-up time of the specimens 47 9 Illustrating the approximate t ' power law relation governing the aging of Nickel 200 at temperatures below 448°K. Data for the 448°K (shown) and 473°K cases do not fit this relation well 48 10 The dependence of the Luders strain on time and temperature in Nickel 200 51

PAGE 10

Figure Page 11 The approximate behavior of the secondary hardening component of the lower yield stress increase (^^H =<^Ext -%) 52 The dependence of Aa and z. on prestrain. Nickel 200 specimens wereprestrained at 273°K, aged for 6000 seconds at 448°K, and restrained at 273°K 54 12 13 (a) The yield return of a Nickel 270 specimen aged for a time to achieve a maximum in ^ofor Nickel 200. (b) Yield return for a Nickel 200 specimen aged only one-half as long 56 14 True stress-true plastic strain curves for Nickel 270 (e = 4.2 X 10-4 s-1) 58 15 Variation of the 0.2% yield stress and the ultimate tensile strength with temperature of Nickel 270 (e = 4.2 X 10-4 s-l) 59 16 Variation of the uniform and total elongation with temperature in Nickel 270 (e = 4.2 x 10-4 s""") 60 17 True stress-true plastic strain curves for Nickel 200 (£ = 4.2 X 10-4 s-1) 62 18 The temperature dependence of the 0.2% yield stress and ultimate tensile strength of Nickel 200 63 19 The temperature and strain rate dependence of the 0.2% yield stress in Nickel 200 on an expanded stress axis .... 64 20 Variation of the stresses at 5, 11, 19 and 30% plastic strain with temperature in Nickel 200 66 21 Variation of the uniform and total elongations with temperature in Nickel 200. Also shown are the approximate temperature ranges over which serrations were observed at the respective strain rates 67 22 Variation of reduction in area with temperature for Nickel 200 and Nickel 270 68 23 The log 0-log a curves of Nickel 270 (e = 4.2 x 10" s" ) . 70 24 The log 0-log a curves of Nickel 200 (e = 4.2 x 10"^ s"^) . 71 25 The variation of m„ and m „ with temperature in Nickel 270 (e = 4.2 x 10-4 s""") 73 IX

PAGE 11

Figure Page 26 The variation of m,, and m,,, with temperature in Nickel 200 (e = 4.2 x lO'^ s"'' ) 74 27 The variation of the work hardening parameter i^c^o^^-^Q 50/) with temperature (g = 4.2 x 10" s~ ) 76 28 The variation of e^ ^"^ *^3 (^^^ approximate strains at which Stages II and III, respectively, begin) with temperature for Nickel 270 and Nickel 200 deformed at a strain rate of 4.2 x 10 s 78 29 The dependence of e^ on temperature and strain rate in Nickel 200 79 30 A schematic illustration of the assumed configuration of the carbon-vacancy pair and the three possible independent orientations that it may assume 87 31 The concentrations of dipoles in each of the three possible orientations (B' 0.3 eV (6900 kcal/mole), A = 0.2 eV (4600 kcal/mole)) 91 32 Schematic illustration of the growth of a saturated carbon-vacancy atmosphere. R is time dependent and the concentration within R is assumed to be a fraction, f, of the carbon concentration 99 33 The aging curves obtained from the model for strain aging in Nickel 200 (Eq. 56); the dashed lines are the experimental data (Figure 7) 109 34 This diagram illustrates the aging stages of Nickel 200. The solid line represents the experimental scope of the present investigation 116 35 The interaction potential u . of a carbon-vacancy dipole with a screw dislocation for r = b and A/b = 0.2 eV 131

PAGE 12

Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy STRAIN AGING IN NICKEL 200 By Walter Raymond Cribb August, 1975 Chairman: Robert E. Reed-Hill Major Department: Materials Science and Engineering Dynamic and static strain aging were observed in commercially available Nickel 200 which contains principally 1000 ppm carbon as an alloying impurity. Static strain aging tests were conducted on annealed tensile specimens which were prestrained at 273°K to a stress level of 265 MPa (approximately 0.05 strain) at a nominal strain rate of 4.2 X 10" s' . Under these conditions, homogeneous plastic flow was guaranteed to occur. Specimens were aged immediately after prestraining for different times at 373, 408, 428, 448 or 473°K and the time dependence of the return of the lower yield stress was observed. The return of the yield experiments indicated that Aa increased as In t or approximately as t kinetically and behaved in accordance with an activation energy of 25 kcal/mole before the observed peak in Aa. It is demonstrated that the defect responsible for this anomalous increase in Aa may be the rotation of carbon-vacancy pairs in the strain fields of dislocations. A quantitative model is derived for the increase in Aa before the aging peak and it is concluded that several important stages in the aging of Nickel 200 may occur: (a) the formation of

PAGE 13

carbon-vacancy pairs and their initial ordering, (b) the migration of vacancies in the strain energy gradients of dislocations and the consequent formation of more carbon-vacancy pairs near dislocations, (c) the growth of an ordered carbon-vacancy dipole atmosphere, (d) depletion of free vacancies in the remainder of the lattice which decreases the flux to the ordered atmosphere and results in a ho maximum, (e) the migration of bound vacancies to dislocation sinks and the resulting decrease in Aa, (f) the migration of carbon atoms in the strain fields of dislocations and the growth of a Cottrell atmosphere, and (g) precipitation of graphite during overaging. Items (f) and (g) are only speculated to occur. This model is different from the CottrellBilby model and can account for the kinetics and activation energy for strain aging observed in Nickel 200. Tensile tests were conducted between 77 and 800°K at nominal strain rates of 4.2 x 10'^, lO"^, 10""^, and 10"^ s"^ The results of these experiments confirm that dynamic strain aging (DSA) in Nickel 200 is exhibited over a temperature interval between 273 and 575°K at -4 -1 4.2 X 10 s . Over the DSA interval, the following phenomena were exhibited and depended upon the strain rate: the Portevin-Le Chatelier Effect, yield stress plateaus, ultimate stress peaks, reduction in area minima and mild ductility minima. An analysis of work hardening indicates that anomalous work hardening over the DSA interval is very weakly exhibited. The mechanism for discontinuous yielding is rationalized to be dynamic Snoek ordering of carbon-vacancy pairs during plastic deformation and can account for the anomalously low temperature interval (with respect to the expected mobility of carbon) over which DSA is observed to occur. xii

PAGE 14

INTRODUCTION Currently, eight aspects of strain aging are recognized [1] as playing a major role in the deformation of polycrystall ine metals (Table 1). The first two are static strain aging phenomena which are obtained by restraining a set of prestrained specimens that have been aged at an elevated temperature. The last six aspects listed in Table 1 are characteristic of dynamic strain aging, i.e., aging which occurs during plastic deformation. Dynamic strain aging can occur in both substitutional and interstitial alloys. The most interesting cases of dynamic strain aging have normally involved interstitial solutes in transition metals. Most research on the role of interstitial impurities in the mechanical behavior of transition metals has been conducted using the body-centered cubic class of metals such as Fe, Nb, Mo, Ta, W and V [2]. The principal interstitial impurities in these metals which are responsible for strain aging are N, 0, C and H. Nickel is the only metal of the commercially important Period IV transition series of the Periodic Table that is face-centered cubic. It is also the only fee transition metal widely used for constructional purposes. The other fee transition metals Rh, Pd, Ir, and Pt are less abundant and have not been used as major construction materials. As such, in-depth investigations of their mechanical properties have not been undertaken.

PAGE 15

TABLE 1 Recognized Aspects of Strain Aging 1. Yield Points 2. Strengthening 3. Discontinuous Yielding 4. Strain Rate Sensitivity Minimum 5. Ductil ity Minimum 6. Abnormal and Rate Dependent Work Hardening 7. Yield Stress Plateaus 8. Flow Stress Transients on Changes in Strain Rate

PAGE 16

The principal purpose of the present investigation was to characterize the strain aging phenomena of commercially available Nickel 200. This alloy contains as its principal strengthening agent solid solution interstitial carbon (0.15 w/o maximum). To the best knowledge of the author, a complete classical static strain aging investigation has never been conducted using an interstitial solid solution fee alloy. A prime goal in the present investigation was to develop a quantitative model that could explain the kinetics and energetics of the return of the lower yield stress. Furthermore, the tensile behavior of Nickel 200 during constant strain rate tests conducted over a wide range of temperatures and strain rates was investigated in order to better define the dynamic strain aging phenomena in Nickel 200.

PAGE 17

CHAPTER I PREVIOUS INVESTIGATIONS 1 . 1 Static Strain Aging 1.1.1 Historical Aspects It has been recognized for a long time that the yield phenomenon in iron and other bcc metals is closely related to the presence of interstitial impurity atoms such as carbon or nitrogen. Most strain aging investigations have centered about the iron and steel industry since the 1930s when the phenomenon in low carbon steel first became a major commercial nuisance. The first metallurgical investigation of aging in mild steel was conducted by Davenport and Bain [3] in 1935, who noted that heterogeneous flow occurred in both annealed and deformed materials after having been "aged" by storing before working. Subsequent work by Gensamer and Low [4] in 1944 related the strain aging and yield point to the presence of trace amounts of nitrogen and carbon. Since the time of these early investigations, much interest has continued to center on iron and other commercially significant body-centered cubic metals such as vanadium [5], niobium [6], tantalum [7], and molybdenum [8]. At the present time, very little research effort has been directed to the study of static strain aging phenomena in face-centered cubic and hexagonal metals containing interstitial impurities.

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1.1.2 Mechanisms of Static Strain Aging in Metals and Alloys Three main dislocation pinning mechanisms have been postulated on the basis of experimental evidence in metals and alloys. These are Cottrell pinning, Suzuki locking and Schoeck locking. In. all of the previously mentioned investigations of bcc metals, the most plausible explanation for static strain aging is due to Cottrell and Bilby [9] who attributed the effect to the diffusion of interstitial atoms in solution (e.g., carbon, nitrogen, oxygen or hydrogen) to dislocations. Their concept relates the increase in flov/ stress and yield point return after aging to the migration of solute atoms to the tensile region about an edge dislocation. The effect of this segregation is to locally lower the strain energy of the system and to consequently stabilize the dislocation to the point where an increased flow stress is required to renobilize the dislocation or to generate mobile dislocations. The Cottrell mechanism is of major importance in causing the return of the sharp yield point in steel while an increase in the steel's ability to work harden and a reduction in du-'.tility (in the later stages of aging) are probably associated with precipitation of carbides and nitrides [10]. The major contribution of the Cottrell -Bilby work was to solve the problem related to the diffusion of an interstitial atom in the stress field of a dislocation. The solution predicts the timetemperature dependence of the rate of impurity migration as inferred from internal friction measurements [11]. Cottrell and Bilby derived the following relationship for n(t), the number of atoms arriving at the dislocation in the time t per unit length, n(t) = 3 (f)l/3 „^(aM)2/3 (,,

PAGE 19

where n is the average number of solute atoms per unit volume and the parameter A is the interaction constant which describes the tendency for a solute atom or center of dilatation to be attracted by an edge dislocation's hydrostatic stress field, D is the diffusivity of the solute and k and T have their usual meanings. The principal characteristics of Cottrell pinning as manifested in the static strain 2/3 aging experiment are (a) a t time dependence of the lower yield stress return, and (b) an activation energy for the yield return approximately equal to that for the migration of interstitial solute atoms. The model assumes long range migration of solute and probably involves about 10 atoms jumps [12] (or a net rms displacement of 30 to 50b). An empirical result is that the increase in stress necessary to free a dislocation from its atmosphere as measured by ^o is directly proportional to the number of atoms, n(t), which have arrived at the dislocation. Thus, the strain energy decrease associated with long range impurity migration is directly proportional to n(t). This model with modifications [13,14] has survived for twenty-five years without its concepts being significantly altered. Excellent reviews of the Cottrel 1-Bilby theory are available in many places [15-19]. Suzuki [20] has pointed out that in face-centered cubic metals containing extended dislocations, a completely different form of interaction between dislocations and impurity atoms can exist. Since the stacking fault has a locally different crystal structure from the matrix, the solid solubility of impurities contained in the matrix can differ appreciably within the stacking fault and outside. Consequently a chemical potential exists across the fault, resulting in the binding of impurity atoms to the stacking fault. Pinning is a result

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of the accumulation of solute at the fault. Thus, this type of interaction should be characterized by an activation energy due to solute migration. Unfortunately, while the magnitude of the locking stresses has been calculated and applied with some success to solid solution alloys, the kinetics of migration to the faults have not been studied [21]. Nickel has a stacking fault energy [22,23] of approximately 400 dynes/cm. The equilibrium separation [24] between two partial dislocations is estimated to be only 3b. The magnitude of the yield point produced by segregation to stacking faults is related to the fault area. Hence, in other metals such as Ag-6 w/o Al [25] where faults are estimated to be 30b wide the effect is more important. Thus, one would not expect Suzuki locking to be a very important pinning mechanism in nickel [26]. The third type of pinning is sometimes called short range order locking and was proposed by Schoeck [27] and later expanded upon by Schoeck and Seeger [28]. Schoeck and Seeger considered a bcc lattice in which the concentration of interstitials is low enough to keep the interaction between interstitials small. Snoek originally proposed [29] that small atom impurities in solid solution occupy the octahedral interstices at the center of an edge or the center of a face of the unit cell in a bcc metal. Such sites have tetragonal symmetry since two of the six solvent atoms surrounding the interstitial site are closer than the other four. As a result, the octahedral sites may be classified into three grcups depending upon which one of three mutually perpendicular <100> directions the two nearest neighbor solvent atoms are aligned along. Thus, the three types of interstitial sites correspond to the three directions of tetragonal ity and if no applied stress is acting, the three kinds of interstitial sites will be occupied by the same fraction

PAGE 21

of interstitials; namely, one -third will be in each of the three types of sites. One may visualize each type of site occupied by an interstitial as a dipole. The principal axis of the tetragonal distortion gives the orientation of the dipole. If an applied (non-hydrostatic) stress is acting, the energy of interaction between the stress and the dipoles will in general depend on the orientation of the dipoles (i.e., the types of sites occupied by an interstitial atom). As a consequence, an applied stress will cause a redistribution of dipole orientations and the population of the sites with lower energy will increase, whereas the population of the sites with higher energy will decrease. This process is known as the Snoek effect [29] and it gives rise to a well established internal friction peak [30,31,32]. The activation energy associated with stress induced ordering of interstitial solute in bcc metals is normally that associated with diffusion of the impurity [33]. Schoeck [27] in 1956 pointed out that a similar redistribution of dipole orientations could be effected by the strain field of a dislocation. By such a process, the energy of the system is lowered in a period of time approximately that required for one interstitial atom jump and, therefore, the dislocation becomes locked. Whereas the locking due to atmosphere formation (Cottrell) requires diffusion of interstitials over long distances, the locking due to stress induced ordering of interstitial dipoles is accomplished merely by atomic rearrangement between neighboring lattice sites and, therefore, takes place in times which are orders of magnitude faster. Schoeck and Seeger [28] examined the process in considerable detail in 1959. Starting with the interaction energy between the interstitial solute atoms and the dislocation and assuming the concentration of solute

PAGE 22

was small, they showed that the line energy of a dislocation surrounded by a Snoek ordered atmosphere is decreased by an amount U given by 2 where c = total concentration of interstitials A = an interaction constant L = cut-off radius kT = thermal energy They next derived that the extra applied stress, At, necessary to pull the dislocation from the ordered atmosphere is given by U kT A more complete derivation of these results is carried out in the discussion relative to the carbon-vacancy pair (Section 4.2.4) and is related directly to pinning in Nickel 200. The possible contribution to the rise in yield stress made by ordering of solute atoms in the stress fields of dislocations has generally been ignored probably because it occurs very quickly at the temperatures that have usually been investigated. Snoek ordering, however, can explain the rapid static strain aging phenomena observed by Wilson and Russell [34] in tensile tests on a low carbon steel and similar observations on a range of materials (for example, Carpenter on tantalum-oxygen [12] and niobium-oxygen [35]; Owen and Roberts [36] on martensite; Rose and Glover [37] in stainless steel). Support for this view comes from an investigation by Nakada and Keh [38] of rapid strain aging in iron-nitrogen alloy single crystals.

