A LITERATURE SURVEY OF THEORIES AND METHODS OF PREDICTING CHARACTERISTICS OF MATERIALS

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• Publication Date: January 1, 1959
• Content Type: REPORT

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AFMDC-TR-59-1
As-IAA Document No. AD 154109
AFMDC-TR-59-1
ASTIA Document No. Al) 154109
A LITERATURE SURVEY OF THEORIES AND METHODS OF
PREDICTING CHARACTERISTICS OF MATERIALS
ERVIN E. UNDERWOOD
MAURICE F. AMATEAU
ROBERT E. MARINGER
GEORGE K. MANNING
BATTELLE MEMORIAL INSTITUTE
COLUMBUS, 01110
JANUARY, 1959
CONTRACT NO. AF 29(600)4557
(PROJECT 6886; TASK 68804)
AIR FORCE MISSILE DEVELOPMENT CENTER
AIR RESEARCH AND DEVELOPMENT COMMAND
UNITED STATES AIR FORCE
HOLLOMAN AIR FORCE BASE, NEW MEXICO
STAT
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NOTICES
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AFMDC-TR-59-1
ASTIA Document No. AD 154109
AFMDC-TR-59-1
ASTIA Document No. AD 154109
A LITERATURE SURVEY OF THEORIES AND METHODS OF
PREDICTING CHARACTERISTICS OF MATERIALS
Ervin E. Underwood
Maurice F. Amateau
Robert E. Maringer
George K. Manning
Battelle Memorial Institute
Columbus, Ohio
January, 1959
Contract No. AF 29(600)-1557
(Project 6886; Task 68804)
Air Force Missile Development Center
Air Research and Development Command
United States Air Force
Holloman Air Force Base, New Mexico
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FOREWORD
This report was prepared by Battelle Memorial Institute, Columbus,
Ohio. It represents the final report under Contract No. AF 29(600)-1557.
The research was sponsored by the Air Research and Development
Command in support of Research Planning Objective 802-A, Research on
Materials, under Project 6886, Task 68804, Theoretical Investigations on
Materials, with the objective to secure a mdre comprehensive understanding
of the physical properties of materials with particular emphasis on those
materials required in ballistic missiles and space vehicles.
flrIssifid in Part SanitizedC
ACKNOWLEDGMENTS
The wise council and encyclopaedic knowledge unstintingly made
available by Professor Egon Orowan has been of inestimable value in the
preparation of this report. Although his opinions and advice have been
given unsparingly, any omissions or errors that may have crept into the
text are not his responsibility.
The kindness of Professor John E. Dorn, Dr. Peter B. Hirsch, and
Professor Eugene S. Machlin in discussing various aspects of this report
is greatly appreciated.
Finally, our thanks are due to the far-sightedness and truly scientific
vision of the Holloman Air Force Missile Development Center, who made
this contribution to basic theory possible.
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ABSTRACT
A literature survey of theories and methods relating to strength and
plastic deformation of materials has yielded almost 1800 abstracted refer-
ences from various publications appearing between 1930 and 1957-1958.
Primary emphasis was given to dislocation theories; absolute reaction rate
theories; thermodynamic theories of fracture strength; relationships based
on equations of state; and empirical relationships and parameters. The
evaluation of the most promising theories and methods indicites that
dislocation-type theories, or the parameter-type expressions, offer the
greatest opportunity for ultimately predicting strength properties.
iv
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TABLE OF CONTENTS
INTRODUCTION
SCOPE OF THE LITERATURE SURVEY
BACKGROUND DISCUSSION
DETAILED PLAN OF THE LITERATURE SEARCH
DISCUSSION OF IMPORTANT THEORIES AND METHODS
RELATING TO STRENGTH PROPERTIES
Dislocation Theories of Plastic Deformation and Strength
Strain Hardening
Alloy Hardening
Precipitation Hardening
Creep
Absolute Reaction Rate Theories
Thermodynamical Theories of Fracture Strength
Relationships Based on Equations of State
Empirical Relationships and Parameters
EVALUATION OF THEORIES AND METHODS
SUMMARY OF CONFERENCES WITH CONSULTANT,
PROFESSOR E. OROWAN
APPENDIX A
ANNOTATED BIBLIOGRAPHY OF THEORIES AND PROPERTIES
OF MATERIALS
AUTHOR INDEX
APPENDIX B
. Page
1
1
3
6
8
9
10
16
21
23
32
34
36
39
42
43
A-1
B-1
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TABLE OF CONTENTS
(Continued)
APPENDIX C
CLASSIFICATION OF THE NUMBERED REFERENCES IN
APPENDIX A BY NUMBER
Theories of Strength
Theoretical Calculations, Mechanisms, and
Discus sions
Review Articles
Experimental and Phenomenological Methods of
Predicting Properties
Experimental Relationships Among Variables
Experimental Observations of Factors Influencing
Strength Properties
vi
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C-1
C-1
C-1
C-2
C-3
C-4
C-5
LITERATURE SURVEY OF THEORIES AND METHODS
OF PREDICTING CHARACTERISTICS
OF MATERIALS
INTRODUCTION
The problem of describing the characteristics of materials is a long-
standing one. When the additional goal is set of predicting material charac-
teristics, almost insuperable difficulties are encountered. Many different
lines of attack have been tried, to solve the problems encountered in de-
scribing or predicting material characteristics. The current approach has
reached a degree of sophistication that leads to the conclusion that the basic
mechanisms, the detailed atomic movements, the various interactions of
elementary particles ? all must be scrutinized in more detail in order to
understand what is happening. This microscopic approach is necessary in
order to understand the gross, macroscopically observed behavior, and
embodies the scientific, long-range view.
On the other hand, the exigencies of the daily demands on engineers
for practical and immediate solutions to his problems leave no alternative
but to use empirical relationships, rule-of-thumb reasoning, or just plain
horse sense. The two points of view on this general problem thus repre-
sent the two extremes that exist today in seeking to arrive at a solution to
the problem of predicting material characteristics. The urgency under which
one must work, then, will dictate the line of approach employed. This
literature survey has attempted to plead both sides of the case ? the plaintiff
may receive the verdict applicable to himself alone. Therefore, for the
purposes of a literature review in which the goal is to ascertain the poten-
tially most useful theories and methods for predicting material prop-
erties, the guiding principle has been to re-examine all available work
pertinent to the general problem. Then, on the basis of current develop-
ments in theory and methods and the opinions of experts in the field, and
from the long-range perspective afforded by this literature review, an
evaluation of the most promising theories and methods has been made.
;COPE OF THE LITERATURE SURVX
Obviously, complete coverage Cannot be afforded a subject as prolific
as characteristics of materials. It is recognized that such factors as
heat resistance, corrosion resistance, strength at low or high tempera-
tures, heat conductivity, etc. , are all integral components of the over-all
problem of material behavior. However, in view of such practical
Manuscript released by the authors, November, 1958, for publication.
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considerations as time, which must be faced in a 6-month literature survey,
and in view of the vast amount of material already printed on the general
subject, it is deemed advisable to limit the scope to the most fruitful areas.
One of the most important: areas would include the mechanical, or strength,
properties; also, the selection should be directed primarily toward metallic
materials. This latter choice is not: as restrictive as it may seem at first
glance, since the crystalline nature of metals is found duplicated in many
nonmetallic substances, and the basic laws should be applicable equally to
both. Furthermore, the term "strength", as used here, would be applied in
the broad sense, in that the yield stress, the critical resolved shear stress,
creep strength, the hardness, etc. ? in fact, almost any index of a ma-
ability to resist plastic deformation ? would be considered.
Some items have been excluded, rather arbitrarily, perhaps, but
from the sheer necessity to draw the line somewhere. Included in a list of
factors not covered specifically would be such items as fatigue, brittle-
ductile transition behavior, age-hardening phenomena, recovery and re-
crystallization processes,, and crack initiation and propagltion. However,
important articles on these subjects have been included in the main appendix.
There is a good reasonforthe emphasis given here to strength properties.
That reason is, if a material cannot pass the basic requirements of
adequate strength under the specified conditions, then there is no great
need to know the other properties.
As to the references themselves, each item deemed worthy of inclu-
sion in this report has been entered in the main appendix, along with a brief
abstract of its contents. The entries are listed alphabetically by the first
author and are numbered consecutively. Here, again, the emphasis in
selection has been on the broader aspects of the problem. Another guiding
principle has been to document the experimental side, as well as the
theoretical, as thoroughly as time and space would permit. This action
was taken because of the great debt owed by theory to experiment, and
because of the great importance such references would have in an experi-
mental continuation of this program. It goes without saying that many
references may have been left out ? either by chance or design ? that might
be considered important. Be this as it may, there is no doubt that the ref-
erences given in the articles covered here will lead to all additional ref-
erences desired and many more. So, it should be a relatively minor task
to locate a particular reference from the other quoted sources.
To facilitate the search for references on any particular subject, the
main appendix of some 1800 entries has been broken down into four sub-
sidiary categories. In addition, there is an author index in which are listed
the references, by number, where the author's name appears. It is
believed that the bibliography contained in the appendix will constitute an
invaluable aid to the research worker in this field, and that the widest pos-
sible dissemination of this report should be made to interested parties.
nitized Coov AIDIDr0V
d for Rel
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BACKGROUND DISCUSSION
Early theories of the strength of metals were restricted mostly to
simple substances, and to a relatively narrow range of temperatures.
Fairly recently, some thought has been given to the case of more complex
materials, as well as to a wider range of temperatures. In spite of the
numerous attempts to develop a comprehensive theory, little progress was
attained until the introduction of the concept of "dislocations" in the early
1930's. The flexibility of the dislocation theories, and the imaginative
and fertile work of the many mathematicians and physicists who became
interested in the possibilities of this concept, led to explanations of many
metallurgical phenomena previously only vaguely understood. Important
as this concept has proved, there still have been many alternative lines of
attack on the basic problem of strength, such as the chemical, thermo-
dynamic, electronic, atomistic, and mechanical. It can be seen that
these various approaches embody tile same goals, but that the points of
view are somewhat different. Thus, all available theories have been re-
viewed so that no possible line of attack has been overlooked that may lead
to a solution of the present problem.
In addition to the theoretical attempts to understand the basic factors
affecting the strength of material, empirical methods have been developed
to describe relationships between significant variables, such as tempera-
ture and stress. Although such ad hoc methods are not completely satis-
fying, they do have the advantages of relating the desired property to
readily measurable experimental data, and, more important, they can be
applied to the complex materials actually used in engineering applications.
The outstanding development in this direction is the parameter involving
time or strain rate and temperature, which correlates the strength prop-
erties rather successfully. However, it cannot be said that such param-
eters are completely empirical, because of their direct connection with the
Boltzmann, or Arrhenius, rate equation, which connects a rate to an acti-
vation energy and temperature exponentially.
'
In the following paragraphs, those theories and empirical relation-
ships that may be useful in describing material behavior at various tem-
peratures will be discussed briefly. Then, areas that have been neglected
or that need further consideration will be specified to some extent.
The outstanding importance of creep in metals stressed at high tem-
peratures has led to a preponderance of theoretical work in this area.
Schoeck(1396)*, Dorn(377), and Weertman(1672) have offered high-
temperature dislocation theories of creep, based on the dislocation climb
*Reference numbers in parentheses refer to the bibliography in Appendix A.
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mechanism suggested by Mott(1081). Parker has claimed that Weertman's
theory is the first to adequately take into account the important experimental
facts. Seeger's ideas on the dislocation mechanisms of plastic deformation
have stimulated those of Wiedersich(1697), as well.as others. Older theories
that have influenced ideas on creep are those due Orowan(1182), mott( 1082),
and Nabarro(1110).
Bleakney(127) and Parker(1217) have examined the creep process in
terms of intercrystalline cohesion, whereas Bochvar( 146) theorized about
intergranular shears. The absolute reaction rate theory has been applied
to creep deformation by Kau zman (357), Nowick and Machlin(1137), and
Deklityar(357), among others. A statistical formulation for creep of metals
has been published recently by Bates, Ree, and Eyring. Other approaches
have been attempted, also. For example, Osipov and Fedotov(1203) have
described creep in terms of self-diffusion coefficients and melting
temperatures.
It can be seen that considerable latitude exists in the various approaches
to a theoretical description of deformation at high temperatures. Theories
that treat deformation from a more generalized standpoint than creep alone
are also very numerous, with the emphasis at lower temperatures where
thermal energy is not an appreciable factor. Such theories are important
in any assessment of the literature on theories of strength, because it is
possible that the basic ideas can be applied,' without too many modifications,
to other temperatures of interest. General theories that have had far-
reaching influence in more modern formulations are due to Becker-(89),
Orowan(1174), Bragg(173), and Mott and Nabarro(1101) to quote a few.
The dislocation concept in plastic deformation was explored thoroughly up
to 1941 by Seitz and Read(1444), ,and the principles of plasticity theory were
invoked by Hollomon and Lubahn(714). Jaswon(758) has considered the
problem of cohesion on an atomic scale. Other, rather extensive, treat-
ments of the theory of metals have been given in books by Wilson and by
Mott and Jones, primarily from the physicists' point of view.
A general review by Matthaes(1054) considered the relationships that
exist among the various physical and mechanical properties of metals and
alloys during plastic deformation. Numerous other authors have suggested
alternative approaches to this problem. The rheological approach, em-
bodied in the expression by Nutting(1140), appears to have extremely broad
applicability. Examples are given by Dekhtyar(356), who established a
relationship between bond energies and energies for diffusion; by Fastov(418)
and Fiirth(502), who utilized the methods of thermodynamics; by Gulyaev(617),
who showed the periodicity of the strengthening influence of solute elements
as related to valency; by Wins. and Kritskaya(733), who relied upon X-ray
measurements of bond strengths to describe the characteristic temperature
of metals and alloys; and by Rovinskii(1360), sipov(1195), and
Dekhtyar(355), who expressed behavior in terms of electron-density
configurations.
*Tech. Rep. 56, ONR (NR-032-168), June 15, 1956.
5
Factors that have definite influence on the realizable strength of met-
als have been systematized by many. Kornilov(854) considered strength and
solubility relationships, whereas Andrews(39) classified properties accord-
ing to a thermodynamic scheme. Ball(60) showed clearly the surprisingly
important effects of subgra.in size, and Biggs and Broom(1") considered
the influence of ordering on strength properties. The hardness of pure met-
als was expressed by Westbrook(169?) as a function of temperature, crystal
structure, specific heat, and the heats of fusion and transformation.
It may be noticed that the theories mentioned above depend, to some
degree, on empirical laws - for example, the Hume-Rothery rules. Un-
fortunately, the science of metallurgy has not yet advanced to the stage
where alloys can be designed on the drawing board, or with a computer.
Thus, it is inescapable that the art of metallurgy today is still dependent on
the rule of thumb, or empirical correlations, which are not based on funda-
mental metallurgical theory. Examples of empirical correlations relating
tensile strength, temperature, and strain rate are due to Kanter(782) and
to Clark, White, and Guarnieri(275). Underwood(1620) has correlated hard-
ness, creep, and tensile data to one straight line.
Zener and Hollomon(1777) and Hollomon and Jaffe(71-2) have proposed
parameters involving strain rate or time and temperature, respectively, to
correlate the flow stress and hardnesses of alloys . Larson and Miller(903) and
Robinson, Tietz, and Dorn(
1341) demonstrated that the time-temperature- ---
parameter could be applied just as well to stress-rupture data as to tensile
data. The "velocity modified" temperature of MacGregor and Fisher(990)
is akin to the parameter of Zener and Hollomon, and Rabotnov's parame-
ter(1300) is useful in the case of isochronous stress-strain curves. The
expressions involving strain rate describe the type of temperature- and
rate-sensitive deformational behavior that is usually designated as viscosity,
relaxation, or, more generally, thermal inelasticity(477). The interrela-
tionship of many of these parameters is obvious, which helps to emphasize
the common origin of such expressions. The range of validity of the time-
temperature parameter has been shown* to extend over rupture times as
short as a fraction of a minute to more than a year and over a 700?F tem-
perature range, for the Alloy 5-590. This is indeed an impressive accom-
plishment for an expression involving only one constant.
