Title: Some aspects of dynamic strain aging in the niobium-oxygen system /
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Title: Some aspects of dynamic strain aging in the niobium-oxygen system /
Alternate Title: Dynamic strain aging in the niobium-oxygen system
Strain aging in the niobium-oxygen system
Physical Description: xiii, 145 leaves : ill. ; 28 cm.
Language: English
Creator: Park, Soon Chun, 1946-
Publication Date: 1983
Copyright Date: 1983
 Subjects
Subject: Strains and stresses   ( lcsh )
Niobium alloys   ( lcsh )
Metals -- Mechanical properties   ( lcsh )
Metals -- Plastic properties   ( lcsh )
Materials Science and Engineering thesis Ph. D
Dissertations, Academic -- Materials Science and Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1983.
Bibliography: Bibliography: leaves 140-144.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Soon Chun Park.
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Bibliographic ID: UF00097422
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000447060
oclc - 11411952
notis - ACK8348

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SOME ASPECTS OF DYNAMIC STRAIN AGING
IN THE NIOBIUM-OXYGEN SYSTEM









By





SOON CHUN PARK


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





UNIVERSITY OF FLORIDA


1983




























To my wife, Nam-He








ACKNOWLEDGEMENTS


I would like to thank a number of people for their contributions

to this study. Foremost among them is Professor R. E. Reed-Hill, my

advisor, whose intellectual stimulation, encouragement, ready assistance,

and warm personal friendship have made my graduate work productive,

rewarding and highly enjoyable.

I am also grateful to Dr. E. D. Verink, Jr., Dr. R. T. DeHoff,

Dr. B. L. Adams and Dr. L. E. Malvern, for their valuable advice and

helpful comments.

I would also like to acknowledge Miss Perrine for her fine

typing work.

I would like to thank my wife, Nam-He, and my parents for their

ever-present support and encouragement.

Finally, I do wish to acknowledge the financial support, partly

by the Department of Energy under contract number DE-AS-05-76ER03262,

and partly by the Engineering and Industrial Experimental Station(EIES),

1981 and 1982, University of Florida.







TABLE OF CONTENTS


ACKNOWLEDGEMENTS ........................

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

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


Page
.. iii

.. vi

. vii


ABSTRACT.......................................................... xii

CHAPTER


I INTRODUCTION....................


II PRINCIPLE OF STRAIN AGING .................................. 5
2.1 Introduction........................................ 5
2.2 Mechanisms of Strain Aging............................. 8
2.3 Theories of Dynamic Strain Aging ....................... 11
2.3.1 The Cottrell Theory............................ 11
2.3.2 The McCormick Theory........................... 14
2.3.3 The van den Beukel Theory....................... 16
2.3.4 The Reed-Hill Theory............................ 19

III PREVIOUS INVESTIGATIONS.................................. 23
3.1 Portevin-Le Chatelier Effect........................... 23


3.2


3.3


3.1.1 Effect of Strain Rate and Temperature...........
3.1.2 Effect of Strain................................
3.1.3 Effect of Solute Concentration.................
Strain Rate Sensitivity (SRS)..........................
3.2.1 SRS at Low Temperature Where Dynamic Strain
Aging is Not Important.........................
3.2.2 SRS at Temperature Ranges Where Dynamic
Strain Aging Becomes Significant...............
3.2.2.1 Effect of Temperature.................
3.2.2.2 Effect of Strain.......................
3.2.2.3 Effect of Base Strain Rate............
3.2.2.4 Effect of Magnitude of Strain
Rate Change............................
Strain Aging Under Stress..............................
3.3.1 Introduction... .................................
3.3.2 Effect of Aging Under Stress on the Yield
Point Return...................................
3.3.3 Effect of Prestrain on the Yield Point Return...
3.3.4 Two Stages of Strain Aging Kinetics Curve.......








CHAPTER
Page
IV EXPERIMENTAL PROCEDURE ................................... 47
4.1 Materials and Materials Preparation................... 47
4.2 Tensile Testing ................................... 51

V EXPERIMENTAL RESULTS ..................................... 55
5.1 Portevin-Le Chatelier Effect Due to Snoek
Dynamic Strain Aging ................................. 55
5.1.1 Tensile Tests.......... ........................ 55
5.1.2 The Ratio of the Strain Rate to the
Diffusion Coefficient ........................ 58
5.1.3 Kinetics of the Snoek Ordering................. 60
5.1.4 Strain Dependence of Serrated Flows............ 60
5.2 Strain Rate Sensitivity of the Flow Stress............ 66
5.2.1 Effect of Temperature ....................... 66
5.2.2 Effect of Oxygen Concentration ................ 69
5.2.3 Effect of Base Strain Rate..................... 77
5.2.4 Effect of Magnitude of Strain Rate Change...... 80
5.2.5 Effect of Prestrain ............................ 83
5.3 Work Hardening Parameter... .......................... 86
5.4 Strain Aging Under Stress............................ 90
5.4.1 Effect of Aging Stress, Prestrain on
the Yield Point Return ......................... 90
5.4.2 Average Strain Rate Dueing Aging
Under Stress.................................. 94

VI DISCUSSION...............................................100
6.1 Strain Rate Sensitivity of the Flow Stress............100
6.1.1 Two Strain Rate Sensitivity Minima............100
6.1.1.1 Effect of Oxygen Concentration ........106
6.1.1.2 Effect of Base Strain Rate...........111.
6.1.1.3 Effect of Prestrain.................. 113
6.1.1.4 Comments on the Relationship
Between Serrations and SRS............117
6.1.2 Strain Rate Sensitivity Peak Above the
Cottrell SRS Minimum Temperature .............121
6.2 Work Hardening ....................................... 132
6.3 Strain Aging Under Stress............................ 135

VII CONCLUSIONS..............................................138

REFERENCES..........................................................140

BIOGRAPHICAL SKETCH............................................... 145







LIST OF TABLES




Chemical Analysis of Marz-Nb-VP-Nb and WC-Nb, ppm

Values of the Parameters Used in Equation (6.2)


Table

4.1

6.1


Page








LIST OF FIGURES



Figure Page

2.1 Schematic representation showing the various stages
in static strain aging. a) No aging period between
unloading and reloading. b) Short aging period.
c) Long aging period. 6

2.2 Logarithm of the aging stress versus logarithm of
the average strain rate over a 35-minute aging period
for V-O aged at 373 K. 9

3.1 Two typical forms of serrated flow. a) Type A serrations
are associated with Luders bands that can travel the
length of the gage section. b) Type B serrations involve
bands that do not propagate. 24

4.1 Specimen dimensions used in this investigation. 49

4.2 Illustration of the extrapolations used for the
calculation of the strain rate sensitivity on various
stress strain curves. 52

4.3 Schematic representation showing the method used to
measure the yield point return when a specimen was aged
at a fixed load. 53

5.1 Stress-strain curve of Nb-0.75 at.% oxygen specimens
for temperatures between 335 K and 422 K. 56

5.2 Details of the serrated curves. 57

5.3 A plot of the maximum serration height observed during
a tensile test as a function of the (e/D) ratio, Nb-0.75
at.% oxygen. 59

5.4 A comparison of the average time between serrations with
the jump time of an oxygen atom. 61

5.5 A plot of the critical strain, Ec, for the onset of
serrations as a function of temperature between 340 and
422 K. Nb-0.75 at.% oxygen. 63

5.6 The strain between successive periodic serrations,
Es, at T = 381 K. 64






Figure Page

5.7 Variation of the strain rate sensitivity, S, with
strain at 346 K. Nb-0.75 at.% oxygen, EL = 8.8 x 10-5 s-1
EH = 4.4 x 10-4 s-1. 65

5.8 Variation of the strain rate sensitivity, Aa/Azni with
temperature. VP Nb specimens, L = 8.8 x 10-5 s-I
(H/L) = 5. 67
5.9 Variation of the strain rate sensitivity, Akno/An,
with temperature. VP Nb specimens, tL = 8.8 x 10- s-1
(H/ L) = 5. 68
5.10 Variation of the strain rate sensitivity, Ao/Akn, with
temperature. Nb-0.24 at.% 0 specimens. L = 8.8 x 10-5 s-
(Hi L) = 5. 71
5.11 Variation of the strain rate sensitivity, AZna/Aint, wi h
temperature. Nb-0.95 at.% 0 specimens. L = 8.8 x 10- s-1
(H/ L) = 5. 72
5.12 Variation of the strain rate sensitivity, Ao/Aknt, with
temperature. Nb-0.95 at.% 0 specimens, tL = 8.8 x 10-5 s-1
(tH L) = 5. 73
5.13 Variation of the strain rate sensitivity, AUno/Aknn, with
temperature. Nb-0.95 at.% 0 specimens, tL = 8.8 x 10-5 s-1
(tH L) = 5. 74
5.14 Variation of the strain rate sensitivity, Aa/Akn, with
temperature for specimens containing three levels of
oxygen. L = 8.8 x 10-5 s-1 ,(tH/ L) = 5. 75

5.15 The effect of the base strain rate on a plot of
(Aa/Ant) versus temperature, VP=Nb specimen. 78

5.16 The effect of the base strain rate on a plot of
(Aino/Aknt) versus temperature. 79

5.17 The effect of the magnitude of the strain rate change
on a plot of (Ao/Aknt) versus temperature.
,L = 8.8 x 10-5 s-1. 81
5.18 The effect of the magnitude of the strain rate change
on a plotof (AZno/Alnt) versus temperature.
tL = 8.8 x 10-5 s-1. 82
5.19 The effect of strain on a pl t of (Ao/Aknt) versus
temperature. tL = 8.8 x 10- s-1 (H/ L) = 25. 84

viii








Figure Page

5.20 The effect of strain on a plot of (Anoc/Aine) versus
temperature. L = 8.8 x 10-5 s-1 (H/ L) = 25. 85

5.21 Variation of the work hardening as a function of the
temperature for three different strain rates.
VP-Nb specimens. 87

5.22 The logarithm of the strain rate as a function of
the reciprocal peak tempearure for each work hardening
maximum. VP-Nb specimens. 89

5.23 The effect of oxygen concentration on the variation of
the work hardening as a function of temperature.
S= 8.8 x 10-5 s-I. 91

5.24 Variation of the work hardening peak as a function
of the oxygen concentration. t = 8.8 x 10-5 s-1 92

5.25 The effect of aging stress on the yield point return.
VP-Nb aged at 371 K for 35 minutes. e = 5%. 93

5.26 Effect of strain on the yield point return. VP-Nb
aged at 371 K for 35 minutes, oa = 96% af. 95

5.27 Logarithm of the aging stress versus the logarithm
of the average strain rate over a 35-minute aging
period. VP-Nb aged at 371 K. Data obtained from
V-O are also shown. 97

5.28 Variation of the average aging strain rate with aging
time at two aging stresses, 98% and 100% f.
T = 371 K. VP-Nb specimen. 99

6.1 The calculated strain rate sensitivity, S as a function
of temperature. a) Curve of S for the dynamic strain
aging independent part of the stress. b) Curve of S
for Snoek strain aging. c) Curve of S for Cottrell
dynamic strain aging. 103

6.2 The calculated strain rate sensitivity as a function
of temperature. Experimental data obtained from
Nb-0.24 at.% 0 specimens are also shown.
L = 8.8 x 10-5 s-1, H = 4.4 x 10-4 s-1. 104

6.3 A plot of the strain aging kinetics curve showing the
effect of the prestrain strain rate (or waiting time)
on the magnitude of the yield point return increment. 107







Figure Page

6.4 Analytical curves of strain rate sensitivity versus
temperature for two relaxation times. T2 = 2T.. 110

6.5 Analytical curves of strain rate sensitivity versus
temperature for three base strain rates.
I = 8.8 x 10-5 s-1, 2 = 5 E', 3 = 25 l1. 112

6.6 Logarithms of the strain rate as a function of the
reciprocal temperature corresponding to the Snoek
SRS minimum and the Cottrell SRS minimum. 114

6.7 Analytical curves of strain rate sensitivity versus
temperature at three prestrains versus temperature at
three prestrains between 1% and 4.5%. The values of
tw at the various strains were determined by the
equation tw = 2.63 (e/4,5)B with B equal to 1.2. 116

6.8 Variation of the strain rate sensitivity with strain
for three temperatures, 395 K, 425 K and 600 K.
EL = 8.8 x 10-5 s-l (H/ L) = 25. VP-Nb specimens. 118