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10 Wilson and Russell [34] verified that the rise in yield stress at 26rK in iron specimens (containing 0.039 w/o carbon) prestrained 4% was 63".' complete in 100 seconds and noted that this time is in reasonable agreement with relaxation times observed in the case of the elastic after-effect due to the ordering of carbon in iron. Thus, the process is complete in roughly the jump time of a carbon atom. It must be noted that parts of their data were taken with a reduced applied load which was generally between 80 and 90% of the load at the end of prestrain. Aging while applying a load has been shown to influence the size of the yield point and depends strongly upon the fraction of the prestrain load that is used in aging [39,40,41,42,43]. A similar case occurs in the data or Nakada and Keh [38] who used single crystal Fe-0.1 a/o C and N in their investigation. Although Quist and Carpenter [35] did not conduct the usual static strain aging experiments, their investigation of dislocation pinning in Nb-0 alloys during internal friction measurements is noteworthy. They conducted their experiments between 273 and 313°K and attributed damping phenomena to the pinning of dislocations by Snoek ordering of oxygen interstitial atoms in the strain fields of dislocation line segments. They observed that pinning was effectively completed in a period of one oxygen atom jump time. 1.1.3 Sunmary of Important Mechanisms of Dislocation Locking During Aging The two most important mechanisms of dislocation locking that may occur in metals containing dissolved interstitial impurities are related to (a) the Cottrell-Bil by model and, (b) the Schoeck-Seeger model .

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11 The principal features of the Cottrell-Bilby model are (1) Solute atoms migrate toward the dislocation over long distances under the influence of the gradient in elastic interaction energy between the dislocation and the solute. (2) t ' aging kinetics are predicted by the model and observed experimentally. (3) The activation energy for the yield return predicted by the model and observed experimentally is that for the diffusion of interstitial solute. The principal features of the Schoeck-Seeger model are (1) Interstitial atoms with tetragonal strain fields reorient in the strain field of a dislocation. (2) The aging process by Snoek ordering is completed in approximately the time required for one atom jump. (3) The activation energy predicted by this model is that for diffusion of the interstitial solute. 1.1.4 Aspects of the Static Strain Aging Experiment Figure 1 illustrates the mechanics of the classical static strain aging test for a specimen deformed in tension (or compression) at a constant strain rate. The initial prestrain and unloading cycle gives the specimen a known deformation history and internal state, i.e., a higher "fresh" dislocation density than that in the annealed specimen. If the material is irmiediately restrained after unloading, the stressstrain curve returns to the curve which would have been attained had the specimen not been unloaded. However, by aging under the proper conditions (e.g., higher temperature and/or longer times) a yield point occurs and is followed by a period of LiJders flow at constant load

PAGE 25

12 STRAIN Figure 1. General aspects of the classical static s aging test. train

PAGE 26

13 before work hardening resumes. In fact, the process of aging results in a gradual transition from a smooth reloading curve for very short aging times to a curve similar to that shown in Figure 1. The important parameters of the reloading curve are a,,, a. and Or , , the upper yield stress, the lower yield stress and the stress increment obtained by extrapolating the post-yielding curve, respectively. For short aging times, a^^ is equal to a , the value of the stress before unloading. The extrapolated stress, Or^t' ^"^ determined by the intercept of the flow curve, i.e., that portion of the stress-strain curve where uniform strain hardening is present, with the pre-yield or "elastic" portion of the reloading curve. The experimental parameter, Aa = a, a , is the parameter that is normally associated with strain aging as it is experimentally the easiest to determine. Accompanying the return of the yield point is the reappearance of heterogeneous deformation, i.e., the passage of a LLiders band down the specimen gage length, at the lower yield stress, which is also characteristic of annealed metal. During the initial stages of aging, the lower yield stress increases with aging time as does the size of the Liiders strain. The rate of change of these properties generally increases with aging time and temperature. After aging for somewhat longer times, depending on the metal and its history, the variation of Aa and the LiJders strain with time in bcc metals becomes much slower, in many cases exhibiting a slight decrease with aging time. 1.1.5 Static Strain Aging Stages in BCC Metals Five stages of aging during static strain aging have been identified for bcc metals containing interstitials. The first stage has been

PAGE 27

14 explained on the basis of observations of yery rapid returns of yield points in interstitial iron alloys [34,38,44] and internal friction experiments in other bcc metals [12,35]. The explanation is that very rapid pinning may be attributed to stress induced ordering of interstitials in the strain fields of dislocations as previously modeled by Schoeck and Seeger [28] (Section 1.1.2). The last four stages have been explained [10,45,46] on the basis of a Petch equation of the form a = a. + 2k d~^^^ (4) T y ^ ' where a is the lower yield stress, a. the lattice friction stress, k a dislocation locking parameter and 2d is the grain size. This equation was developed originally by Petch [47,48,49] in order to provide a method of separating the factors contributing to the lower yield strength of polycrystal 1 ine iron. During the LLiders band propagation, it was believed that unpinned sources release many dislocations which pile up at the grain boundaries. Thus, a feature of the model is the grain size and the boundaries are pile-up sites which act as stress concentrations. The pile-ups are controlled by the grain size and act in conjunction with the applied stress to unpin nearby dislocations in a neighboring grain. The friction stress is represented by a. and is the stress to move an unbound or free dislocation through the lattice. Rosenfield and Owen [50] formulated the aging phenomena in terms of an equation of the form Aa = ^a^ + 2k d"^/^ (5)

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15 where Aa is the gain in the lower yield stress after aging, AOl, the gain in the hardening component of the lower yield stress increase, as determined by extrapolating the load-time curve after the Liiders strain back to the reloading curve, and k and d have the usual meanings. Szkopiak [6] performed static strain aging experiments on niobiumoxygen alloys and separated the two components of the yield stress increase of Eq. 5 as shown in Figure 2. In Figure 2a, the typical return of the yield stress experiment on a bcc metal shows that at small aging times the increase in lower yield stress is very rapid (depending upon temperature of aging) and approaches a maximum. At longer aging times, the lower yield stress increment shows a slight decrease. In Figure 2b, the two components of Aain Eq. 5 are shown separately. The five stages of aging that have been observed in alloy systems such as Nb-0 [6,35], Fe-C [34,45,46], and Fe-N [38,51,52] alloys and probably occur in Ta [12,18], V [5], and Mo [8] as well are: Stage I : This stage is observed clearly only at low temperatures since locking occurs by stress induced ordering of interstitial s in the strain fields of dislocations and occurs within the time span of approximately one solute atom jump. The strength of pinning agrees reasonably well with the Schoeck-Seeger model. Stage 11 : In this stage, k reaches a maximum and remains constant. The lower yield stress reaches a maximum and the Liiders strain increases very rapidly. The rationale for this stage is that the formation of Cottrell atmospheres takes place during aging and upon reloading dislocations become unpinned from their atmospheres. Stage III : Further increases in the lower yield stress are due to an increase of the Aa^ parameter. In this stage, the Liiders strain remains

PAGE 29

16 <3 I 2 (O Oxygon, ppm D600 • O300 10 100 1000 Aging Time, Minutes (b) 10000 Figure 2, An example of the stages of the yield return in Nb-0 alloys [8]; (a) the increase in Aa with time, (b) the components of Aa and their dependence on time,

PAGE 30

17 nearly constant as the yield stress increases and enhanced strengthening occurs. The principal rationale for this hardening is that dislocations have been aged to the extent that they tend to remain immobile or pinned upon subsequent reloading. Thus, new or additional dislocations are created and the yield stress continues to increase. Stage IV : During this stage, solute continues to be accomodated in the strain field of dislocations but no longer effects an increase in Aa . Hence, Aa remains approximately constant. Stage V : As more and more interstitial solute segragates to dislocations, a condition of "overaging" is satisfied and precipitates may form; hence, the loss of a coherent strain field or the robbing of solute near dislocations and a mild decrease in the hardening component. The above stages of static strain aging appear to hold true for most of the body-centered cubic metals containing interstitial oxygen, carbon or nitrogen. However, in the case of face-centered cubic metals containing interstitial impurities no complete investigations of the behavior of the return of the lower yield stress have been carried out. 1.1.6 Static Strain Aging in Nickel and FCC Alloys Among the fee conmercial alloys, nickel containing carbon is probably the most significant where an interstitial (carbon) is deliberately added to improve mechanical properties. Other than pure nickel, only fee multicomponent alloys such as the austenitic stainless steels [37,53,54] contain carbon for similar reasons and exhibit mechanical properties similar to nickel-carbon alloys. There exists some experimental evidence related to the static strain aging of nickel-carbon alloys. In particular, two short notes were

PAGE 31

18 published by Macherauch et a1 . [55] and by Macherauch and Vbhringer [56] regarding static strain aging in Ni-0.05 w/o C after prestraining slightly beyond the initial yield plateau. Their data were plotted by this author and an approximate t or t time dependence of the lower yield stress return was exhibited. They did not speculate on the kinetics; however, they determined an activation energy of 10.2+2 kcal/mole in agreement with the activation energy for diffusion of hydrogen in nickel (see, for example, Boniszewski and Smith [57]). The only other investigation relating to nickel-carbon alloy is due to Sukhovarov [58] and Sukhovarov et al . [59]. Using compression specimens deformed at room temperature and aged between 433 and 493°K for various times, they deduced with apparent difficulty (because serrated flow occurred) that the average activation energy was 30.7 kcal/mole, somewhat lower than the carbon migration energy in nickel. This author plotted their lower yield stress data and noted that Aa/Ao ,„ varied approximately as t * . They conclude (incorrectly, max it is believed) that the Cottrell-Bilby model explains the rise in Aa and that probably the formation of precipitates eventually occurs; they never observed this aspect. In addition, since serrated flow occurred in the investigation of Sukhovarov [58], it is probable that the data were scattered because the lower yield stress was not as clearly defined as in the present investigation where yield point measurements were made under conditions precluding serrated flow. On this basis, their data should be used judiciously. Hydrogenated nickel exhibits strain aging behavior [57,60-55] when deformed below room temperature. Much effort has been directed toward

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19 understanding fracture, ductility, and other embrittlenent related phenomena attributable to hydrogen. Also, serrated flow [57, 60-65] is exhibited in hydrogenated nickel between approximately 130 and 225°K [65] at a nominal strain rate of 10 s . The kinetics of static strain aging in hydrogenated nickel were very briefly investigated by Boniszewski and Smith [57] and they concluded that the Cottrell-Bilby model can account for static strain aging of charged specimens. However, they did not speculate on the exact kinetics that the experimental data may have followed. Marek and Hochman [66] have demonstrated the existence of static strain aging effects in AISI 316 alloy and related it to the approximate activation energy for diffusion of interstitial carbon in austenite. However, the effect was most marked in the micro-yield region (0.01% proof stress) with no effect on flow stress after yielding, UTS, or elongation, which is indicative of a low interstitial/dislocation interaction energy. 1 .2 Dynamic Strain Aging Dynamic strain aging (DSA) is a feature exhibited in most commercial metals and alloys [1]. In general, aspects of DSA have created very little interest in the past, probably since in steel it exhibits its most significant effects over a temperature range around 450°K where steel is not normally worked. Other bcc metals such as titanium, tantalum, niobium and vanadium exhibit DSA over a temperature range where these metals are most needed [1]. As with steel, nickel containing carbon exhibits its effects at relatively low temperatures (300 to 500°K, roughly). Early investigations by Sukhovarov and Kharlova [57] confirmed that dynamic strain aging

PAGE 33

20 occurs in nickel when alloyed with small amounts of carbon. In a subsequent investigation Popov and Sukhovarov [68] indicated that the apparent activation energies associated with the appearance and disappearance of serrated flow are 20 and 33 kcal/mole, respectively. No conclusion regarding the very low activation energy for the onset of serrated flow was ventured. On the other hand, Nakada and Key [51] have indicated that the onset activation energy in Ni-C alloys is 15+2 kcal/mole and that for the disappearance of serrations is 26±4 kcal/mole. It is a well established experimental fact that the activation energy associated with the diffusion of carbon in nickel is approximately 35 kcal/mole (see Table 2). Generally, in interstitial alloys the activation energy for the onset of serrated flow is associated with the diffusion of impurity atoms and is made on the basis that when the velocity of dislocations is approximately equal to that of the velocity of the diffusing impurity atoms, a drag or pinning of dislocations occurs. Thus, the pinning as observed through the serrated flow phenomenon is assumed to be controlled by the diffusion of impurity atoms just as in the static strain aging experiment. By plotting log £ versus 1/T, where £ is the strain rate at temperature T where serrated flow is first observed, an activation energy may be deduced. The values of 15+2 and 20 kcal/mole for the onset of serrated flow determined by the preceding authors are much too small to be related to the diffusion of carbon in nickel . Popov and Sukhovarov [68] made no conclusions regarding this apparent anomaly. Nakada and Keh, however, ventured that pipe diffusion of carbon along dislocation cores controls serrated flow in nickel.

PAGE 34

21 TABLE 2 The Diffusivity of Carbon in Nickel ^

PAGE 35

22 Regarding the activation energy associated with the disappearance of serrations, Kinoshita et a1 . [74] proposed that this value may represent the suni of the activation energies for the diffusion of solute plus the binding energy of solute atoms to dislocations. On this basis, Nakada and Keh [51] have deduced a binding energy of 11.0 kcal/mole (0.5 eV) for a carbon atom to a dislocation in nickel assuming that serrated flow in Ni-C alloys is caused by carbon directly. Popov and Sukhovarov [68] attributed their value of 33+3 kcal/mole (1.4 eV) for the disappearance of serrations to a combination of creep processes coupled with Cottrell atmosphere formation. Other than the above nickel-carbon studies and the experimental work on hydrogenated nickel [57,60-65] which shows strain aging, no other investigations of the effect of interstitials on the stress-strain and work hardening behaviors in pure face-centered cubic metals have been conducted. However, some face-centered cubic ferrous alloys appear to possess mechanical properties similar to those of nickel-carbon alloys. In an investigation by Jenkins and Smith [54] complications due to substitutional alloying elements such as Cr occurred. Nevertheless, AISI 330 stainless steel (Fe-15Cr-33Ni-0.4C) , exhibits similar dynamic strain aging trends. A calculation of the energies for the onset and disappearance of serrations revealed that 26.6 and 62.0 kcal/mole are the onset and termination activation energies. They indicated that the onset activation energy is very close to that for vacancy migration in Fe-30 Ni. As the Portevin-Le Chatelier effect is absent for low carbon content, they conclude that vacancies alone are not responsible and that carbon-vacancy pairs account for the observed activation energy. Mention was not made of the exact mechanism for the pinning during serrated flow. A similar argument appears to apply in the case of Nickel 200.

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23 Other strain aging effects occur in face-centered cubic alloys but these arise mainly from the diffusion of substitutional solutes and are outside the scope of this dissertation. 1 .3 Work Hardening in Metals and Alloys In studying the work hardening of metals and alloys it is desirable to determine the mechanisms that control the rise in flow stress. In general, this involves relating the macroscopic behavior to changes in the microscopic structural features of the metal. For example, observations of slip line lengths or dislocation structures can supplement an explanation of work hardening. From the macroscopic point of view, polycrystal line stress-strain curves have been shown to be generally piece-wise continuous [75-81]. For example, Figure 3 shows schematically that a polycrystal 1 ine facecentered cubic metal may deform so as to show discontinuities in its stress-strain behavior. Zankl [75] and others [76-81] have shown that these stages can be correlated very well with deformation processes. According to his experimental work [75] the identifiable stages are related to the following processes: 1. The Transition Stage. This extends from zero plastic strain to approximately 0.1%. In this stage, multiple slip starts first in the largest grains and then spreads into neighboring grains. 2. Stage I. This begins when all grains have begun to deform with slip still involving multiple slip systems. Stage I in polycrystal s is thus basically different from the easy glide Stage I of face-centered cubic crystals. It ends at approximately Cp. 1.0%.