1043) have advocated their empirical eXpression
Manson and Haferd(
that defines stress as a function of time and temperature. Since their f
parameter does not stem from the basic Arrhenius rate expression, it does
not appear to have the significance of the other parameters listed above.
Stowell(1527) attempted to lump together, in one equation, factors dependent
on stress and strain, heating rate, and the time-temperature type of defor-
mati9n. Under specified conditions, the expression can be simplifiAd to a
parameter similar to some mentioned above. It may be useful for rapid-
heating rate tests.
*Underwood, E. E., unpublished research at Battelle Memorial Institute.
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The preceding discussion was designed to give a brief survey of vari-
ous kinds of theories and methods that have been advanced to describe plas-
tic deformation. Conspicuous gaps in the coverage afforded by these theo-
ries are evident. For example, the marked influence of very small
additions of a solute is not explained, nor is the existence of solution soften-
ing generally recognized. When several variables, such as strain, tempera-
ture, and aging, all change simultaneously, the problem of expressing such
behavior analytically becomes hopelessly complex. No present theory can
account for a creep rate that is governed by the instantaneous arrangement
of dislocations, impurity atoms, and vacant lattice sites(227). Even
N. F. Mott, who has done so much to advance dislocation theories of creep,
says that they are "extremely tentative and may well have to be revised or
abandoned(3Z)."
In spite of such shortcomings, progress in the theory of plastic defor-
mation and strength has increased greatly during the past years, and fur-
ther advances can be expected in the ensuing years.
DETAILED PLAN OF THE LITERATURE SEARCH
In order to get the most significant coverage of the literature, it was
decided to start with articles appearing between 1930 and 1957. Most of the
references of value today would fall in this period, and, when exceptional
cases Were noted, these could be included in the survey. Also, special vol-
umes on meetings, reports, symposia, etc., and textbooks and specialized
treatises dealing with strength of materials were included in the literature
search, to insure that no important papers were missed.
The following sources were consulted:
Chemical Abstracts
Metallurgical Abstracts
Abstracts of the Journal of the Iron and Steel Institute
ASM Review of Metal Literature
Metals Review
Abstracts of Metallurgy of the USSR
Transaction of the AIME
Transactions of the ASM
---
7
(9) Acta Metallurgica
(10) Applied Mechanics Reviews
(11) NAGA Bulletins, General Electric Company Bulletins,
ASTM Bulletins, etc.
(12) Science Abstracts
(13) Proceedings from conferences on deformation, strength,
creep, high-temperature behavior, etc.
(14) Textbooks and treatises on creep andistrength
(15) Personal files already available at Battelle Memorial
Institute.
Items (1), (2), (5) , (7), (8), and (9) were checked as far into 1958 as possible, also.
Other sources of information were consulted during the search period, in-
cluding translations from Battelle Memorial Institute, National Research
Foundation, Henry Brutcher, etc.
In addition to the above list, reports of special value to this investiga-
tion should be mentioned:
Report of a Conference on the Strength of Solids (1948)
A Symposium on the Plastic Deformation of Crystalline Solids
(1950)
Imperfections in Nearly Perfect Crystals (1952)
Dislocations and Plastic Flow in Crystals (1953)
Creep and Fracture of Metals at High Temperatures (1956)
Dislocations and Mechanical Properties of Crystals (1957).
Collecting titles to references of possible value is a relatively easy
and minor part of this task. More time-consuming is the next step ? that
of obtaining a suitable abstract for the reference. Depending on its impor-
tance, either a routine annotation was prepared for the record; a more
thorough abstract was obtained from any of the abstract sources listed
above; or the original article was read and evaluated, and an abstract pre-
pared therefrom. The most time-consuming process of all is, of course,
the full analysis given those articles of outstanding importance. All entries
selected, regardless of importance, were then assembled in the main appen-
dix, alphabetically by first author, and numbered consecutively. Evaluation
of the important theories and methods is covered elsewhere in the report.
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DISCUSSION OF IMPORTANT THEORIES AND
METHODS RELATING TO STRENGTH PROPERTIES
Before plunging into detailed discussions of the theories and methods
that are considered to be important to this literature survey, a brief outline
of the main subdivisions will be given:
(1)
(2)
(3)
Dislocation Theories. These are the results of the most
modern theoretical developments, and are being exploited
to an increasingly greater extent by scientists all over the
world. They deal with detailed mechanisms, are very
flexible, and are applied to a great diversity of physical
phenomena. Quantitative predictions are now becoming,
possible in simple cases.
Absolute Reaction Rate Theories. These more general-
ized theories do not depend so much on a detailed mecha-
nism as do the dislocation theories. A "unit of flow" is
postulated that conforms to thermodynamic criteria, and,
depending on the detailed interpretation given, this "unit
of flow" can be applied to the phenomenon of plastic flow.
Thermodynamic Theories of Fracture Strength. These
are based on the idea that the breaking of atomic bonds
by fracture is analogous to the melting process. Great
generality is to be expected from an approach along such
lines, but the very fact that such generality is achieved
makes it impossible to account for effects due to varia-
tions in microstructure, crystalline defects, etc.
(4) Relationships Based on Equations of State. Equations
of state for solids are better known for the case where
the selected variables are volume, temperature, and
hydrostatic pressure. When variables such as stress,
strain, strain rate, and temperature have been chosen,
their functional relationship is known as a "mechanical
equation of state". These equations are particularly
attractive because they offer the possibility of predict.:,
ing the mechanical properties of materials. However,
the main stumbling block to the acceptance of such func-
tions has been their dependence on the past history of
the material.
(5)
Empirical Relationships and Parameters. These parame-
ters usually express a functional relationship between
variables, such as strain, stress, strain rate, tempera-
ture, and/or time. Their greatest usefulness lies in the
ability to predict behavior for the complex materials used
9
for engineering purposes. However, their range of applica-
bility is unknown, and calculated values extrapolated too
far are uncertain.
Dislocation Theories of Plastic Deformation and Strength
Relatively speaking, the dislocation concept is new to the fields of
plastic deformation and strength. However, the facility with which hitherto
poorly explained observations could be rationalized; the extreme versatility
with which the properties of dislocations could be applied to detailed mecha-
nisms; the amenability of dislocations to mathematical analysis; and the
experimental detection of dislocations have all helped in making this line of
attack very popular nowadays. Unfortunately, the values of yield stress,
for example, given by equations derived from dislocation models are indi-
cated only to within orders of magnitude. The engineer desires numbers he
can apply to the actual problems of everyday life. Furthermore, there are
no all-encompassing dislocation theories involving strength properties, but
mostly fragmentary explanations of mechanisms applicable only to isolated
processes. Different types of failure (yielding, brittle fracture, shear frac-
ture, fibrous fracture, fatigue fracture, etc.) have different underlying
mechanisms, so that an "all-encompassing" theory is inherently impossible.
However, at the present expanding rate of effort in this field, major break-
throughs can be anticipated in the next 10 years or so.
1175)
Modern dislocation theories of plastic flow stem from the papers of
Orowan(1174) Taylor(1575) and Polanyi( in 1934. They de-
) 1272)
scribed whatis now known as an edge dislocation, which is characterized by
the fact that its dislocation line, which separates the slipped portion of the
crystal from the unslipped, is normal to the direction of slip. Depending on
the orientation of the edge dislocation, they may be called either positive or
negative. Another type of dislocation, known as the Burgers or screw dis-
location, has the feature that its dislocation line is parallel to the direction
of slip. Excellent review articles are available on the nonmathematical as-
pects of dislocations (see Appendix C), numerous papers have been published
with a mathematical approach to the problem (see Appendix C), and many
symposia held by specialists in the field of dislocations have been printed
(see the six reports listed in "Detailed Plan of the Litei'ature Search").
Earlier contributors to this field include Orowan, Mott, Nabarro, Frank,
W. T. Read, Jr. , and Cottrell. More recent workers are Seeger, Schoeck,
Friedel, Leibfried, Hirsch, and Eshelby, among others.
For the purposes of this literature survey, only the most important theo -
ries, in the light of present-day knowledge, will be reviewed here. \For con-
venience, they are considered under the headings of theories that peitain
primarily to strain hardening, alloy hardening, precipitation hardening,
and creep. Although this may seem to be a rather arbitrary classification,
it is not intended as a hard-and-fast separation of distinct theories. Rather,
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it is found that these theories overlap and duplicate one another in varying
degrees. The basic features or mechanisms, such as intersection of dis-
locations, jog formation, pile-up at barriers or obstacles, dislocation
climb, and stacking faults, may appear in a few or all of the theories in
the above-listed categories.
Strain Hardening
With increasing amounts of slip, further deformation becomes more
and more difficult (i.e., the yield stress rises); this is strain hardening.
The earliest quantitative dislocation theory that applied to strain-hardening
phenomena was proposed by G. I. Taylor(I575). The Taylor theory is dis-
cussed here in some detail because it has introduced into the theory of
plasticity several concepts and mathematical methods that are of great im-
portance, independent of the particular model of strain hardening to which
they were first applied. Also, Taylor's model, although not giving a
realistic picture of strain hardening, does contain the basic elements that
are responsible for the additional hardening due to obstacles such as grain
boundaries and inclusions. His idea was that the yield stress depends on the
internal stresses opposing the movement of dislocations and that the dis-
locations create the stresses during plastic deformation. Taylor suggested
that the dislocations do not pass completely through a crystal but interact
elastically with one another and with obstructions such as mosaic boundaries,
which prevent further motion. These immobile dislocations cause the in-
ternal stresses that raise the yield strength. Since plastic flow results from
the movement of dislocations, any mechanism that causes a decrease in the
mobility of dislocations causes work hardening. An expression relating
stress and strain was derived for the case where slip begins at random
points throughout the crystal and occurs by the separation of one positive and
one negative, dislocation at each of these points. (However, no separation
occurs if the dislocation can arise at a surface or even an interface.) If the
average length along the slip plane through which the dislocations move
apart is L, and if the yield stress is assumed to be the same as the highest
internal stress acting upon a dislocation when it is set in motion, then
= aGOn1/2,
L
(1)
where cr is the tensile stress; a. is a constant; G is the shear modulus; y is
the plastic strain; and b is a vector (called the Burgers vector) that specifies
the direction and distance by which atoms above the slip plane have moved
relative to those below. This parabolic relation between stress and plastic
strain was confirmed experimentally for metals crystallizing in the cubic
system. L was hypothesized to be of the order of 10-4 cm, in agreement
with the order of magnitude of faults found in metals and rock salt. The
crystallographic nature of the faults is immaterial from the point of view of
the theory. However, in Taylor's theory, no provision is made for
1
11
crystalline deterioration or the development of imperfections. It is a pure
barrier theory; thus, if the strain is reversed, the dislocations should go
back to their source, and there should be no strain hardening. This is con-
trary to observation. Further detailed analyses and criticism of this theory
have been presented by Orowan(1189) and Cottrell(293).
Although Taylor's theory is no longer accepted, some of the general
ideas are basic to later theories. Mott(1085) suggested a theory based on
piling up of edge dislocations against obstacles in their slip plane. These
obstacles were thought to consist primarily of sessile dislocations (a type
of fixed dislocation that can move only by the transport of atoms to or from
the perimeter of the fault by diffusion) randomly distributed in the crystal.
At each obstacle, a group of n dislocations of the same sign is piled up and
anchored by combining with a few dislocations on an intersecting slip system.
The stress field of each group acts through large distances, n times farther
than that of a single dislocation. An expression for the internal stress,
on each group of dislocations is derived, which, when combined with that for
the plastic strain, gives a parabolic hardening law
= (G/Z?Tr) (n-yb/L)1/2 , (2)
where the terms are as before. A value for L, the spacing of the obstacles,
can be obtained by postulating a dynamic generation (later discarded) of dis-
location loops, from a source of length 2, which move a distance L. This
gives
_
o-? = G(-y13/2.7r2)1/2? (3)
The following characteristics of this result may be seen. When 2 is
constant, a parabolic stress-strain curve is obtained; with a reasonable
value of 2 (about 10-4 cm), the resulting coefficient of work hardening is of
the right order of magnitude; and the stress-strain relation is independent
of L, which is one df the more serious deficiencies of the theory. Even
though the intensity of strain hardening is roughly the same in many crystals
of different orientations, strain hardening is probably mainly due to the
decrease of the spacing between obstacles.
Mott's theory resembles that of Taylor, but differs in several respects:
the newer theory pictures the dislocations as being locked; the strength of
each group of dislocations is n times larger; and the spacing between them
is correspondingly larger.
Other theories have contained alternative explanations of work harden-
ing(173,218,815,911), but they either have been unable to stand the test of
time, or the number of assumptions introduced has prevented their accept-
ance. However, two rather comprehensive reports have emerged lately,
which attempt to give a more unified picture of the present status of ex-
periment and theory. These are the papers by Orowan(1189) in 1954 and by
Seeger(14Z5) in 1957. Orowan points out the generality of the phenomenon
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of strain hardening, and the attendant uncertainty as to its cause. He points
out that a number of factors can contribute to strain hardening, and that the
pure barrier pile-up theories cannot account for more than a small fraction
of the hardening. In the case of polycrystalline metals, strain hardening
can arise from the mutual interference of neighboring grains. However,
the mechanism for intracrystalline hardening, e.g., in single crystals, is
by no means clear.
A mechanism of strain hardening was first outlined by Taylor, and
this has been treated above. Another possibility of strain hardening, dis-
cussed by Orowan, is that arising from the removal or inactivation of lattice
defects that make the crystal soft. This possibility is not too probable.
Without the presence of a multiplication mechanism, a crystal would harden
extremely rapidly as the dislocations present initially were used up by slip.
However, the number of dislocations needed for producing large amounts of
plastic deformation is far too high to be present initially in the crystal. An
alternative scheme embodies the idea of an initial stock of Frank-Read
sources (dislocation double mills) with different widths. As deformation
proceeds, the double mills with widest spacing (and correspondingly lowest
operating stresses) are used up first, and the mills with smallest spacings
would go last. Although this mechanism for the origin of strain hardening
cannot be ruled out completely, it is not entirely satisfactory.
More probable is the creation of lattice "injuries" acting as obstacles
to slip in the course of deformation. One way of producing lattice imper-
fections other than dislocations during slip is the generation of vacancies;
similarly, excess (interstitial) atoms may come into the lattice. Twin and
kink(1180) lamellae may, and in general will, arise locally, particularly
in a polycrystalline material. Other mechanisms of hardening are the
cutting of dislocations by slip taking place in a plane pierced by them, and
the intersection of stacking faults by crossing slip planes. These latter two
mechanismth may play a more significant role in the hardening process than
the others, so they will be described in more detail later.
The lack of satisfactory explanations for multiplication mechanisms
of dislocations was one of the major problems in the theory of plasticity.
Either new dislocations must be created during slip, or a dislocation must
be able to produce large amounts of slip. This difficulty was circumvented
when it was realized that a dislocation loop need not lie entirely in one plane.
Whenever a dislocation loop does not lie wholly in one slip plane, any part
of it that lies in a certain slip plane may sweep over the plane many times
without disappearing. Sources that can supply many dislocations have been
called dislocation mills. Various types of dislocation mills are described
by Orowan, including the slip-deflection mechanism and the "Z-mill", a
dislocation sheared through the middle, which resembles the letter Z. The
present status of the problem may be summed up in the following way:
Most crystals of moderate size contain flaws from which slip can start at a
low stress. Once slip has started, there are numerous ways, topological
or dynamical, in which new dislocations or dislocation mills can arise.