6.9 a) A schematic diagram showing the effect of two base
strain rates on the SRS versus temperature relation.
This diagram may predict the variation of the stress
strain curve at two strain rates. b) Variation of the
stress-strain curves at two strain rates. T = 470 K,
488 K and 520 K. Nb-0.24 at.% 0. 119

6.10 The effect of oxygen concentration on the strain
rate sensitivity peak at 600 K. EL = 8.8 x 10-5 s-l
(iH/) = 5. 122

6.11 A schematic diagram of the drag stress as a function
of tensile strain rate. The calculated value of
critical strain rate at T = 600 K is 2.02 x 10-4 s-1. 125

6.12 A schematic diagram showing the effect of concentration
on a plot of the drag stress versus tensile strain rate. 127

6.13 Variation of the stress strain curves upon increasing
the strain rate from EL = 8.8 x 10-5 s-1 to
H = 4.4 x 10-4 s-1 at several temperatures.
Nb-0.95 at.% oxygen. 128

6.14 a) Variation of the flow stresses with temperature at
two strain rates, F L = 8.8 x 10-5 s-1 and
EH = 4.4 x 10- s- Nb-0.07 at.% 0. =-4.5%. 131








Figure Page

6.15 Superimposed are the variations of work hardening
and strain rate sensitivity with temperature. An
arrow points to the temperature range over which
serrations were observed. 4L = 8.8 x 10-' s-1
(tH/ L) = 5. VP-Nb specimens. 134
6.16 Variation of (Wa/~ ) with (ca/af). T = 371 K.
& = 8.8 x 10-5 s- t = 35 minutes. VP-Nb specimens. 137
p a







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



SOME ASPECTS OF DYNAMIC STRAIN AGING
IN THE NIOBIUM-OXYGEN SYSTEM


by

Soon Chun Park

December, 1983



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



Some important aspects of dynamic strain aging (DSA) in niobium

containing oxygen between 0.01 and 0.95 at.% were investigated between

77 K and 971 K. These aspects include negative strain rate sensitivity,

occurrence of the serrations on a stress strain curve and high work

hardening rate.

The strain rate sensitivity (SRS) of the flow stress was measured

as a function of temperature by changing the strain rate. It was observed

that there were two temperature intervals within which the SRS becomes

negative. These occurred near 390 K and 500 K for Nb-O.O1 at.% oxygen

specimens prestrained 4.5% when a (5:1) strain rate change and a base

strain rate of 8.8 x 10- s-1 was used. It was found that the SRS

minimum temperature is a function of the base strain rate, the magnitude

of the strain rate change, the prestrain and the oxygen concentration.








The first two cause the SRS versus T curve to shift towards higher

temperatures whereas the other two have an opposite effect. The two

SRS minima were explained by using the phenomenological theory. They

were found to be due to Snoek and Cottrell DSA.

It is proposed that the serrated flow observed in a higher oxygen

specimen in the low temperature range, between 355 and 422 K, is due to

Snoek dynamic strain aging. This is supported by 1) the occurrence

of the serrations at much lower temperatures than those normally associ-

ated with Cottrell dynamic strain aging, 2) the observation that the

time between the Type A serrations equals the jump time of an oxygen

atom, and 3) the observation of a negative strain rate sensitivity in

this region.

The Cottrell SRS minimum was observed to be closely related to

the work hardening peak for Nb-0.01 at.% oxygen specimen.

Above the region of the Cottrell SRS minimum, the SRS passes

through a sharp maximum. This peak increases sharply in height with

increasing oxygen concentration. It was deduced that this peak is

probably associated with the viscous dragging of solute atoms by dis-'

locations.


xiii






CHAPTER I
INTRODUCTION


It is well known that the mechanical properties of bcc metals are

extremely sensitive to the presence of interstitial solute atoms in solid

solution. One of the best examples is strain aging. The strain aging

phenomena in the bcc metals containing interstitials such as oxygen,

carbon and nitrogen, are due to the interactions between dislocations

and interstitial solute atoms.

The strain aging may take place after or during plastic deforma-

tion. The case of aging after straining is the more normal one and is

referred to as "static strain aging." If the aging takes place during

plastic straining it is called "dynamic strain aging." The best known

aspect of dynamic strain aging is the so-called "blue brittle"

effect in iron or low carbon steel strained in tension in the range of

150-2000C. Other examples of dynamic strain aging are well documented

in the literature [1]: these are serrated yielding (Portevin-Le

Chatelier Effect), abnormally high work hardening rates, a low (or

negative) strain rate sensitivity, yield stress plateaus in a plot of

yield stress versus temperature and flow stress transients on changes

in strain rates.

Among the features of the dynamic strain aging, the most visible

is the appearance of serrated flow in stress-strain curves. Serrated

yielding has been widely investigated in both bcc alloys [2-7] and fcc

alloys [8-12]. Most of the investigations in substitutional alloys

have been concerned with rationalizing the onset of serrations through








the enhanced diffusion of solute atoms due to vacancies produced by

plastic deformation. In the case of bcc interstitial alloys, it is

considered that the dynamic strain aging may take place when the dis-

locations are pinned by the interstitial solute atoms during deforma-

tion.

It is generally agreed that there are two forms of dislocation

pinning processes in bcc interstitial alloys. The first is due to

"Snoek ordering" of the interstitial atoms in the stress fields of the

dislocations. This was analyzed originally by Cochardt et al. [13]

and later in more detail by Schoeck and Seeger [14]. It has also been

treated more recently by Evans and Douthwaith [15] and by Reed-Hill [16].

The second form of pinning process of the dislocations involves the long

range drift of interstitial solute atoms to form solute atmospheres

at the dislocations. These are called "Cottrell atomspheres." This

area was originally treated by Cottrell and Bilby [17] who derived a

kinetics law for the time dependence of the arrival of the solute at

the dislocations.

A study by Bradford and Carlson [2] on dynamic strain aging in

the vanadium-oxygen system is interesting because serrations were

observed in two different temperature ranges when specimens were

deformed at the same strain rate. A strong effect of oxygen concen-

tration was also noted.. Thus, for specimens containing between 47 and

265 ppm of oxygen, well defined serrations were observed only between

623 and 723 K.

However, specimens containing higher oxygen concentrations, i.e.,

955 and 1800 ppm, also showed serrations in a lower temperature range

between 423 and 448 K, with the largest serrations appearing in the









specimens with the higher oxygen concentration. They interpreted the

high temperature serrations as being due to Cottrell dynamic strain

aging by using the relationship, E/D = 109, which was obtained by

Cottrell [18], using Manjoine's experimental data [19] for the condi-

tion of the onset of serrations. However, they were not able to explain

which mechanism was responsible for the low temperature serrations. It

is now believed that the low temperature serrations are due to Snoek

dynamic strain aging [20-23]. The results of Bradford and Carlson also

suggest that a high solute concentration is needed to make the Snoek

serrations apparent.

There is a good correlation between the appearance of serrated

yielding and the strain rate sensitivity of the flow stress; i.e., a

negative value of the strain rate sensitivity of the flow stress is

necessary for the occurrence of serrated yielding [24-26].

In the present study, work on Snoek and Cottrell dynamic strain

aging has been extensively done in the niobium-oxygen system. It was

again shown that the low temperature serrations are associated with

Snoek dynamic strain aging by using different approaches. These include

1) Measurement of the average time between serrations on the

stress-strain curve,

2) a plot of the maximum serration height of low temperature

serrations as a function of the (e/D) ratio, and

3) measurements of the strain rate sensitivity as a function of

temperature.









Since no strain rate sensitivity data,as a function of temperature,

including both Snoek and Cottrell dynamic strain aging are available,

an extensive study of the effects of the experimental parameters on the

plot of strain rate sensitivity versus temperature were made and analyzed

by using the phenomenological theory developed recently [27-29]. An

attempt was also made to relate the strain rate sensitivity to the work

hardening.

The material employed was niobium containing between 0.01 and 0.95

at.% oxygen. A study of Snoek dynamic strain aging was largely made with

Nb-0.75 at.% 0. Niobium is an ideal bcc metal to use in a dynamic strain

aging study since the solubility limits of interstitial atoms, such as

oxygen and nitrogen, are very high [30].







CHAPTER II
PRINCIPLE OF STRAIN AGING


2.1 Introduction


Strain aging is a phenomenon involving strengthening or hardening

due to aging after some plastic deformation. This can occur either in

the static or dynamic mode.

In static strain aging, as illustrated in Figure 2.1, the specimen

is first prestrained to a point X on the stress-strain curve. It is

then unloaded and allowed to age for some period of time. At the end

of the aging period, the specimen is again deformed. If the aging time

is very short, the restraining curve will rise up to the flow stress at

point X and then follow the same stress-strain path as that of an unaged

specimen. This is indicated by curve (a) in Figure 2.1. Aging for an

intermediate time will result in the appearance of a small yield point

upon reloading. This indicates that a higher stress level is needed to

initiate yielding. However, after the yield point the stress-strain

curve again follows the path of a specimen that was not subjected to the

aging. This is indicated by the curve marked (b). If the specimen is

allowed to age for a long period of time, not only the magnitude of the

yield point but also the level of the stress-strain curve will be raised,

as may be seen in curve (c). Thus, the primary results of static strain

aging are twofold. First there is a return of the yield point and then

at long aging times there may be a basic strengthening of the material.










V)
U)

H-




(a) STRAIN



CU

CO



(b) STRAIN



CO






(c) STRAIN
Figure 2.1. Schematic representation showing the various stages in
static strain aging. a) No aging period between
unloading and reloading. b) Short aging period.
c) Long aging period. After Beckerman [23].








In static strain aging, since the aging occurs while the specimen

is unloaded, the dislocation can be considered to be effectively at

rest while they are being aged. However, in the case of dynamic strain

aging, the interaction between the mobile solute atoms and the disloca-

tions occurs with dislocations that are not at rest but that are moving.

However, if the concepts of thermally activated dislocation motion are

assumed, then the dislocations are periodically immobilized as they move.

Sleeswyk [31] was the first to propose that during this arrest time the

dislocations could be subject to strain aging. Thus, during the waiting

time both reordering of the interstitials and the formation of Cottrell

atmospheres could occur. Therefore, it can be considered that two factors

determine the degree of dynamic strain aging. These are

1) How long does it take for a dislocation to wait at an obstacle

for thermal activation?

2) How fast do solute atoms diffuse to the dislocations while

they dwell at the obstacles?

In this regard it can be argued that the waiting time should be

inversely related to the strain rate; the faster the strain rate, the

shorter the aging time. On the other hand, the mobility of the solute

should vary directly as the diffusion coefficient of the solute. Thus,

the solute mobility should increase rapidly with increasing temperature.

From this it can be concluded that dynamic strain aging phenomena should

exhibit a strong interrelationship between the strain rate and the

temperature.

Bolling [32] pointed out that during a strain aging under stress

experiment the specimen usually deforms plastically as it is aged.










Recently, Beckerman and Reed-Hill [23,33] have measured, as a function

of the applied aging stress, the average strain rate over a fixed aging

period of 35 minutes using vanadium specimens. Thus, in Figure 2.2

data [23,33] corresponding to the average strain rate in a 35-minute

aging period are plotted as a function of the fraction of the pre-

strain stress that was applied during the aging period. It may be

seen from the figure that the strain rates in typical strain aging

under stress experiments are not significantly different from those

in slow tensile tests and in fast creep tests. In other words, the

strain aging under stress experiments allows the strain aging process

in a metal to be observed under conditions approaching those existing

during dynamic strain aging.



2.2 Mechanisms of Strain Aging


It is generally agreed that there are two primary mechanisms

involving interactions between the interstitial solute of atoms and dis-

locations in bcc metals. The first of these merely involves the re-

ordering of the interstitial solute atoms around the dislocations.

This can be explained in terms of the Snoek effect [34]. An intersti-

tial atom in bcc lattice will produce a local tetragonal distortion.














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There are three types of interstitial sites in a bcc unit cell so that

the tetragonal distortion can occur in three mutually perpendicular

directions. In the absense of an applied stress, each of the three

kinds of site will be occupied by the same fraction of interstitial

atoms and there will be no overall tetragonality. However, when a

stress is applied, some sites are preferred since the interaction

energy between the stress and the interstitial will depend on the

direction of the tetragonal distortion. If the interstitials can

diffuse they will tend to move preferentially to those sites with

the lower energy. The diffusion distance will be less than one atomic

spacing and therefore no long range diffusion is required.