PAGE 37

24 3. Stage II. Here slip tends to occur predominantly on a single (primary) system but with interaction from secondary systems. The deformation is accordingly analogous to that in Stage II of a fee single crystal. Large grains may break down into several regions [80] with different primary systems. This stage extends to about £ =5. OX in pure fee metals such as copper and nickel. 4. Stage III. As in fee single crystal deformation, this stage is controlled largely by dynamic recovery and has been associated with cross slip [82]. In general, polycrystal 1 ine stress-strain curves appear to be continuous in shape and the stages difficult to identify on such curves. This is in marked contrast to single crystal stress-strain curves which often exhibit well-defined stages. A sensitive empirical method [83] for detecting polyerystall ine stage behavior has been developed as a logical projection of previous empirical analyses [84-92]. This is based upon the assumption that each stage of stress-strain behavior can be reasonably described by a modified Swift [92] equation: e = So+co (6) where a is the true stress, e the true plastic strain, m the work hardening exponent and £o and c are constants. One may solve yery simply for the parameters in Eq. 5 by plotting log Q versus log o where Q = 7j^ • ^ schematic example of such a plot is shown in Figure 4 for a typical face-centered cubic metal such as copper or nickel. Any straight line on this type of plot has an equation of the type: log = (1-m) log a log cm (7)

PAGE 38

25 Figure 3. Schematic example of stage behavior in polycrystalline fee metals. LOG cr Figure 4. Schematic description of the work hardening behavior in a metal using a log Q-log a plot.

PAGE 39

26 Thus, the value of m typically characterizes the power law relationship of Eq. 6. An m equal to one is a linear stress-strain curve and a log versus log o plot would show a line with a zero slope; a parabolic stress-strain curve would show a (l-m)-value of -1.0 (i.e., m=2) and so on. High m values correspond to curves with a great deal of curvature, i.e., wery rapidly decreasing work hardening rate with increasing stress as in Stage III when dynamic recovery processes reduce the work hardening rate very rapidly with continued deformation. All other parameters held constant, a high value of m for a single stress-strain curve would, in general, imply that the material has low ductility, even though the material might possess a reasonably high ultimate strength. However, it should be noted that a three or four stage stress-strain behavior could well lead to a combination of both high strength and high ductility depending upon the m values of the various stages and the extent of a particular stage during deformation. It should be noted that the stage behavior observed by using Eqs. 6 or 7 is within limits independent of the empirical power law equation used. For example, an analysis based upon the Crussard and Jaoul method [87-89] using a log a log e diagram shows that discontinuities occur at the same places on the stress-strain curve as determined by using a log log a diagram. In the current investigation, interstitial solute concentration was the principal alloy variable known to affect stress-strain behavior. Interstitial elements tend to significantly increase the strength of a metal while generally decreasing its ductility. This in turn affects the stage behavior of the parent metal.

PAGE 40

27 Another factor influencing the stage behavior of metals and alloys is the stacking fault energy. This intrinsic property of a metal or alloy determines the separation distance of the two partial dislocations of an extended dislocation [93]. For high SFE metals like nickel [22,23,94] or aluminum [93] the separation distance is small and mobile dislocations may be assumed to approach total dislocations. In low SFE metals and alloys such as Ni-40Co and Ni-60Co [94] this distance becomes appreciable and the partial s may be separated by a wide band of stacking fault. By effectively varying the geometry of the dislocation by lowering its stacking fault energy, the deformation behavior might be expected to change as well. Thus, it would appear that in a low stacking fault energy metal a dislocation effectively loses a degree of freedom of movement by virtue of its assuming a planar character. This should reduce the ability of the material to undergo dynamic recovery involving either cross-slip or climb. One may also view a stacking fault as a building block for a deformation twin. In general, twin boundary energies are lower in lower stacking fault energy metals. This observation is in agreement with the fact that twinning plays a greater role in the defomiation behavior of metals and alloys [95,96] with low stacking fault energies. Twinning was not observed in the present investigation. 1.4 Anelastic Phenomena in Nickel and FCC Ferrous Alloys Considerable research has been performed for many years on the effects of interstitial solutes on the strain aging of body-centered cubic metals and alloys [2]. In addition, much emphasis has shifted

PAGE 41

28 toward examining a variety of internal friction effects (such as the' Snoek effect, the cold-work peak and dislocation damping behavior) that can be correlated closely to strain aging phenomena [97]. In contrast, however, there exists but a dearth of strain aging and internal friction studies of interstitial solid solutions of facecentered cubic metals. The reason for this lack of interest probably stems from the fact that in terms of the mechanical behavior of these alloys, the interstitial s apparently cause a less dramatic effect on the mechanical properties. In addition, fee metals are not expected to exhibit a Snoek peak because of site symmetry [30,32,98,99]. Very different specific mechanisms for the observed relaxation peaks in fee metals and alloys have been suggested by various authors. Adler and Radeloff [99] reviewed the types of defects which could account for internal friction in fee metals and alloys: (1) interstitial-solute clusters which have noncubic strain fields [30,93]; (2) interstitial-substitutional solute clusters in which the interstitial reorients preferentially under stress if it is a near neighbor to an immobile substitutional solute [98,99]; (3) interstitial -vacancy complexes of different types, i.e., Wu and Wang [100] suggested a defect consisting of one interstitial occupying a vacancy with a nearest neighbor interstitial in its normal site. Within each category listed above are many possible specific combinations which could in principle cause a relaxation effect. There is evidence for relaxation due to carbon pairs in both Ni and the fee allotrope of Co, and for oxygen pairs in Ag [101]. The

PAGE 42

29 existence of an internal friction peak associated with dissolved C in Ni was first reported by Ke and Tsien [102] and further investigated by Ke, Tsien and Misek [103]. The peak is quite small and occurs at 523°K for a frequency of 1 Hz. By application of a saturating magnetic field, the existence of the peak was shown to be unrelated to the ferromagnetic nature of the sample. Further, the peak was found to decline in strength as the carbon precipitated from solution. K§ and Tsien pointed out that unassociated C atoms, located at the body-centered position of the fee structure and in the equivalent positions midway along the cube edges, could not be responsible for the relaxation, since the symmetry of such defects is cubic. After Tsien [104] found in later work that the relaxation strength varied essentially as the square of the carbon content in solution, it became evident from mass-action considerations that carbon pairs were the responsible defect. Diamond and Uert [69] also investigated the diffusion of carbon in nickel utilizing elastic aftereffect measurements above and below the Curie temperature and noted no discontinuity in carbon diffusivity. Hence, magnetic transformation effects do not affect carbon diffusivity. In addition, they concluded that the elastic aftereffect is due principally to interstitial diffusion of C-C pairs in agreement with the results of the previously mentioned group of Chinese investigators [102-105]. The simplest interstitial pair configuration in their view consists of two atoms occupying nearest neighbor octahedral sites which are the largest for fee metals, e.g., the two sites JjOO and OJ5O. The stress induced reorientation of such pairs (designated 110 pairs) which gives rise to anelastic effects, is the result of one of the atoms jumping into an unoccupied nearest neighbor interstitial site. A summary

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30 of Ni-C diffusion data is presented in Table 2. It should be noted that all methods agree reasonably well with an activation energy of 35 kcal/mole for the migration energy of carbon in nickel. No distinction is made in the last four references cited in the table concerning the nature of the mobile species , e.g. , dicarbon complexes. C.T. Tsien [104] considered the effect of impurities on an internal friction peak in a carburized 18.5 w/o manganese steel. The principal experimental observation was that the internal friction peak height varies linearly with carbon content. It was proposed that the addition of Mn to Fe-C alloys may reduce the opportunity of forming carboncarbon atom pairs and there may be greater probability of forming Mn-C pairs instead. Thus, in high manganese steels the internal friction peak is not attributed to rotation of the carbon atom pairs but due to that of carbon-manganese pairs. As a result, the height of the internal friction peak was observed to be directly proportional to the carbon content. This brings us to the point of the possible mechanism involved in Nickel 200 which has 0.18 w/o Mn and smaller amounts of iron and copper. Presuming a pair mechanism (in order to obtain a tetragonal defect which would account for an internal friction peak) one might presume a possibility of C-C atom pairs or C-Mn atom pairs causing the static strain aging effects. However, in the investigations previously sited, not one was conducted using a deformed metal and this factor may be an important consideration. Hence, a third and possibly fourth speciesmay be involved, namely, the vacancy and the self-interstitial. It has been demonstrated by Seitz [106] that in fee metals the predominant defect produced during plastic deformation is the vacancy. Table 3 shows the data of

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31 a number of authors [107] who have described the vacancy concentration during plastic deformation for a variety of fee metals and alloys as c, = Ke" (8) . where c = vacancy concentration (atom fraction) e = true strain K = proportionality constant n = 1 The principal experimental technique used to determine c involves monitoring resistivity changes during deformation and subsequent annealing of the specimens [106,108]. The identification of the defects annealing out during each recovery stage has been the subject of extensive previous work in nickel. Studies have been made of the resistivity recovery spectrum following neutron irradiation, electron irradiation and quenching from high temperatures [109,110]. In addition, changes in magnetic properties have also been studied in conjunction with the resistivity recovery process [73,111,112]. On this basis [108] cold rolled nickel behaves as Cy = 2.1 X 10"\ (9) In addition, it has been estimated that approximately eight times as many vacancies as self-interstitial s are generated. Point defects are generally believed to be generated during plastic deformation by two basic mechanisms [31,113]: (a) nonconservative motion of jogs on screw (or mixed character) dislocations and (b) recombination of dislocations containing edge type components. The second mechanism is due to annihilation of edge dislocations and may

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32 TABLE 3^ Estimates of the Rate of Vacancy Production During Plastic Deformation Material K Type of Defo r mation Cu

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33 not be important relative to the first mechanism in the production of vacancies until Stage III deformation occurs. The relation between vacancy concentration and strain is also probably only valid to strains of the order of 10%. It has been noted that the vacancy concentration tends to approach a saturation value at large strains [108]. Thus, vacancies may play an important role in deformed and aged nickel. As concerns the manganese impurity in Nickel 200, a qualitative argument may be made against the existence of Mn-C complexes as opposed 3 to C-V complexes. The gram atomic volume of Mn [114] is 7.39 cm while that for Ni [115] is 6.59 cm . Thus, a Mn atom is only approximately 4% oversize on a nickel lattice site. In an interstitial octahedral lattice site, a carbon atom is about 14?^ oversize. Relative to binding to a vacancy, the carbon atom should provide a stronger compressional center of dilatation than the manganese atom even though in the former case an octahedral interstitial site is occupied and in the latter a normal lattice site is occupied. Hence, the binding energy of a vacancy to a carbon atom should be higher than that of a manganese atom to a carbon atom or a vacancy. In addition, Nickel 200 contains approximately three times less manganese than carbon on an atom fraction basis. The value for the diffusion energy of a vacancy through a lattice is yery sensitive to the presence of impurities which tend to slow down a freely migrating vacancy because of binding to impurity atoms [116,117]. In pure nickel, the vacancy migration energy is between 0.8 and 0.9 eV and for impure nickel [117,118] (99.9% pure plus an unspecified amount of carbon) is approximately 1.1 eV. Thus, one may deduce that the approximate binding energy of carbon to vacancies is between 0.2 and 0.3 eV. In short, the carbon-vacancy interaction could cause the formation of a

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34 defect complex which might cause strain aging in Nickel 200. It is not clear whether or not dicarbon-vacancy defects may be completely ruled out as a possible complex causing strain aging. An excellent discussion concerning internal friction and strain aging to carburized ferrous austenite by Ulitchny and Gibala [107] suggests that the relaxation phenomena in these alloys are attributable to the rotation of carbon-vacancy pairs. Their conclusion is based upon experiments in which vacancies were produced in stainless steels by (1) quenching, (2) deformation and (3) irradiation. These processes have in common that they (a) increase the vacancy concentration in the metals, (b) increase the observed peak heights of the bound pair peak and (c) increase the peak heights in proportion to the relative numbers of vacancies they are anticipated to produce. These alloys possess mechanical properties quite similar to nickel-carbon alloys. In addition, the diffusivities of both carbon atoms and vacancies in the austenitic stainless steels are similar to those in nickel-carbon alloys [107] suggesting that a similar mechanism in the present investigation of Nickel 200 should not be ruled out.

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CHAPTER II EXPERIMENTAL PROCEDURES 2.1 Materials Nickel 200 bars of 0.75 inch diameter were obtained through a local supplier and from the International Nickel Company. These two heats had slightly different compositions (Table 4); the mechanical properties as a result were somewhat altered, albeit small. All static strain aging experiments were conducted utilizing Nickel 200 . The standard tensile tests were conducted using Nickel 200^. This procedure was followed in order to minimize possible scatter of the data, particularly in the static strain aging experiments. In addition, Nickel 270 was also purchased; its composition also appears in Table 4. Nickel 200 contains approximately 0.18 w/o Mn and 0.10 w/o C. The highest equilibrium solubility limit [1 19] for the nickel-carbon system is 0.27 w/o carbon. The room temperature solubility limit of carbon in nickel is only 0.02 w/o [120]. Also, nickel carbides are unstable in Ni-C alloys [119,121]. The development of visible graphite in nickel during cooling is generally agreed to occur very slowly. The present specimens, which were furnace cooled from the annealing temperature (1073°K) at a rate of approximately 2.0°/s, are believed to have retained all of the carbon in excess of the equilibrium solubility in solution since graphite was not observed either by optical or transmission electron microscopy of the annealed specimens. 35

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36 Ni 270 c

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37 2 . 2 Experimental Techniques 2.2.1 Swaging The 0.75 inch (19.1 mm) bar stock of all the materials were cold swaged in a Model 3F Fenn rotary swaging machine to a diameter of 0.25 inch (6.4mm). Intermittent annealing was not necessary. The resulting swaged bars were machined into threaded-end specimens with a nominal reduced section of 0.8 inch (20mm) and a gage diameter of 0.15 inch (3.8mn ). 2.2.2 Annealing Annealing was accomplished in a Vacuum Industries Minivac furnace assembly. This unit utilizes a resistance heated tantalum element. Pressures as low as 10 millitorr can be maintained. No cold trap was used. All Nickel 200 specimens were annealed for 30 minutes at 800°C (1073°K); Nickel 270 specimens were annealed for 32 minutes at 595°C (868°K). These treatments resulted in specimens with a mean grain intercept of approximately 22ijM. Annealing twin boundary intercepts were not counted in obtaining this result. 2.2.3 Specimen Profile Measurements A Jones and Lamson Optical Comparator capable of measuring to 0.0001 inch (2.5yM) in the vertical and 0.001 inch (25yM) in the horizontal directions was used to measure the profile of as-annealed specimens. For all tests a specimen gage length was assumed to be identical to its reduced section and was determined to within ± 0.005 inch (0.13mm) with experience.

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38 2.2.4 Tensile Testing All tensile testing was performed on two Instron machines (Model TT-C and Model FDL of 10,000 and 20,000 pound capacities, respectively). The standard crosshead speed was 0.02 inch/minute resulting in a nominal specimen strain rate of 4.2 x 10" s~ . Three additional strain rates were also employed with the Nickel 200 specimens. All tests were conducted between 77 and 900°K. Above 297°K tests were carried out in a capsule using commercial purity argon gas. At no time was oxidation visible on the specimen surfaces. Below ambient temperature, liquid nitrogen (77°K), dry ice-acetone (196°K), or ice-water baths were employed. Load-time curves were processed to yield true stress versus true plastic strain curves as well as the slope of these curves as a function of stress or strain. The stage behavior of the specimens was analyzed [83], In brief, the procedure involves the plotting of log versus log a and identifying portions of the curves through which straight lines may be passed. Each linear interval is assumed to represent a stage. This assumption was tested against results of Zankl [75], Schwink and Vorbrugg [77] and others [76,78-81] and a good correlation was obtained between stages detennined in this manner and the method of Zankl and others using different plotting and metallographic procedures. To each linear region a different set of parameters (m,c,£ ) may be deduced corresponding to the Swift [92] equation e-e^+co"" (10) 2.2.5 Static Aging Experiments Annealed tensile specimens were prestrained approximately 5% to a stress level of 38.5 ksi (265 MPa) at 273°K and immediately unloaded.