13
The principle of strain hardening due to dislocations cut across by
slip may be illustrated in a simple way. Consider two blocks sliding upon
one another. The surface of contact is the slip plane. Now, an elastic
thread (dislocation) is imagined to pierce both blocks perpendicular to the
slip plane, and, if slip occurs, the upper and lower halves of the thread will
be displaced correspondingly. The connecting portion of the thread which
lies in the slip plane represents the connecting dislocation between the ends
of the two segments. The tension of this connecting dislocation increases
the shear stress required for further slip. If the number of immobile dis-
locations traversing a slip plane is n per unit of its area, their intersection
gives rise to an additional shear stress n times greater than the force
needed with one. Since the number of dislocations traversing any slip plane
increases in the course of plastic deformation, the yield stress must also
rise, i. e. , strain hardening must occur.
If the mechanism by which n increases with increasing deformation
were known, and a relationship between n and the plastic strain could be
established, the rate of strain hardening due to the intersection mechanism
and the corresponding stress-strain curve could be calculated. However,
ak present, any such calculation would require a number of highly arbitrary
assumptions that could hardly be justified by any agreel-nent of the result
with observed stress-strain curves.
The intersection mechanism can explain the difficult problem of why
latent slip planes, intersecting the operative slip zones, harden before they
would be expected to. Another phenomenon explicable on the above basis is
the Bauschinger effect in single crystals, although it now appears to be due
to other causes. Another mechanism of hardening, pointed out first by
Seitz(1443) and also by Nabarro(1113), deals with the mechanism of inter-
section of screw dislocations, which produces strings of vacancies or of
interstitial atoms.
A newer mechanism of strain hardening, which may be of considerable
imi)ortance in heavily deformed face-centered-cubic crystals, involves a
type of defect known as a stacking fault*. The contribution of stacking faults
to the general theory of slip, and in particular to that of strain hardening,
is believed to be of great importance. A stacking fault, as its name im-
plies, is merely an irregularity in the sequence of stacking of octahedral
planes in a close-packed-cubic crystal. Calculations show that the extensive
development of stacking faults can occur only at fairly high applied stresses,
in a strongly distorted state of the crystal. The effect of intersecting slip
upon a stacking fault is to generate two mixed edge-screw dislocations, which
move apart as the intersected edges of the fault move apart by slip. No slip
across the fault can occur until the shear stress is high enough to produce
these two dislocations and move them apart. This process probably requires
the piling up of dislocations at the stacking fault before it can be broken
through.
*W. T. Read, Jr., Dislocations in Crystals, McGraw-Hill Book Company, Inc., New York (1953), 110-113.
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IT a strongly cold-worked face-centered-cubic crystal contains several
intersecting sets of stacking faults, it will be strain hardened in a way that
resembles Taylor's theory of hardening. The difference is that the obstacle
walls are not pre-existent, but are generated by slip, and that they are not
impermeable to slip, but can be penetrated at a sufficiently high stress.
Furthermore, it is not necessary to assume the piling up of dislocations in
the cells between the fault walls, as in the Taylor theory; intersecting walls
of stacking faults could produce internal stresses high enough to explain the
observed yield stress. It should be repeated, however, that hardening by
the intersection of stacking faults is likely to be important only in strongly
deformed face-centered-cubic metals. Under these conditions it may be the
dominant factor in hardening.
An ambitious attempt to explain theoretically the dislocation processes
governing the plastic properties of close-packed metals was undertaken by
Seeger(1425) in 1957. He stated that the present state of the theory would
permit only a semiquantitative interpretation of the experimental facts.
Furthermore, there would be no attempt to derive the stress-strain curve
of a crystal from first principles because of the complexity of dislocation
patterns and behavior. Rather, the goal of devising a theory of cold work
and work hardening would be directed toward finding the fundamental proc-
esses hidden behind the individual behavior of crystals.
The first step in a theory of work hardening is to develop a theory of
flow stress, i.e. , to determine the resolved shear stress, T, under which
extensive plastic flow in an unstrained crystal begins. The next step is to
determine how the dislocation arrangement changes with strain, and, in
some cases, with time. Combining this with the theory of flow stress gives
the slope 0 =.dT/de of the stress-strain curve as a function of strain, c, and
strain rate, c; i. e. ,
dr/dc = 0 (c, ) .
(4)
The equation of the stress-strain curve, T = T(c), for a given strain rate is
obtained by integration. Equation (4) is considered to be the fundamental
relation of the theory of work hardening. It should be noted here, however,
that Equation (4) is incorrect because the slope of the stress-strain curve is
not determined by the instantaneous values of strain and strain rate. Also,
integration of Equation (4) cannot give the equation of the stress-strain curve.
The program outlined above by Seeger cannot be carried through com-
pletely because of insufficient knowledge. However, the individual features
of the dislocations in various metals, even though different, can be rational-
ized, at least for the face-centered-cubic metals. This is done in terms of
the magnitude of a single quantity, namely, the stacking-fault energy, I/.
The dislocations with which we are mostly concerned are "extended" ones,
and the degree of extension is determined by a quantity containing y. The
stacking-fault energy, y, is defined as the surplus free energy of a stacking
fault of unit area. Theoretical arguments based on the increase in energy
n T Part Sanitized C
15
of the conduction electrons and measurements of the energy of coherent
twin boundaries give order-of-magnitude estimates of the stacking-fault
energy and show rather good agreement.
The "surface tension" of the stacking-fault energy opposes the elastic
repulsion between the partial dislocations and binds them together at a finite
equilibrium distance. The dimensionless parameter that determines the
equilibrium extension is ?yc/GbZ, where G is the shear modulus, c is the
separation between neighboring glide planes, and b is the distance between
neighboring atoms in the glide plane. If the parameter is larger than 10-2,
the metal is considered to have a high stacking-fault energy (e.g., alumi-
num) and the partials are close together; if less than 10-2, the stacking-
fault energy is considered to be low (e.g., copper), and the partials are
well separated.
Next, Seeger turns to an account of the principal experimental results
on stress-strain curves of face-centered-cubic single crystals. The results
.9
can be summarized in terms of three separate regions of the stress-strain
curve. There may, or may not, be a region of easy glide (Stage I), de-
pending on the crystal orientation and the impurities present. Most charac-
teristic for these crystals is the stage of rapid work hardening (Stage II),
which follows the easy-glide region. In this region, the ratio of the work-
hardening coefficient to the shear modulus is practically independent of the
applied stress and temperature, not very dependent on the crystal orienta-
tion, not sensitive to the impurity content, and of the same order of magni-
tude for all face-centered-cubic metals. The understanding of the rapid
work hardening in this stage seems basic to an understanding of the plastic
deformation of face-centered-cubic metals. At larger strains (Stage III),
the slope of the stress-strain curve diminishes with increasing strain. The
stress, T3, at which this stage begins depends markedly on temperature and
on a kind of "dynamic recovery" ? a term coined by Diehl to emphasize the
effects of temperature during deformation.
The explanation of the easy glide observed in Stage I is that no Lomer-
Cottrell dislocations, a particular form of immobile dislocation, are formed
in some crystal orientations, permitting unperturbed easy glide. Easy glide
is assumed to take place primarily in only one glide system, but there may
be some dislocation movement in other glide systems. Most dislocations
during easy glide slip out of the crystal and only a small fraction is held to
contribute to work hardening. The Lomer -Cottrell dislocations act as
obstacles to the glide of screw dislocations. In other orientations of the
crystal, the resolved shear stress in certain secondary glide systems is
large enough to result in some Lomer-Cottrell dislocations being formed,
and, with increasing plastic deformation, the slip distance in all directions
is limited by Lomer-Cottrell dislocations. Stage II, rapid work hardening,
is fully developed if Lomer-Cottrell dislocations of all (three) types possible
in the primary glide system are formed in significant numbers. The
linearity of Stage II is explained as a result of continued formation of
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Lomer-Cottrell barriers. This gives rise to a decrease of the slip dis-
tance with increasing strain and to small dislocation groups of essentially
constant size, piled up against the barriers and distributed randomly. The
geometrical arrangement of the groups is thought to be more or less
statistical, in a pattern such that groups of opposite sign stabilize each
other to some extent. As a consequence, the flow stress is proportional
to the density of dislocations intersecting the glide plane. It should be noted
here that the very fact that stacking-fault formation does not occur in this
picture shows how very special the assumed mechanism is. In all
probability, Lomer-Cottrell dislocations are only one of many types of
obstacle produced by deformation.
The general picture of the dynamical recovery occurring in Stage III
starts with the dislocation arrangement built up in Stage H (which is essen-
tially independent of temperature and strain rate). At a sufficiently high
stress and temperature, the dislocations can undergo processes that had
been suppressed at lower stresses and temperatures. For example, they
may allow the dislocations to climb around obstacles that held them up in
Stage II; another possibility is that these processes enable some disloca-
tions of opposite sign to annihilate each other and to reduce the internal
stress fields. This results in a reduction of the work hardening.
Although the work-hardening coefficient decreases, this effect is
merely one of superposition on the continuing hardening process from
Stage II. The flow stress is found to be composed of the temperature.
dependent contribution, Ts, of the dislocation forest and the temperature-
independent contribution of the stress fields of the dislocations lying in the
primary slip plane, TG. The increase in the flow stress, i. e., the hardening
of the primary slip systems, is mainly due to the increase in TG, rather than
in Ts.
This general discussion of the stress-strain curve, although couched
primarily in terms of face-centered-cubic metals, is applicable to hexagonal
close-packed metals and alloyed single crystals, when their specific
characteristics are considered. Although instances of discrepancies from
this general treatment may arise, they can be accounted for, except when
the data are as yet too limited to permit any clear-cut explanations.
Seeger's picture drawn for the hardening mechanism still lacks certain
details. They can best be filled in by theoretical and experimental studies
of additional properties of the deformed crystals.
Alloy Hardening
So far, the most promising approach to an understanding of hardening
in alloys is one employing dislocation models. The "hardening" considered
in this section will include that due to alloying in single-phase solid solu-
tions and not that due to particle effects in polyphase alloys. Actually,
fl-Iiifir1 in Part d Coov AP
17
depending upon the degree of dispersion, these two cases can be treated from
a common point of view, and this will be done whenever applicable.
There is at this time no theory that will permit, a priori, the pre-
diction of mechanical properties of an alloy; the available theories are
actually rather primitive, since-direct observation of dislocations is still
rather limited. Theories cannot predict, nor explain fully, the effects of
temperature, stress, or chemical species on strength properties. How-
ever, there are certain general trends, confirmed experimentally, available
to guide theoretical developments. These behavior patterns may be listed
as: (1) the addition of a second element to a pure material almost always
increases its strength; (2) the strength increases with increasing amount of
the added element, then may decrease later; and (3) the scale of dispersion
of a second phase has an effect on the strength of an alloy.
The more important dislocation theories of alloy hardening may be
considered on the following basis:
(1)
Those theories that assume that solute atoms cause internal
stress in the matrix, and that hardening is due to the
resistance that these stresses offer the movement of
dislocations
(2) Theories that are based on interactions between solute
atoms and dislocations
(3) Theories in which the passage of dislocations is hindered
by discrete aggregations of atoms, inclusions, or
particles.
The basic treatment upon which theories assigned to Section (1) above
are based is that due to Mott and Nabarro(1 100), as modified later by
Mott(1082). The case considered is that for which the wavelength, A, of the
internal stress field, cri, is small compared with the limiting radius of
curvature of the dislocation. The local stresses around dissolved atoms
have a "wavelength" A - c 1/3 where d is the interatomic spacing and c is
the solute concentration. In this case, the dislocation is hardly bent by
these local stresses, so that the appropriate average value for 1:1.e internal
stress turns out to be
o-i Gec log (1/c) ,
(5)
where c is the strain around the dissolved atoms, and the other terms are
as before. The length of the dislocation line is estimated from the relative
magnitudes of the amplitude of nZ loops of the dislocation line and the
wavelength of the stress field. The positions of equilibrium of the disloca-
tion line occur at intervals of about A in the solid solution. From statistical
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considerations of the force on single loops of the dislocation line, an equiva-
lent uniforria stress, crijn, is derived for the dislocation line. This stress
should be the zero-point yield strength, a-, of the material. Substituting for
cri and n, the approximate expression is obtained:
2.5 Ge4/3
(6)
for solutions with solute concentrations varying from 0.001 to 0.01.
The equation predicts a stronger solution-hardening effect than is ob-
served experimentally, and this may be due to an underestimation of the
length of the coherent piece of dislocation. In their earlier treatment, Mott
and Nabarro assumed a much greater length, which leads to a smaller yield
strength
(7)
There is some evidence that the hardness of polycrystalline solid solutions
changes as the square of the change of lattice spacing, but this relationship
is not always clearly defined. Some of the disagreement between theory and
experiment can be explained on the basis that a valency effect exists in
addition to the size-factor effect. Thus, it has been shown that solute atoms
whose valencies differ from that of the solvent have a greater hardening
effect, for the same size factor, the greater the difference in valencies.
Theories pertaining to Section (2) above are based on interactions
between solute atoms and dislocations. There are at least four important
types of interactions that are found between solute atoms and dislocations.
They are called, after Cottrell(295), elastic interaction, electrical inter-
action, chemical interaction, and geometrical interaction. When a solute
atom replaces a solvent atom in a substitutional solution, or enters a
vacant site in an interstitial solution, the hole it enters may be the wrong
size or shape. Work must be done to alter the dimensions of both the hole
and the atom; and, if a stress field alters the dimensions of the hole, the
work done in inserting the atom may also be altered. Calculations of this
interaction for the case where the dislocation provides the stress field
have been made for several cases(110,933).
Mobile solute atoms will drift, in the presence of an inhomogeneous
field such as the stress field of a dislocation, toward those places where
they have lowest energy. The segregation produced in this way is an
equilibrium state of distribution for the solute atoms, since its formation
involves a lowering of the free energy of the system. One of the more im-
portant theories stemming from the segregation of solute atoms around dis-
locations is the theory of the sharp yield point due to Cottrell(286).
19
Under an external force a dislocation surrounded by an atmosphere of
solute atoms starts to move and leave the latter behind. A force is then
exerted, pulling the dislocation back to its original position at the center of
the atmosphere. If the applied force is less than this anchoring force, the
dislocation cannot move. The dislocation can move, however, if the
atmosphere moves with it, but only at a speed equal to the speed of migra-
tion of the solute atoms in the atmosphere.
In many cases the rate of straining is too high or the temperature is
too low for the atoms to keep up with the dislocation. However, to produce
plastic flow, the dislocations must first be pulled away from their atmos-
pheres, and to do this the applied force must exceed the anchoring force.
Because of the strong affinity between a dislocation and its atmosphere, this
applied force will be larger, in many cases, than the force needed to keep
the dislocations in motion once they have escaped from their atmospheres.
The rAterial can then exist in either of two condition ? the strain-aged
condition or the overstrained condition. In the former case, the dislocations
are anchored and the deformation is purely elastic, whereas in the latter
case, the dislocations are free to move. The conversion from the first to
the second condition coincides with the beginning of plastic deformation in
the material. When the dislocations escape from their atmospheres, the
force needed for movement is smaller, so the material suddenly becomes
softer and yields under a decreasing stress .at the beginning of plastic
deformation. This is the explanation of the phenomenon of the sharp yield
point observed in body-centered-cubic metals with certain impurities,
notably carbon or nitrogen.