A similar redistribution will occur in the stress field of a

dislocation and will produce a lowering of the energy of the system.

As a result, an additional stress will be required to move the dislo-

cation. This is called Snoek aging and has been critically analyzed

by Schoeck and Seeger [14]. Since no long range diffusion is required

this reordering can occur very rapidly and is normally essentially

completed in about the time required for an average interstitial atom

to make one jump.

The second form of strain aging is called Cottrell aging and was

originally treated in detail by Cottrell and Bilby [17]. It involves

the slow drift of the interstitial atoms to the dislocations so as to

form atmospheres of the solute around the dislocations. This process

is much slower than that of Snoek aging since it involves a long range

migration of the interstitials requiring many atomic jumps. Both the

Snoek and the Cottrell form of aging result in lowering of the energy










of the system. This effectively locks the dislocations in the sites

which they occupy while they are being aged. Consequently an increase

in the stress is required in order to move dislocations after they are

aged.



2.3 Theories of Dynamic Strain Aging [27]

2.3.1. The Cottrell Theory [35,36]

Cottrell's model of dynamic strain aging for the onset of the

Portevin-Le Chatelier effect (or serrated yielding) suggests that

solute atmospheres form about, and are dragged along, by dislocations

moving at less than a critical velocity, vc, given by [35,36]

4D
c T (2.1)


where D is the solute diffusion coefficient and k is the effective

radius of the atmosphere. The radius k is equal to A/kT where A is

a parameter that depends on the elastic constants, the volume change

caused by the solute atom, and the strength of the dislocation. Since

at this velocity the stress to move a dislocation decreases as the

velocity increases, it is logical to conclude that vc also represents

a critical condition for the appearance of serrations on a stress-strain

curve.

If it is now assumed that the Orowan equation is valid, we have


S= pmbvy
m c


(2.2)










4D
S= pb
m T


(2.3)


where e is the applied strain rate, pm is the mobile dislocation density,

b the Burgers vector, D the solute diffusion coefficient, and Z the

radius of the dislocation atmosphere. The dislocation density is

normally considered to be a function of the strain, e, so that we can

write


P NEB


(2.4)


where N and B are constants. In addition, in the case of substitutional

alloys, the diffusion occurs by vacancy motion and the diffusion coeffi-

cient is given by


Q + Qf
D = D exp [-( m kT )]


(2.5)


where Qm is activation energy for the movement of vacancies and Qf is

the work to form a vacancy. The diffusion equation may also be written

QM
D = DoC exp(-k ) (2.6)

Qf
where C* = exp(- k) is the thermal equilibrium concentration of

vacancies. During plastic deformation vacancies are formed at a rapid

rate and it is generally agreed that the vacancy concentration increases

with strain according to the empirical relation


C = Kem
v


where C is the vacancy concentration and K and m are constants.


(2.7)








The vacancy concentration created by the plastic deformation

normally exceeds the thermal equilibrium concentration by many orders

of magnitude so that the diffusion equation becomes

Q
D = D C exp(- ) (2.8)

or

D = D0Kem exp (- ) (2.9)

With the aid of equations (2.4) and (2.9) we may now write the Orowan

equation as

4b m m
e = NEN KEm D exp(- k) (2.10)

or

e 4b N K D (+m) exp(- k (2.11)

Solving for the strain term gives
Q
(B+m) exp(T (
c 4bNKD (2.12)
c 4bNKD

where ec is the critical strain for the start of serrations on the

stress-strain curve of a substitutional alloy. Equation (2.12) suggests

experiments for the determination of the parameters (6+ m) and Qm"

While the Cottrell theory is able to predict both the strain rate and

the temperature dependence of the critical strain, it fails when it

comes to predicting the critical strain itself [37].








2.3.2 The McCormick Theory

An alternative model for the Portevin-Le Chatelier effect in

substitutional alloys has been proposed [37-39]. McCormick's model

is based on the assumption that dislocation motion is a discontinuous

process. For such a process the average dislocation velocity may be

expressed in terms of an arrest or waiting time at the obstacles, tw,

and a time of flight time through the lattice to the next obstacle,

tf, as

-- L
Stf (2.13)
tw + t

where L is the average distance between arresting obstacles. In most

instances, the average dislocation velocity is determined primarily by

the arrest time, so that we may write

L
v= t- (2.14)
w

McCormick uses a suggestion originally due to Sleeswyk [31] that during

the time when a dislocation waits in front of an obstacle, mobile solute

atoms could be drawn to it and result in strain aging. Thus he proposed

that the initiation of serrations on the stress-strain curve occurs when

the time required to age or lock a moving dislocation, ta, becomes equal

to the time that the dislocation has to wait at an obstacle for a ther-

mally activated event that will allow it to pass through the obstacle;

i.e., tw = t,. If tw is less than ta at the start of plastic deformation

dislocations arrested at obstacles will not be locked and the stress-

strain curve will be continuous. However, during straining, ta will








decrease due to vacancy production while tw increases as a result of

dislocation multiplication so that at the critical strain, c' ta

becomes equal to tw. At ec the few remaining unlocked dislocations

will multiply rapidly, causing the formation of a Luders front and the
start of serrated yielding.

The arrest time, tw, may be expressed in terms of the strain rate

and dislocation density as

p mbL
tw (2.15)

where it is assumed that t >> tf. The time required to age or pin an

arrested dislocation, for elastic solute-dislocation interactions is

given by, for short aging times, by [17]

t Cl %3/2 kTb2
ta oC U-m (2.16)
o m
where C1 is the solute concentration at the dislocation line which is

required to lock it, C0 is the solute concentration of the alloy

(C1 > C ), Um the binding energy between the solute and the disloca-

tion, Q is a constant equal to about 3 and D is the solute diffusion
coefficient.

On the assumption that both the dislocation density, pm, and the
vacancy concentration are functions of the strain as expressed in

equations (2.4)and (2.7), McCormick arrives at a critical strain equation

for strain aging involving a substitutional solute of the form

m+a Cl 3/2 UKTb exp(Qm/kT)
S ( ) NKU D (2.17)
o mo








In bcc interstitial alloy systems diffusion occurs by a mechanism

involving the jumping of interstitial atoms from one interstitial site

to another. Since this form of diffusion is independent of the vacancy

concentration, the equation for the critical strain in bcc interstitial

alloy systems becomes


S (C1 )3/2 skTb exp(Q/kT) (2.18)
c C 0 NLU D
o mo

where D is the interstitial diffusion frequency factor and Q is the

activation energy for the diffusion of interstitial solute atoms.


2.3.3 The van den Beukel Theory

While the Cottrell and McCormick models are specific to the

conditions controlling the start of serrations on a stress-strain

curve, the van den Beukel's theory [40] is more general and tries

to treat most of the well recognized aspects of dynamic strain aging [1].

The van den Beukel's theory is based on the idea that the moving

dislocations "see" a solute concentration which depends on the waiting

time and the solute diffusion coefficient. In effect, he assumes that

the activation enthalpy H in the thermally activated strain rate equation

is modified by the concentration variation due to dynamic strain aging.

Thus, he starts with the equation


E = Eo exp (-H/kT) (2.19)

and the assumption that


H = H (a*,C)


(2.20)








were o* is the effective stress and C is the solute concentration that
it "sees." The value of C will be a function of the time that a dis-

location waits at an obstacle, that is tw, and the rate at which the
solute diffuses to the dislocation. Thus, he writes

C = C (Dtw) (2.21)

He shows further that the quantity Dtw is a function of strain, strain

rate and temperature as follows.

(m+-) Qm
Dtw exp[- -] (2.22)

where m and B are constants, and Qm is the migration energy of vacancies.
2/3
For small values of Dt he uses a t2/3 relation between the concentration
and the time formulated by Friedel [41].

C Co = K(Dtw)2/3 (2.23)

where Co is the nominal solute concentration of the alloy, and K is

given by

3U
K = m (TCo)3/2 (2.24)
b2kT
and Um is the solute atom-dislocation binding energy.

By considering H to be a function of both the local concentration
of the solute at the dislocation, as well as of the effective stress,

he obtains some very significant relationships as follows.

First of all, the activation enthalpy equation he obtains has the
form of

Ho* = H dC 2.5
H = TV + T (2.25)
1T aC dT








where V is the activation volume. Equation (2.25) shows that the

activation enthalphy can be expressed as the sum of two terms, the

first of which represents the Conrad-Wiedersich relationship [42]

for the activation enthalpy, which assumes that a single thermally

activated mechanism controls the flow stress. This term is con-

sidered to give the activation enthalpy in the presence of dynamic

strain aging. The second term on the right hand side of the above

equation is considered to represent the component of the activation

enthalpy associated with dynamic strain aging.

Second, he shows the strain rate dependence of flow stress,

o kT 1 aH dC (2.26)
Dtw -w t^- (2.26)
FV (V _aC W dtDtw )

The first term on the right hand side is again the normal one in the

absence of diffusion; the second is due to dynamic strain aging.

Third, he also obtains the relationship of the strain rate

sensitivity by assuming that, when the strain rate is changed from

EL to EH at a constant strain, the change of activation enthalpy is
given by
H
AH = kT 9n =- (2.27)
EL
Thus, he writes
AC kT 1 8H dC (2.28)
nF ~V V aC dini
kT
The first term on the right, -K, corresponds to the strain-rate

sensitivity in the absence of dynamic strain aging, and the second term

is considered to be due to dynamic strain aging.


Ao ) should read (
I (A- ) should read (An u ) throughout the text.
Aing A(Zn E))








Van den Beukel extends further his investigation and shows that

the temperature dependence of the flow stress as well as the work

hardening can be expressed by two terms; one of these is that which

would exist in the absence of dynamic strain aging. The other is the

part due to dynamic strain aging.


2.3.4 The Reed-Hill Theory [27-29]

The more general theory of van den Beukel obviously represents

a major advance over the earlier theories of Cottrell and McCormick.

However, the theory of van den Beukel does have some shortcomings.

The most important of these are

1) Quantitative predictions of the theory depend largely upon a

knowledge of just how the activation enthalpy H varies with

changes in the solute concentration near the dislocation.

Data to evaluate this subject are not generally available.

2) With the theory, it is difficult to visualize the simple

physics of dynamic strain aging due to its basically

mathematical approach.

3) The theory deals only with the long range diffusion of solute

to dislocation. It is thus specific to Cottrell form of

strain aging. However, in bcc interstitial alloys, Snoek

dymamic strain aging is significant [14,16,20,23,33].

All of the shortcomings led Reed-Hill to propose a phenomenological

theory of dynamic strain aging.

It is assumed in the phenomenological model that the total flow

stress, ot, consists of two parts. The first being the stress a that









would exist in the absence of dynamic strain aging and the second oD that

is due to dynamic strain aging. Thus


Ot = o + oD (2.29)

It is also shown that the dynamic strain aging independent part of the

flow stress a has two parts, the internal stress aE and the effective

stress a*. Thus it becomes


at = aE + a* + OD (2.30)


He developed a new technique [28] to evaluate the internal stress. He

also demonstrated that in commercial purity niobium the effective stress

can be approximately by a power law of the form

(- kT/H
a* a= a ( -) (2.31)
0 60

where a* is the effective stress, a* is the effective stress at 0 K,
o
E is the nominal strain rate, o and Ho are constants, and k and T have

their usual significance.

At the same time the dynamic strain aging component of flow stress,

OD, can be represented by an equation deduced from the strain aging under

stress studies of Delobelle, Oytana and Varchon [43]. In this equation,

which follows, the first term on the right hand side represents the

contribution of Snoek aging and the second term that is due to Cottrell

aging.








t t
D = S {1 exp[-( )]}+o { exp[-(w)2/3]} (2.32)
D max TS max TC


where OSmax and Cmax are the isothermal maximum obtainable magnitudes

of the Snoek and Cottrell contributions of the flow stress, respectively,
tw is the waiting time of a dislocation at an obstacle, and TS and TC are
empirical relaxation times for Snoek and Cottrell aging, respectively.

These parameters are strongly temperature dependent and can normally be

expressed by


S = TS exp(Q/RT)

TC = TC exp(Q/RT) (2.33)

where Q is the activation energy for the diffusion of the solute atom,
and So and TCo are constants that can be determined experimentally.