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39 removed from the testing jig and immersed in a silicon oil bath at 473, 448, 428 and 408°K or in a boiling water (distilled) bath for times as long as 2 x 10 seconds (approximately two weeks). Control of the constant temperature baths was held to within approximately one-half degree. Upon completion of the aging treatment, the specimen was removed and quickly quenched into cold water and tested immediately. The reloading temperature for the static strain aging tests was 273°K as in the prestrain. It was recognized early in the investigation that at room temperature and a strain rate of 4.2 x 10~ s" (the rate corresponding to a crosshead speed of 0.02 inch/minute, standard at this laboratory), discontinuous flow occurred in Nickel 200. To alleviate this feature it was decided to conduct the prestrains in ice-water baths at a rate of 4.2 x 10' s' . This had the additional benefit of producing a stable and reproducible lower yield stress plateau and, in addition, the ice-water bath assured that specimens were at the test temperature after removal from the high temperature aging baths. Quite apparent from the start was the fact that specimen alignment offered a problem. Upon removal and replacement of a specimen in the tensile jig, it was evident that exact repositioning was difficult and uncertain. Therefore, on restraining a specimen after aging it, a small bending moment nomally develops which may cause yielding to occur nonuniformly across the gage section of the specimen. The result is that the upper yield point as observed on the machine chart was usually absent. As a consequence all data reported are lower yield stresses. Compilation of the data included interpreting the Luders extension as the chart displacement which occurred at constant load

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40 (Figure 5). This method proved to provide the most consistent set of data and tended to alleviate apparent alignment or reloading displacements, Occasionally (perhaps 10% of the time), a yield point was observed and the data fell consistently in line with other LLiders extension data recorded by the foniier method. The latent hardening achieved during long term and high temperature aging treatments was computed by linearly extrapolating the postyielding curve back to the reloading line as demonstrated in Figure 5. This method also proved consistent. However, in most cases of short aging times or low temperature aging, the extrapolated stress fell below the prestrain value. This is physically unreasonable and is attributed to alignment effects which generally eliminate the yield point as mentioned previously and may cause the extrapolations to come back to the reloading curve somewhat low. However, the resulting data, again, proved to be consistent. As an example of typical experimental results. Figure 6 shows a series of load-time curves which were obtained after prestraining a series of specimens at 273°K to 5%, aging at various times at 408°K, and restraining at 273°K. Figures 6b and 6d show examples of partial yield points that were obtained in a few cases during the present investigation.

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41 Aged

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42 Figure 6. Selected load-tine curves obtained after prestraining a series of specimens 5% at 273°K, aging at 408°K for the times indicated, and restraining at 273°K.

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CHAPTER III EXPERIMENTAL RESULTS 3.1 The Behavior of the Lower Yield Stress Increase, Aa Figure 7 illustrates the dependence of the lower yield stress increment, Aa, on time and temperature over five decades of time and at five temperatures. The principal features of Figure 7 are (1) The curves appear to approach a common value at small times. It would seem that the data obtained at the higher aging temperatures and for very short aging times are influenced by the time required to heat the specimen to temperature. This was confirmed by assuming a fixed heat-up time and displacing the curve (at each temperature) to shorter aging times. Thus, a heating time of approximately 40 to 60 seconds straightens out the start of the higher temperature curves to approximately the same linear dependence as exhibited by the 373°K curve. In addition, a simple heat transfer calculation indicated that a time constant of approximately 40 to 60 seconds should describe the specimen heat-up time. The dashed lines in Figure 7 show the approximate corrections necessary to account for heating the specimens in the baths. (2) Each Aa curve shows a roughly linear increase with log t for times before reaching the maximum in Aa. (3) All curves show a well-defined peak whose height increases slightly, the lower the aging temperature. 43

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44 10 10 Time (sec) Figure 7. The time and temperature dependence of the return of the lower yield stress in Nickel 200. Specimens were prestrained to a stress level of 265 MPa. The dashed curves are approximate corrected curves which account for specimen heat-up in the aging baths.

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45 TABLE 5 Least Squares Parameters for Static Strain Aging Data Assuming Aa Is a Function of In t Temperature t (sec) 373 480 ^ t ^ 1.43 x 10^ 408 240 ^ t ^ 6.62 x 10^ 428 60 ^ t ^ 2.16 X 10^ 448 30 ^ t ^ 3.4 X 10^ 473 not fitted Slope (MPA/ln sec)

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46 (4) Aa decreases significantly after Ao is passed. Just max after the peak, the decrease in Aa is almost linear with log t. However, at 473°K, the highest temperature investigated, the data at •^ery long aging times show that the lower yield stress decreases to a constant value of approximately 17 MPa. Table 5 lists values obtained by the method of least squares for the slopes and intercepts (at t=ls) of the aging curves in Figure 7. A Ao versus In t relationship is assumed to hold for times before Aa reaches a maximum. In addition, Figure 8 shows a set of Aa curves which were normalized to their respective maximum peak heights, Aa ,„. An approximate value max of 28.5 MPa for Aa was assumed for the 373°K curve since, for the max aging times investigated, a peak was not attained. This diagram has the effect of making the aging curves more nearly parallel. Figure 9 is a plot of the increase in lower yield stress versus t ^^ using data from the 373, 408, 428 and 448°K aging curves of Figure 7. Least squares analysis of log Aa versus log t curves indicated that Aa varies approximately as the 0.14 and 0.15 power of time. See Table 6 for the complete results. This represents an approximate time dependence of t . It is important to note that this author prepared a specimen which had been aged for approximately 6 hours at 525°K (specimen taken from the undeformed threaded end of a deformed tensile specimen) and observed it carefully in a transmission electron microscope. No evidence of precipitates or free graphite in the grains, at dislocations or at grain boundaries was observed. Thus, it is highly unlikely that precipitation of carbon occurs during aging between 373 and 473°K in Nickel 200.

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47 n-m — I I 1111111 — I I I iinii — i i r iiiiii 1 i iiiiiii — r-rr rmn 1 i r iiiiij I I I II rJ tilt mil I I I tinil ! I I mill I r II Hill 10' 10' 10^ , iO^ t(sec) 10^ 10' Figure 8. Normalized aging curves for Nickel 200. The 373°K curve was normalized to an assumed maximum of 28.5 MPa. The dashed curves reflect approximate corrections for the heat-up time of the specimens.

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As^') Figure 9. Illustrating the approximate t ' power law relation governing the aging of Nickel 200 at temperatures below 448°K. Data for the 448°K (shown) and 473°K cases do not fit this relation well.

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49 TABLE 6 Least Squares Parameters for Static Strain Aging Data Assuming a Log Aa-Log t Linear Relationship Temperature

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50 3.2 The Liiders Extension, c Figure 10 illustrates the dependence of the LLiders extension on log t. The behavior is similar to that exhibited in Figure 7 for curves of ^o versus log t. However, note that the peak in the LiJders extension occurs (at a given temperature) at an earlier time than the corresponding peak in Aa. This may be indicative of the onset of hardening and is consistent with results obtained in bcc metals. Note also that at long times the extensions tend to return to their short time values; that is, they do not remain constant after reaching their maximum values. An interesting point is that the highest temperature curve (473°K) reaches an apparent minimum at approximately 1500 minutes. 3.3 The Hardening Component, iso^ Figure 11 is a plot of ^a^, the hardening component of the increase in lower yield stress. This parameter was deduced as noted in Figure 5 by using the equation ^o^, = Or ^. o . Figure 11 shows that this n txt hardening component appears only at discrete times. Note also that Aaj, peaks at approximately the same time as Aa and decreases to a value higher than that observed at very short aging times. It should be noted that the values of Aa^. cannot be taken as exact due to alignment problems which affected the choice of o^ ^ (see Figure 5). 3.4 Activation Energies In order to establish a mechanism for static strain aging, the apparent activation energies associated with particular time dependent aging events were deduced. The activation energy for the return of the lower yield stress where in the interval an approximate logarithm of

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51 O ir>

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52 -r—r O O a> o CM o ro CO CO CO '^ NO r
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53 time behavior is exhibited was calculated on the basis of the respective times to achieve a stress increase of 15.0 and 20.0 MPa. On this basis the activation energy for the return of the lower yield stress in Nickel 200 is 25.2 ± 3.2 and 26.4 ± 2.7 kcal/mole. The activation energy for the development of a 0.6% Luders extension is similar, 24.3 kcal/mole. In addition, the shift of the peaks of Aa versus log t is consistent with an activation energy of approximately 22 kcal/mole. The downward trend of the aging curves behaves in a manner corresponding to an activation energy of 29.0 kcal/mole on the basis of the method of cuts at Aa= 23 MPa. 3.5 The Dependence of Aa and e^ on Prestrain To more fully characterize static strain aging, a series of specimens were prestrained various amounts and then aged at 448°K for a fixed time of 6000 seconds, the approximate time required to achieve the maximum Aa at this temperature when the prestrain was 5% (see Figure 7). Figure 12 illustrates the dependences of Aa and e^ on the amount of deformation at a constant aging time and temperature. The Aa curve shows that this parameter increases with prestrain, exhibits a broad maximum and then decreases slowly. It is interesting that Stage III of the work hardening behavior (see Section 3.8), as detennined from log versus log a plots, begins at approximately 18% true strain at 273°K. Stage III is normally associated with dynamic recovery and in view of the broad peak and subsequent decrease in Aa, it is possible that Aa is reflecting the dynamic recovery.

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54

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55 The Liiders extension increases continuously with prestrain as indicated in Figure 12. The Liiders strain is determined not only by the size of the lower yield stress but also by the magnitude of the work hardening rate. The latter decreases continuously with strain and tends to make e, increase with strain. The fact that e. continues to increase with £ in Figure 12 is probably due to this cause. This is similar to the case of Type A Liiders bands which exhibit an increase in Liiders strain during plastic deformation [122]. 3.6 Comparison of Nickel 270 and Nickel 200 Static Strain Aging Several experimental observations indicate that the higher purity Nickel 270 does not contain sufficient carbon to give rise to measurable dynamic strain aging phenomena. Specifically, the Portevin-Le Chatelier effect was not observed in this metal. Also, even at the highest temperatures investigated, yield points or yield plateaus were not observed in annealed material. Thus, Nickel 270 may be nearly representative of pure nickel in terms of its mechanical properties. To test if static strain aging is weakly exhibited in Nickel 270 a specimen was prestrained 5% at 273°K and aged for 1200 seconds at 473°K. The resulting curve shown in Figure 13a appears to indicate some aging since a short yield plateau is exhibited. However, the lower yield stress increase for this specimen was only 4.76 MPa which is small compared to the 23.00 MPa value for the commercial purity Nickel 200 specimen (Figure 13b) which was achieved in only one-half this aging time. It is also possible that a significant portion of the 4.76 MPa yield effect that has been observed in face-centered cubic metals [17,123]. Thus, one may be reasonably assured that strain aging phenomena in Nickel 270 are generally weak.

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56

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57 3.7 The Stress-Strain Behaviors The details of the basic mechanical behavior of the high purity nickel. Nickel 270, are shown in Figures 14-16. Those for commercial purity Nickel 200 are shown in Figures 17-22. Nickel 200 unlike Nickel 270 exhibits serrations and yield points. As indicated in Figure 21, the Portevin-Le Chatelier effect was observed over four orders of magnitude of strain rate in Nickel 200. The figure also shows the approximate temperature intervals over which Types A, B and C serrated flow were observed. Type C serrations were sudden load drops appearing at regular intervals on the load-time curve. The serrated flow intervals correspond closely to those observed by Nakada and Keh [51] in nickel-carbon alloys indicating that the presence of the manganese in Nickel 200 does not appreciably affect the dynamic aging' effects. The data of the present investigation are not extensive enough to calculate the apparent activation energy for the onset of serrations in Nickel 200 with accuracy. However, the data appear to be consistent with the apparent activation energy 15+2 kcal/mole calculated by Nakada and Keh for the somewhat purer alloys [51]. In addition, the apparent activation energy for the disappearance of serrations calculated by Nakada and Keh [51], 26+4 kcal/mole, also appears to be reasonable for Nickel 200. Figure 14 shows representative stress-strain curves for Nickel 270 obtained at several temperatures. The 0.2» offset flow stress and the ultimate tensile strengths of Nickel 270 are plotted in Figure 15 as functions of the temperature. Note that the ultimate stress decreases monotonically with temperature without any undue irregularity. This type of stress-temperature variation is characteristic of a metal which

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58 (2"J/NW)20I *ss3JiS 9nJl

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59 1400 1200 1000 800 600 400 200 \ \ 1 1 : 1 NICKEL 270 • Ultimofe true stress n Engineering ultimate o 0.2% Offset ^ a , °-o. '°-c ^OdOO-o— <3 O O—c/ O 100 200 300 400 500 600 700 800 900 T CK) Figure 15. Variation of the 0.2% yield stress and the ultimate tensile strength with temperature in Nickel 270 (e = 4.2 X 10""^ s"^).

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60 110 100 90 8070 t:6o c *o ^50 403020101 r NICKEL 270 • Total o uniform 100 200 300 400 500 600 700 300 900 T('K) Figure 16. Variation of the uniform and total elongation with temperature in Nickel 270 (e = 4.2 x 10" s~ ).

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61 does not exhibit pronounced dynamic strain aging. The total and uniform elongations are illustrated in Figure 16 and show no anomalies. A minimum in ductility was observed at 835°K and surface cracking was noted on the specimens. Cracking also appeared on the specimen tested at SSO^K. The Nickel 270 specimens were annealed at 868°K and this temperature was the upper testing limit. The elongations are reasonably constant over a wide range of temperature (approximately 200 to 650°K). A representative sample of Nickel 200 stress-strain curves are shown in Figure 17. It should be noted that the curves at 300 and 525°K were serrated. Only average stress-strain behavior can be shown in these cases. Note that at 525°K the curve shows an anomalously high ultimate strength. The enhanced strengthening during work hardening is best illustrated in Figure 18 which shows dependence of the 0.2% flow stress and the ultimate stress on the temperature. Because of the scale of the drawing in Figure 18 the 0.2o offset stress appears to decrease monotonical ly with temperature. However, a plot of the 0.2% offset stress for Nickel 200 at two strain rates with the stress axis expanded as in Figure 19 shows that there is a small yield stress plateau between approximately 300 and 475°K. This is generally characteristic of dynamic strain aging in bcc and hep metals. The stresses were not normalized with respect to the elastic modulus, as is customary, because nickel exhibits a large magnetostriction [124,125] and the choice of modulus is uncertain below 626°K, the Curie point [126]. This plateau is weakly exhibited compared to that of titanium [127-129], for example. In comparison to the 0.2% stresses observed in Nickel 270, Nickel 200 exhibits a larger temperature dependence (compare Figures 15 and 18).

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62 {^^Wti'/i)^Q\ *=C0JiS snJi

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63 1200 1000 800 600 400 200 True Nickel 200 4.2 X 10 ^ sec"-'-1 0.2 t— O B fl^ . 200 600 Figure 18. The temperature dependence of the 0.2/o yield stress and ultimate tensile strength of Nickel 200.

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64 200 150 s? 100 CM 50 T ' 1 — "» r Nickel 200 • 4.2 X IC'^s"' o 4.2 X 10"^ s"' 200 400 T 600 800 Figure 19. The temperature and strain rate dependence of the 0.2% yield stress in Nickel 200 on an expanded stress axis.