The theory has been applied quantitatively to the yielding of iron con-
taining carbon or nitrogen(302), and has stimulated the investigation of
yielding in other metals. In spite of several modifications to the theory(289),
some discrepancies are apparent, as Orowan has pointed out(1189), and
some experimental contradictions have arisen (see discussion to
Reference 289). However, the basic ideas appear to be widely accepted and
applied to many diverse phenomena. Schoeck and Seege.r, in a paper pre-
sented at the Fall Meeting of the Metallurgical Society of the AIME, in
October, 1958, proposed an alternate mechanism of interaction based on
the Snoek effect, which had been applied to the yield phenomenon previously
by Nabarro(1109) and Crussard. Here, the interstitial atoms jump to
neighboring sites that arc now preferred because of the stress field of the
dislocation. These jumps can take place in times that are orders of magni-
tudes smaller than it would take to form Cottrell atmospheres. Excellent
experimental verification is claimed for the magnitude, temperature de-
pendence, and concentration dependence of the flow stress calculated from
this model.
*C. Crussard, Metaux et Corrosion, 25 (1950), 203.
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Electrical interactions, between a solute atom and a dislocation, are
known to play a strengtheniAg role in metals, and calculations have been
made of the interaction effects(306). A comparison of electrical and elastic
interactions shows, in alloys of copper, that the latter are 3 to 7 times
larger than the former. So, unless the effective charges on solute atoms
are substantially larger than the values used in the calculation, the elec-
trical interaction is relatively unimportant, except possibly at the center
of a dislocation.
Chemical interaction is the name of the mechanism suggested by
Suzuki(1546) to account for the heterogeneous distribution of solute in faulted
and unfaulted regions. The heterogeneous distribution of the solute atoms
results in a hardening atmosphere, similar to the Cottrell atmosphere in
body-centered-cubic metals. In a face-centered-cubic solid solution,
thermodynamic equilibrium will in general require the concentration of
solute to differ from that in the close-packed hexagonal structure of the
faulted layer. From consideration of the free-energy curves of the matrix
and the faulted material, and assuming that the proportion of the phases
remains fixed, the equilibrium concentrations can be found. Cottrell has
extended Suzuki's theory by removing the constraint on the proportions of
the phases. Further experimental verification of Suzuki's theory has been
obtained by Hibbard(683) and Flinn(451). His theory appears to be an im-
portant extension of the ideas on interactions of dislocations.
The fourth type of interaction with dislocations is called "geometrical
interaction", and is associated with the presence of a dislocation in an
ordered alloy. The interaction is a result solely of the slip displacement
caused by the dislocation. In a crystal with long-range order, a unit per-
fect dislocation in the lattice of atomic sites is oily a half-dislocation in
the superlattice. It must therefore always be attached to a stacking fault
in the superlattice, i.e., to an antiphase domain boundary. The fault has
surface tension due to the energy of its "wrong" bonds and it exerts a force
on the dislocation line at its boundary. Quantitative aspects of the hardening
expected from this source have been discussed by Ardley and Cottrell(43).
For the /3-brass superlattice, the stress to move a dislocation would have
to be about 3 x 109 dynes/cm2.
The motion of dislocations through short-range ordered solutions has
been discussed by Fisher(435). Since there is no superlattice in these, unit
dislocations in the lattice of atomic sites are perfect and there are no long-
range faults to pull these dislocations in pairs. However, because the dis-
tribution of neighboring atoms is not random, the passage of a dislocation
along a slip plane will destroy the short-range order between the atoms
across the plane. As in the case for long-range order, the stress, T, to
move the dislocation is given by
T = yib
(8)
21
where -y refers to the energy, per unit area of the plane, associated with the
degree of order across it, and b is the length of the Burgers vector.
Although -y is smaller for short-range than for long-range order, in an alloy
with intense short-range order the stress to move dislocations can approach
the magnitude of 108 dynes/cm2. Both types of ordering should play a
definite role in the development of a theory of hardening, but the magnitude
of their effects is not expected to be too important.
Precipitation Hardening
The third major subdivision of dislocation theories pertaining to
alloy hardening deals with precipitation hardening, or the hardening
associated with hard, discrete particles. Two papers by Mott and
Nabarro(1100,1101) have provided a detailed treatment of solid-solution
hardening and precipitation hardening, i. e. , of cases where the solute
atoms are dispersed at random, either singly in solution or as clusters in
precipitates. Later, these authors emphasized that the dislocation line
passing through the alloy is flexible(1074,1107); it is akin to a stretched
string, which vibrates and radiates elastic energy as it moves and is
"plucked" by the solute atoms as it passes. The amount of energy dissi-
pated by the plucking process depends on the distribution of the solute
atoms and increases when they are clustered together, as in precipitates.
Orowan pointed out* that, when the distance between clusters increases
beyond a certain range, it becomes possible for sections of the dislocation
line to bulge between neighboring clusters and then expand into the slip
plane on the other side, thus bypassing the clusters. For a fixed compo-
sition, there is thus a critical distance between clusters at which the
hardening is greatest, as is observed in precipitation-hardening alloys.
Theory gives the distance as typically about 50-atom spacings, which is
reasonable.
The approach. adopted by Mott and Nabarro in their theory of disper-
sion hardening was based on residual stresses. The stresses are con-
sidered to arise from the misfit of the precipitate in the matrix and the
resultant elastic accommodation. The dislocations producing slip must be
pushed through those regions in the slip plane where the stress is adverse
to their motion. They estimate the average magnitude of the stress and
identify this with the yield stress due to the precipitate.
The value of the stress/ crl is
a.= Gef ,
(9)
*E. ?rowan, Discussion in Symposium on Internal Stresses in Metals and Alloys, Institute of Metals, London
(1948), 451-453.
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22.
where G is the rigidity modulus, c is the resultant strain in the precipitate,
and f is the volume fraction of the solute atoms in the alloy. The implica-
tion in their original theory is that the dislocations may pass completely
through the precipitate particles, but later work favors the idea of flexible
dislocations, mentioned above.
For the case where the radius of curvature of the flexible dislocation,
p, is approximately equal to the mean spacing of the precipitate particles,
A, the argument is that each segment of the dislocation of length A must be
separately forced through the local adverse stress regions between the pre-
cipitate particles. At this stage of hardness, the relation describing the
critical dispersion is given as
A 13/c f .
(10)
Several criticisms can be directed at this theory in its present
state(645). In the first place, this method would require that the strength
remain high at the value Gef, even when the spacing exceeded the critical
spacing. This is contrary to the customary behavior. Furthermore, this
residual-stress theory is too qualitative, since it is rather difficult to
assign values to the parameters. Also, Equation (9) must be judged deficient
from the value of c which is necessary for a sufficiently large yield stress at
maximum hardness.
?rowan* proposed a particularly simple and straightforward method
for computing the yield stress due to precipitates in terms only of the
spacing of the particles. He considered that dislocations would be held up
by internal stresses at precipitate particles and that it would be necessary
to force them between the particles. Loops of dislocations would then be
left encircling the precipitate particles as the dislocation line moved past
them. The yield stress estimated from this model is
cr = 2 Gb/X ,
where X is the spacing between two particles (or obstacles) and the other
terms have the same meanings as before. Orowan concluded that, since
X increases with increased aging time, this might explain the decrease in
yield stress of overagecl alloys. Equation (11) is the same in form as that
developed by Mott and Nabarro; however, the assumptions and interpreta-
tions are distinctly different. One important distinction is that the A of
Equation (10) represents the length of an independently mobile dislocation
segment, and belongs essentially to a two-dimensional picture, whereas
the X of Equation (11) is the spacing between particles in three dimensions.
Orowan's treatment implicitly denies the validity either of Equation (9) or
of its interpretation as the yield stress and replaces Equation (9) with
'E. ?rowan, loc. cit.
?
23
Equation (11) entirely. However, t:hc applicability of the Mott-Nabarro
criterion of a dislocation being forced across a region of adverse internal
stress (which is really G. I. Taylor's original idea) is not denied. At
sufficiently high internal stresses, the Taylor -Mott-Nabarro criterion
(yield stress = adverse internal stress) would apply, although not in the
form of Equation (9).
A minor extension to ?rowan's theory was suggested by Geisler to
the effect that coherency strains around particles make the effective
spacing smaller. ?rowan's theory can then be applied over the entire
hardening range, instead of only in the overaging range. The drop in yield
stress at early aging times is explained on the basis that X is larger because
of the smaller amount of precipitate available. The calculated yield stress
at maximum hardness agrees in order of magnitude with that observed.
Also, the yield stress in the overaged region is of the same order as in
the solid solution, as would be expected.
Another theory of dispersion hardening was advanced by Fisher, Hart,
443). However, in this case, they are concerned with incremental
and Pry(
strain hardening, and not the yield strength, as in the previous two theories.
They show that an appreciable hardening increment is obtained, as a result
of strain, due to the stresses from trapped dislocation loops around
particles, as originally described by ?rowan.
The loops encircling the precipitates are shown to exert a shear stress
on the rest of the surrounding slip plane, which stress is of opposite sign to
that which caused the slip. By certain simplifications, the final expression
for the maximum value of the stress increment is obtained,
Th = KTc f3/ ,
(12)
where K is a constant, Tc is the limiting shear strength of either the pre-
cipitate or the matrix, whichever is weaker, and f is the volume fraction of
the precipitate. .Some agreement with experimental results was claimed,
but actually the data used, for particle radii and volume fraction, were cal-
culated and not measured. Under its present state of development, this
theory seems in need of further clarification.
Creep
Creep may be defined as the plastic deformation that can occur under
a constant applied stress. It is an important part of any study of plastic
deformation, and is not, as sometimes thought, confined only to high tem-
peratures. In discussing theories of creep, we will be concerned primarily
with dislocation theories that explain macroscopic behavior in terms of the
structural processes on an atomic scale. There are two main processes
to consider: transient creep and viscous creep. The former is a
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decelerating component, the rate of which depends on the applied stress and
disappears with time; the Litter is a constant-rate process that depends
primarily on the applied stress. This important distinction was made clear
by Andrade(22) in 1914. The term "steady-state creep", however, is
ambiguous in that either viscous or transient creep may appear to be a
steady-state creep. Normally, the term "steady-state creep" is meant to
apply to viscous creep.
The basis upon which all formal theoiies of creep have been built was
laid down by Becker(89), starting in 1925. Becker's purpose was to use
thermal stresses for explaining the discrepancy between the high theoreti-
cal value and the low observed value of the yield stress. The contributions
of Orowan(1182) led to the present activation theory of transient creep, and
the main ideas embodied in this treatment are:
(1) At the beginning of creep, the applied stress is equal to
the yield stress, so the activation energy is vanishingly
small. This is the reason for an infinitO.y high initial
creep rate, even at the lowest temperatures.
(2) The flow rate during creep is limited because thermal
fluctuations are needed to supply the difference between
the applied stress and the yield stress.
(3) As the creep strain increases, the yield stress also in-
creases, rising progressively farther above the applied
stress, and the activation energy increases accordingly.
The increasingly large thermal fluctuations that are thus
needed cannot be accomplished as frequently as the small
ones that were sufficient earlier, and the rate of flow
slows down. If a stage is reached where the yield stress
no longer rises, the activation energy becomes constant
and steady-state creep is observed.
Becker proposed the following formula to relate the creep rate, E
with the stress, a-, and temperature, T:
Cexp
[ V(cro-g)2
2GkT
(13)
where C is a constant, go represents the theoretical yield stress of the
crystal, and V is the volume in which a stress fluctuation occurs.
Equation (13) does not represent viscous creep; Becker emphasized very
sharply that this is an entirely different process of atomic rearrangements
("amorphous plasticity"). For this reason, it cannot explain the transient
component of creep, the rate of which falls from infinitely large to zero in
the course of time at constant stress and temperature. The deceleration
occurs because of strain hardening or some other progressive change of
the material during flow.
^
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Orowan(1182) introduced strain hardening into Equation (13) by
assuming that go increases during deformation by an amount OW, which is
a function of the strain, C; the revised formula gives
exp [ V(a- 0 + 0(c) - qa-)2
2GkT
where q is'a stress-concentration factor. Equation (14) was subsequently
found to be unsatisfactory, and this was attributed to the failure of the
"mechanical equation of state", of which Equation (14) is a special case.
At the present time(1191), the mechanism of creep is held to be as
follows. Although the details of the mechanism are not yet clear, there is
no doubt that transient creep is a consequence of thermal vibrations, super-
imposed on a sufficiently high applied stress, which result in slip. During
further creep, the material hardens and thermal vibrations are then less
and less frequently able to produce local slip; this is the cause of the
gradual disappearance of transient creep. The applied stress must always
be high enough to cause at least a small amount of sudden plastic strain
before transient creep can be observed ? and this is why transient creep
occurs even at the lowest temperatures. Then, if the applied stress is
sufficient to cause slip without any thermal contribution, additional local
slip can occur, at points where the applied stress is not quite high enough,
by means of very slight thermal fluctuations.
(14)
Viscous creep is believed to be produced by at least two different
mechanisms, and often the two act simultaneously. The first type of vis-
cous creep is called recovery creep. After the load is applied, the rapid
plastic deformation produces strain hardening, which raises the yield stress
to the level of the applied stress and thus resists the load. However, at a
high enough temperature, thermal recovery gradually reduces the strain
hardening. In order to carry the applied load, therefore, the material
must strain harden further until the amount of strain hardening lost by re-
covery is replaced. This means that additional plastic strain is continually
required to make up for the strain hardening removed by recovery.
The second important type of viscous creep is due to sliding between
the grains of a polycrystalline metal when a stress acts at a sufficiently
high temperature. At low temperatures, the grain boundary is a strong
part of the structure; it resists the slip in the grains. At high temperatures,
however, the boundary becomes soft and viscous and is an element of weak-
ness. Of course, this aspect of the creep process is bypassed when single
crystals are dealt with, which may be necessary for purposes of simplifying
the experimental and.theoretical problems.
Mott and Nabarro(1101) used the elastic properties of dislocations to
develop a theory of creep exhaustion ? that is, creep in which the slowing
down of the extension is due to the exhaustion of easily moved dislocations.
They point out that dislocations move in a potential field that changes during
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work hardening and stress application, and that the internal stresses that
must be overcome for plastic flow are primarily determined by the energy
a dislocation must acquire to pass a precipitate particle. They thought that
the potential field should undergo less change with strain in age-hardened
materials, which should be simpler to work with, providing, of course, that
no metallurgical changes occur during the creep test.
Their analysis indicates that the average distance between precipitated
particles is of prime importance. The dislocations will be retarded by the
particles but will advance beyond the particles in the regions between them,
giving rise to a wavy dislocation. The wavelength of the dislocation line
would be equal to the distance betweeil particles. An approximate calcula-
tion of the energy, E(cr), required to activate one loop of the dislocation, so
that it could escape from the particle that anchors it and thus move freely
forward until another particle is encountered, is given by
= C (cr)3/2 , (15)
where cri is the internal stress. The exponent 3/2 arises from the assump-
tion that the internal stress field is sinusoidal rather than constant, as was
assumed in the Becker-Orm,van theory. The creep rate works out to be
c = NaApv exp [ C(cri )3/2 I
?
kT
(16)
where N is. the dislocation density; a is the elongation per unit slip process;
A is the distance between precipitated particles; p is a multiplicity factor
to account for an avalanche of dislocations; and v is the frequency of oscilla-
tion of a dislocation loop. It was found that experimental values of yield
strength were not in accord with predictions based on Equation (16). Mott
and Nabarro concluded that the internal stress must be dependent on
temperature.
Since cri is not necessarily constant, but may represent a range of
values, the initial part of the creep curve could be accounted for by the
theory, because regions requiring low stresses to activate dislocations
would gradually be exhausted as the strain proceeded. The exhaustion
theory of transient creep based on this concept gives
c = CT 2/3 (ln vt)2/3
where E is the strain and t is the time.
(17)
The exhaustion theory was further developed by Mott(1082) and com-
pared with the results of Davis and Thompson(339) on precipitation-
hardened polycrystals of copper with silver. The data conformed closely
with the time law expressed in Equation (17), but a serious discrepancy
was noted in the value of v, the frequency of vibration of a loop of dislocation
against its obstacle. Orowan(1188) pointed out the theoretical insufficiency
s_
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27
of the exhaustion type of theory, which was later recognized by Mott(1086),
who said, "...following Orowan, we consider that exhaustion creep, though
it may occur, will mainly be included in the instantaneous extension: the
phenomena observed...by Davis and Thompson occur because creep is
slowed down due to work hardening".