The parameters aSmax and GCmax are expected to be temperature dependent

and to decrease with temperature due to an increase in dynamic recovery
with increasing temperature. Thus,

a = ao0 exp[-a(T-300)]
max max

aC = o exp[-c(T-300)] (2.34)


where Sax and OCmax are the maximum amplitudes of the dynamic strain
Smax Cmax
aging stress components at 300 K, and a is a dynamic recovery parameter.
Thus, the total stress may be written








ot = a E+o (-)kT/H + {exp[-a(T-300)]}

o ( {1 exp[- ( )]}
S max S
+ o (1 exp[-(t)2/3]} (2.35)
max TC

If the above equation is differentiated with respect to kne, one obtains
the following equation for the strain rate sensitivity of the flow stress.


dot o*kT (EikT/HO
dt T ( kT/Ho exp[-a(T-300)]
din H o 0

om ) exp[-(t )]
Smax S TS
+2 0 t .2/32/3 36)
tw 2/3 exp[-( )2/3] (2.36)
3 Cmax C TC /

It has been shown that the phenomenological theory can model the
temperature dependence of the flow stress and the strain rate sensi-
tivity of niobium-oxygen specimens [27] as well as those of a substi-

tutional solid solution of Cu-3.1 at.%Sn [29]. It has also been
demonstrated that the theory can model both the temperature as well
as the strain dependence of the strain aging under stress kinetics

curve of niobium specimens containing oxygen in solid solution [44].








CHAPTER III
PREVIOUS INVESTIGATIONS


3.1 Portevin-LeChatelier Effect

Probably the best known aspect of dynamic strain aging is serrated

yielding which is commonly called the Portevin-LeChatelier effect because

in 1924 these authors reported [45] stress-strain curves showing serrations.

Serrated flow can take a number of forms or shapes. The two most

important forms were shown by Reed-Hill [1] and reproduced in Figure 3.1.

In both of the cases the yield point associated with a serration is

related to the formation of a Liders band. In the Type A bands shown

in Figure 3.1a the Liders band nucleates at one end of the gage section

and then moves progressively along the gage section to its other end.

When the Liders band front has completely crossed the gage section,

plastic deformation effectively stops and the load rises elastically

until another band is formed. This new band then forms at the same

end as where the previous band was formed. Its formation causes the

load to drop. Note that the load rises as the Liders band front

traverses the gage section. The rise in load is a result of a strain

gradient along the gage section which developed as the first band began

to form. The LUders bands thus propagate into material that is progres-

sively harder and harder.

Type B bands, as illustrated in Figure 3.1b, are LUders bands that

form but do not propagate [8]. Both Type A and B bands can form in the

same specimen. Type B bands are normally observed at higher temperatures










































Figure 3.1


(a)


(b)


Two typical forms of serrated flow. a) Type A serrations
are associated with LUders bands that can travel the length
of the gage section. b) Type B serrations involve bands
that do not propagate. (After Reed-Hill [1].)








or slower strain rates than are required for Type A bands; thus they

occur where the diffusion rate of the solute atoms is higher. In this

case, the dislocations in the bands become pinned at the bottom of the

load drops. Succeeding LUders bands may form sequentially next to each

other.

The Portevin-LeChatelier effect [45] or serrated yielding has been

observed in a large number of alloy systems, including steels [3,5,6],

vanadium [2,46], aluminum alloys [8,11,12,45], Ni alloys [24,47] and

titanium alloys [48].

The effect can therefore operate in bcc, fcc and hexagonal crystal

structures. It is generally attributed to a dynamic strain aging process

which occurs when solute atoms are diffusing sufficiently rapidly to

dislocation. If the solutes are interstitials, for example C, N, 0, etc.

in bcc metals, serrated yielding may be observed at two temperature

intervals: one being due to Snoek dynamic strain aging while the other

due to Cottrell dynamic strain aging. Thus, in this system, it may be

possible to observe serrations even at temperatures close to room

temperature. For substitutional alloy systems, on the other hand,sincethe

Snoek dynamic strain aging effect does not operate, the Portevin-

Le Chatelier effect is normally seen only at elevated temperatures,

unless diffusion has been intentionally accelerated, for example; by

quenching from a high temperature to retain excess vacancies or

generating vacancies during plastic deformation, or by radiation damage.

In the following sections, the effects of strain rate, temperature,

strain and solute concentration on the Portevin-Le Chatelier effect will

be briefly reviewed.








3.1.1 Effect of Strain Rate and Temperature

The Portevin-Le Chatelier effect is usually observed in certain

strain rate-temperature regions. Thus, a large number of authors, in-

cluding Keh et al. [5], Nakada and Keh [47], Pink and Griberg [3],

Mulford and Kocks [24],' Kinoshita et al. [49] and Dadras [50], show

plot of the strain rate-temperature domains for serrated flow.

On the basis of Manjoine's results [19], Cottrell [18] related the

strain rate and temperature corresponding to the onset of serrations due

to Cottrell dynamic strain aging by the relation

= 109 (3.1)

This relation has been conformed by a number of investigators 2,20,23

A study by Bradford and Carlson [2] of dynamic strain aging in the

vanadium-oxygen system shows that there are two temperature different

ranges at the same strain rate. The specimens were deformed at a strain

rate of 1.67 x 10- s- between room temperature and 773 K. Five alloy

compositions were used containing 47, 150, 265, 955 and 1800 ppm oxygen.

For specimens containing 47, 150 and 265 ppm oxygen, serrations were

observed between 623 K and 773 K. The specimens containing 955 and 1800

ppm oxygen showed serrations in the temperature range 423 K to 448 K.

They interpreted the high temperature serrations as being due to Cottrell

dynamic strain aging by solving Cottrell's relation E/D = 109 for their

data. The low temperature serrations were not identified at that time.

Beckerman [23] solved for the ratio L/D corresponding to the low

temperature serrations, and obtained a value of /D in the tange of

3.11 x 1012 to 2.3 x 1014 with a value of Z-/D = 1.54 x 1013 corresponding








to the maximum height of the serrations observed. On the basis of these

values, Beckerman concluded that these low temperature serrations could

be due to Snoek dynamic strain aging. Evans and Douthwaite [15] calcu-

lated the strain rate corresponding to the maximum drag force exerted on a

moving dislocation by aSnoek atmosphere. They compared this with a simi-

lar calculation for the maximum drag force exerted on a moving dislocation

by a dynamic Cottrell atmosphere and determined that Snoek ordering should

occur at a strain rate three orders of magnitude faster. Baird [20], in

his review article on strain aging, also suggests that these low temper-

ature serrations are due to Snoek dynamic strain aging.

It can be concluded from the above published results that serrations

may be observed in two temperature ranges in bcc interstitial alloys when

a single strain rate is used; one being due to Snoek dynamic strain aging,

the other due to Cottrell dynamic strain aging.


3.1.2 Effect of Strain

It has been reported in the literature by many authors that there is

a critical strain, ec, at which serrated flow begins to appear. It is

often emphasized that the critical strain increases with strain rate and

reciprocal temperature. McCormick [8] found in an Al-Mg-Si alloy that

at higher strain rates, 1.7 x 10 s the critical strain increases

with decreasing temperature. MacEwen and Ramaswami [11] demonstrated

in both single crystal and polycrystal of an Al-Mg alloy, that the

critical strain decreases as strain rate decreases when tested at 295 K.








However, the critical strain decreased in both single and polycrystal

specimens with increasing temperature when deformed at a rate of ~ 10-5
-1
s They had, however, some difficulties in defining the critical

strain. For single crystals, the small irregularities in the stress-

strain curve prior to the development of a regular background made the

precise definition of critical strain impossible. However, there was no

ambiguity in defining the critical strain in polycrystalline specimens.

Pink and Grinberg [3] reported that they could not distinguish

between the onset of Type A and and onset of Type B serrations. They

plotted the critical strains for different types of serrations as a

function of the reciprocal temperature at several strain rates. The

results show that, for pure Type A serrations, the slope is positive,

while for combined Type (A + B) serrations it is positive or zero.

At higher temperatures, and especially when Type C serrations appear,

the slope is negative.

Recently, Jovanovic, Djuric and Drobnjak [51] observed several

types of serrations in a Cu-Be-Co alloy. They distinguished the critical

strain for the onset of Type A, Ec(A) from that for Type B serrations,

Sc(B). They plotted both critical strains, Ec(A) and Ec(B) as a function
of strain rate at several temperatures. It was found that at 150 C and

180 C, e~(A) decreases initially with increasing strain rate down to the

critical strain rate, about 1 x 10- s and 5 x 10- sI for 150 C and

180 C, respectively, and then an inverse behavior was observed; i.e.,

Ec(A) increases with further increase in strain rate. At lower tempera-
tures, 90 C and 120 C, the critical strain, E (A) increases with decreasing








temperature for all strain rates employed. However, at slightly higher

temperature tests, 150 C and 180 C, the strain rate dependence of criti-

cal strain, ec(A) becomes more complicated. E (A) at 150 C becomes

smaller than that at 180 C at slower strain rates. However, as the

strain rate increases, the situation becomes reverse: ec(A) at 150 C

is larger than that at 180 C. In contrast to the case of Type A serra-

tions, e (B) has a linear dependence of strain rate. It was also

observed that at all test temperatures e (B) becomes smaller as the

temperature increases.

A recent study made by Qian and Reed-Hill [29] in Cu-3.1 at.% Sn

also shows that the critical strain is a function of temperature as well

as strain rate. Thus, they reported a plot of critical strain versus

temperature at three different strain rates, 1 x 10-5, 4 x 10-4 and

4 x 10 s-. They observed that the critical strain decreases rapidly

as temperature increases from about 320 K to 650 K for all strain rates.

The critical strain at higher strain rates was always greater than that

at slower strain rates. From the plot of c versus T at several strain

rates, they were able to evaluate the activation energy by plotting the

strain rate needed to obtain a fixed critical strain versus the recipro-

cal temperature.


3.1.3 The Effect of Solute Concentration

In order to study the effect of carbon content on several serrations,

Nakada and Keh [47] used Ni-C specimens with six different carbon concen-

trations at various strain rate-temperature combinations. The critical








strain for initiation of serrations was used to classify the degree of

serrations; the smaller the critical strain, the more pronounced are the

serrations. They plotted the critical strain versus and also versus

strain rate for specimens containing six different carbon concentrations,

between 0.022 and 0.2 at.%. It was concluded that the critical strain at

a given temperature decreases with increasing carbon content. It is thus

shown in Figure 4 in Nakada and Keh's paper [47] that at 273 K the

critical strain observed at 0.048 at.% C was approximately 1%, which

should be compared with 0.6% e for a specimen containing 0.16 at.% carbon.

Bradford and Carlson [2] observed serrated yielding at two temperature

ranges in vanadium alloys containing different levels of oxygen between

47 ppm and 1800 ppm. At the lower temperature (~ 423 K), the frequency

and height of the serrations increases with increasing oxygen content.

In the higher temperature range (623 to 673 K), however, the serrations

diminish in frequency as well as magnitude with increasing oxygen concen-

trations and virtually disappear at the higher oxygen contents. In other

words, increased amounts oxygen tend to diminish the intensity of the

serrations and do not give the expected increase in the size of the

maximum in the Cottrell dynamic strain aging temperature range. A high

interstitial content is necessary for the observation of serrations due

to Snoek dynamic strain aging.

Roberts and Owen [52] investigated the serrated flow in martensite

and ferrite with various concentrations of carbon. The martensitic

specimens were Fe-21% Ni-C alloys with different carbon concentrations

between 0.015 and 0.12%. They observed two types of unstable flow at








constant strain rate tests of ferrite iron-carbon alloys; jerky flow

due to Snoek dynamic strain aging and serrated flow due to Cottrell

dynamic strain aging. At any strain-rate and carbon concentration,

jerky flow occurs at a lower temperature than serrated flow. However,

in martensitic alloys, only the higher temperature serration was observed.

In both cases of plastic flow, an increasing carbon concentration decreases

the temperature at which unstable flows are observed. It can be said that

the purer the specimen, the higher the temperature required to get the

same degree of serrations. This has also been pointed out by other

authors [5,20,53,54].

Russell [10] obtained in Cu-Sn alloys the relationship between the
-1/2
critical strain and the Sn contents as c C CSn by assuming that the

strain aging is due to the movement of a Sn,.atomto a dislocation by a

vacancy mechanism.