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65 An important feature to note in Figure 18 is that anomalous strengthening is exhibited between 300 and 600°K. In Figure 20 the strain dependence of the flow stress is shown for 5, 11, 19, and 31% plastic strain. This figure demonstrates that only above approximately 11 to ]9% plastic strain does the anomalous strengthening become significant. That is, the strengthening effect is not due to anomalous work hardening as Sukhovarov and Kharlova [67] previously suggested. Note that unlike the behavior of the ultimate tensile strengths plotted in Figure 18, the stress levels attained at 3]% strain exhibit a rate dependent shift in the peaks. These peaks have a rate dependence corresponding to an activation energy of approximately 38 kcal/mole. Figure 21 illustrates the temperature dependence of the uniform and total elongations in Nickel 200 at several strain rates. The total elongation at a strain rate of 4.2 x 10' s~ shows a mild ductility minimum (blue-brittle effect) between approximately 300 and 450°K. This minimum is not well-defined. However, a well-defined but small reduction in area minimum does occur in Nickel 200 that is strain rate dependent as shown in Figure 22. This minimum was noted to have shifted in accordance with an apparent activation energy of approximately 25 kcal/mole. A reduction in area minimum is not always manifested in dynamic strain aging [130]. It is interesting to note, however, that 25 kcal/mole is approximately the activation energy for vacancy migration in nickel [109,118]. Not only is the loss in reduction in area not marked, but the lowest value recorded is still above 75%. Adjunct observations of the fracture surfaces under a low-power microscope did not reveal any striking difference in fracture mode at the reduction in area minimum.

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66 c/i O O) O 1/1 CM (Dd^v) ss3dis

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67 100 80 ?60 UJ c 20 1 \ 1 T" Nickel 200 4.2 X 10"^ sec ' • 4.2 X lO^'^sec"' o 4.2 X 10'^ sec"' 4.2 X 10"^ sec"' ^ Total Uniform 200 400 600 T 800 Figure 21. Variation of the uniform and total elongations with temperature in Nickel 200. Also shown are the approximate temperature ranges over which serrations were observed at the respective strain rates.

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60 100 98 96 94 92 90 < SS ^o 85 84 82 80 78 76 74 Ni 270 4.2 X lO'V Ni 200 4.2 X iO'^s ' * 42 X IO"^s"| ° 4.2 X lO'^'s"' o 4.2 X 10"^ s"' J U_ i L 200 400 600 SCO 1000 Figure 22. Variation of the reduction in area with temperature for Nickel 200 and Nickel 270.

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69 Note that the higher purity Nickel 270 shows more ductility at all temperatures than does Nickel 200. Although a mild reduction in area minimum does occur in Nickel 270, it is spread over a wide interval between approximately 425 and 750°K and is not as pronounced as that in Nickel 200. Even the lowest reduction in area observed in this investigation is considerably higher than that in most commercially available bcc metals and alloys. The strain rate sensitivity in Nickel 200 and Nickel 270 was not investigated because the temperature interval of serrated flow in Nickel 200 was some 300°K wide and it was believed that strain rate changes conducted during discontinuous plastic flow would prove inconclusive. It was noted, however, that at room temperature during moderately heterogeneous plastic flow, changes in rate resulted in the appearance of flow stress transients in Nickel 200 and a steady state strain rate sensitivity yery close to zero was observed. In summary, whereas the higher purity Nickel 270 shows no anomalies in ultimate strength and elongation with increasing temperature. Nickel 200 between 300 and 600°K shows anomalous strengthening, serrated flow, a small yield stress plateau and a mild elongation minimum. 3.8 The Work Hardening Behavior of Nickel 270 and of Nickel 200 Figures 23 and 24 show a cross-section of the log versus log o curves for Nickel 270 and Nickel 200, respectively, deformed at a -4-1 nominal strain rate of 4.2 x 10 s . These curves satisfactorily represent the general trend of work hardening at all strain rates investigated. No attempt has been made to draw in the straight lines representing the stage behavior in order to reduce the complexity of the figures.

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70

PAGE 84

71 T

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72 The work hardening behavior of Nickel 270 and of Nickel 200 are similar to those found earlier in pure face-centered cubic metals. The stages appear in the same manner as Zankl [75], Schwink and Vorbrugg [77], and others [76,78-81] have shown to be the case for pure face-centered cubic metals. It should be noted that hexagonal close packed metals such as zirconium and titanium [131] as well as body-centered cubic metals such as iron and niobium [132] show much higher m values (order of 7 to 40) than the fee alloys presently being investigated. That is, all log e versus log o plots for these metals show much steeper slopes. Also, these metals tend to show only one or two stages of deformation behavior indicating that the deformation in these metals is possibly controlled by a different set of deformation phenomena. Figures 25 and 26 show the m values obtained for Nickel 270 and Nickel 200 by measuring the slopes of log Q versus log a plots at different temperatures. The error of each particular m.. value is approximately +0.1 as measured by the plausible maximum and minimum slopes that might conceivably characterize a particular work hardening stage. It should also be noted that Stages I and III are difficult to characterize in many cases. The parameter m^^r is plotted in the figures only to show trends in the third stage as a function of temperature. They are not accurately defined since in Stage III log 0-log a plots are not linear but curved. However, Stage II is generally uniquely defined by a straight line on log G-log a curves. Nickel 270 and Nickel 200 possess m^j values which are close to 1.5 as shown in Figures 25 and 26. Note that in Nickel 200 m.. remains constant over a wide range of temperatures and is strain rate independent.

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73 o

PAGE 87

74 1 1

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75 It is important to emphasize that the apparent work hardening peak observed in Nickel 200 (Figures 18 and 20) is not a true work hardening peak in the sense that Nickel 200 stress-strain curves show anomalous increases in work hardening over a uniquely defined temperature interval. The peak is associated with an increase in the uniform elongation around the temperature of the ultimate stress peak. Figure 27, a plot of (Oj-^^-a ) versus temperature, readily shows this behavior. This parameter (where Oro, and a^ no/ ai^e the true stresses at 5 and 0.5% true plastic strain, respectively) confirms that there is little temperature dependence of the work hardening. The constancy of m^-, in Figure 26 also substantiates this observation. Figures 28 and 29 show ^ery clearly that the onset of Stage III is delayed over exactly the same temperature interval as develops the peak in ultimate stress. These facts imply that there is no prominant work hardening peak in nickel as earlier authors have indicated [67]. There is, however, a significant delay in the onset of Stage III or of dynamic recovery. In Stage III deformation in both a polycrystall ine or single crystal fee metal or alloy, the stress-strain curve shows a high curvature, i.e., work hardening decreases ^ery rapidly with continued deformation. Thus, Stage III m values tend to be high. Figures 25 and 26 show that m-... increases with increasing temperature. This is consistent with the concept that dynamic recovery becomes more important with increasing temperature. Now consider the extent or length of Stage II which is not illustrated well in the log 0-1 og a diagrams. In purer metals such as Nickel 270 it is observed that Stage II, which is governed primarily by slip on single slip systems within each grain [75], is lengthened by

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76 -

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77 the presence of interstitial carbon as in Nickel 200. Figure 28 compares the approximate strain at which Stage II deformation begins, e^. and that at which Stage III begins (and Stage II ends), Cy ^°^ Nickel 200 and Nickel 270. The principal effect of interstitial carbon in the DSA interval in this respect is to postpone Stage III dynamic recovery processes to a later time in the deformation history. Also the general level of the Nickel 200 curve is higher than that for Nickel 270. It is not clear at this time how the carbon accounts for the general delay in Stage III. Nickel 200 exhibits an e^ maximum over the dynamic strain aging interval. This maximum is closely related to the maximum exhibited in the uniform elongation (Figure 21) and accounts for the strengthening observed over this temperature interval. This maximum in e^ ""S strain rate dependent as shown in Figure 29. Not only do the peaks shift to higher temperatures with increasing strain rate but the peak heights decrease as well. This is similar to the behavior of the flow stresses described in Figures 19 and 21. The shift in peaks follows an approximate activation energy of 37 kcal/mole, similar to that obtained for the flow stress peaks.

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78 50 40^ ^1 NICKEL 270' ^3 NICKEL 200 (All slroins ore true plostic strains) 30 20 10 100 ^°° ^^rC^^f ^ ^ 700 800 900 Figure 28. The variation of e^ and £3 (the approximate strains at which Stages II and III, respectively, begin) with temperature for Nickel 270 and Nickel 200 deformed at a strain rate of 4.2 x 10"^ s"^

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79 50

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CHAPTER IV DISCUSSION 4 . 1 Ratio nale for Static Strain Aging in Nickel 200 The static strain aging data for Nickel 200 aged between 373 and 473°K after a 5" prestrain indicate: 1. The activation energy for the return of the lower yield stress for times before the maximum in Aa is approximately 25+3 kcal/mole. 2. The kinetics of the rise in the lower yield stress can be described by a In t time law. According to the previous discussion three principal causes for strain aging in other metals have been identified: (a) Cottrell pinning, (b) Suzuki locking and (c) Snoek pinning. It would appear on the outset that aging in Nickel 200 is such that Cottrell pinning must be ruled out as a candidate mechanism. First, 2/3 the kinetics for aging as described by Cottrell and Bilby [9] are t ' . In Nickel 200 the rise in Aa with time is much slower, approximately t ; this is very different from the Cottrell-Bilby time law. Secondly, the activation energy deduced to govern the early stages of pinning in Nickel 200 is not consistent with the migration energy of carbon, the principal impurity in Nickel 200, as the Cottrell-Bilby model would suggest. The migration energy of carbon in nickel is approximately 35 kcal/mole (see Table 2) and thus significantly different from the observed activation energy. Accordingly, the random migration distance 80

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for carbon at all temperatures investigated during the time required to cause maximum strengthening is only about 8b which is a very short diffusion distance compared to distances involved in the formation of a Cottrell atmosphere. Suzuki pinning is not believed to be responsible for the observed strain aging behavior in Nickel 200 primarily because the stacking fault energy of pure nickel is very high. Hence, the probability of forming stable faults in nickel is very low. It is believed that carbon impurities do not lower the stacking fault energy enough to have a significant effect with regard to this mechanism. Snoek pinning by itself cannot account for the rise in Aa unless anisotropic defects exist in the material in sufficient numbers and have a sufficiently high interaction energy with dislocations. As will be demonstrated the carbon-vacancy pair can satisfy these requirements. However, Snoek pinning should occur in times, very short at the temperatures investigated, compared to the times observed for the rise in Aa. The formation of ordered carbon-vacancy pair atmospheres near dislocations can account for the observed static strain aging behavior. The yery long times required to reach the maximum in Aa can be rationalized in terms of a stress assisted migration of vacancies toward dislocations where enhanced trapping by carbon occurs. The result is an accumulation of ordered carbon-vacancy pairs near the dislocation and the concommitant growth of an ordered carbon-vacancy pair atmosphere. In short, the proposed model utilizes aspects of both the Cottrell-Bilby and the Schoeck-Seeger theories of static strain aging. In summary the following stages are envisioned: (a) At the end of deformation vacancies diffuse to and become trapped in large measure

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82 by adjacent carbon atoms, (b) The carbon-vacancy pairs thus formed undergo Snoek ordering within the stress fields of the dislocations. (c) Then a slow migration of vacancies toward the dislocation develops an atmosphere of carbon-vacancy pairs near the dislocation. (d) Eventually the vacancy concentration decreases because of losses to sinks. This reduces the pair concentration and causes an eventual loss of strengthening. These stages will now be considered in detail. 4 . 2 The Mechanism of Static Strain Aging Exhibited in Nickel 200 4.2.1 The Distribution of Vacancies, Carbon Atoms and Dislocations After Plastic Deformation After the prestrain of 5%, the vacancy concentration in nickel is approximately 10' atomic fraction as given by Eq. 9. The particular heat of Nickel 200 used in this investigation contains approximately 0.1 w/o or 0.5 a/o carbon. Thus, at 5% strain the carbon to vacancy ratio is approximately 500 to 1 . If the carbon and the vacancies are distributed at random throughout the nickel lattice, then the mean carbon atom spacing will be approximately 6b; the mean vacancy spacing will be approximately 50b. After plastic deformation to 5% strain, the dislocation density [133,134] 9 -2 is estimated to be between 3 and 8 x 10 cm . Assuming an equidistant array of straight dislocation lines as a first approximation to the dislocation configuration, then their mean spacing is approximately 450 to 730b. 4.2.2 Vacancy Trapping by Carbon Atoms In the absence of other defects such as grain boundaries or dislocations, vacancies in a metal are attracted to oversized impurity

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83 atoms by a hydrostatic pressure gradient. The strain energy released when a vacancy is moved from an infinite distance away from an impurity to the impurity is known as the binding energy of the vacancyimpurity pair. This phenomenon is commonly called trapping and it is generally accepted that impurity atoms can trap vacancies in metals. For example, it has been shown that the presence of carbon in austenitic stainless steel forestalls radiation induced swelling in certain critical nuclear reactor parts. This indicates that void formation is hindered by the presence of carbon due to trapping of vacancies [135]. Trapping of carbon by vacancies in irradiated mild steel has been demonstrated as well [136]. 4.2.3 The Concentration of Carbon-Vacancy Pairs After deformation in Nickel 200 and before aging of a specimen, it is reasonable to assume that vacancies become trapped by carbon atoms initially. This probably occurs very rapidly even at 373°K, the lowest aging temperature used, since the vacancies do not have to travel far to become trapped by carbon atoms. Vacancies in nickel are very mobile as evidenced by their approximate 0.8 to 0.9 eV migration energy and assuming that an equilibrium is established between the carbon-vacancy pairs and free vacancies, their concentrations may be calculated on the basis that the binding energy of the carbon-vacancy pair can be estimated to be 0.2 to 0.3 eV per pair from diffusion data (Section 1.4). For the moment, it is assumed that the dislocations in the metal do not affect the concentrations. The concentrations of single (free) vacancies (c ) and bound vacancies in carbon-vacancy pairs (c ) vary with carbon concentration (c ), temperature T, and carbon-vacancy binding energy (B)

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84 according to the equations [116] and Ic' c' e'/'' (11) c = c^ + c (12) V V cv where c is the vacancy concentration generated during plastic deformation, c is the free carbon concentration and Z is 6, the nearest neighbor coordination number for a carbon atom in an octahedral site of the fee lattice. Substituting Eq. 12 into Eq. 11 gives c = Z(c -c )(c -c )e^/'^"^ (13) cv ^ c cv'^ V cv For c =5x10' and c = 10' one may write on the basis that =cv = Z=c
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85 where Q^ is the vacancy formation energy. At 408°K and a Q^ of approximately 39 kca1/mole (1.7 eV) [118] the vacancy concentration -21 is approximately 1.8 x 10 atom fraction. This is many orders of magnitude smaller than the free vacancy concentration calculated above and suggests that the deformed metal is not in its lowest energy state with regard to numbers of vacancies. Thus, there exists a distinct tendency for vacancies to migrate to sinks. Because of the high binding to carbon atoms, vacancies spend a large portion of their migration time bound to carbon atoms and, hence, the process of annealing to dislocations and grain boundaries is sluggish and takes a considerable time. Consider next the influence of a dislocation on the carbon-vacancy pair concentration. It has been demonstrated that the carbon-vacancy pair may be visualized as an elastic dipole which can reorient in the strain field of a dislocation. In Eqs. 11-14, the binding energy, B, is completely general. That is, in the specific case which includes Snoek ordering, B becomes position dependent. Specifically, B = 8' + u(r) (16) where B' is the binding energy of a carbon-vacancy pair and u(r) is the position dependent interaction energy of a carbon-vacancy pair with a dislocation. Thus, the effect of u(r), which varies as r" , is to enhance the binding of vacancies to carbon atoms near the dislocation. This will be explained in detail presently. 4.2.4 Theory of Schoeck Locking by Carbon-Vacancy Pairs In the case of nickel-carbon alloy, the carbon-vacancy pair acts as a dipole (Section 1.4). Appendix A presents a calculation of the

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86 approximate interaction energy between an assumed carbon-vacancy dipole and a screw dislocation in a fee lattice. The interaction calculation related to an edge dislocation is more complicated [137]. However, as indicated first by Nabarro [138] and later expanded upon by Cochardt et a1 . [137], the strength of interaction between a dislocation and impurity atoms may be assumed to be very similar whether the dislocation is in the screw or edge orientation. Let us consider the case of locking of a screw dislocation by an ordered atmosphere of di poles in the fee lattice. The reasoning is analogous to that for the calculation made by Schoeck and Seeger [28] for carbon (or nitrogen) in alpha iron. However, the interaction energy for a dipole in a fee lattice as shown in Figure 35 in Appendix A is somewhat different. The carbon atoms in nickel are assumed to occupy octahedral interstitial sites, e.g., the body-centered position of the fee unit cell. The six nearest neighbors are face-centered atoms. A dipole is formed when a vacancy is situated on a face-centered position as schematically shown in Figure 30. Thus, the carbon-vacancy dipole may be oriented in one of three possible 100 type directions. These positions are denoted by an orientation number 1, 2 or 3. The dislocation line is assumed to lie along one of four 101 type directions. It is shown in Appendix A that for the specific case of a screw dislocation, each of the three possible orientations of the carbonvacancy dipole interacts differently with the dislocation. This gives rise to three possible interaction energies and one of the three orientations is most stable. That is, a carbon-vacancy dipole in, say, the 1-orientation can flip to another orientation (say, 2) and the result is that the interaction energy between that particular dipole

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87 Figure 30. A schematic illustration of the assumed configuration of the carbon-vacancy pair and the three possible independent orientations that it may assume.