With the further development of the dislocation theory, more and more
details of plastic deformation could be explained. It now seems possible(1396)
to give a detailed structural interpretation of the different processes that are
rate controlling during creep in simple structures and in the early stages of
creep. Since creep is thermally activated, those deformation processes
will be considered in which thermal activation takes place and which may be
rate controlling. With respect to dislocation mechanisms, there are three
possibilities conceivable in which this could be the case. The rate-controlling
process can be
(1)
The formation of dislocations, or
(2) Their movement through the lattice, or
(3)
Their rearrangement or annihilation in a recovery process.
The formation of dislocations has been a major problem in dislocation
theory of plastic deformation. Now, it is generally believed that dislocations
are always present and that they form three-dimensional nets in annealed
metals. The dislocation segments between the nodes of the net are potential
Frank-Read sources, which can produce a large number of dislocation loops
under a critical stress. Since thermal fluctuations over the distances the
dislocation loop has to move are extremely unlikely to occur, only those
sources can operate for which the applied stress is practically equal to the
critical stress. Therefore, the production of dislocations will generally be
an athermal process, although there are certain cases where secondary
processes, which are thermally activated, may make the formation of dis-
locations possibl from Frank-Read sources(293, 1546). Early theories
based on the generation of dislocations were proposed by Kochendc3rfer(817)
and Laurent( 9 1 n, but did not obtain popular acceptance because other
mechanisms appear more likely to be rate controlling. Their theories
have the merit that most of the important variables were considered,
whereas other theories tend to be oversimplified.
The movement of dislocations through the lattice can be impeded by a
wide variety of obstacles, and many explanations are available to explain
different experimental observations. Some of these have been discussed
previously, so will not be dealt with again.
A dislocation encounters a frictional resistance to movement, even in
a perfect lattice. This Peierls for ce(1250) is relatively slight in soft
metals and is usually neglected, although it has been tentatively identified
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dislocations described by Lomer and Cottrell( 288) cannot move and there-
fore form obstacles for the glide movement of other dislocations in the same
glide planes. There exists the possibility that these sessile dislocations,
which stop piled-up groups of dislocations, may break down under the com-
bined action of stress and thermal fluctuations, having the same effect as
a recovery process.
In the preceding sections, possible rate-controlling processes during
creep were examined. There are thermally activated deformation proc-
esses that can be rate controlling during creep; on the other hand, the move-
ment of a dislocation itself can be thermally activated; furthermore, re-
covery processes may be rate controlling. At high temperatures, viscous
creep is controlled by self-diffusion-type processes; at low temperatures,
it does not exist. At low temperatures, only transient creep exists, and its
mechanism is some kind of slip activation, without diffusion being essen-
tially involved. In the following paragraphs, some of these processes will
be examined in terms of their application to actual theories of creep, or to
specific mechanisms in the cases where theories do not exist.
The primary structural change observed during creep of pure metals
at high temperatures consists of subgrain formation. With increasing de-
formation, the angle between the subgrains increases, whereas the disloca-
tion density in the grains stays constant. In. viscous creep, the subgrains
obtain an equilibrium size and no work hardening occurs. Another feature
of viscous creep is the equality of activation energies for creep and self-
diffusion.
Dorn(376) has proposed an expression to describe the observed stress
and temperature dependence of the creep rate;
= Cexp [-Alid/kT] 0(cr)
(18)
where q5(cr) = C'e13?- at high stresses, and = Co at low stresses, and
ATrld is the activation energy for self-diffusion. Polygonization is explained
as the climb of dislocations out of their slip planes and the formation of
small-angle boundaries. If this takes place during viscous creep, the rate-
controlling process is expected to be climb, which, in turn, is dependent.
on the rate of self-diffusion.
Theoretically, the jog energy, AHi, should be included with Alld,
giving the activation energy for creep Alic = LHd + LHj. However, if LH is
small compared with (Mid, it can be neglected, and this is generally found
to be the case. Climb under creep conditions in face-centered-cubic metals
is then thought to occur in the following manner: Vacancies given off by
jogs will diffuse to other dislocation lines, where they become elastically
attached (1415) to a half dislocation. There they will diffuse along the dis-
location line with considerably higher speed until they are annihilated at a
jog. If the distance between intersection jogs is comparable to the distance
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with the low activation energy for creep in aluminum single crystals below
400 K(379). In harder crystals, however, it is in many cases the high
value of this frictional stress that makes a hard crystalline material hard.
There are essentially three other types of obstructions that are important
in hindering dislocation movement: the stress fields of other dislocations
in parallel glide planes; the forest of other dislocations that the moving
dislocation has to cut and the jogs formed thereby; and the impurity atoms
that are dragged along by the dislocation.
The stress fields in the first type above interact with one another at
fairly long range, whereas in the second type the characteristic feature is
that they exert no long-ranged force on the moving dislocation. Thermal
fluctuations in the first type of obstruction generally do not help to over-
come the internal stresses, whereas, in the second type, thermal activation
can contribute considerably toward the intersection process. Also, the
activation energy to intersect obstacles should decrease linearly with the
applied stress. The third possibility of impeding moving dislocations by
interaction with impurity atoms or point defects has been discussed pre-
viously. Movement by this process is helped by thermal fluctuations.
The discussion of rate-controlling processes during plastic deforma-
tion leads next to a consideration of the recovery process(216). In the pre-
ceding paragraphs, the movement of dislocations in their glide planes was
described. During deformation, the crystal work hardens and finally all of
the dislocations may get stopped at some obstacles. If the back stress of
the piled-up groups on the Frank-Read sources is high enough to prevent
further formation of dislocations, deformation will stop, unless the applied
stress is increased or unless recovery takes place. This recovery can be
due to a rearrangement of dislocations whereby a redaction .of the internal
stresses may be obtained or to an actual annihilation of dislocations by the
combination of two of opposite signs. For this to happen, the dislocations
must leave their glide planes, and this is done differently by edge and screw
dislocations. The edges can leave their glide planes by climb(1081) and the
screws by cross slip(
1085).
The movement of an edge dislocation perpendicular to its glide plane
is called climb. To move in one direction, the dislocation absorbs vacancies,
and to move in the other, it gives off vacancies (or absorbs interstitial
atoms). Since these defects must diffuse away, climb can be observed only
at temperatures where self-diffusion occurs with reasonable speed. On the
other hand, no diffusion is necessary for a screw dislocation to escape
from its glide plane, since it can escape by cross slip (i. e., slip from one
slip plane to another). The dislocation-climb hypothesis of recovery seems
very far from certainty. The climb process is very likely to be a factor,
but it may well be a relatively secondary factor.
Another recovery process has been proposed as a possible explana-
tion of "work softening "(308) in face-centered-cubic metals. The sessile
Jr
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over which the vacancies have to diffuse to the dislocation line, the latter
time will be long compared with the time to find a jog. The rate-controlling
process will then be only volume diffusion, and hence we find that AFIc =
AFId?
Dorn's work at lower temperatures revealed the existence of discrete
plateaus in the activation energies for creep in aluminum single crystals(379).
Over the entire temperature range of 78 to 800 K, three plateaus of activa-
tion energy were found, which were constant over a range of temperatures,
stresses, and strains. These plateaus are thought to correspond to three
unique processes, which have been ascribed tentatively to three rate-
controlling mechanisms: (1) the 35,500-cal/mole activation energy, ob-
tained at the highest temperatures, is identified with a dislocation climb
process; (2) the 28,000-cal/mole activation-energy process, occurring over
the intermediate temperature range, is attributed to a cross-slip mecha-
nism; and (3) the 3400-cal/mole process observed below 400 K may be
identified with the Peierls energy. Similar values of activation energies
were found for polycrystalline samples of aluminum, but displaced to lower
temperatures. This suggests that turbulent slip in polycrystals can block
the easier creep processes, forcing creep to continue by the more difficult
higher activation-energy processes. Undoubtedly, this is why the more
difficult creep processes are found to occur over lower ranges of tempera-
tures in polycrystalline aluminum than in single-crystal aluminum.
The most complete analysis of viscous creep based on dislocation
climb has been made by Weertman(1672) 1677, 1679). The first paper(1672)
uses Mott's mechanism of dislocation climb, and makes the assumption that
the rate-controlling process is the diffusion of vacancies between disloca-
tions that are creating vacancies and those that are destroying them. The
obstacles are assumed to be created by the Lorner-Cottrell mechanism. In
the second paper(1677), Weertman considers the case where dislocation
climb does not require the production of immobile dislocations. Instead,
the pile-up is assumed to occur between dislocations in the same slip
system, in the manner suggested by Mott(1093). Climb by the leading dis-
location of each group will lead to their annihilation; viscous creep will
occur through their continual replacement. In the third paper(1679), two
processes other than dislocation climb are assumed to be rate controlling.
In the first, the dislocations are considered to move in a viscous manner,
with their velocity of motion proportional to the force exerted on them. The
second mechanism makes use of the Peierls stress, where the motion of
dislocation lines over Peierls stress hills is the rate-controlling process.
The equations developed for these particular cases have some features
in common, with the creep rate being proportional to stress at a power of
about 3:
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Case 1 (Dislocation climb):
= C(cra/kT) exp (-411-Id/kT) ,
where C is a constant, a, = 3 to 4, and the other terms are as before.
Case 2 (No immobile dislocations produced):
= Ao-3 sinh (Bo- 1. 5/kT) exp ( -AI-Id/kT) .
Case 3a (Viscous motion):
- 3
c = b2 /icLAB ,
(19)
(20)
( 2 1 )
where b.is the length of the Burgers vector, p. is the shear modulus, A is
a temperature-dependent constant, and B is a constant.
Case 3b (Peierls stress):
D crZ? 5 exp (-46,1-1d/kT) exp (7ro- Alid kT)
where T is the Peierls stress.
(22)
Equation (19) applies particularly to face-centered-cubic metals,
Equation (20) to hexagonal crystals, Equation (21) to metal alloys at low
stresses, where a microcreep mechanism may be rate controlling, and
Equation (22) to brittle materials having a large Peierls stress (such as in
nonmetallic crystals). The results agree fairly well with the appropriate
material, especially at lower stress; for example, in Equation (19) the
relationship between E and cr is obeyed over about 3 orders of magnitude of
the stress out of 4, whereupon c increases more steeply at higher stresses.
Possible explanations for this behavior are that at higher stresses the
sessile dislocations may break down or that piled-up screw dislocations
may escape by cross-slip.
The comparison of Equation (22) with experiments on high-purity zinc
single crystals shows good qualitative agreement at all stresses; at higher
stresses the experimentally observed increase of with cr is also seen in
the theoretical curve, because of the increased importance of the exponen-
tial cr term at higher stresses.
Weertman's analyses are deficient in several respects, but these
shortcomings are not necessarily permanently fatal to the theory. Such
factors as primary creep are disregarded; the necessity for mechanical
jogs is overlooked; node formation is neglected; and the assumption of
constant Frank-Read sources is unrealistic ? it does not give the observed
stress dependence. More serious, perhaps, is that the currently fashionable
dislocation-climb process is far from having been proved, and that theories
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based on this mechanism are yet on shaky ground. Although his models need
more detailed analyses, some agreement can be said to have been attained
with experiment. As more pertinent experimental observations are made of
the actual dislocation mechanisms, more realistic assumptions can be made
in theoretical treatments.
Absolute Reaction Rate Theories
A number of creep theories have been proposed that are based on the
rate theory developed by Eyring(415) and his associates. The theory of
rate processes utilizes the methods of statistical mechanics and involves
the concept of "activated" complexes or units, and has been successfully
applied to a wide variety of chemical and physical rate processes.
Kauzmann(787) and Dushrnan Dunbar, and FIuthsteiner(392) working
independently, were the first to apply rate theory to the flow of metals. In
their theories, which are essentially identical, flow of metals is considered
to take place by the movement of flow units in a periodic potential field.
The "units of flow" are regarded as generalized elementary structures within
a solid, whose motions constitute the shear process. It is assumed that, in
order for two units of flow to pass one another, an energy barrier must be
overcome, so that the unit shear process cannot take place unless the units
of flow become activated. These theories are quite similar in final form,
the logarithm of the strain rate being proportional to the stress. The
difference lies mostly in the description of the flow units and other details of
the postulated mechanism. Only a few representative theories will be dis-
cussed here because of their similarity.
When an external shear stress is applied, Kauzmann assumes that
the activation energy for motion of units of flow in the direction of shear is
lowered by an amount proportional to the applied stress, and that in the
opposite direction is raised by an equal amount. The theory predicts a
viscous creep rate
= CT exp [-AF/kt] sinh [q.A.e cr/kT ,
(23)
where AF is the free energy of activation, q is a stress-concentration factor,
A is the area of the projection of a flow unit in the slip plane, and .8 is half
the distance between two potential minima. The term qA.e must vary with
temperature and stress in order to provide agreement between theory and
experiments. The Kauzmann theory goes no further than the Becker theory
in elucidating the mechanism of plastic flow in metals. It is worthy of note,
insofar as it presents another approach to the subject, but the interPretation
of various terms involved (e.g. , the entropy term contained in AF, and
qA.e) is vague.
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Saibc1( 1383) applied reaction-rate principles to such phenomena as
creep and fracture. He obtained expressions for the creep rate, as well as
for the rate of propagation of fracture cracks in metals. Seitz and Read(1444),
Machlin and Nowick( 1137 , 1138), and Feltham (428) applied the Kauzmann
theory to the motion of dislocations, which they specified as the units of
flow. Using theoretical reasoning similar to that used by Kauzmann, Seitz
and Read derived an equation for transient flow and obtained
= NX2/t exp (-6,H/kT) sinh (qA2 cr/kT) ,
(24)
where N is the density of dislocations cutting across a plane that is normal
to the slip plane and extends in the slip direction; X is the slip distance
associated with the passage of a dislocation; t is the time required for the
transition of an activated dislocation to pass from one equilibrium position
to the next; and /NH is the change in energy required for the unit process.
The viscous flow is given by
= A exp (-AH/kT) ,
(25)
where A is a constant. They consider the creep curve to be resolved into
these two component curves ?a transient and a viscous part. However, they
believed at that time that there was good evidence against this type of res-
olution being generally applicable. Their modification of the Kauzmann
theory does not lead to a satisfactory expression for the creep rate; also,
values cannot be assigned to some of the terms, nor can they be determined
experimentally.
The expression developed by Nowick and Machlin(1138) takes into
account the work hardeningof the lattice that has already occurred before
the steady-state stage of creep. The internal stress fields resulting from
such hardening are assumed to reduce both the effective shear stress and the
rate of generation of dislocations. The reduced form of their equation (as
well as Feltham's) for the creep rate, under reasonably large values of
stress, is
in = C + ln T - A/T + Bcr/T ,
(26)
and has the same form as that derived by Kauzmann.
A theory of fracture in creep-rupture tests, proposed by Machlin and
Nowick(1026), is also based on the application of the general reaction-rate
theory. They assume that creep to rupture is a rate process governed by
the maximum shear stress, but do not speculate as to the nature of the unit
process or physical mechanism involved. An expression relating the time
to rupture to the applied stress and temperature is derived and gives
A + BT-Dcr
log tr =
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where tr is the time to rupture, T is the absolute temperature, 0" is the
applied stress, and A and E are material constants. D is defined as
log D = E + FT, where E and F are material constants. The evaluation of
the numerous material constants requires that many tests be made. The
authors report that a given material may have three different sets of con-
stants, depending on the temperature and stress and the occurrence of
microstructural changes during tests. This would appear to be a severe
limitation to the practical application of the derived expression.