McCormick [37] also showed the concentration dependence of the

critical strain for the onset of serration, eC = C 3/2(m+, assuming

the obstacle spacing is independent of solute concentration. The above

relationship was shown in section (2.3.2) in Chapter II. If solute atoms

act as the controlling obstacles, however, then the average distance

between obstacles would be expected to decrease with increasing solute

concentration, thus decreasing the concentration dependence of E

It was reported in carbon steel [55,56] that the presence of substi-

tutional solute suppresses the serrations in the stress-strain curve.

Bratina et al. [56] found in Fe-C steel that adding 0.95% Mn in Fe-0.095 C

alloys suppressed the serrations at room temperature in tests made at a

rate of 7 x 10 s However, serrations became pronounced at elevated








temperatures, 360 K. It was concluded that manganese is responsible for

the delay of the serrated flow by either reducing the amount of inter-

stitials in solution by forming carbide or impeding the motion towards

dislocations.


3.2 Strain Rate Sensitivity (SRS)


The strain rate sensitivity is a useful parameter for studying

different aspects of deformation. First, since the strain rate sensi-

tivity is related to the thermal component of the stress, it can be a

tool for studying the thermally activated deformation mechanism [57,58].

Second, the variation of the strain rate sensitivity with strain can

determine the validity of Cottrell-Stokes law [59] in a metal. According

to Cottrell-Stokes law, Ao/o = constant, where Au is the change in the

flow stress upon changing the strain rate or temperature. Third, there

is a considerable amount of evidence in the literature [24,27,40]

involving many metals showing a strain rate sensitivity-temperature

diagram with a minimum in the dynamic strain aging temperature interval.

Therefore, the temperature variation of the strain rate sensitivity can

be used to detect regions where dynamic strain aging is significant.

There are two forms of parameters in common use to evaluate the

dependence of the flow stress upon the strain rate. The first of these

is


n d- nu (3.2)


where n is known as the strain rate sensitivity, a is the flow stress and








is the strain rate. It is often possible to write a power law to

express the stress in terms of the strain rate; i.e.,


a = cr(o)fl


(3.3)


where o is a constant. Note that in this case, n, the strain rate

sensitivity, is the exponent in the power law equation. Under the

assumptions that 1) the Orowan equation holds, 2) that the thermally

activated component dominates the flow stress, and 3) that the mobile

dislocation density does not change during a strain rate change, n

defined in this manner is equal to the reciprocal of the dislocation

velocity-stress exponent m, defined by the Johnston-Gilman empirical

relationship [60]


V1 = )
()m
V2 U2

The second form of the strain rate sensitivity parameter, S, is

S = d
s W


(3.4)




(3.5)


This is based on the assumption that the deformation can be described by

an equation


(3.6)


= o exp(-H/kT)


where is the strain rate, o is a constant, H is the activation enthalpy,

and k and T have their usual meanings.

The parameter S is normally preferred to be used because it is

primarily easy to measure from the strain rate change test and it can

also be related directly to the activation volume.








3.2.1 Strain Rate Sensitivity at Low Temperatures Where Dynamic Strain
Aging is Not Important

In the low-temperature region where deformation is thermally activated,

it has been observed that the strain rate sensitivity (SRS) passes through

a maximum at a certain temperature, Tm [29,58,61-65] The magnitude of

the maximum in SRS varies with material and other experimental parameters,

such as base strain rate, grain size and concentration of the alloying

elements.

Michalak [65] investigated the influence of temperature on the

development of long range internal stress during the plastic deformation

of high purity iron. Differential strain rate tests were performed during

the tensile deformation of single crystal and polycrystalline specimens

of two zone-refined irons in the temperature range 78-300 K. He observed

that below about 175 K the strain rate sensitivity of iron was independent

of strain. The magnitude of the SRS maximum for the single crystal was

smaller than that of polycrystalline irons. It was also found that the SRS

maximum temperature, Tm, shifted to higher temperature for the polycrystall-

ine specimens; the maximum temperature falls at about 190 and 225 K for

the single crystal and polycrystalline specimens, respectively.

Ravi and Gibala [61] reported that the addition of small amounts of

oxygen to niobium resulted in alloy softening and an increase in the

strain rate sensitivity peak temperature. An inverse effect of impurity

content was observed Tseng and Tangri [64] on iron specimens with

different impurity levels. They found that Tm increases with decreasing

impurity content. However, their results [64] of the magnitude of the









strain rate sensitivity maximum agree with the data in the literature:

SRS increases with increasing impurity content.

Christian and Masters [58] have reported data on strain rate sensi-

tivity of the flow stress for polycrystalline niobium, vanadium and

tantalium over the temperature range 4.2 to 373 K. They found that the

SRS peak temperatures fall at about 90 K, 75 K and 220 K for niobium,

vanadium and tantalum, respectively. It was also observed that, above

90 K, the strain rate sensitivity was dependent on the purity of the

niobium; the higher the purity content, the higher the magnitude of

the strain rate sensitivity. However, at 90 K and below, this depen-

dence disappeared.

The SRS peak temperature, Tm, the magnitude of the strain rate sensi-

tivity maximum and the flow stress at Tm appear to be very useful experi-

mental data in order to estimate the internal stress. This technique has

been recently uniquely developed by Reed-Hill and Qian [281. The internal

stress at 0 K, OEo' is evaluate with the aid of the equation
ao eam
S 0 m (3.7)
E 0 1 eEm/Eo

where ao is the flow stress at 0 K, e is the base of the natural logarithms

and Em and E are Young's modulus at Tm and 0 K, respectively. This

technique has been successfully tested in Nb-0 [28] and Cu-3.1 at.% Sn [29].


3.2.2. Strain Rate Sensitivity at Temperature Ranges Where Dynamic
Strain Aging Becomes Significant

3.2.2.1 Effect of Temperature. It is well documented that in a

number of materials in the intermediate temperature regions the strain








rate sensitivity becomes abnormally low or negative due to dynamic strain

aging. Bradford and Carlson [2] showed a plot of strain rate sensitivity, n,

as a function of temperature for a vanadium specimen containing 265 ppm 0.

They found that strain rate sensitivity decreases gradually as temperature

increase and has a minimum value of -0.023 at about 610 K. At this

temperature, the intensity of the serration and the yield stress curve

reached a maximum. Negative values of the strain rate sensitivity were

observed in the temperature range between 408 K and 698 K. At that time,

they were not able to explain the significance of this negative strain

rate sensitivity. A close examination of Figure 6 in the paper of Bradford

and Carlson [2] reveals that there is another minimum in strain rate

sensitivity at around 408 K, which is close to the temperature of the

serration peak observed in the higher oxygen specimens. This minimum is

due to Snoek dynamic strain aging. They ignored this low temperature

minimum because they did not realize that in bcc metals Snoek dynamic

strain aging also contributes negatively to the strain rate sensitivity

of the flow stress.

Thompson and Carlson [46] made a similar study of nitrogen in

vanadium. They plotted the strain rate sensitivity, n, as a function

of temperature for vanadium specimens containing 210 ppm of oxygen.

They observed that the strain rate sensitivity was negative between 538 K

and 753 K and reached a value of -0.012 at the minimum. The strain rate

sensitivity minimum occurs at about 673 K, which is approximately 60 K

higher than in the vanadium-oxygen specimens.








The strain rate sensitivity minimum has also been recently reported

by Qian and Reed-Hill [29] in Cu-3.1 at.% Sn alloy. Ao/Aknt was measured

at a fixed strain, 2%, using a tenfold change in strain rate at tempera-

tures between 77 and 800 K. The base strain rate was 4 x 10-4 s1. They

observed that the strain rate sensitivity becomes negative between 390

and 620 K with a minimum at about 500 K.

3.2.2.2 Effect of Strain. The strain rate sensitivity (SRS) of

the flow stress plays a crucial role in determining macroscopic behavior.

Any solute mobility makes a negative contribution to the total rate sensi-

tivity and this contribution increases with strain [27,37,43,44,66]. When

the total rate sensitivity becomes negative, plastic flow becomes unstable

in the characteristic form of serrated flow [26,66-69]. This assumption

has been supported by the experimental observation [24,26]; a negative

strain rate dependence of the flow stress is a necessary condition for

the occurrence of serrated flow.

The strain dependence of the SRS has been reported by a number of

authors [24-27,44,67]. Van den Brink et al. [26] investigated the effects

of strain and composition on the strain rate sensitivity of Au-Cu alloys.

For alloys containing more than 1% Cu, Ao/Aan decreased with increasing

strain, becoming negative at sufficiently large strains. The strain Eo,

at which Ao/Adn = 0, decreased with increasing composition and serrated

yielding was observed to initiate at somewhat larger strains in the

negative strain rate sensitivity region.

Mulford and Kocks [24] noted that the SRS in normal alloys is more

appropriately described in terms of the flow stress than the strain. Thus,

they measured the strain rate sensitivity of Inconel 600 as a function of








flow stress at several different temperatures. At 300 K or lower,

Aa/Ahne increases with increasing flow stress. At 400 K, however, the

behavior is anomalous: Ada/An decreases linearly with increasing flow

stress but still remains positive. At higher temperature (e.g., 700 K),

Aa/Ahin decreases rapidly with flow stress and as the deformation con-

tinues a negative strain rate sensitivity is observed. When the defor-

mation is continued into the region of negative strain rate sensitivity,

the Ao/Akne versus flow stress plots are no longer linear.

At 960 K, it was observed that there is a rapid drop of Ao/Akne

through zero, followed by a subsequent increase to positive values again.

Serrated flow was observed during the portions of the stress-strain curves

where the rate sensitivity was negative.

Recently, Reed-Hill documented [27,44] the effect of the dynamic

strain aging. He showed., that dynamic strain aging phenomena are

basically functions of the quotient of the waiting time of dislocation

at an obstacle, tw and a temperature dependent relaxation time, T.

The waiting time is actually a function of both mobile dislocation den-

sity, m and strain rate, e: i.e.,


t = (pmbL)/ (3.8)

Accordingly, as pm grows with increasing strain, tw must also increase.

Thus, for a given condition such as the onset of serrations, an increase

in tw can be compensated by a corresponding increasing in the relaxation

time; i.e., a decrease in the temperature. Therefore, the increase in

pm with strain may be expected to expand the lower temperature boundary
of the temperature range in which serrations are observed.








3.2.2.3 Effect of Base Strain Rate. As equation (3.8) indicates,

since t, is inversely proportional to the strain rate, increasing the

base strain rate should decrease the waiting time. Therefore, it is

expected that changing the base strain rate will modify the temperature

or strain dependence of the strain rate sensitivity.

Van den Brink et al. [26] made measurement of SRS versus strain

at four different base strain rates in Au-4.5% Cu alloy at 20 C. They

observed that SRS decreases with increasing strain at all base strain

rates used. Decreasing base strain rate causes Ao/Aknt to be negative

at much smaller strains. At a strain rate of 4 x 10-4 s-1, A/An

remains positive up to about 9% of strain. However, it becomes negative

at only 4% when tested at 4 x 105 s1

The effect of base strain rate on the SRS has also been observed

by Mulford and Kocks [24] in an Al-l% Mg alloy. They made a strain rate

sensitivity measurement at 250 K at three base strain rates, 1 x 10-5

5 x 10 and 1 x 10 s and plotted the Ao/Aknt versus flow stress

data. They observed that, for all three base strain rates, Ao/Aint

decreased as the flow stress increases and increases with a further

increasing flow stress. The strain rate sensitivity, S, at a higher

strain rate, is always greater at all flow stresses than that at a

slower strain rate.

McCormick [25] also studied the effect of base strain rate on the

strain rate sensitivity in low carbon steel at 353 K. He plotted Ao/AInt

versus strain for four strain rates. Measurements of Ao/Aknt were found

to decrease with increasing strain for all strain rates. The rate of

decreasing of the strain rate sensitivity increased with decreasing base

strain rate.