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and the dislocation is lower after one jump of a vacancy. Thus, the strain energy is lower by u-, u„ where u, is the interaction energy of a type 1 dipole with the dislocation and u^ is that associated with the type 2 dipole. This is the energy associated with the stress induced ordering of a dipole in the dislocation strain field. The three possible dipole orientations for an assumed dislocation are shown in Table 8 of Appendix B and possess the following interaction energies: U-, = (sin + /? cos 0) = u (17a) u^ = (17b) U3 = -u • (17c) where A is an interaction constant assumed approximately equal to 0.2 eV-b (7.97 x 10"^^ dyne cm^). The concentration of dipoles in each of the three orientations can be calculated easily. Let c be the total carbon-vacancy pair concentration expressed as an atom fraction. The concentrations of pairs in the three possible orientations are designated (c ),, (c )„ and (c )_. The total concentration of carbon-vacancy pairs after initial ordering at the beginning of the strain aging process is (^cv>l * (=cv'2 * (=cv>3 (18) where (^cv)3=2<< exp(^^) (19C)

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and c and c are the initial free carbon and free vacancy concentrations, calculated from Eqs. 11 and 12. Recall that u depends upon position. In order to simplify the solution of the equations and to demonstrate the principles involved, assume that u = A/r and is independent of 0. The more general case where u is given by Eq. 17 will then be treated. Combining Eqs. 18 and 19, one obtains Note that for large r and f'^cv)! ' <=cv'2 ' (<=cv>3 ' ^cv/3 (22) Another way of writing the concentration of carbon-vacancy pairs -u./kT (c ). = c — 7t;y (i=1,2,3) (23) ^ cV^i cv -U./kT \ ' > ' \ / I e ^ -U./kT ^TTTkT Z e ^ (24) where n. and u. are the number per unit volume and the interaction energy, respectively, of dipoles in the i orientation and n is the initial number of dipoles per unit volume. Thus, the population of the three possible orientations approaches equality at large distances from the dislocation and Eq. 21 becomes identical to Eq. 11. The principal consequence of the Snoek energy

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90 is to increase the equilibrium concentration of carbon-vacancy pairs in one orientation, near a dislocation, at the expense of the other two. Figure 31 illustrates the results of a calculation using Eqs. 18 and 19 and shows that 1-type orientations are favored near the dislocation. Note also that c , the total carbon-vacancy pair concentration (equilibrium), remains almost constant with distance from the dislocation. At distances greater than 200b the concentrations of type 1, 2 and 3 dipoles are almost equal, in agreement with Eq. 21. It should be kept in mind that up to this point no long range redistribution of vacancies has been assumed to have occurred. VJhat is illustrated is the situation shortly after plastic deformation has ceased and vacancies have migrated short distances of the order of 4 or 5b to form carbon-vacancy pairs which then reorient. The result of the stress induced ordering of dipoles shown in Figure 31 is to lower the strain energy of the dislocation thereby causing pinning. The Schoeck-Seeger [28] model can be utilized to calculate the initial strength of pinning as measured by Aa due to the ordering of carbon-vacancy pairs. The number of carbon-vacancy pairs dn between r and r + dr (assuming circular synmetry about the dislocat

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91 CM o o 00 O E O O
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92 dU = u^ d(n^^)^ + U2 d(n^^)2 + U3 6{nJ^ (26) or substituting Eq. 18 and Eq. 25 into Eq. 26 Assuming that u, = A/r, u_ = and u_ = -A/r and substituting these values into Eq. 27 gives The total energy change is given by l/" ^ C "=cv>, (=cv'3) ^'(29) where r is an assumed core radius of approximately lb and R is approximately 400b. It is a very simple matter to numerically evaluate the integral from the graph in Figure 31. Such an integration using 10b spacings and the trapezoidal rule gives .400b ,-, (c ), dr 3.55 x 10""cm (30) J lb ^^ ' ,400b ,, . (c )-, dr = 3.03 X 10 cm (31) Jib ^"^ -^ These results when substituted into Eq. 29 give an approximate value for the energy decrease associated with Snoek ordering of existing carbon-vacancy pairs: U„ (0.52 X 10"^^ cm) ^^(0.2 eV) (32) = 1.50 X 10^ eV/cm = 2.40 X 10"^ dyne-cm/ cm

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93 In order to calculate the additional stress needed to remobilize a dislocation from its Snoek atmosphere one may use the following simple relation derived by Schoeck and Seeger [28] which relates the maximum force necessary to free a dislocation from its ordered Snoek atmosphere to U , ^max \ (33) where R = A/kT. At 450°K, A/kT = 1.28 x 10'^ cm and mU 2Fr A0 == or§ (34) where m is an orientation factor which relates shear stresses to tensile stresses and is between 2 and 2.75. Assuming m = 2.5, Eq. 34 simplifies to Aa == 9.4 X 10^ dynes/cm^ (35) = 131 psi = 0.90 MPa The above simple calculation is only an estimate of the strength of pinning due to carbon-vacancy pair ordering since the angular dependence of the interaction energies were not taken into account. A more exact expression for U and Aa may be calculated in the following analogous fashion. Due to local ordering of the dipoles the line energy of the dislocation is decreased by an amount U compared with a dislocation surrounded by randomly oriented dipoles. The decrease in energy U is the sum of all interaction energies between individual atoms and

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94 the dislocation and is given by ,2tt ,R U = \ I n.u.rdrdG (36) Jo J,^ ' ' where R is approximately one-half the mean dislocation spacing. Substituting Eq. 24 into Eq. 36 gives r2u.^o u/kT -u/kT where u is defined by Eq. 17. Noting that u«kT, one may simplify Eq. 37: 2n fZu r o % = 3kf 1 I " '•'"•* ^ lo I " 2n A^ .2tt .Ro o h. -^ \ (sine + /ZcosO)^ ^ do Sirn A^ R T^ m ^ (38) where r is approximately the core radius. This expression is quite similar to that derived by Schoeck and Seeger [28] for bcc metals which is 2 irn A -, where L = A/kT. It should be noted that the form of the interaction potential used by Schoeck and Seeger [28] for carbon in alpha iron is different from that used to obtain Eq. 38 and because of this, their analog to Eq. 37 becomes more complicated and not integrable in closed form. Hence, "the integration is carried out... by dividing in different

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95 angular sectors and taking average values of u. and c, the error involved being less than a few percent." The results of Eq. 38 were obtained without any similar assumptions concerning the integration. Using Eq. 38 to calculate U results in U^ = 4.90 X lO-" In -^ (eV/cm) (40) = 2.94 X 10^ eV/cm = 4.70 X 10"^ dyne-cm/cm Substituting this result into Eqs. 33 and 34 one obtains 7 2 Aa ^ 1.84 X 10 dynes/cm (41) = 258 psi =1.78 MPa The inclusion of the angular dependence of the interaction energy approximately doubles the previous result of Eq. 35. The aging curves in Figures 7 and 8 at the lower temperatures investigated begin at approximately 5 to 6 MPa. These aging curves only commence after approximately 60 seconds. Qualitatively then, the 5 to 6 MPa value is high relative to Aa caused by Snoek ordering because the carbon-vacancy dipoles were probably ordered an order of magnitude in time before the experiments began. It is suggested that the value for Aa calculated by Eqs. 35 and 41 are in reasonable agreement with the data if one qualitatively extrapolates the Aa versus log t curves to short times consistent, at each temperature, with a vacancy migration distance of approximately 6b. At 373°K, for example, the jump time of a vacancy is approximately 0.06 seconds. Thus, the formation of pairs and their ordering is complete in approximately 1 second.

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96 The Schoeck-Seeger estimation of approximately 1 to 2 MPa is in fair agreement with this conclusion. In short, one result of vacancy trapping by carbon atoms after plastic defoniiation is to create a large number of elastic dipoles which form very quickly because of short range migration of vacancies to adjacent carbon traps. The ordering of the dipoles in the strain field of a dislocation can account for the initial ao obtained on restraining a specimen which has been aged for a \/ery short time at the temperatures investigated. Hence, the curves of Aa versus log t in Figure 7 appear to approach a non-zero value for short times and low aging temperature. 4.2.5 The Mechanism Controlling the Increase in Aa with Time It has been demonstrated [109,118] that the migration energy of a single vacancy in impure nickel during annealing experiments is approximately 1.1 eV (25.3 kcal/mole). In pure nickel, the migration energy has been deduced to be approximately 0.8 to 0.9 eV (18.4 to 20.7 kcal/mole), The difference in energies can be assumed to be due to a binding energy of impurities to vacancies [116]. Thus, the apparent migration energy of vacancies as observed in annealing experiments in nickel is very sensitive to the presence of impurities [117]. The early stages of the static aging experiments give an activation energy consistent with a mechanism involving the migration of vacancies in impure metal. Carbon atoms possess a migration energy of approximately 35 kcal/mole (Table 2) and at the temperatures investigated may be assumed to be immobile for times less than the times to achieve the maxima in Aa, For example, at 408°K a carbon atom will have jumped only approximately 75 times in 1.2 x 10 seconds [69], the time required

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97 to reach Aa^^^. A random walk distance of only approximately 8 lattice constants will be achieved and this should not result in significant redistribution of the carbon (to dislocations, for example). A vacancy on the other hand with a Q^ approximately equal to 25 kcal/mole will have jumped approximately 10 times. The rise in Aa, during aging, after the initial Snoek ordering of carbon-vacancy pairs can be rationalized on the basis that stress assisted diffusion of the vacancies toward the dislocation occurs. This aspect can be described by the Cottrell-Bilby model assuming a modulus interaction [139]. Pinning by vacancies alone can result in a small yield point, for example, as obtained in quenched and aged high purity aluminum [140,141]. The maximum yield return is less than 6.9 MPa (1000 psi), however. Nabarro [142] and others [143] have discussed vacancy pinning and indicate that there is considerable doubt that single vacancies can cause significant pinning of dislocations. Thus, it is assumed here that vacancies produced by plastic deformation do not play an important pinning role insofar as direct pinning is concerned. The net effect of vacancy migration toward the dislocation is to form an ordered carbon-vacancy Snoek atmosphere, which because of more pairs being closer to the dislocation causes a significant reduction in the dilatation produced by the dislocation in the metal. As a first approximation, it is assumed that in the early stages of strain aging (i.e., before the peak inAa), yery few vacancies reach the core of the dislocation. This assumption is within reason since the effective binding energy of a vacancy to a carbon atom is the sum of the Snoek energy and the binding energy of a vacancy to the

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98 carbon atom. Since the Snoek energy is assumed to vary as r , then very close to the dislocation the net effect is for the vacancy to be tightly bound to the carbon atoni. As free vacancies continue to migrate toward the dislocation, it is assumed that a saturated Snoek atmosphere is formed; that is, a fraction, f, of the carbon atoms near the dislocation have single vacancies as nearest neighbors. To relate the increase in Aa to time, it is assumed that a "saturated radius" may be defined which grows radially with time. The principal consequence of this assumption is that the carbon-vacancy pair concentration is limited to fc radius within R where c^ is the atomic fraction of carbon atoms and R is the "saturation radius." The nature of the "saturation radius" may be visualized more clearly in Figure 32. At the beginning of the second stage of aging (after initial Snoek ordering) the concentration profile of the most favored carbon-vacancy pair orientation, as a result of continued vacancy migration, becomes constant for distances closer to the dislocation than r. The value of f must certainly be much less than 1 since it is evident that all carbon atoms cannot be paired to vacancies because of equilibrium considerations. For convenience f is assumed independent of r. As vacancies continue to accumulate, R must increase with time. The growth of the saturated Snoek atmosphere will cause the energy of the dislocation and its carbon-vacancy pair atmosphere to decrease with time. Relating the energy decrease to the increase in the yield stress gives the kinetic dependence of Aa. In addition, the activation energy associated with this process is that for long range vacancy migration in the presence of carbon impurity

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99 O rt3 M-

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100 since the diffusion process, in conjunction with trapping at carbon atoms, controls the supply of carbon-vacancy pairs. One should recognize the existence of two components of Aa which overlap during aging. The first component is attributable to the initial ordering of carbon-vacancy pairs while the second accounts for the increase in Ao caused by the growth of the Snoek atmosphere with time. Thus, ^^ " ^"^initial ordering ^ ^'^atmosphere ^^^^ Most of the contribution to Aa, particularly after time for significant vacancy drift toward the dislocation has elapsed, should be contained in the second term. In view of this, the development of the present model neglects the first term. Quantitative model of the static strain aging in Nickel 200 From the Cottrell-Bilby model and assuming that the vacancy interaction energy with a dislocation behaves as Ar (modulus interaction) [15,138,142], one may deduce the number of vacancies that have arrived at a dislocation in time t as n^(t)=n;.b(M)l/2tl/2 • ,,3, where n is the average initial vacancy concentration per unit volume, A is the effective interaction constant of a vacancy, D is the apparent diffusivity of a vacancy in the presence of carbon impurity atoms, and t is the time in seconds. Equation 43 may be rewritten in terms of the atomic fraction of vacancies, c , instead of n , as

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101 where Q. is the atomic volume of the solvent. Assuming that saturation of carbon atoms occurs within a radius r of the dislocation and that the growth of the "saturation radius" is described reasonably well by a two-dimensional growth model, then: dn , -^=fn^ (2.rbf) (45) where dn /dt is the accumulation rate of vacancies moving into a cylinder of radius r and depth b between r and r + dr in time dt. The concentration of vacancies in this cylinder is assumed to be fn 3 where n is the number of carbon atoms per cm and is assumed to be unchanging and independent of position as indicated in the previous section. Integration of Eq. 45 results in n (t) = fTTR^bn V s c or n (t) = f^R^bc /Q (46) where c is the atom fraction of carbon. Equation 46 can be interpreted as the expression which describes the number of carbon-vacancy pairs accumulated within r at time t since each available vacancy is assumed to be attached to a carbon atom. Equating Eq. 44 to Eq. 46 gives S ^ ^ f8ADJ/2 J/2 . ^^^s^^c .47^

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102 The growth of the ordered Snoek atmosphere is thus approximately given by 1/4 Thus, the atmosphere grows approximately as t . Equation 44 is, strictly speaking, the approximate accumulation of vacancies near the dislocation with time. The meaning of Eq. 46, however, is the number of vacancies that have accumulated within R ; that is, the vacancies are not arriving at the core of the dislocation as Eq. 44 assumes but are accumulating as R grows. Thus, the exact meaning of Eq. 44 is relaxed somewhat to allow for the estimation of the growth of a Snoek atmosphere. It has been shown (Section 4.2.4) that the decrease in energy associated with Snoek ordering of carbon-vacancy pairs around a dislocation after a time span approximately equal to that required for one reorientation event is approximately Sun A^ R where n is the average volume concentration of the ordering species (dipoles) and r is approximately the dislocation core radius, lb. -1/2 R is the upper limit of integration on U and is p /2, i.e., onehalf of the mean dislocation spacing. U in this case applies strictly to a quasi-static array of dipoles. That is, no long-range migration of dipoles is assumed to have occurred. -In order to estimate the kinetics of pinning for the present case, it is assumed that only those dipoles (carbon-vacancy pairs) close to the

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103 dislocation are significantly influencing U . That is, only those carbon-vacancy pairs within the "saturation radius" are assumed to contribute to the decrease in energy. This is reasonable since the assumed interaction energy between a dislocation and a carbon-vacancy pair decreases as r~ and at further distances from the dislocation the contribution to pinning of a dislocation by a carbon-vacancy pair reorientation event is small relative to the same event occurring closer to the dislocation. Thus, the upper limit of the integration is approximately R rather than one-half the mean dislocation spacing and Eq. 38 becomes "o ' 3^kT '" r 8ufc A^ R^ U„ = ^?^UTIn ;:^ (49) where r =; b and n becomes fc /f2. c' In order to calculate the increased stress necessary to free a dislocation from its Snoek atmosphere, one needs to know the change in energy associated with moving the dislocation a short distance away from its Snoek atmosphere. That is, one needs to know the behavior of U(x) where x is the displacement of a dislocation from the center of its Snoek atmosphere. U(0) = U is the energy decrease caused by stress induced ordering around a dislocation at the center of its Snoek atmosphere. U(x), unfortunately, can only be evaluated numerically. Appendix B illustrates, however, the basic set-up of equations needed to evaluate U(x), the energy of a dislocation displaced a distance x«R from the center of its Snoek atmosphere, and is based upon the work of Evans and Douthwaite ["52]. To calculate the maximum additional force required to free a dislocation from its Snoek atmosphere one needs to know the position x at which the maximum force is experienced.