Eyring and his school are still working in this area and have come out
recently with a statistical mechanical, theory of plasticity involving the virial
theorem and absolute reaction rates.* Their statistical formulation for
plastic deformation is based on the relative displacement of a "system of
atoms". These domains or patches are displaced along a shear surface as
a series of relaxations obeying a generalized absolute reaction rate
equation. The interpretation of the exact nature of these domains, the
cause of their creation, and the manner in which they slip are matters for
conjecture, but, to conform to current concepts, they are called "disloca-
tion domains". These dislocation domains involve the dislocation and the
atoms that are associated with it in its movement from one relaxation site
to another occurring during the deformation process.
The equation resulting from the application of their theory gives, at
higher stresses,
(13/2) exp (cr/2gcri.n)
(28)
where c is the secondary creep rate and a- is the applied stress, p is pro-
portional to the relaxation time and is essentially constant, Crm is the local
microstress that is influencing the dislocation domain movement, and g is
related to the velocity of sound. 2gcrin is the important variable, and its
variation with tensile strength, hardness, alloying, and cold working, as
well as with annealing and recovery, is claimed to conform to expected
behavior. However, the objections raised with regard to the theories dis-
cussed previously still seem directly applicable to this later theory.
Thermodynamical Theories of Fracture Strength
These types of theory are based on the idea that the breaking of atomic
bonds by fracture is analogous to the loosening that occurs during melting.
Born(159,161) and Fiirth(502,505) suggested an approach to the problem of
fracture in terms of melting that is closely dependent on thermodynamic
quantities. Assuming the maximum normal-stress criterion of fracture, and
*Bates, J. L., Ree, T., and Eyring, H., "A Statistical Formulation for Creep of Metals", Tech. Rep. 56,
ONR, June 15, 1956.
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no plastic deformation, Fiirth(502,503) developed a relation between breaking
strength and the energy of melting that yields values of strength in fair agree-
ment with those observed experimentally. His equation for the tensile
strength, F, of an isotropic body at low temperatures is
1 -Zbt
F = Qp
3-51/
(29)
where 0 is heat of melting per unit mass, p is the density, and I/ is Poisson's
ratio. Experimental data for ten metals showed agreement with theory
within 7 per cent. Later, Firth attempted to relate his theory with that
proposed by Bragg(173), by assuming that the block structure in an intrinsic
feature of the crystal lattice. This idea was attacked by Patterson(1232) on
X-ray grounds, but defended by Wood(1717), who stated that Patterson's
objections were not conclusive. Aroesta(46) points out that Firth's results
have been criticized because rupture strengths are associated with surface
phenomena. By assuming that melting is dependent not only on block size,
but also on the degree of atomic misfit between the blocks, Aroesta relates
the theory to yield strength, which is less surface dependent. Suzuki(1545)
applied a thermodynamic model to the tensile breaking strength of an internal-
stress-free polycrystalline specimen at 0 K. The formula derived is the
same as Firth's.
Saibel(1379,1380,1381) has formulated a thermodynamic theory of
fracture, assuming a relation between fracture and the latent heat and
volume change of melting. The criterion of fracture was that of a critical
strain energy per unit volume. His theory is based on the assumptions that
all of the strain energy is available for the abolition of cohesive strength,
the heat of fusion is uniformly partitioned throughout the material, and the
energy required is that part of the energy of fusion associated with the
change in volume on passing from the solid to the liquid state. Thus, the
criterion for fracture can be expressed in the form
U = JQ AV/V ,
(30)
where U is the strain energy per unit volume, J is the mechanical equivalent
of heat, Q is the latent heat of fusion, and AV/V is the change in volume per
mole on passing from the solid to the liquid phase. Saibel's calculations in-
dicated that, if no plastic deformation occurs prior to fracture, the breaking
strength will correspond to the value obtained from the theoretical calcula-
tions. If plastic deformation occurs, the fracture stress is reduced to the
magnitude experimentally observed, and fair agreement is obtained between
his calculations and the experimental values. Consequently, it was concluded
that plastic flow precedes all fractures.
In a recent paper by Osipov and Fedotov(1203), it is shown tha?ti several
mechanical properties of metals are related to the energy required to melt
them. This quantity, AW, differs from that used by FUrth, and others, in
that to the heat of melting is added the additional energy required to bring
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the specimen from the test temperature to the melting point. Apparently,
this additional energy is responsible for the good linearity demonstrated
between hardness at higher temperatures and W. Although this paper is
not on as theoretical a level as the papers by Firth, for example, they are
close enough in principle that a suitable theory should be readily developed.
Further work along this line should be instigated, since the results appear
quite promising.
Whether such theories can correctly embrace the influence of metal-
lurgical structure, by which means the strength of some alloys can be
varied widely, is not apparent. It is also not clear whether the thermo-
dynamic approach can account for the surface effects noted by Griffith(593)
and others. Zener( 1771) has criticized the thermodynamic criterion for
fracture of metals on the grounds that the strain energy absorbed prior to
fracture is dependent on test conditions and is also structure sensitive. The
thermodynamic quantities are not dependent on these factors. Therefore,
he believes that these theories contradict the established principles re-
garding fracture. The interesting findings of Osipov and Fedotov may alter
these conclusions somewhat, if further experimental or theoretical justifi-
cation for their results is found. '
Relationships Based on Equations of State
From the engineering standpoint, the equation of state probably repre-
sents the most desirable solution to the problem of predicting material
behavior. If an expression were valid that related the instantaneous values
of the variables strain, strain rate, temperature, and stress, for example,
then new data could be calculated for any other set of conditions or type of
test. Expressions relating this particular set of variables have been called
"mechanical equations of state" by Hollomon(708), in obvious analogy with
the equations of state for gases or solids(391,549), examples of which are,
respectively,
PV = nRT ,
(31)
where P is the pressure, V is the volume, n is the number of moles, R is
the gas constant, and T is the absolute temperature; and(1488)
P = Pcc(T) + Pi (T) + P2( T) +???
Vo Vo
Vo-V Vo -V
(32)
where P is the hydrostatic pressure on the solid, with V and T as before,
and Po is the pressure that must be applied to a solid to reduce its volume
to Vo, the volume at absolute zero under no pressure. For the more com-
plicated case where a solid undergoes stresses other than simple pressure,
or strains other than a mere change in volume, the equation of state is a set
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of relations giving the stress at every point of the solid as a function of the
strains and the temperature.
The existence of a mechanical equation of state means that the flow
stress depends only on the instantaneous strain, strain rate, and tempera-
ture, and not on their past history. The range of validity of a mechanical
equation of state yet remains to be established, although data already re-
ported cast considerable doubt that it will be very widely applicable. How-
ever, the influence of the changes on the flow properties may be sufficiently
slight in some cases that the concept of a mechanical equation of state will
be of some practical utility.
One of the more comprehensive critical revi.ews of the literature per-
taining to the mechanical equation of state was written by Zener and
Hol1omon(1779) in 1946. Their conclusions on the pros and cons of this sub-
ject were hampered by the lack of suitable experimental evidence, and this
situation has not improved materially in the intervening years.
Hollornon and Lubahn(714/715) attempted to derive a general relation
for the combined effects of strain, C, strain rate, c, and temperature, T,
on the stress, a-, required for plastic flow. They deal, in turn, with the
relationships between c and a-; E, c, and a-; and c, c, T, and a-. They then
suggest that all these variables are related in the following way:
a- = CGT (Lico)DT
exp (E - FT ln L/Lo) , (33)
in which the constants, C, D, E, F, G, and co are independent of all the
variables concerned, and the strain rate and temperature are constant in
arriving at that strain at which the stress is determined. At constant strain
rate and strain, Equation (33) predicts that the logarithm of the stress
should vary linearly with the temperature, and this behavior appears to be
followed very generally. Also, the available data indicate thaf there are two
distinct ranges of temperature in which the material constants in Equation (33)
are different - i.e. , a low-temperature range and a high-temperature
range. This behavior suggests that there are two different mechanisms of
deformation - at low temperatures, deformation occurs primarily by slip,
and at high temperatures, the deformation occurs primarily by rotation at
the grain boundaries.
For the case where the strain rate and temperature are not held con-
stant during the test, the validity of the empirical relationships obtaining
under variable conditions was investigated. Hollomon and Lubahn conclude
that their general relation should be an equation of state in which the stress
at a given strain should not depend on the past history of the temperature or
the strain rate. However, they did not expect an equation of state to be valid
whenever structural changes such as phase transformation and recrystalli-
zation occur, as is pointed out also by Sylwestrowicz(1561).
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Many other proposals have been advanced for relating the important
variables in plastic deform.tion. Andersen(20) introduced the time variable
explicitly in his general relation denoting stress as a function of strain,
time, and strain rate. He attempted to show that the phenomena of various
mechanical tests conform to the laws of dynamics, and that all equations
describing special testing operations are derivable from one equation.
Wyatt( 1735) showed that transient creep at low temperature and stresses
obeyed an equation of state and confirmed this by experiment; and
Graham(563) gave an expression that is a special case of the equation due to
Hollomon and Lubahn. Further, he obtains an alternative expression re-
lating stress, time, strain, and temperature that is akin to the Nutting
equation(1140), generalized to include temperature,
frs = ern (tsionT ,
(34)
where o-s, ts, and n are constants. The results of Carreker(232) with
platinum were described satisfactorily by Equation (34). Equations of state
for zinc single crystals in creep were reported by Thompson(l558) and
Gilman(542), and Wiedersich(1697) showed that a modified form of Seeger's
equation relating strain rate to temperature and stress described the data
obtained by Dorn, et al., with high-purity aluminum(1603).
Many such examples as above are available in the literature. In
spite of the well-known arguments against the existence of a general equa-
tion of state for solids( 380, 1182), it appears that there is good experimental
evidence for the validity of more limited relationships. Further work with
'the Boltzmann superposition theory(771) and with incremental- rather than
total-strain equations may extend the validity of these equations to a point
where they are of real practical use in forecasting the results of any given
experiment.
Equations of state have been confirmed either at low temperatures,
where recovery is not appreciable, or at high temperatures (in the absence
of phase changes, etc.), where recovery proceeds rapidly along with
hardening. Perhaps this line of demarcation between the high- and low-
temperature regions of plastic deformation has not been drawn too clearly
in formulating equations of state for solids. In any event, it appears that a
mechanical equation of state represents a possibility of great theoretical
and practical importance. Although the basic postulates are extremely
attractive in principle, the verification has lagged in practice. The frequent
reports in the literature of cases where an equation of state is obeyed over
restricted ranges of variables lends credence to the idea that such an ex-
pression may exist for the more general case, could we but see it and
formulate it correctly.
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Empirical Relationships and Parameters
A generally valid equation of state and its experimental verification
has not been obtained; however, relationships have been found in many
cases over a more limited range of the variables stress, strain, time, strain
rate, and/or temperature. Frequently, two variables are involved in such
a way that they appear in the form of a combined function or parameter.
Thus, for example, Zener and Hollomon(1778) proposed that the stress-
strain relation fr(c) in steels at low temperatures depends upon strain rate
and temperature only through a single parameter P, such that
o- = cr (P,c) ,
and that this parameter has the form:
p = eQ/RT
where Q is the heat of activation for the deformation process.
Other parameters can be systemized in functional notation according
to the following scheme, which emphasizes the interrelationship of these
expressions.
Z ener -Hollomon parameter(1778)
Hollornon-Jaffe parameter(712)
Larson-Miller parameters(903)
(35)
(36)
Dorn 0-parameter(134)
MacGregor -Fishr velocity-
modified temperature(99u)
Rabotnov parameter 300)
The Hollomon-Jaffe
both have the form
Z = Z (L,T)
p P (t, T)
pl = pl (L,T)
P2 = P2 (t)T)
0 = 0 (t,T)
Tensile tests
Hardness tests
Creep and tensile tests
Creep and tensile tests
Tm = Tm (,T) Tensile and creep tests
Pc = Pc (cr,t)
Creep and tensile tests
parameter(71Z) and Dorn's 0-parameter(1341)
p = te -Q /RT
(37)
Equation (37) was applied by Hollomon and Jaffe to the room-temperature
hardnesses of steels tempered for different times and temperatures,. and by
Dorn to the creep strain obtained under constant stress for various combina-
tions of time and temperature. Larson and Miller(903) proposed parameters
of the form
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P1 = T (C - log L)
P2 = T (C + log t) ,
(38)
(39)
which were derived simply from the extremely general Arrhenius rate ex-
pression.,
Rate = Constant exp (-Q/RT) . (40)
MacGregor and Fisher(989/990) developed an expression, similar to
that by Zener and Hollomon, in terms of a parameter Tm, termed the
"velocity-modified temperature". Their derivation was based on the
Kauzmann(787) analysis of the dependence of creep rate on stress and
temperature. MacGregor and Fisher proposed that in general the flow
stress is a function of the strain and the strain-rate-modified temperature
of flow:
where Tn.." has the form
= cr (T e)
Til
Trn = T(1 - K ln L/L0) ,
(41)
(42)
where K and co are constants. This relation also holds for either tension
tests or creep tests and over a wide range of temperatures. The similarity
of Equation (42) to Equation (38) is apparent, and is also seen when
Equation (36) is rearranged.
Rabotnov's parameter(1300) relates creep data to the tensile test in
the form of isochronous stress-strain curves. In essence, he postulates
that the tensile stress is a function of the tensile strain and a creep function
Pc, such that
and
cr = cr(e,Pc)
Pc = ?c(1 + at/) ,
(43)
(44)
where crc is the (constant) creep stress, t is the time to a selected creep
strain, and a is a material constant.
These parametric relationships have been applied in many cases to
creep, creep-rupture, tensile, and hardness data, and very satisfactory
results have been obtained (see, for example, the discussion to Reference
510 by J. Miller). Those cases where unfavorable results were ob-
tained(510, 869) can usually be traced to an inapplicable constant, for
example, a blanket use for all materials of C = 20 in Equations (38) and (39).
Other expressions and parameters have been proposed(1043,1527) that
appear to have only empirical significance. It is believed that the
_1
41
parametric expressions listed above offer the greatest probability of success-
ful application to practical problems; and furthermore, through the close
interconnection with the Arrhenius rate expression, they may possibly achieve
a more respectable theoretical status at a later date.
Empirical expressions devised to account for material behavior under
various combinations of stress, strain, strain rate, temperature, and time
are extremely numerous (see Appendix C). One of the more recent empiri-
cal attempts to correlate high-temperature creep and rupture data is that
due to Conrad*. For the rupture time, tr, and the minimum creep rate,
cs, he gives
s ?K A exp (-LM-ID/RT) sinh [cr/o-o(T)]n , (45)
tr
where K, A, cro(T), and n are constants, and AHD is an activation energy fair
diffusion. Better fit of this expression with experimental data is claimed
than with the Larson-Miller parameter, but four adjustable constants are
employed versus one in the Larson-Miller expression, and in addition an
arbitrary value of 20 is assumed for their constant. Under these circum-
stances, it is not surprising that Equation (45) should appear in the better
light. Kanter(782) proposed a relation between high-temperature tensile
tests such that
(L/Lo)n = cricro
(46)
where n is a function of temperature, and *co and 0-0 are constants deter-
mined at the convergence of the curves for different temperatures. In-
terestingly enough, he found that activation energies found by his approach
are of the magnitude of the heat of melting, rather than of the heat of
vaporization.
Correlations between different mechanical properties are also quite
numerous (see Appendix C). Examples of these include direct, experi-
mentally observed relationships between, for example, hardness and tensile
strength or rupture strength and tensile strength. Periodic variations of
strength with atomic number of solute also appear to be useful(617). How-
ever, cognizance of these relationships in the present report stems pri-
marily from their potential practical applications, in the event the theoreti-
cal approach is not followed in any subsequent research problem. To
illustrate what can be done with the variables hardness, tensile strength,
and rupture strengths, attention is called to the correlation proposed by
Underwood(1620). By comparing tensile and rupture strengths with hot
hardnesses, at the same parameter value [Equation (39) ], a straight-line
*H. Conrad, Westinghouse Research Laboratories, Scientific Paper 6-94701-1-P9, March 26, 1958.