3.2.2.4 Effect of Magnitude of Strain Rate Change. Van den Brink

et al. [26] studied the effect of the magnitude of a strain rate change

on the strain rate sensitivity at a temperature of 293 K in a Au-4.5

at. % Cu. alloy. The base strain rate was 8.33 x 10-5s-1 and three

different strain rate ratios were used (( H/ L) = 5, 10 and 20). The

strain rate sensitivity decreases with increasing strain for all strain

rate change ratios. It was found that increasing the ratio at constant

base strain rate resulted in an increase in the strain, Eo, where

Ao/Aane = 0. The values of o were found to be approximately 4.6%, 5.2%

and 5.9% for the ratios of 5, 10 and 20, respectively. Except at small

strains, less than 1.5%, where Ao/Akne was independent of Akne, the

strain rate sensitivity increases with increasing the magnitude of the

strain rate change.


3.3 Strain Aging Under Stress

3.3.1 Introduction

Strain aging under stress is known to enhance the strain aging

process. This effect is commonly observed when strain aging monitored

by observing the return of the yield point. When a specimen is aged under

an applied stress, a greater yield point return is obtained than when it

is aged in a stress-free condition. Most of the strain aging under stress

experiments were conducted under the condition of relaxation of the speci-

men; i.e., aging was carried out by merely stopping the testing machine

for a period of time and then continuing the deformation. Alternatively,

the specimen may be unloaded from the flow stress to desired aging








stress and then aged. In both cases, the specimen is allowed to relax

during aging. Since the stress changes continuously it cannot be said

that the specimen is aged at a constant stress. This problem was largely

eliminated by Beckerman and Reed-Hill [23,33] by utilizing an electroni-

cally controlled feedback loop between the Instron strip chart and cross-

head. The advantages of this method are twofold. The first is that it is

possible to maintain a nearly constant stress during aging. The second

is that the amount of strain as well as the average strain rate during

aging can be measured.

Beckerman and Reed-Hill [23,33] found that the strain rate during

aging was a maximum when agin occurred at stress levels approaching the

prestrain flow stress, and decreases as the aging stress level decreased.

Thus, the average strain rate in a 35=minute aging period at 92% af in

the V-O system, where of is the prestrain flow stress, was found to be

only about 2.5 orders of magnitude slower than the prestrain strain

rate. This strongly suggests that a materials deforms with a finite strain

rate during strain aging under stress.


3.3.2 Effect of Aging Under Stress on the Yield Point Return

The effect of the aging under stress on the yield point return

has been investigated by a number of authors. Wilson and Russell [70]

observed that aging a low carbon steel under stress resulted in a greater

yield point return than aging in an unstressed condition.

Mura and Brittain [71] studied the effect of the application of

stress during aging on the yield point return of ingot iron. They aged

a set of specimens for a constant time of 2800 seconds and observed the








development of heterogeneous yielding as the aging stress was increased

from 0 to 212 Mpa. Their results indicated that the yield point became

more fully developed.

Almond and Hull [72] studied the effect of applying a tensile stress

during the strain aging of iron, niobium and tantalum. They observed

that the upper yield stress, the lower yield stress and the yield point

elongation are allincreased when tensile specimens are aged under stress.

Duval and Dickson [73] investigated the influence of the aging

stress on the yield point phenomena in very low carbon steel. Specimens

were restrained at a nominal strain rate of 6.6 x 10-4 s-1 and unloaded

to the desired aging stress, whereupon the crosshead was stopped, allowing

the stress to relax for the desired aging time; after this the specimen

was reloaded at the nominal strain rate. The procedure was repeated at

intervals of 1 or 2% of the strain.

One set of specimens was prestrained 2%, 6% and 14%, and aged at

266 K and 300 K for 30 seconds at several stress levels. For specimens

aged at 300 K after restraining 6%, the magnitude of the yield point

return was observed to initially decrease with increasing aging stress

to 40% of the prestrain flow stress. Upon aging at stress levels greater

than 40%, the magnitude of the yield point return increased until it

attained a maximum value at 60% of. Aging at stress levels above 60%

of caused the magnitude of the yield point return to decrease until a

minimum was attained at 90% of. A further increase in the aging stress

caused the magnitude of the yield point return to increase to a value

at 100% of the flow stress, about equal to that obtained at 60% of.









Only a minor difference was observed in the manner by which the yield

point return varied with aging stress when the specimens were prestrained

2% and 4% instead of 6%.

On the other hand, for specimens aged at 266 K, raising the aging

stress increased the magnitude of the yield point return up to a maximum

value at an aging stress level approximately 70% of the prestrain flow

stress. A further increase in aging stress up to 80% of the prestrain

flow stress caused the yield point return to decrease and a value about

equal to that obtained upon aging at 60% of was found. Upon increasing

the aging stress up to 100% of, the magnitude of the yield point return

continuously increased.

The most extensive study of the dependence of the yield point

return on aging stress was performed by Beckerman and Reed-Hill [23,33].

A number of vanadium specimens were prestrained to 9% at a rate of

6.7 x 10-5 s and aged for 35 minutes at 353 K, 363 K and 373 K at

several stress levels between 27% of and 98% of. They observed that

raising the level of the aging stress caused the yield point return to

increase continuously up to a maximum value at 92% of. At this aging

stress level, the magnitude of the yield point return is approximately

three times that obtained at the lowest stress level (27%). Aging

above 92% of results in the magnitude of the yield point return dropping

off rapidly. At 98% of, it approaches or falls below the value obtained

upon aging at 27% of. It was also observed that the effect of raising

the test temperature is to increase the magnitude of the yield point

return at all aging stresses.








They also showed the aging stress dependence of the yield point

return in tantalum. Tantalum specimens were prestrained 9% at 378 K
-5 -1
using a strain rate of 6.7 x 10 s- and then aged for 25 minutes at

a number of stress levels between 20% of and 95% of. Their results

indicated that the effect of raising the aging stress level was to

increase the magnitude of the yield point return continuously up to

a maximum at 85% of. Upon aging at stress levels above 85% of, the

magnitude of the yield point return decreased slightly. The yield

point return obtained by aging at 85% of was approximately four times

greater than that obtained upon aging at 20% of.



3.3.3 Effect of Prestrain on the Yield Point Return

The results of Szkopiak and Derby [74] who strain aged commercially

pure niobium for 10 hours at 350 C show that the magnitude of the yield

point return increases rather rapidly as prestrain increases to 6%.

With further increase in prestrain, the magnitude of yield point return

gradually falls off.

In contrast are the results of Rosinger et al. [75] for strain aging

of annealed Ferrovac E iron strain aged at 21 C and 40 C for aging times

from 5 to 3600 seconds at various strain intervals. They observed that

the yield point return remained essentially independent of strain up to

300 sec at 21 C and up to 30 sec at 40 C. For longer aging periods,

1800 and 3600 sec, the magnitude of yield point return decreased slightly

with strain.

Duval and Dickson [73] reported the influence of the strain on the

magnitude of yield point return in low carbon steel aged at various








temperatures for 30 seconds. It was found that for a small homogeneous

strain (< 1%), the yield point return increased rapidly with strain.

However, for greater strains, it decreased slowly with increasing strain.

The prestrain dependence of the yield point return was also investi-

gated by Delobelle, Oytana and Varchon [43] for the niobium-oxygen system.

They determined the variation of the magnitude of the yield point return

with time at 344 K, 371 K and 398 K at different amounts of prestrain,

2%, 4%, 8% and 12%. At all test temperatures the magnitude of yield

point return increases with prestrain.

Beckerman [23] also reported the effect of prestrain on the yield

point return for vanadium specimens prestrained to 2.8% and 9% at a rate

of 6.7 x 10- s- and aged at 363 K for 35 minutes at different levels

of aging stress. It was found that different amounts of prestrain did

not significantly affect the magnitude of the yield point return at all

aging stress levels.


3.3.4 Two Stages of the Strain Aging Kinetics Curve

When a strain aging experiment is performed in a bcc interstitial

alloy at a proper temperature for a reasonable period of time, one nor-

mally observes two plateaus in a plot of the magnitude of the yield point

return versus aging time. The magnitude of the first plateau appearing

at shorter aging time is much smaller than that of the other plateau at

longer aging times. It was pointed out by Wilson and Russell [70] for

the Fe-C system, later by Carpenter and Baker [76] for the Ta-0 system

and Nakada and Keh [21] for the Fe-N system, that the first stage aging








occurs too quickly for long range diffusion of interstitial atoms to

take place. Nakada and Keh [21] compared the time to reach the first

plateau to the interstitial atom jump time and concluded that the first

aging stage is the result of Snoek ordering of nitrogen atoms in the

Fe-N system. This was supported by Owen and Roberts [22], who studied

the deformation kinetics and the effects produced by aging iron-nickel-

carbon martensite during relaxation under stress. They also found that

the time to reach the first plateau value agrees reasonably well with

the time for a single jump of a carbon atom in ferrite.

Rosinger et al. [75] plotted the magnitude of the yield point

return versus time to the 2/3 powder for Ferrovac E iron. For each

test temperature, they observed that the data appeared to define two

stages: each stage being linear with t/3, but having a different slope.

They then assumed that the first stage corresponds to Snoek strain aging,

and the second to Cottrell strain aging. They further assumed that the

time corresponding to the inflection between the two stages represented

the time required for a carbon atom to undergo a single jump. They

checked this assumption by plotting the logarithm of the inflection time

versus the reciprocal of the absolute temperature. This curve was then

compared to a plot of the logarithm of the carbon atom jump time versus

1/T, and they concluded that the agreement was good.

It is generally accepted that the second stage of the strain aging

is due to Cottrell strain aging. It involves the long range diffusion

of solute atoms to form so-called "Cottrell atmospheres."









CHAPTER IV
EXPERIMENTAL PROCEDURE


4.1 Materials and Materials Preparation


The two purity grades of niobium used in this study were initially

obtained from the Materials Research Corporation in the form of 6.35 mm

diameter annealed rods and are designated as Marz Nb and VP Nb in

Table 4.1. The two grades of niobium were used for a survey of the

tensile properties of low oxygen niobium. The rods were swaged directly

to a diameter of 3.18 mm and then machined into tensile specimens having

dimensions shown in Figure 4.1. These tensile specimens were annealed in

groups of 10, in a "Precision Scientific Minivac" vacuum furnace under

a dynamic vacuum of 10-4 Torr at 1073 K for 1 hour. The finished

specimen had a mean linear grain intercept of 65 pm. The tensile

specimens were measured using an optical comparater for average gage

section diameter and gage section length.

VP grade niobium rods were also used as a base material for

producing the higher oxygen specimens. The rods were first swaged

to a diameter of 3.18 mm. Then niobium alloys containing 0.24, 0.75

and 0.95 at.% oxygen were prepared in the following manner. First,

the rods were cut into 3.5 cm lengths. Sets of four of these sections

were then inserted, along with 6 gm of a 325 mesh Nb205 powder, into

the open end of a 1.59 OD thin walled niobium tube whose other end had

been crimped and welded closed. Following this, the open end was

crimped and welded to form a totally enclosed packet. These enclosed









TABLE 4.1
CHEMICAL ANALYSIS OF Marz Nb, VP Nb and WC Nb, ppm




Marz Nb VP Nb WC Nb

C 25 40 35

0 15 20 120

N <5 <5 50

H <1 <1 <5

Fe 3.4 20 <50

Si 17 25 <50

Ta 200 250 750

Ni 1 <30 <20

Cr 1.8 30 --





49













LO


7 )















CL

If)







EE
(I,,


C~L c\'


a, -,










Q)
a, =





a)



QO
E

ri)









a)
S.-

U-








packets were next heated in a tube furnace under a flowing argon atmosphere

for 4.5 hours at 1173 K to produce the 0.24 atomic percent composition and

for 10 and 12 hours, respectively, at 1273 K to yield the 0.75 and 0.95

atomic percent alloys. The packets containing the oxidized specimens

were allowed to cool in the furnace. The specimens were then removed

from the packets and lightly etched in a 30% HF-70% HNO3 solution to

remove a thin surface scale left from the oxidizing step. After de-

scaling they were given a homogenizing anneal in a vacuum furnace under

a dynamic vacuum of 10-4 Torr at 1273 K for 18 hours.

Some of the sections were then mounted, polished, and etched to

provide a suitable surface for a microhardness survey. All microhardness

measurements were made on a Kentron microhardness tester using a DPH

indenter and were correlated with published DPH versus oxygen concentra-

tion surveys [77]. It was possible to maintain microhardness values

over all tensile specimens of this composition to 4 DPH units, which

corresponds to a variationof oxygen concentration of approximately 0.04

at.%. Each rod section was then machined into cylindrical threaded

tensile specimens, with the dimensions given before.