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104 Then one may compute Fmax = -('^"(»)''<"<>max (5°) U(x) was not evaluated. However, Schoeck and Seeger [27,28], as noted previously, have estimated the maximum force to be U kT f.ax = ir (51) where U is the total interaction energy of the carbon-vacancy di poles with unit length of a dislocation at the center of the ordered atmosphere (x==0) and A is the interaction constant of the carbon-vacancy pair/dislocation. An estimate of the increased applied tensile stress necessary to free the dislocation from its Snoek atmosphere is approximately given by F .V = ^ Aa (52) max m Thus, Aa^|Iu^ (53) where m is an orientation factor between 2 and 2.75. Using the result of Eqs. 49 and 53, one deduces that ? 4m7Tfc A R Ao --tob^'n^ (54) Thus, the increase in Aa is related to the size of the ordered atmosphere. Substituting the time dependent relation for R , Eq. 48, into Eq. 54 results in

PAGE 118

105 Qm/RT m' 4ni7Tfc A c , ,SAD e v.nI/2 / 'c '^' '^ -^ 1" (ff )^'' (-\t )^'' (^) t^/' (55) It should be noted that this model neglects (a) any migration of carbon atoms, and (b) movement of vacancies within the "saturation radius", R , to the dislocation where annihilation might be assumed to occur by elimination of jogs or rapid "pipe" diffusion to other sinks. Equation 55 may be rewritten as mirfc A mTTfc A ^^ 4mTTfc A C . /^ 8AD , ,, C Equation 56 is the increase in flow stress with time caused by the stress enhanced diffusion of vacancies toward the dislocation to form a growing saturated and ordered carbon-vacancy pair atmosphere. Accurately speaking, Eq. 56 is the upper yield point increase which is caused by breakaway of dislocations from their atmospheres. As an approximation, however, the lower yield stress increment, Aa, should follow the time dependent behavior of the upper yield point much in the same way as in bcc metals which obey Cottrell -Bilby kinetics [2]. The time dependence of Aa is reflected only in the first term of Eq. 56, i.e.. In t. Furthermore, the first term also predicts that the slopes of Aa versus In t curves should be independent of temperature. This feature is reasonably well exhibited in both the Aa versus log t and '^cf/Aa versus log t plots of Figures 7 and 8. The temperature dependence of static strain aging from this model lies in the second and third terms of Eq. 56. The predominant temperature dependence is reflected in the second term. Thus, the model predicts (for times less

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106 than those to achieve maximum hardening) straight Aa-ln t aging curves which are parallel and are simply displaced in time depending upon the size of Q and T. Quantitative predictions of the model As an example, the 428°K aging curve of Figure 7 was chosen to demonstrate the surprising accuracy of the Ao predicted by Eq. 56. The values assumed for the various parameters to obtain a reasonable fit to the data are listed below: 2.5 9. =

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107 Table 7 lists the values for the slopes and intercepts (at t=l second) of the Aa versus In t curves calculated from Eq. 56. These values may be compared to those of Table 5 obtained from the experimental data of Figure 7. In addition. Figure 33 illustrates the comparison between the model and the experimental data. The agreement is reasonable. Note that the principal unknown which could not be estimated was f the maximum fraction of carbon atoms which can be occupied by vacancies. The value assumed for f is 0.065 and would imply that the solubility limit of carbon-vacancy pairs near a dislocation, that is, the saturation limit, is considerably less than the total number of carbon atoms. This is a reasonable result in view of the comments made by Cochardt et a1 . [137] relating to the saturation concentration of carbon in iron near a dislocation. They estimate a maximum carbon concentration of 0.07 atom fraction near edge dislocations and about twice as much near screw dislocations. In addition a variation in f could account for the difference between the upper and lower yield points since the model presented relates to the upper yield point. f -9 The value for the average free vacancy concentration, c , of 10 atom -8 fraction is in fair agreement with the 7x10" value obtained in Section 4.2 which assumed a binding energy of approximately 0.3 eV between carbon atoms and vacancies. Thus, the model as derived can rationalize the aging behavior for times less than the time to achieve a maximum in Aa. 4.2.6 Regarding the Behavior of Ni 200 After the Peak in Aa The quantitative model presented in Section 4.2.5 rationalized that the yield point increase with time is a result of long range

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108 TABLE 7 The Slopes and Intercepts (at t=ls) of Aa Versus In t Curves Calculated from Eq. 56 Temperat

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109 Time (sec) Figure 33. The aging curves obtained from the model for strain aging in Nickel 200 (Eq. 56); the dashed lines are the experimental data (Figure 7).

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no migration of free vacancies toward the dislocation principally because of the dislocation-vacancy interaction. When the vacancies approach the dislocation, however, they are shown to be more tightly bound to carbon atoms and as a result more carbon-vacancy pairs are formed and via Snoek ordering in the dislocation strain field, the dislocation/carbon-vacancy pair total strain energy is decreased. Hence, additional energy in the form of an increased applied stress is necessary to remobilize the dislocation. This energy was found to be time dependent and, as a first approximation, the vacancies were assumed to saturate a fraction of the carbon atoms near the dislocation. The result is that an ordered Snoek atmosphere grows as a power law in time. The energy, however, was shown to vary as the logarithm of time. The assumption that the flux of vacancies into the core of the dislocation is small at early times is reasonable on the basis that the binding of vacancies to carbon atoms is stronger near the dislocation and, hence, the carbon \/ery near the dislocation essentially slows the flux of vacancies into the dislocation. Also, the flux of vacancies into the saturated atmosphere is probably very large at the beginning of aging, and dominates during the early stages of aging. Regarding the time required to achieve the maximum in Aa, it is probable that one of two things occurs. The strengthening is assumed to be a result of the carbon-vacancy pair concentration near the dislocation. A decrease in the number of carbon-vacancy pairs in this region would cause a decrease in Aa and may be the result of depletion of vacancies far away from the dislocation. This would slow the flux to the Snoek atmosphere and, hence, R would cease to increase the size. In addition, it is probable that with increasing time vacancies attached to carbon

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ni atoms close to the core of the dislocations are migrating to the core in significant numbers relative to the flux into the atmosphere at R . Thus, vacancies probably are migrating at this time to the core at a greater rate than can be supplied by the matrix. The net result is that a loss of carbon-vacancy pairs occurs and thereby decreases the energy U of Eq. 38 by decreasing the number of dipoles and, hence, Aa decreases. The rate controlling process responsible for the decrease in Aa should be the separation of vacancies from ordered carbon-vacancy pairs and their subsequent migration to the core of a dislocation. The energy associated with this event is B' + u(b) + E^ where B' is the binding energy of a vacancy to a carbon atom, u(b) is the Snoek energy at approximately r=b and E^^ is the migration energy of a free vacancy. Thus, the apparent activation energy for this process is higher by an amount u(b) than the energy controlling the rise in Aa during the time to reach Aa (which is B' + E^). Typical values for B', u(b) and E are 0.3, 0.2, and 0.8 eV, respectively. The conclusion is that the activation energy associated with aging at early times is different from that during "overaging" by approximately u(b), the Snoek ordering energy near the dislocation. Although the data are not as complete for the "overaged" state as during the aging stage, an activation energy was calculated on the basis of the method of cuts at Aa=23MPa and a least squares value of 29.0 kcal/mole (1.26 eV) with a correlation coefficient of 0.973 was obtained. The early aging curves gave an activation energy of approximately 25 to 26 kcal/mole. Thus, the difference in energies is

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112 approximately 3 to 4 kcal/mole (0.13 to 0.17 eV). This is in good agreement with the Snoek energy calculated in Appendix A. It is very important to point out that the carbon atoms during overaging may be partially responsible for the increased activation energy just calculated. This is because the carbon has had some time to start redistributing by stress assisted drift as predicted by the Cottrel 1-Bilby model. As previously noted, a carbon atom will have jumped only approximately 75 times during the time for Aa to reach Ao . For Cottrell pinning one would expect a considerable number of jumps [12] (of the order of 1000) per interstitial atom for completion of the pinning process since diffusion over fairly large distances is involved. Thus, during overaging some migration of carbon toward dislocations may have occurred. Evidently it is not significant in the time span of the aging times that were investigated because Aa never increased again. It is curious to note, however, that the 473°K curves of Figures 7 and 8 show that Aa has slowed its decrease beginning at t = 2 x 10 seconds. This may be an indication that carbon may be beginning to cause Cottrell type pinning. In addition, the Liiders extensions shown in Figure 10 for 473, 448 and 428°K may be foreshadowing another possible peak in Aa due to, perhaps, the carbon. Generally, in bcc metals the Luders extension maxima tend to lead in time the Aa maxima [2] and this may be a case here for the existence of a second peak. The secondary hardening curves l\a^, versus log t of Figure 11 also suggest a similar interpretation. Secondary hardening in bcc metals has been associated with the ease of remobilizing dislocations. When the

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113 reloading stress-strain curve no longer extrapolates back to the prestrain stress as in Figure 5 it is generally interpreted as an indication that dislocation multiplication has occurred. This increases p above that obtained during normal strain hardening and, hence, the stress level is higher after the passage of the LiJders band down the gage section when normal work hardening resumes, i.e., the metal suffers permanent hardening. This could be the case for Nickel 200 as well. For example, the trend for the 448 and 473°K curves of Figure 11 is for a maximum in ^o^ to occur and afterward a significant decrease occurs. This is indicative of dislocation multiplication as hypothesized in previous bcc metals investigations. AQh probably decreases because of the loss of vacancies (and, therefore, carbon-vacancy pairs) to the cores of dislocations and, hence, remobil ization is again favored. Note that for the 473°K AQm curve a possible minimum in Aom is achieved hi hi and this may be an indication that carbon atoms have begun to migrate in significant numbers to the dislocations and may indicate the beginning of Cottrell pinning by carbon atmospheres. As suggested previously the austenitic stainless steels are quite similar to nickel in their mechanical behavior. Also, the carbon diffusivities are very similar. Marek and Hochman [66] conducted a \/ery brief survey of static strain aging in as-received Type 316 stainless steel. Their aging temperatures were higher (see their Figure 1) by approximately 150 to 200° K than the temperatures used in this investigation. The interesting point, however, is that they observed peaks in the 0.2% yield stress (of as-received, i.e., prestrained material) during aging with an apparent activation energy of 31 kcal/mole which is close to the diffusivity of carbon in Type 316 stainless. Curiously,

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114 their 0.2% offset curve at 350°C (623°K) shows an initial decrease in yield stress for short aging times. Thus, there is a strong possibility that another peak may have occurred at an earlier time at 350°C (623°K) which they missed because of their higher aging temperatures and long aging times. This may be an aging peak caused by carbon-vacancy pinning such as that observed in Nickel 200. Rose and Glover [37] have attributed room temperature aging under load experiments of similar material to the rotation of carbon-vacancy pairs. Thus, there exists reasonable evidence obtained in some austenitic stainless steels which suggests that aging peaks similar to those observed in Nickel 200 might be observed in these steels if they were aged over an appropriate temperature range. For extremely long aging times in Nickel 200 it is probable that precipitation of graphite can occur since this alloy is supersaturated. This factor was not observed. 4.3 Summary The static strain aging model developed in this chapter may be ' summarized briefly by listing the steps which are believed to occur during static strain aging in Nickel 200: (a) Formation of carbon-vacancy pairs and initial ordering which causes an initial Aa very quickly (Stage I). (b) Migration of free vacancies in the strain field of the dislocation toward the dislocation and the formation of new carbon-vacancy pairs near the dislocation. (c) Fontiation of a saturated Snoek atmosphere around the dislocation which can be described by a size parameter called the "saturated radius."

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115 (d) Growth of the "saturated radius" and the consequent rise in Aa (Stage II). (e) Depletion of vacancies in the remainder of the lattice thus decreasing the flux of vacancies into the ordered radius and slowing the rise in Aa (Stage III). (f) Concomitant migration of vacancies in significant numbers to dislocation cores which causes a decrease in number of carbonvacancy pairs and, hence, a decrease in Aa (Stage IV). (g) Probable long range migration of carbon atoms to dislocations; Cottrell-Bilby model may be applicable. There is the possibility that another peak in Aa may occur at much later times that in the present case (Stage V). (h) Probable precipitation of supersaturated carbon as graphite and final overaging (Stage VI). A supplementary schematic illustration of the probable behavior of Aa with time during aging of Nickel 200 is shown in Figure 34. 4.4 Comments on the Relationship Between Static Strain Aging and Dynamic Strain Aging in Nickel 200 The results of static strain aging experiments on Nickel 200 indicate the yery strong possibility that carbon-vacancy pairs interact in a strong way with dislocations and can cause significant pinning. Discontinuous yielding is now considered to arise, in bcc metals particularly, when a low density of free or mobile dislocations develops at a stress sufficient to move dislocations large distances. Dynamic strain aging occurs in a temperature interval where interstitials are mobile enough to cause dynamic pinning so that homogeneous plastic deformation is no longer possible. As a result, a specimen yields

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116 S / en O H ' -OV

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117 locally and produces load drops. Each serration represents a local yield return event. Also, as listed in Table 1, other dynamic effects occur and may be attributed to this critical interstitial mobility. Internal friction phenomena should be related directly to effects occurring during the deformation of a metal containing interstitial s. For example, the use of the torsion pendulum has established that in static strain aging experiments on bcc metals (see, for example, the work of Szkopiak [ 8 ] on Nb-0 alloys), the Snoek peak (a measure of solute concentration) decreases and the cold work peak (attributable to atmosphere pinning of dislocations) increases. Thus, evidence related to the segregation of impurities to dislocations is available for the bcc metal s. Another important feature of internal friction measurements is that these types of experiments give direct information relating to the mobilities of impurities (e.g., jump times). The mobility of the impurities as deduced by internal friction must be related to dynamic strain aging. Gibala [144] has suggested that during a typical tensile test (10" s~ ) if T 0.1 second, then a specimen should experience inhomogeneous plastic deformation. This has been shown to be a quite accurate estimation [130]. For example, in alpha Fe at room temperature, the jump time for a carbon atom is about one second and, indeed, serrated flow is observed. Although one does not expect an internal friction peak due to single interstitials in fee metals, one may still estimate an approximate temperature where serrations should begin to be observed; that is, the temperature where x = 0.1 second . For the case of hydrogen [145] in