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relation is obtained between hardness and strength, which accounts also for
the effects of time and temperature of test. Such correlations may ulti-
mately be shown to have theoretical justification, but, until then, they may
be used for engineering purposes. A very definite role of these various
property interrelationships is to guide theoretical developments, since, in
the last analysis, theory must conform to experimental observations.
EVALUATION OF THEORIES AND METHODS
From the detailed discussion given in the preceding text, it can be
seen that there have been many theoretical attempts to describe the strength
of metals. These theoretical attempts may be considered as falling into the
following broad classifications: dislocation theories, absolute reaction
rate theories, and thermodynamic theories. Because of their inability to
account for specific effects, theories based on thermodynamic criteria and
on absolute reaction rates are rejected as being too general to afford a
basic understanding of the underlying processes occurring during plastic
deformation. Therefore, dislocation-type theories are held to represent
the most promising avenue for further theoretical and experimental research
in this general area. However, this does not imply that early results can be
expected in the prediction of material properties. A general description of
the present situation is that dislocations are recognized as representing the
most important element in the mechanism of plastic deformation; their
properties, and the way of interaction between different dislocations, are
well known, but the arrangements and interplay of dislocations in different
processes are largely unknown as yet.
Having selected those theories based on the dislocation concept as the
most promising, a preference rating of specific theories will be made. It is
extremely difficult to subdivide theories in an arbitrary manner and then
consider their parts as separate entities. However, an attempt will be made
to indicate the best portions of those theories according to the organizational
scheme adopted in the text.
The G. I. Taylor theory, as modified by Mott and Orowan, is believed
to be the fundamental theory at the present time for explaining strain harden-
ing. Details and interpretations have changed, but Taylor's basic ideas are
still valid. Seeger, Friedel, and others have suggested improvements and
have stressed new mechanisms (some of which are unproved), but these are
still within the basic framework.
The theories of Mott and Nabarro, the Cottrell-Suzuki interactions,
and Fisher's short-range ordering effect appear to be the most applicable to
those various aspects of alloy hardening covered by their theories. There
is no all-inclusive theory that is applicable to the entire subject.
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Precipitation hardening is closely linked with the preceding categories,
but the treatment afforded by Orowan of precipitation hardening seems to be
the most realistic. The modifications and extensions contained in the Mott
and Nabarro treatments, and the Fisher, Hart, and Pry theory should also
be considered.
In the field of creep theories, the 1947 paper by Orowan on the ther-
mal activation of transient creep and his 1956 paper appear to give the basic
treatment. The theories of Weertman for viscous creep need modification,
but follow experimental observations most closely. Numerous other con-
tributions of merit are found in papers by Mott, Nabarro, Schoeck, and
Dorn, but these also include items not applicable today, or else not firmly
established.
Of the methods available for predicting strength properties, the most
promising appear to be those expressions employing parameters. The
Larson-Miller parameters are equivalent to the others listed in the text
that relate either strain rate and temperature or time and temperature.
Rabotnov's relation using a parameter for stress and time should also
prove useful.
Also very promising is the periodicity, first investigated systematic-
ally by Gulyaev, of the strengthening due to .solute elements as a function
of their atomic numbers. Closely related to the periodic behavior are the
relationships demonstrated by Osipov and Fedotov. Since both mechanical
properties and latent heats in general vary with atomic number, perhaps
some relationship between the two should not be unexpected.
A final recommendation of a prediction method is based on the idea of
a mechanical equation of state. The success enjoyed by the parameter-type
expression gives reason to hope that a more inclusive relationship may be
found. The equation developed by Hollomon and Lubahn seems the most
general, but may need overhauling when sufficient experimental data are
obtained. The general rheological approach contained in the Graham-
modified Nutting equation should also prove worthy of more extensive
testing.
SUMMARY OF CONFERENCES WITH CONSULTANT,
PROFESSOR E. OROWAN
In three full-day conferences with Dr. Underwood in Pasadena, in
La Jolla, and in Columbus at Battelle Memorial Institute, the preseht status
of the physical understanding of the phenomena of plasticity and strength
was reviewed and separation of the comparatively secure fundamental lines
from hypotheses in the workshop stage was attempted. As a side line,
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Professor Orowan read a large number of Russian papers supplied in trans-
lation by Dr. Underwood and gave written or oral comments on them. The
main points that emerged from the conferences are as follows:
(1) Plasticity
The usual reason for the great discrepancy between the high molecula
cohesion and the low observed yield stress of crystalline materials is the
presence of dislocations. These represent the most important element of
the structure of real crystals as far as their plastic behavior is concerned.
The structure and interaction of dislocations are fairly well known; to
understand the plastic properties of materials, however, the arrangement
of the dislocations and their interplay during the process considered must
be known.
The yield stress (stress required for plastic deformation after a cer-
tain preceding deformation) may be determined either by the stress re-
quired for moving dislocations in the absence of obstacles (frictional stress
driving stress, Peierls-Nabarro force) or by the obstacles in the way of the
dislocations. Foreign atoms in interstitial solution may be adsorbed at
dislocations and anchor them to the lattice (Cottrell locking), or one type
of atoms in a substitutional solid solution may be concentrated in certain
parts of a dislocation, with similar effects. Such adsorption phenomena
seem to be the cause of most, but hardly all, yield phenomena. The ob-
stacles hindering the movement of dislocations may be hard precipitated
particles, stress fields around them, dislocations piercing through the
slip plane, sessile dislocations, stacking faults, etc. They may be over-
come by the dislocation cutting through them, or by dislocation bulges
being extruded between obstacles.
Strain hardening is due in general to the increase in the number of
obstacles produced by plastic deformation and the increasing difficulty of
driving slip across them. Its quantitative treatment will be possible only
if the rate of increase of the number of obstacles during slip, as well as the
exact nature of the obstacles, is known. A special type of hardening is
that upon which the hypothesis of G. I. Taylor was based; it is due to the
piling up of slip at relatively impenetrable barriers. Such a component is
present probably in most cases of strain hardening, but its relative signifi-
cance is usually small.
Solution hardening seems to have two main causes: first, the internal
stresses around larger or smaller atoms that do not fit between their
neighbors; second, valency or ionic types of bonding forces between neigh-
boring atoms, which also increase the driving stress of dislocations.
Precipitation hardening is due to the resistance of the precipitated particles
to being cut or sheared by dislocations, and to the internal stresses in the
lattice around them. Overaging is a consequence either of easier disloca-
tion-bulge extrusion with increasing spacing of the particles or of the
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a-
45
reduction of internal stresses by the breaking away of particles from the
matrix. The adsorption of atoms at dislocations has been mentioned already;
similar adsorption may take place at stacking faults also ("Suzuki effect").
Transient creep is due to thermal-stress fluctuations superposed on
the applied stress; its slowing down with time is a consequence of increasing
strain hardening. Viscous or hot creep is due to atomic-rearrangement
processes of the self-diffusion type. These processes may lead to thermal
softening and, under stress, to recovery creep; or to viscous flow at the
grain boundaries; or to the movement of interstitial atoms or vacancies or
of atoms from or into grain boundaries, all of which may give rise to
viscous creep when taking place under stress.
(2) Fracture
Brittle fracture in fully brittle materials is a crack-propagation proc-
ess by cleavage; it is usually governed by the Griffith equation, which is an
expression of the second law of thermodynamics. If the material has some
ductility, internal stresses developed by plastic deformation (e.g., where
dislocations pile up) may be superposed on the applied stress and may cause
cleavage fracture. Finally, fracture-like separation of bodies into frag-
ments may take place exclusively by plastic.deformation, as when a wire of
pure gold or indium necks and separates at sharp needlepoints in the
center of the neck.
(3) Russian Work on Mechanical
Properties of Materials
Perhaps the only interesting work found was that concerned with
empirical relationships between the thermal and mechanical properties of
materials. Most of the other papers were second or third rate; in general,
the present level of Russian work in this field is incomparably lower than
was that in the 'twenties and 'thirties. Dislocation theory is practically
nonexistent; it seems to have been scorned for political reasons until a few
years ago, and the recent publications are mostly either unimpressive
second-hand reviews or amateurish concoctions. Many papers are published
on hot creep testing, mostly without serious substance. It seems that the
good workers have been drafted into industrial production, and scientific
work has come to a near standstill.
EEU:MFA:REM:GKM/all
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ANNOTATED BIBLIOGRAPHY OF THEORIES AND
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^
(1) Accra. L.. II
Elasticity and the Transition Zone of Steels (In Spanish)
Inst. hierro y acern 4 (1951) 220-226
Factor? affecting elasticity anti transition temperature. Results of
experiments on the relationship between values of elasticity, ductility, awl
tensile strength.
(2) Achter, M. 11., and Smoluchowski, R.
On the Role of Grain Boundaries in Creep
Acta Met. 411956) 331-332
Rate of creep is Influenced by the availability of dislocations and the
ease with which they can move across the grain boundary In blery?tala, a
grain boundary parallel to a plane of high density would provide a better fit
between the lattices of the two grates than a low-density plane, and would
facilitate creer
(3) Adams, M, A.
The Plastic Behaviour of Copper Crystals Containing Zinc In the Surface
Layer
Acta Met. 6(1958) 327-338
Tensile teats have been made on Cu crystals containing various con-
centrations of Zn in the surface layer Crystals containing .mall amounts of
Zn, after a small prestrain and an anneal at 200 C, showed a yield point,
while in crystals of higher surface Zn content the yield point was absent.
Previous deformation affected the yielding behavior of both lightly and heavily
zincified specimens. Critical shear stress measurement, showed that a
lightly zincified surface strengthens a Cu crystal more effectively than a
heavily zincified surface; removal of the surface layer from a lightly zinc'.
fled crystal caused a large decrease in strength and a disappearance of the
yield point. The results are consistent with the idea that the first Frank-
Read sources to become active in Cu crystals are those which Ile near the
surface.
(4) Adams, M. A., and Cottrell, A. It.
Effect of Temperature on the Flow Stress of Work-liardened Copper Crystals
Phil. Mag. 46 see 7)19551 1187-1193
Changes in the flow etre.; of copper crystals due to changes in the
temperature of deformation have been measured. The change of flOW stress
with temperature Is closely proportional to the flow stress itself. These re-
sults are discussed briefly in terms of recent ideas of dislocation processes
senaltive to temperature. The effect of increasing the temperature of de-
formation is to produce a yield drop, similar to those observed in aluminium
crystals during work softening.
(5) Akidov, N. S., and Galenko, P. P.
Theory of Plastic Deformation of Metals (In Russian)
Dok/ady Akad, Nauk S.S S.R. 103 (July 1955) 387-390 (Translated by British
ACRE Lib/Trawl. 672, August 1956)
Uses concept of "blocks" as elements which behave as units in plastic
aleforrnation. They ronnider the case where displacement of blocks is
irreversible and calculate the area of the hysteresis loop using the Rayleigh-
Preiaach theory for magnetic hysteresis.
(6) Akulov, N. S., and Miryasov, N. Z
A New Method of Studying the Plaatic Deformation of Ferromagnetic Cryatals
(In Russian)
Zhiir. Tekh. Fix. 18(1918) 389-394 (Physics Abstr 53, 80)
The "deposition lines" on ferromagnetic crystals are connected in a
definite way with the crystallographic directions. These lines also determine
the degree and character of the residual stresses in the crystal and, con-
sequently, enable the distribution of the latter to be referred to the crystallo-
graphic directions. The existence of two independent systems of deposition
lines wan proved, each of which characterizes a definite mechanism and de-
gree of residual deformation. When the plastic deformation of the crystal Is
Increased, qualitative and quantitative changes of the character of the deposi-
tion figures are found. The results of these observations make possible a
systematic investigation of the magnitude of the plastic deformation for ferro-
magnetic single- and polyerystals under a complex aystem of stresses The
new method was found to be much more sensitive and to aupply a far more
complete description of the plastic deformation than any other method.
A-I
(7) Alexander. 11 IL , Dawarin. M. II., and Kling, II. P.
(8)
The Deformation of Gold Wire at High Temperature
Anni PhY 22 (1951) 439-443
Gold wires were subjected to .mall tensile ?t at high tempera.
tore, and the strain-time relationahip, coefficient of viscomity, and surface
energy were determined. The creep corve? obtained were similar to those
for metals at lower temperature except that the mininium creep rate a ap-
plied stress, i.e., the flow was viscous. Theories to explain vi?cou? be-
havior of metals were discwised
Alexander, IL II , Kuczynski, G. C., and Dawson, M.
Relations Between Diffusion and Viscous Flow lii Metals
Article in Physics of Powder Metallurgy, McGraw-llill Book Co., New York
(1951) 202-213
When Cu wires wound on heavy Cu cylinders are heated, the radius of
the sintered neck which forms bears a 5th power relationship to the wire
dianieter at temperatures up to 1070 C, indicating that sintering occurs by
volume diffuidon In creep teids on Au wires at 970 C, however, deformation
occur? according to a vi?cous-flow law, It is a?iiiimed that there is concen-
tration of vacancies near the curved surfaces of the necks and that the volume
diffusion is caused by the diffusion of vacancies The same atomic mecha-
nism leads to oleic.u, flow in the large mosaic blocks in the Au wire under
creep loading,
(9) Allen, N P.
Cohesion and Fracture in Metals
Article in The Fracture of Metals, Inet.of Metallurgists, London (1950) 5-28
Crystal structure, plastic and brittle fractures, and fractures due to
their combination, Discusses the structure of metals, metallic bowie, and
dislocations. Modern research uses knowledge of the forces within a metal-
lic lattice to find out how a crack of atomic dimensions ahnuld behave Work
I. also being done on factors which influence the rate of the speed of the
crack
(10) Allen, N P
Introductory Paper
Proc. N P.L Symposium on Creep and Fracture of Metal, at High
Temperatures (1956) 1-18
Progress made since 1946 in the study of plastic deformation is re-
viewed. The behavior of creep-resisting alloys over long periods, the in-
fluence of method of manufacture, and atudlea of new alloys are discussed
(II) Allen, N P., Hopkins, B. E , and McLennan, J. E.
The Tensile Properties of Single Crystals of lb:el-Purity Iron at Tempera-
tures from 100 to -253 C
Proc. Roy, Soe (Loisilon) 234A (1956) 221-246
Single crystals of high-purity Fe of various orientations were tested in
tension at temperatures from 100 to -253 C. The reeling ?110Well that; ( I) in
the early stages of deformation slip lines were straight, and corresponded to
one of the planes 11101, 1112), and 1123) wavy slip Imes developed at a
later stage of deformation; (2) the values of the critical shear atreonee for
the three operative slip systems were very similar, the ratios being close to
unity stall teinperaturea; (3) down to -124 C all the crystals were ductile,
giving chisel-edge fractures and 100 per cent reduction in area; at -196 C,
depending on the orientation of the stress axis, the behavior covered the
whole range from fully ductile to completely brittle; between these limits,
mixturea of slip, twinning, and cleavage were obtained; at -253 C most of the
crystals broke by cleavage without prior deformation except near 1011),
where twinning, or a small amount of slip, preceded cleavage; and (4) the
yield point and the critical shear stress for slip increased with decreasing
temperature of test.
((2) Allen, N P., Schofield, T. 11., and Tate, A.E.L.
Mechanical Properties of a-Solid Solutions of Copper, With Zinc, Gallium,
Germanium, and Arsenic
Nature 168 (September I, 1951) 378-379
The ultimate tensile strength, the resiatance to plastic deformation,
and perhaps also the 0.5% proof strews of these alloys appear to be almost
entirely goveroed by their electron-atom ratio.
?