Initially, the research plans called for the work to be done with

VP gradeNb as a base material. As testing proceeded, and the VP niobium

stock neared depletion, another batch of niobium was obtained from the

Wah-Chang Corporation. This will be designated by WC Nb. The major

impurity concentrations of this niobium are also listed in Table 4.1.

The oxygen concentration is about six times larger than that of VP

niobium. The Wah-Chang niobium was directly used for the measurement

of the tensile properties. Tensile specimens were prepared in the same

manner as for the VP niobium.








4.2 Tensile Testing


Tensile tests were performed.inan Instron Testing Instrument Model

1125 which allows the recorder chart to be driven in conjunction with

crosshead commands. Tensile tests were conducted over the temperature

range between 77 K and 971 K. For the tests up to 473 K, specimens

were tested by using an inverted tensile fixture suspended in tempera-

ture controlling baths. Below room temperature, the specimens were

tested in conventional cryogenic baths. Tests between room temperature

and 473 K were carried out in an agitated silicon oil bath. For the

temperatures above 473 K, the tests were performed in a vertical resis-

tance furnace under a flowing high purity (99.99%) argon atmosphere.

During the test period the temperature was controlled to within 1 K

at each test temperature.

The strain rate sensitivity was determined by changing the strain

rate at a fixed strain and measuring the change in flow stress.

The extrapolation technique was used to measure the flow stress difference,

as defined in Figure 4.2. When jerky flow was present, the extrapolations

were drawn tangent to the stress peaks. Most of the strain rate sensitivity

measurements were made by changing the strain rate by five times with a base

strain rate of 8.8 x 10-5 s

VP grade Nb was used to investigate the strain aging at a constant

stress. A technique [23,33] for aging under load is shown in Figure 4.3.

Here the specimen is unloaded to some fraction of the flow stress

developed during the prestrain. This is accomplished by holding the














Nb-0.95 at. %0


=8.8 x

= 4.4x


5 -1
10 S

io-4 SI


300 K









355 K

30MPaf 0.5E 3
IE


394 K


0
E^ L


STRAIN


Figure 4.2


Illustration of the extrapolations used for the calculation
of the strain rate sensitivity on various stress strain
curves.






53

















0




Ln
0

0
w

w


















4-)
C)













G0")
0



0 -

0






Q) w

u -a

4)u,

0 C)


lS-C
S)
o -












(CU
SE-
C)














U-








specimen between two closely spaced load limits. Within these load

limits the specimen undergoes a number of small relaxation and reloading

cycles. Thus, as it deforms plastically during the load relaxation

part of a cycle the load falls until the lower load limit is reached,

then the machine is turned on. The load then rises to the upper load

limit at which point the machine is again turned off and the cycle is

repeated. Figure 4.3 also shows the manner in which the yield point

return data were measured in a strain aging test. In this method, the

yield point return, Aaa, is defined as the stress difference between

the upper yield point and the point oa at which the flow curve obtained

after yielding extrapolates back to the elastic part of the stress-strain

curve.

The average strain rate during an aging period was measured at

each test.








CHAPTER V
EXPERIMENTAL RESULTS


5.1 Portevin-Le Chatelier Effect Due to Snoek
Dynamic Strain Aging

5.1.1 Tensile Tests

Figure 5.1 shows a set of stress strain curves for specimens with

0.75 at.% oxygen deformed at a strain rate of 8.8 x 105 s- between

335 K and 422 K. The pertinent details of these stress strain curves

are reproduced in Figure 5.2. Serrations were observed on these curves

as described in the following. Note these serrations occur at temperatures

too low to be identified as the usual Cottrell type of serrations. Serra-

tions of this type have not been previously reported except in a very few

cases [2,15,56]. At 335 K (curve 1), no serrations were observed. At

340 K (curve 2), on the other hand, a single serration appeared on the

stress-strain curve near the point of maximum load. With increasing

temperature the number of serrations increased. At 355 K (curve 4)

periodic embryo Type A serrations appeared. Definition of this type of

serrations has been reviewed in section 3.1. With further increase in

temperature the number of serrations increased while the magnitude of

the serrations decreased. In curves 6 and 7 in Figure 5.2, it can be

seen that between 381 K and 394 K smaller irregular serrations appeared

between the regular larger Type A serrations. These smaller serrations

are probably associated with nucleation of secondary LUders bands. At

a still higher temperature, 409 K, curve 8 in Figure 5.2, both the Type A

serrations and the closely spaced irregular intermediate serrations dis-

appeared while Type B serrations began to appear just before the maximum













































5 10 15
STRAIN (%)


Figure 5.1


Stress-strain curve of Nb-0.75 at.% oxygen specimens
for temperatures between 335 K and 422 K.


600


500




400


300


200


I00




0















































Figure 5.2 Details of the serrated curves shown in Figure 5.1.








load was attained. These serrations reached maximum size just after

necking began and then decreased in size and disappeared. Note that,

at point A in curve 9 in Figure 5.2, the chart speed was increased by

a factor of 10.


5.1.2 The Ratio of the Strain Rate to the Diffusion Coefficient (e/D)

Cottrell [18] was the first to propose that the ratio of the strain

rate to the diffusion coefficent, e/D could be used as an index for the

conditions controlling the start of the serrations on a stress-strain

curve. On the basis of Manjoine's data for mild steel [19], he proposed

that the t/D should be about 109 for serrations associated with Cottrell

dynamic strain aging. Beckerman [23] has also used the //D ratio to

characterize the range of strain rates and corresponding temperatures

where dynamic strain aging dueto the Snoek effect is expected to occur.

She showed that, for the onset of serrations due to Snoek dynamic strain

aging, the e/D ratio should lie between 1012 and 1014. This is generally

three orders of magnitude greater than that for Cottrell dynamic strain

aging.

A plot of the magnitude of the largest serrations observed on the

stress-strain curves of theNb-0.75 at.% 0 specimens between 335 and 422 K

is given as a function of the /D ratio in Figure 5.3. The e/D ratios

were calculated using the diffusion coefficient for oxygen in niobium as

determined by Boratto and Reed-Hill [78]. Figure 5.3 shows that these

serrations were observed at e/D ratios between 1011 and 1015 with the

largest serrations appearing at an 6/D equal to 1014






59













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IC)~
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(A ,








x>



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*~( )(N 1r0D00()4
N WU - -






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(D U-).) U

(0 LC) I'O C8 J 0
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DIAJ 'S J IS NiVHB .VA
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co I r)
U-








5.1.3 Kinetics of the Snoek Ordering

Since in this temperature interval the serrations are most likely

to be related to Snoek aging, it was felt that the period between suc-

cessively larger peaks on the stress-strain curve might be related to

the jump time of the oxygen atoms.

The average time between Type A serrations was determined for each

test temperature between 346 and 394 K. The solid line in Figure 5.4

shows this average time when plotted against the test temperature. The

dashed line in this figure gives the jump time of the oxygen atom versus

the temperature as determined by the equation below [79].


(5.1)


where T is the mean time of stay of an oxygen atom in an interstitial

position, a is the lattice constant of the niobium crystal, and D is the

diffusion coefficient for oxygen in niobium. Figure 5.4 shows that

between 346 and 381 K, where relatively periodic Type A serrations appear,

the experimentally determined average time between serrations agrees well

with the jump time of the oxygen atom.


5.1.4 Strain Dependence of Serrated Flows

In addition to the magnitude of the serrations and the number of

serrations, three other parameters are also used to characterize these

low temperature serrations; the critical strain, EC, for the onset of

the serrations, and the strain occurring between successive serrations,

gs, and the strain rate sensitivity, S.


a
24D










Average time between
serrations


1000







100


atomic


/
/
/


381


2.7


2.6


1000/ T


362 355
I


2.8
-I
K


Figure 5.4


A comparison of the average time between serrations with
the jump time of an oxygen atom.


O*-


o---o Calculated
jump time

0.75 at.% 0


394


2.5


346 K
I


2.9


-d
-r I








The critical strain, ec' for the onset of serration, is plotted in

Figure 5.5 as a function of temperature. The critical strain falls

sharply from about 8.8% to 1% as temperature increases from 340 K to

355 K. At the intermediate temperature range, between 355 and 394 K,

the critical strain becomes small and relatively constant, i.e., the

serrations at this temperature range begin to appear as early as the

deformation proceeds ~ 1 % strain. With further increase in temperature,

the serrations appear near the maximum load and the value of the critical

strain increases rapidly. In other words, no visible serrations were

observed during uniform plastic deformation.

The strain between successive serrations, es' was measured from a

stress-strain curve obtained at 381 K. It appears that at this tempera-

ture serrations become regular and consistent. Figure 5.6 is a plot of

Es as a function of strain. It can be seen that es increases linearly

with increasing strain. A definition of s is shown in this figure.,.

The strain dependence of the strain rate sensitivity at this low

temperature range where the Portevin-Le Chatelier effect is observed

was studied. As may be seen in Figure 5.2, the stress-strain curve at

346 K shows only two well defined Type A serrations, the first appearing

at about 6.1 % e. Just below or at this strain homogeneous deformation

breaks down into discontinuous plastic flow. Thus, 346 K appeared to be

an ideal temperature for studying the interrelationship between the

strain, the strain rate sensitivity, and the initiation of the Portevin-

Le Chatelier effect. The results of the strain rate sensitivity measure-

ments made on a 0.75 at.% oxygen specimen as a function of the strain




















c4-

0 0 U-)

O
IO

S* -0 0"




(0 o
X 4-

a UL C"iS


O O CS )

q 0
C,
Y- o"





r-o
S o4-



r()c C)

LU
a .+S




'- u,



0
4-C


3 *
CO
0r-- a




O n
0 0f (0 CJ 0 O
r-
Sx



0 co cC L0



% 3 ) 'NIVdIS -73O1813
r-
i,





64







00



II
Q)
0 11
--



w
UD


S-
-



O1 LO

O
O





U o u



O 'O
0 \ I













4 xY 0

co g

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< W









\ I IO -
w 3 . .
z F-



H-
4- O L-




\L4-









d o o o d d


%' S~



















TEMP. = 346 K

0.75 % 0


4 5.2 6
STRAIN, %


Figure 5.7


Variation of the strain rate sensitivity, S,
with strain at 346 K. Nb-O.75 at.% oxygen
EL = 8.8 x 10-5 s-1, H = 4.4 x 10-4 s-1.


0r


.t4


O
0



- I


I0


I


L


i\








at 346 K are shown in Figure 5.7. The strain rate sensitivity parameter

that was used in this figure is


S = (A .)T (5.2)

where a is the flow stress and the strain rate. The strain rate sensi-

tivity was measured using the extrapolated steady state change of stress

rather than the initial transient stress peak, as defined in Figure 4.2.

As may be seen in Figure 5.7, the strain rate sensitivity, S, is positive

up to e ~ 5.2%. Above this strain S becomes negative. A minimum occurs

at about 6.1% strain followed by a gradual increase with further defor-

mation. The effect of strain on the strain rate sensitivity will be

discussed in detail in the discussion chapter.


5.2 Strain Rate Sensitivity of the Flow Stress


5.2.1 Effect of Temperature

The strain rate sensitivities, S and n, are plotted against tempera-

tures between 77 K and 773 K for VP Nb in Figures 5.8 and 5.9. The speci-

men was deformed to 4.5% strain at a strain rate ( oL) of 8.8 x 10- s

and then the rate change to tH, where H = 5 L, was made. The difference

in flow stresses at two strain rates was measured. Note that a horizontal

dotted line indicates zero strain rate sensitivity. The results show that

the strain rate sensitivity is strongly dependent on the temperature. At

the temperatures below 200 K, there is a very high peak in the strain rate

sensitivity. Although the experimental data are limited, it is seen that

the peak appears at around 150 K and that the magnitude of the peak is





67







O

0



II--------II------- 0".
SI 0 E

00 0
o 0
(0
O -o I-
co LO.If

0 0
co -

\(31 1




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dil *> Ul LJ /?^ -







Sn w


-0 a

.- f./
ru ,O
ai
I /^ .w







co o-
I A 4



LI

DdAJ ? U|1V/oV





68














0
O
-0

co







-r


0 V -i
I 0 ow
D > CO






o 'L ) 00
o n- -.-
O L


=I
o >



0 C- E -

^r^ W W
4-')







I I

1_o a lI- W l


ro CXj 0 \J
5- r



0 0 0 0 0 0



9 U| V /-. U| V








about 14 Mpa. As the temperature increases, the strain rate sensitivity

drops sharply and shows a minimum at T = 370 K. At this temperature, the

Aa/Aknt = 0. As the temperature increases further, the strain rate sensi-

tivity increases and reaches a small peak at around 440 K with the value

of approximately 1.5 Mpa, followed by a sharp drop as temperature rises.