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118 nickel D = 0.0070 exp ( " ^^ ). Assuming x = a /12D then one expects to observe serrations beginning at approximately 148°K. Serrations are observed in hydrogenated nickel between approximately 130 and 230°K [57,63]. Thus, for hydrogen in nickel, reasonable agreement is obtained for the onset temperature. Consider the approximation related to carbon atoms in nickel. The onset temperature assuming [69] x = 3.57 x 10' exp (34800/RT), is -4 -1 523°K. Figure 21 shows Mery clearly for a rate of 4 x 10 s , that serrated flow occurs between approximately room temperature and 575°K. The conclusion is that carbon atoms alone are not responsible for serrated flow between these temperatures. That is, if carbon were solely responsible for DSA in Nickel 200, then one would expect on the basis of Gibala's estimation [144] that serrated flow should begin at much higher temperatures than are observed. Evans and Douthwaite [52] have previously concluded that Snoek ordering of carbon atoms at room temperature in alpha iron could be responsible for serrated flow in agreement with the x 0.1 second estimation. Thus, it is very possible that dynamic Snoek ordering of carbon-vacancy pairs is occurring during DSA in Nickel 200. In addition to the observations that the onset of serrated flow can be described using a 15 to 20 kcal/mole activation energy [51,68], the present aging investigation of Nickel 200 indicates an activation energy for the return of the yield point at 5% strain of approximately 25 kcal/mole. This activation energy is not consistent with an explanation based solely upon diffusion of carbon atoms to dislocations and the ensuing pinning process. Nakada and Keh [51] suggested that the low

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119 value of the activation energy for the onset of serrations is different from that normally associated with carbon diffusion (Table 2) because of dislocation pipe diffusion. However, this is not a very tenable explanation [130,146]. Blakemore [147] and Blakemore and Hall [148] have suggested that the Portevin-Le Chatelier Effect could be due to carbide precipitation. However, no indications of carbide or graphite were observed either in their investigation or the present investigation (see Section 3.1 ). The very wide range of temperature (300 to 575°K at 10' s" ) over which serrated flow is observed in Nickel 200 suggests that interactions between carbon atoms and vacancies may occur over a wide span of temperatures. The Portevin-Le Chatelier Effect has been observed in Type 330 stainless steel [53,54] between approximately 550 and 950°K and serrations were rationalized to occur over such a wide range of temperature because of overlapping mechanisms caused by interactions between point defects. The behavior from 550 to about 750°K was attributed to the carbon-vacancy interaction and above 750°K chromium was rationalized to have caused dynamic pinning of dislocations. When no apparent defect interactions occur as in alpha iron, the serrations are observed over a narrower range of temperatures, 340 to 475°K [149,150]. It is possible that in Nickel 200 at the lower end of the DSA temperature interval, predominantly the flipping of carbon-vacancy pairs may be responsible for serrated flow since this, the flipping process, should occur fairly rapid. As the temperature is raised, it is possible that the migration of vacancies is fast enough to increase the carbon-vacancy pair concentration near moving dislocations and may cause stronger pinning as evidenced by Type B serrations. No real

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120 distinction between these two processes occurring at the lower and upper ends of the serration spectrum can be made but this could account for the wide spread in temperature over which the Portevin-Le Chatelier Effect is observed compared to the relatively narrow temperature range of serrations observed in, for example, iron containing carbon. Additional evidence for the carbon-vacancy pair mechanism relative to serrated flow in Nickel 200 indicates that both carbon and vacancies must be present for aging to occur. Materials in which interstitials alone are responsible for hardening, such as low carbon steels [149] and Ni-H [57,63] have serrations starting at the onset of plastic flow. Nakada and Keh [51] observed that a finite strain was required before serrations initiated in Ni-C alloys. Initial smooth regions of plastic flow have only been found in fee alloys in which substitutional solute atoms cause serrations. This is generally believed to result from the diffusion rate of substitutional atoms in these materials being too slow to enable them to cause serrations during tensile testing. As straining proceeds, the vacancy concentration increases and the solute atoms can move just fast enough to initiate jerky flow. Hence, the smooth region of plastic flow is a measure of the concentration of vacancies needed to increase the diffusion rate at a given temperature to a level where the solute is mobile enough to cause serrations [151]. The observation by Nakada and Keh [51] of a small but finite incubation strain supports the existence of the carbon-vacancy pair in Ni-C alloys and in Nickel 200.

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121 CONCLUSIONS AND OBSERVATIONS 1. The carbon-vacancy pair is a defect which can be rationalized to cause strain aging in Nickel 200. 2. The rise in Aa during static strain aging is very slow in Nickel 200 and may be described in terms of a In t time law. An approximation for comparison to aging phenomena in other metals and alloys is at time dependence. 3. The activation energy controlling the rise in Aa is about 25 kcal/mole (1.1 eV/atom) and apparently represents the migration energy of a vacancy in the presence of carbon impurity. 4. Pinning in Nickel 200 can be rationalized as being due to the stress induced ordering of carbon-vacancy dipoles in the strain fields of dislocations. 5. Six stages are believed to occur during static strain aging of Nickel 200: a. the formation of carbon-vacancy pairs and their initial ordering (Stage I); b. the migration of vacancies in the strain energy gradient of a dislocation, the formation of new carbon-vacancy pairs near dislocations and the growth of an ordered carbon-vacancy dipole atmosphere (Stage II); c. depletion of free vacancies in the remainder of the lattice due to movement towards dislocations and to sinks in general which

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122 decreases the flux to the ordered atmosphere and therefore slows the rise in Ao (Stage III); d. eventual loss of vacancies from carbon-vacancy pairs due to migration to dislocation sinks which causes Ao to decrease at long times (Stage IV); e. the migration of carbon atoms in the strain fields of dislocations and the growth of a carbon Cottrell atmosphere (Stage V); f. possible precipitation of graphite and overaging (Stage VI). 6. Nickel 200 exhibits dynamic strain aging between approximately 300 and 600'^K depending upon strain rate. 7. Dynamic Snoek ordering of carbon-vacancy pairs can account for serrated flow in Nickel 200. The carbon-vacancy pair possesses the necessary mobility to cause DSA over a wide range of temperatures unlike single carbon atoms in alpha iron, for example. 8. Nickel 270 does not exhibit significant strain aging. 9. Enhanced strengthening as measured by the ultimate tensile strength is not caused by anomalous work hardening but to a prolonging of Stage II activity, i.e., single slip, which gives rise to a ductility maximum at 525°K. 10. Nickel 200 does not exhibit abnormal rate dependent work hardening as defined by log versus log o plots. 11. Nickel 200 exhibits a 0.2% yield stress plateau between approximately 300 and 475°K. 12. Discontinuous yielding in Nickel 200 is in agreement with the results of Nakada and Keh [51] who tested purer Ni-C alloys and obtained onset and termination activation energies of 15+2 and 26+4, respectively.

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123 13. Nickel 200 exhibits a well-defined reduction in area minimum in the DSA interval which has an apparent activation energy of approximately 25 kcal/mole.

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APPENDICES

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APPENDIX A COMPUTATION OF THE INTERACTION ENERGY BETWEEN A CARBON -VACANCY PAIR DEFECT AND SCREW DISLOCATIONS IN FCC METALS The technique used for obtaining the interaction energy is based on the work of Barnett and Nix [152]. Briefly, this method assumes that a tetragonal defect can be created by permitting the material within a small spherical volume, 9., to undergo a pure shear transformation strain. 2 0\ e.] . c I -1 ,o -1 / (58) This equation is written for a coordinate system aligned with the principal distortion axes of the defect. The total interaction energy between the tetragonal defect and an internal stress pi. is (ijj = 1, 2, 3 with summation over repeated indices implied). Since there is no hydrostatic component of the stress tensor for any of the screw dislocations (p! . = 0), one may split the transformation strain, £. ., into two parts •1 0\ n 0-1 + 3e 0-1/ Vo then 125 E.^^ = e|0-l 0) +3e|0 o) (60)

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126 E.^, = -J^ p! . e.J dv = -a^ep^l (61) where p-,-, is the 11 component of the dislocation stress field aligned with the principal axes of the tegragonal defect (carbon-vacancy pair). This is quite convenient because the interaction energy is easily obtained by calculating only one component of the dislocation strain field. Let X,, xl, and x-1 be the defect coordinate axes and x-, , x^, and xbe the coordinate axes for a right handed screw dislocation. The xl axis corresponds to the largest principal distortion of the defect. The dislocation lies along x_ and glides in the x,-xplane. Let P^g(cx,6 = 1,2,3) represent the dislocation stress field expressed in the dislocation coordinate system. The stress component needed for the computation of the interaction energy is obtained by transforming Pll ^° PaB ''^""3 where '<.-, and £,_ are the direction cosines between the x-J and the la IB 1 x^ and x axes, respectively. The only non-zero components of p are p-,^ and p^^ so that ^int^^l'^2^ = -(3ne) 2{il^^£^3P^2(^r^2^'^^12^13P23^^r^2^^ ^^^^ Next consider the specific case of an fee metal with a <100> tetragonal defect. Barnett and Nix [152] in their discussion utilized a <111> tetragonal defect which possesses four possible orientations. For the case of <100> defects (carbon-vacancy pairs) only three orientations are

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127 possible and the interactions are different. The dislocation coordinates are taken to be: x^ =l//6' [121], x^ = 1//3 [111], x^ = 1//2 [Toi]. The three possible orientations of the carbon-vacancy pair are: x^ = 1//2"[001], 17/2" [010], 17/^ [100]. The interaction energies for each of these orientations are given in Table 8. The components of the stress field of a screw dislocation in the fee structure are given by Hirth and Lothe [153] for the general case of anisotropy. For isotropy ^4^ '13 2u ^2 ^ 2 ^1 ^2 C44b (64) (65; nz 2TT 2 ^ 2 ^1 ^2 Thus, the interaction energy for a [001] defect is ^int = -;|^^^ (sin . /2 cose) (66) where f^ = a 74 (the atomic volume of a solvent atom). The value for AV the dilatation, due to carbon in nickel can be deduced from the lattice parameter measurements of Ni-C alloys determined by Zwell et a1 [115], Their result is a(A) = 3.5238 + 0.74c^ (67) where a is the lattice parameter in Angstroms and c is the carbon concentration in atomic fraction. The dilatation caused by carbon can be estimated from [154]

PAGE 141

128 >>

PAGE 142

129 AV=ff" (68) Substitution of the appropriate values into Eq. 68 gives Y-=0.63 (69) Thus, as an average e = 0.21. This value can be rationalized from the hard ball model as well by assuming that the carbon atom in o the lattice is about 1.54A in diameter (its size in the diamond lattice [155]) and can be squeezed into the body-centered position of the fee unit cell. This approximation gives a misfit of 0.15, less than the lattice parameter estimate. The vacancy on a 100 site should relax the strain somewhat. However, exactly how much it should relax the carbon dilatation is unknown. For the purposes of this discussion a value of e = 0.1 has been assumed for the carbon-vacancy pair. Thus, for the [001] defect -21 ^int " " ^'^^^ r ^° ^^'" ® " ^ ^°^®) ^^9-cm (70) In addition, it is assumed that = in order to estimate the size of the interaction constant. On this basis P 6.62 x 10"^^ erg-cm Q. 1 7eV-b ,7,x ''int "" ' r " r ^"^ At r = b = 2.49 x 10"^ cm, E.^^ 2.66 x lO"^"^ ergs = 0.17eV. Thus, the interaction energy is estimated to be about 0.2 eV. From Table 8 it is evident that the [001] defect possesses the lowest interaction energy. Hence, it is possible that reorientation of the other defects into [001] directions can lower the total

PAGE 143

130 interaction energy between the dislocation and defects. This type of ordering could probably occur in the strain field of an edge dislocation as well. However, as for the case of bcc metals [137] the form of the interaction energies of the three classes of defects with an edge dislocation is probably more complicated than for the screw dislocation. The magnitude of the interactions are probably similar as pointed out by Cochardt et al . [137]. Note that for the case of a 001 dipole interacting with a screw dislocation in an fee metal u^ = ^ (sin + /2 cos 0) = u (72a) U2 = (72b) (72c) where the subscripts indicate the three possible dipole orientations. Figure 35 illustrates the behavior of the interaction energies of Eq. 72 on angular position of the dipoles around the screw dislocation.

PAGE 144

131 0.4

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APPENDIX B DETERMINATION OF U(x), THE ENERGY OF A SCREW DISLOCATION DISPLACED A DISTANCE x FROM THE CENTER OF ITS SNOEK ATMOSPHERE The purpose of this appendix is to briefly present the method of Evans and Douthwaite [152] used to calculate U(x), the strain energy increase which a dislocation experiences when moved a small distance x from the center of its equilibrium Snoek atmosphere. The maximum force on a dislocation moving away from its static, equilibrium Snoek atmosphere is found by calculating the interaction energy of the dislocation as a function of its displacement, x, from the center of the atmosphere. The logic used is based upon the Schoeck and Seeger [28] calculation of U , the total interaction energy of dipoles with a screw dislocation and the assumption made by Evans and Douthwaite that as a consequence of displacing a dislocation to x, repopulation of the three possible dipole orientations does not occur appreciably. However, the energy, U(x), does change. Figure 1 of Evans and Douthwaite [152] demonstrates the geometry of the situation. When the dislocation is displaced a distance x (x«L) from the center of the equilibrium atmosphere, the coordinates of any point (r,0) relative to the dislocation at x = become (r',0') relative to the dislocation at x and the interaction energy is given approximately by (^""f^ .... U(x) = z n.u.r dr de (i = 1,2,3) (73) h h i ^ ^ 132

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133 where n. is the number of di poles per unit volume in the i possible orientation at a position r and is given by the probability function in the text (Eq. 24). In Eq. 73 the u'. have the same form as the u. given by Eq. 72 and the new coordinates are related to the old coordinates by r = r + X + 2r x cos , sin = r sin /r (74) Simplifying Eq. 73 gives r2u rL U(x) = (n^u^ + n2U2 + n^u^) r dr d0 (75) Thus, in new coordinates (i.e., after the dislocation moves to x), r is replaced by r'. Using Eq. 24 for the n. and substituting into Eq. 36 gives r27T rL , u/kT , -u/kT A reasonable assumption is that u«kT. As a result, the exponentials may be expanded and the integral becomes U(x) =1 Hq pj=[ [ (sin0' + /2.cos0')(sin0 + /I cos0) ~d0' (77) Utilizing transformation equations relating r to r' and to 0', Eq. 74, the author has concluded that Eq. 77 is not integratable in closed form. Thus, a numerical approach is in order. Due to time considerations, Eq. 77 was not evaluated numerically. Using the results of a tabulation of U(x), however, one may deduce F = -dU(x)/dx and the equivalent maximum applied shear stress required to free the dislocation from its

PAGE 147

134 Snoek atmosphere. At = (dU(x)/dx) /b. Using a suitable max max orientation factor one may calculate the tensile equivalent of the shear stress. Best estimates [28] indicate that this maximum stress occurs when At = U /2R (78) where U is the total energy decrease caused by stress induced ordering of dipoles and R = A/kT.

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BIOGRAPHICAL SKETCH Walter Raymond Cribb was born on August 29, 1949, in Columbia, South Carolina. His father was a member of the U.S. Army and as a result, he travelled quite extensively with his family throughout his entire elementary and high school years to numerous places within the United States and also overseas to Panama and West Germany. The author began his undergraduate work at the University of Florida in September, 1967. He entered the Department of Engineering Science and Mechanics and received the degree Bachelor of Science in Engineering Science (with honors) in December, 1972. January, 1973, the author began his work toward the Ph.D. degree in the Materials Science and Engineering department at the University of Florida. He received a Master of Engineering degree from this department in June, 1974. He is a member of the American Institute of Metallurgical Engineers as well as Tau Beta Pi, Sigma Tau and Alpha Sigma Mu honor societies. He is a First Lieutenant in the Un.ited States Air Force Reserves. He is married to the former Katherine Ann Morlock who is presently expecting a child. 144

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. R,lQaJ'lkl^ R.E. Reed-Hill , Chairman Professor of Materials Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ^M.A. Eisenberg '" Professor of Engineering Science and Mechanics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. C.S. Hartley Professor of Materials Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 'J^ jTk Hren Professor of Materials Science and Engineering

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This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August, 1975 Dean, College of Engineering Dean, Graduate School

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