As the temperature passes around 455 K, the strain rate sensitivity

becomes negative again. The negative character of the strain rate

sensitivity grows with further increasing temperature. When the temper-

ature reaches ~ 500 K the strain rate sensitivity has a deep minimum with

a value of approximately 2.5 Mpa. With further increase in temperature,

the strain rate sensitivity increases but it still has a negative value.

At temperatures around 550 K, the strain rate sensitivity again passes

the dotted horizontal line, where Ao/AAnt = 0, followed by another sharp

peak at T = 600 K. At the temperatures above 600 K, the strain rate

sensitivity decreases gradually as the temperature increases and the

values remain positive. In summary, there are two temperature intervals

within which the strain rate sensitivity is a minimum. One is centered

near 370 K, the other around 500 K. At the higher temperature minimum,

the strain rate sensitivity becomes negative. Following the high temper-

ature minimum, the strain rate sensitivity passes a sharp positive peak

at about 600 K. This will be discussed in detail in the discussion

chapter.



5.2.2 Effect of Oxygen Concentration

The measurements of the strain rate sensitivity for specimens

containing higher oxygen concentrations are shown in Figures 5.10 through








5.14. Here again the higher oxygen specimens were deformed to a pre-

strain of 4.5% and the strain rate sensitivity was measured by varying

the strain rate between 8.8 x 105 sI (eL) and 4.4 x 104 sI (e).

Figures 5.10 and 5.11 are plots of the strain rate sensitivity

S and n, for niobium specimens containing 0.24 at.% oxygen. The overall

picture of the strain rate sensitivity versus temperature for 0.24 at.%

oxygen specimens are not much different from that of the VP grade speci-

men; i.e., for 0.24 at.% oxygen specimen, there are also two temperature

intervals within which the strain rate sensitivity becomes a minimum and

a sharp positive peak at T = 600 K. Even for the higher oxygen specimen,

0.95 at.% oxygen,the temperature dependence of the strain rate sensitivity

remains basically unchanged. Plots of the strain rate sensitivity, S and

n, versus temperature for 0.95 at.% oxygen specimens are shown in Figures

5.12 and 5.13. In these figures, a dotted horizontal line again indi-

cates zero strain rate sensitivity.

Effect of oxygen concentration on the strain rate sensitivity-

temperature curve are clearly seen in Figure 5.14 in which Figures 5.8,

5.10 and 5.12 are superimposed. It may be seen from Figure 5.14 that

increasing oxygen concentration from 0.01 to 0.95 at.% does not change

the basic plot of strain rate sensitivity versus temperature. In other

words, for all three composition specimens, there exists a low temperature

peak, two minima at two different temperature intervals and another peak

at high temperature around 600 K.

Even though the data are limited it can be seen that the strain

rate sensitivity peaks at low temperature for two different compositions,

0.01 and 0.95 at.% 0, fall almost at the same temperature, around 150 K.













0


Eo


4- J 11

I O. "-
.w
4 C )
-l


4- <--Co

- X







N*
On \n LO



0







z cA
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0- o c0

MI J





c J -! -






DdVl U|V/OV





72






O
O



I 4-)
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<-,
Ir II
o _
1 -- ~-------------- 0 -





1 .w



L *
c- r-


44
r- x





- II
^ 0r *s3c: VF
11
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0 0" 0 0





6 6O




u/
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0
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co <
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<
-cm

C*r- U





ta- O r o



o 10 L -Ood
So I
0 "
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0 i l *_
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0 0 0 0 O
eI 1 Ul I
NI) N\J *- v

D~J 3U L
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Ddl^Jl 3 UV/^)















































































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0 0 0 0 ~ 0
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COJ
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> _j0 4) C

DdW 0 Iro .C








The peak temperature seems to be independent of the oxygen concentration.

It is observed, however, that the peak height increases with oxygen con-

centration. The positive dependence of concentration on strain rate

sensitivity continues until the temperature reaches around 335 K.

For all three different oxygen contents, there exists two temperature

intervals where the strain rate sensitivity is a minimum. Note that the

low temperature strain rate sensitivity minimum becomes negative as oxygen

concentration increases from 0.01 to 0.24 at.% oxygen. Furthermore, the

negative character of the strain rate sensitivity increases with increasing

oxygen concentration. In other words, the strain rate sensitivity of

higher oxygen specimens is more negative than that of lower oxygen speci-

mens. This is not true at the higher temperature minimum. At the higher

temperature minimum the strain rate sensitivity is negative to about the

same degree at all three concentrations within the experimental error.

The exact change of the stress with change in strain rate is not capable

of being measured because deformation occurs discontinuously when serra-

tions occur and the strain rates are not the same as the applied strain

rates.

A careful examination of Figure 5.14 apparently reveals that the

strain rate sensitivity minimum temperature, or the temperature at which

Ao/Aqn = 0 is! shifted to lower.temperatures as the oxygen concentration

increases. For instance, the temperatures where Ao/Azn = 0 at the higher

temperature side are 545 K, 530 K and 510 K for the specimens of 0.01, 0.24

and 0.95 at.% oxygen, respectively. A 35 K shift is observed by changing

oxygen contents from 0.01 to 0.95 at.%.








Following the higher temperature minimum, the strain rate

sensitivity passes a sharp peak at all oxygen concentrations. It is

observed that the peak temperature is not affected by the oxygen con-

centrations; i.e., it is about 600 K for all specimens. However, the

magnitude of the peak is a strong function of the oxygen concentration.

It grows sharply with increasing oxygen concentration.



5.2.3 Effect of Base Strain (L )

The effect of base strain rate on the plot of strain rate sensitivity

versus temperature is shown in Figures 5.15 and 5.16. The VP grade Nb

specimens were prestrained to 4.5% at three different base strain rates,
-5 -1 -4 -1 -3 -1
8.8 x 10 s 4.4 x 10 s- and 2.2 x 10 s and then a 5:1 rate

change was made. Here again a dotted horizontal line indicates zero

strain rate sensitivity.

It is seen from Figures 5.15 and 5.16 that the basic strain rate

sensitivity versus temperature dependence is not changed by changing the

base strain rate. For all three base strain rates, there exist two

temperature ranges where strain rate sensitivity reaches a minimum

and a strong peak at higher temperatures.

It is also found that, while the low temperature strain rate

sensitivity minima for the first two slow base strain rates remain

positive, the corresponding value ata faster base strain rate, 2.2 x 10-3

s-1 becomes negative. At the high temperature minimum the strain rate

sensitivity is negative to about the same degree at all three base strain

rates. It is generally seen from the figures that the curves are shifted



















I


S- I
-\u .\i/


^~ \


I
u) C/, U),


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X X X
^- C\J -



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lDdAl ul V/-oV


0
0





C



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r0
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)< ro \J
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co o ,o
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I I


? Ul V /-O U V


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o
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0
0
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rlL)
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o
o
o


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4 -
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LUJ



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4-
0a



cW
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to higher temperature as the base strain rate increases. Thus, for

example, the temperatures where Aa/Akn n 0 at the higher temperature

side are 545 K, 570 K and 595 K for the base strain rates of 8.8 x 10-5
-1 -4 -1 -3 -1
s 4.4 x 10 s1 and 2.2 x 10 s respectively. Increasing the base

strain rate by 25 times results in the strain rate sensitivity curve

shifting by about 50 K to highertemperatures.

It might be practically necessary to know the temperature range

where the plastic flow becomes unstable at a certain value of the strain

rate. The temperature ranges in which the strain rate sensitivity

becomes negative are measured fro all base strain rates: 450 545 K,

480 570 K and 505 595 K for 8.8 x 10- s1, 4.4 x 104 s1 and

2.2 x 10 s respectively. It appears that the temperature intervals

for all three strain rates are about 95 K, regardless of the value of the

base strain rate.

The peaks at temperatures above the strain rate sensitivity

minimum are also observed at all three base strain rates. As the base

strain increases, the peak temperature moves to the higher temperature

side. The height of the high temperature SRS peak does not vary signifi-

cantly with base strain rate.


5.2.4 Effect of Magnitude of Strain Rate Change

The effect of magnitude of strain rate change on the plot of strain

rate sensitivity versus temperature is shown in Figure 5.17 and 5.18.

VP Nb specimens were strained to 4.5% at a base strain rate of 8.8 x 10-5
-I
s andthena 25:1 strain rate change was made at each test. Data for

(tH/ L) = 5 are also shown in the same figures.














O
I 0

ININ I
II II





I

8I




.,
1 ,


CN 0


'0
-o

x


CO
II

*-J
jj


o



o
r--



CO

C'-
*r- CO
t3

SII
c 0


(Q4

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*- E





4- a'


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i <


QLU


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cr-
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C.,
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Ic


ODcJAl 3UIV/-

















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* N


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'*,

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II
I '
* J


I I


3? Ul V/.OUl V


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0





0




O
OL





It

D

OF-
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0 -

O <
O









0
0
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h -




















O
O


QC-



SCO


L)
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(_ I
!= 0







0 m3
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OI




_0 C-
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J= <
4- 4-
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4-( 1
S-

,=

4-.
*0
oc-



4-<

-- O-







r--
t.)








It may be seen that increasing the ratio five times does not affect

significantly the magnitudes of both the low and high temperature minima.

However, it shifts the overall curve towards higher temperatures. Thus,

the two minimum temperatures and the peak temperature increase.

Accordingly, the temperature at which the strain rate sensitivity becomes

negative also increases with increasing magnitude of (tH/tL).



5.2.5 Effect of Prestrain

In all of the previous strain rate sensitivity results prestrain

was held constant at 4.5%. In Figures 5.19 and 5.20, the effect of

prestrain on the strain rate sensitivity as a function of temperature

is shown. Specimens were prestrained to 1% at a base strain rate of
-5 -1
8.8 x 10 s and then a 25:1 rate change was made at each test.

Superimposed on this curve are data for the 4.5% strain. Figures 5.19

and 5.20 reveal that decreasing prestrain from 4.5% to 1% brings the

curve to the higher temperature side. However, the magnitude of both

low and high temperature minima are not apparently influenced by the

amount of strain imposed when the rate change is made. It is seen from

Figure 5.19 that the magnitude of Aa/AZnt at higher temperature is

reduced by 40 ~ 50%. It is also observed that decreasing the strain

causes the high temperature peak temperature to move to higher tempera-

ture. This is better seen in Figure 5.20.























II II
II

'O --

q) i

\. "'X

^'^ '1


10



0()
JI
cO



II



a.


Ca;
4-


4->











L o
0 a







0 01
o ra



<


4--





o .



0--
0 O
0 4-




II
L .- .W






















































K) 0 c\
0 0 0 0 0 0
6 d 6 d 6 6
I I
U I v/.o Ul V


0
o,

0.


4->


0
0

4-
o



O 0



OH
4 --



oJ



w *


on





4-)
0 Q co
4 -







L.-
O O






0
U- 1




j








5.3 Work Hardening Parameter


The parameter taken as a measure of the work hardening rate is the

change in stress between the flow stresses at 4.5% and 4% e. This

quantity represents an average value of work hardening for a region near

where the strain rate change was made. The results of the work hardening

parameter of VP Nb specimens are plotted in Figure 5.21 as a function of

temperature for three strain rates.

At each strain rate, a large work hardening peak is apparent. This

figure reveals that increasing the strain rate increases the peak temper-

ature of each maximum. The peak temperatures are approximately 510 K,

540 K and 580 K for strain rates of 8.8 x 105 s1, 4.4 x 10-4 s- and

2.2 x 10 s respectively. This effect can be understood by employing

a relationship suggested by Cottrell [18]. He proposed that the tempera-

ture at the onset of serrated yielding, a dynamic strain aging feature,

could be related to the applied strain rate by the relation


= K D (5.3)


where K is a constant and D is diffusion coefficient of the solute atom.

The above equation can be rewritten as


S= K D exp( -) (5.4)

and taking the natural logarithm of this equation gives


Zn = kn(KD ) H (5.5)
0 RT




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