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(13) Altmann, S. Coulson. r A , and Iiiime-Rothery, W.
On the Relation Between Bond Hybridc and the Metallic Strut ture?
Proc. Roy Sm (London) 240 (1957) 145-159
The failure of existing thenrie? to account for the type of crystal struc-
ture of a given meta/ is emphasized It is suggested that, particularly when
the number of bonding electrons is high, the metallic bond ha? greater diret -
Ilona( characteristics than are generally assumed, and that those tan be re-
lated to the symmetries of known hybrid bonds. Consideration Is given to
the fcc, bee, aml cph structures of the transition metals, and it is shown that
hybrids suggested by the crystal structures can be correlated with the known
electron characteristic, of these metals,
(14) Amelinckx, S.
Slip Line? and Etch Pits
Phil. May 44 (1953) 1048-1049
Experiments with relatively large crystals of pure Al which were re-
crystallized, then pollahed electrolytically, deformed and etched, lead to the
conclusion that slip line? are characterieed by the occurrence of rows of di.-
locations along them, and provide evidence for the validity of the Frank-Read
mechanism [Phys. Rev. 79(1950) 7221. Dislocations were found to be clue-
tered and not distributed uniformly along slip lines, as Is to be anticipated in
accordance with Mott's theory of work hardening Phil. May, 43(1952) 11511.
(15) Amelinekx, S.
The Direct Observation of Dislocation Pattern? in Transparent Crystal.
Paper from Dislocations and Mechanical Properties of Crystals, John Wiley
and Sons, Inc? New York (1957) 3-51
Methods which allow the direct microscopical observation of disloca-
tion lines aro described and the decoration mechanism Is discussed. The
nature and crystallography of the precipitation are considered. The method
was used to study the geometry of dislocation patterns in deformed and
annealed NaCI single crystals. Singularities In the patterns of the disloca-
tion nets were analyzed In terms of Burgers vectors. Direct evidence was
found for the climb of single dislocations and of networks. The pinning of dis-
locations by precipitates and by locked dislocations was observed.
(16) Amelinckx, S.
Dislocation Patterns In Potassium Chloride
Acta Met. 6(1958) 34-58
Dislocation patterns observed In "single crystal." of KCI, as grown
and after deformation, are analyzed. Large areas covered by regular
square and hexagonal networks are found. Singularities in the networks
are explained in terms of Burgers vectors and the genesis of the nets is
coneldered. In particular, evidence is presented for the stability of a (100)
dislocation with respect to dissociation into two a/2 [110] dislocations. It
is found that a small driving force coming either from a third crossing
dislocation, or from purely geometrical circumstances, and which tends to
bring the two combining a/2 ( 1101 dislocations into a parallel position, is
sufficient for the formation of a segment of a (100) dislocation. Evidence
for the formation of "holes" at the intersection point of two singular dislo-
cations with the same Burgers vector Is also presented. It is Suggested
that the network. are sometimes decorated more than once A systematic
use of this phenomenon might be of some value in studying the evolution of
networks during anneal. Dislocation, which moved during the anneal
accompanying the decoration leave traces of particles behind them, suggest-
ing that certain parts of the dislocation line are more effective in causing
precipitation. The density of the decorating particle, depends on the
orientation of the line with respect to the Burgers vector; screws seem to
be less easily decorated than edges.
(17) Amelinckx, S., and Maenhout-van der Vorst, W
A Slip Source in Potassium Chloride
Paper from Dislocations and Mechanical Properties of Crystals, John Wiley
and Sons, Inc., New York (1957) 55-56
A photograph of a slip source is shown, and its structure is
analyzed.
(18) Amelinckx, W., Bontinck, W., Dekesyaer, W , and Seitz, F
On the Formation and Propertie. of Helical Dislocations
Phil. Mag. 2(1957) 355-378
The factors influencing the development of helical dislocations in
CaF2 (Bontinck and Amelinckx, ibid , 94) are studied at temperatures
near and below 700 C. Climbing dislocations having a strong screw com-
ponent assume a helical or spiral prismatic form Thr model of F-center
A-2
I,, Carz and their equilibriem t liar.,, te met. s are ilieciesacal, Ilellral
ilislotat ions should he generated in many material. In a variety of circum-
stances and, if they ormir near the surface, they iihnuld be capable of
generating whiskers vatic), grow froin the tia?e Controlled climb in any
material may be produced by appropriate temperature gradient It the pro-
duction or annihilation of lattice defects at the ;Relocations engender climb.
(19) Aniker, B., lieziett, IL , and Parker, E it.
Relationship Between Sniall-Angle Dislocation Buendaries and Creep
J Appl Phy 27(1956) 331-310
The relationship between the amall-angle boundary density, creep
strength, and room-temperature tensile properties of 99-95t. Ni Was in-
vestigated. Specimen bare were given controlled prestrain., followed by
recovery at 800 C for I hour Creep tests under constant stress at 700 C
were halted at strains of I and 2-1/2.76. Half of the bars were then tensile
tested, and the duplicate series used for X-ray examination of the sub-
structure by an adaptation of the Laue method (Barrett, frail. AIME 161
(1945) 19 The linear relation of flow stress to mean density per grain of
small-angle boundaries was confirmed. Increased amounts of prestrain re-
sulted in decreased initial creep-strain rates. The shape of the creep
curve may be made to vary as a function of the initial density small-angle
boundaries; primary creep can be eliminated by introducing small-angle
boundaries of a certain type of density The creep-induced substructure is
characterized by the large number of parallel, small-angle boundaries,
and differs from that developed by an equal amount of rooni-teniperature
deformation plus recovery. Consideration of the processes of creep, self.
diffusion, stress-induced generation of vacancies and their migration sug-
gests that creep deformation at high temperatures involves dislocation
climb. A mechanism for the production in creep of short parallel bounda-
ries is proposed in terms of the interaction of dislocatinna from a Frank-
Read source with those piled up at a barrier Removal of these boundarlea
by tensile tests is discussed.
(20) Andersen, A 0.11
Stress-Strain-Time Phenomena in Mechanical Testing A Study of the
Stress-Strain-Time Functions of Metals in Simple Monotonic Tension
USAEC Publ., ORNL-1114 (1952) 93 pp
It is shown that stress can be expressed as a function of strain 6 and
time t according to;
S
t?
v(r6)v (rovi di or ?1.?)-Yc ((4) 4
c16 Y v
where v is the strain rate, (fo)v is the part of the total change in street, that
can be assigned statistically to effect of strain alone, tiering an infinitesi-
mal period, and (fdv is the part of the total change in stress during an in-
finitesimal period that can be assigned to the relaxing effect of Hine alone
The total change in stress is given by d(e)v 1y)? t16 (fdvdt. The hy-
pothesis is developed that stress is a functional of strain that reduces to
functions of the independent variables: strain, time, strain rate, and varia-
tions In strain rate, i e
a0) F[6(t)1 = Stn (ft vf. afv a(a 4 ..) dt, where a e
dt
0
It is shown that the phenomena of various mechanical tests conform to the
laws of dynamics, and that a general equation representing stress as a
function of the independent variables, strain, time, and strain rates can
represent the phenomena All equations describing special testing opera-
tions are derivable from one equation:
. .
F (Mtn r ,:rtt) = (b, 6, 6, 6 . t) f(ft vf o arv I at.,t ) dt
fl-kiifir1 in Part d Coov AP
6
(SI) Andrade, F.. N. da C.
Viscous. Flow in Metals
Proc. Roy. Soc. (London) 04A (June 9, 1110) 1-12; Pliyink. J. II
(August IS, 1)10) 709-715
The flow of a lend wire loaded heynnil its elastic limit becomes vis.
onus after a time, i.e., the extension is proportinnal to the time. The var-
iation of the rate of flow with the load was studied by the use of a "hyper-
bolic weight" /sinking into n liquid, the form of the weight being so designed
that the upthrust at any moment varied inversely aa the length, and directly
an the sectional area of the wire, then maintaining conetent stress. Experi-
ments were carried out on Ph, fuse wire (Sn LI 5, Ph 78.5%). and Cu, the
results showing that all these metals are subject I,, vinrolie flow right lip to
their breaking points F.xteneinn is divided into three parte: M Immediate
extension on loading, (2) an initial Dow which gradually disappears ?
termed the p-flow, and (3) the viscous flow ? a constant flow throughout the
extension. In Cu the immediate extension is large and the viscous flow
small in comparison with Ph. The formula
I a tott + ft tinle"
where si Is a measlier of the (1-flow and k a measure of the viernus flow,
represents the extension-time curves very closely. For very large values
of the etre,e ft tends to become constant. The "fluidity" (k/stress) curve is
roughly hyperbolic with one asymptote parallel to, and the other steeply
inclined to, the stress axis
(22) Andrade, E, N da C
Flow in Metals Under Large Coneiant Stresses
Proc. Roy. Soc. (London) 90A (July l, 1914) 329-342
The 1910 work is continued to determine if the empirical laws for Ph
are applicable to other metals, and also to investigate the effect of tempera-
ture on the flow. The metals investigated were Pb, Sn, Fe, Ce, lig, a Pb-
Sn alloy, brass, and German silver. It was found that, without exception,
all the pure metals gave extension-time cerves which closely correspond to
the eipintion
lull p tli3)ekt
with the deviations less than 2%. In the equation, I. represents the
immediate length on loading, ft is a measure of that part of the flow whiner
rate decreases as the time increases, and k inerieures the final or viscous
flow, The values of these constants depend upon the temperature and at a
given temperature with increasing stress, /3 tends toward a constant value
while k increases at a rate which itself increases to a constant value. The
main result of this part of the work shows that metals of widely different
nature obey the same general laws of flow over a range of temperature.
Small quantities of impurities do not affect the general type of extension-
time curves. Duplex alloys show an anomalous behavior At a certain
stage of the flow produced at certain temperatures the wire softens instead
of showing the hardening which Is common to the pure metals. At other
temperatures, however, alloys were found to behave normally. By atib-
jetting to a high preliminary stress either in tensinn or torsion, wires can
be put into a state In which they will flow viernuely from the start of sub-
sequently applied smaller stresses Photomicrographs of strips of soft Fr
extended at different temperatures showed that slip bands wen formed with
equal distinctness and frequency, regardless of whether the extension took
several minutes or was produced immediately On the basis of the co-
existence of the crystalline and amorphous phases an explanation of these
various phenomena has been worked out, When a wire is loaded, there is
an immediate stretching owing to a slip between, and inside, favorably dis-
posed crystals An immediate redistribution of stress then takes place,
with the result that other less favorably placed crystals begin to slip and
give rise to the fl flow There is a limit to the ultimate number of lamallae
which can be produced, corresponding to the production of elementary
crystals This is represented by the limit to which (3 tends with increasing
stress If the amorphous material is capable to flow it gives rise to the
vlscoes or k flow, At low temperatures the aniorpholis phase is hard, but
as the temperature rises, it softens at a relatively greater rate than the
crystalline material This is obviously no limit to the viscous flow
123) Andrade, E N. da C
Flow of Metals
J. Inst. Metals 60 (1937) 427-445
Flow in most easily observed in the liquid state, but the structure of
the liquid state is still obscure. Nevertheless, it is poe?ible on simple
lines to formulate a theory of liquid viscosity, which both gives the observed
viscosity of simple liquids and accounts for the temperatiire variation. In
this connection, viscosity of liquid metals is important. because of the
simplicity of the molecular state The flow of solids is best formulated,
from which it Is clear that the velocity of flow is of fundamental significance.
The perfect crystal lattice does not offer a basis for any theory of the flow
of solids; the only successful attempts to construct a mechanism to explain
plastic behavior arc based on the assumption of flaw.. or clinlocations, hi
the lattice, are propagated along prefereetial directions when external
stresses are applied Thermal fluctuations lllll st also be invoked. Forth.,
investigation of the nature of the inherent flan. It. of importance, tint only
for physical theory but also her rnetallurgit al practice
A-
(24) Anil, ide, 1?, N da ( .
Creep of Metals
Engineering (61.1March 8. 19461 231-245
Ilsotorical aill"Viy and review of empirical laws of creep Discusses
familiar r. pl-P law and Stiggr?lii A f (a -4,)" - he rail") where
5, and n are principal stresses
(25) Andrade. F. N. (la C.
Metal Crystals Anil Metal Strength
Proc. Roy lion. (London) 34 (1947) 217-250
Thr breaking strength of a perfect crystal is considered, and the
met hanistu of plastic deformation of single crystal* Is described in terms
of structure.
1261 Andrade, E N. da C.
The Flow of Metals
.1. phys. radium 8 119471 311-326
The phenomena of flow and creep In metals are reviewed To
simplify the experimental results a constant stress is essential and two
ways of achieving this are deecribed. Under these conditions, metals
exhibit two flow regimes, one temporary, the other permanent, and their
relative importance in the case of any metal depends upon the temperature
The flow of single-crystal metals is rharactcrised by certain piastre and
crystalline directions of slip, by alight shear stress, and by marked hard-
ening due to strain. These phenomena are explained by a theory ba iced
upon dislocations propagated through the crystals. The state of the surface
of the single errant plays a considerable role in the theory, and it is con-
sidered that the deformation of single crystals can Angela a mechanism
for the flow of ordinary metals
(27) Andrade, E N da C
The Creep of Metals
Paper from Report of a Conference on Strength of Solids, Phys. Sac
(London) 11948) 20-26
A brief review is given of creep phenomena, with special referent e In
creep under constant strese, the effects of ',crystallization, "martentiltic"
phase changes, rotation nf t ryntallites, grain-bounclary flow. and the creep
of single crystal.. Three methods for maintaining ronatant stress on
specimens under axial tension during testing are proposed
(28) Andrade, E. N da C
Creep of Metals and Recry.tallization
Nature 162 (September II, 1948) 410
Results of some experiments on creep of pure Pb which show that
recrystallisation during creep has a fundamental effect upon Ilse form of the
creep curves This metal, all) flu normally statute at atinosplierit tem-
perature, recrystallizaes tinder titres. Posaibility of an analogous expla-
nation of the creep behavior of other metale
(29) Andrade, K N da C.
The Physics of the Deformation of Metall
Endeavour 9(1950) 165-177
Discusses in general terms structure-seeettive and etrecture-
ineensitive properties of metals. A description of rtastuc flow, erre'',
dislocations, etc , is given
(30) Andrade. E. N da C
The Flow of Metals
J Iron Steel hint 171 (1952) 217-228
Reviews knowledge about rfrip of metal. ender Mere's Creep be-
havior of Sn, Cr,, Fe, Cd, solid lig and litany other materials is very
sornslar if allowance is made for temperature, and can be expressed by a
simple cquuitiurn 1 he phynical mechanism of creep us disc...teed
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(31) Andrade, E N ila C
Critical Shear Stress and Temperature
Phil Mag. 13 tier 7 (1952) 1218'1221
A. regards the onechanit at behavior of single ? rystals, there ii a eon
leant between metals of hexagonal crystal strut lure, oho ti in general
one act of glide planes, and the metals of titbit estatei strut, hire,
having many glide planes It has, however, it, en (mind Hist the pare fee
metals, Au and Ag. show the hexagonal type ot "easy glide" at low tamper,-
lure when the glide is not large. With amiably i bonen condition., glide in
cubic crystals ran, apparently, be restricted to one set of glide planes with-
out even local interference by other sets, and then the behavior is hexago-
nal in character, The critital shear stress shows no distinction between the
two types of metals, Graphs of critical shear sterna if the metals Ili, Zit,
Cd, Cu, Mg, Ag, Au, P1,, and lig againnt the ratio of temperature, Ti In
Tn.?, the melting point of the metal, i e , T/T,? (r 0), show no nuggention
grouping according to crystal structure, telt follow a very similar course,
A plot of a, the shear stress, against 0 shown that for An, Ag, Cti, and Zn,
?rowan's laws =s - BT1/2 holds within the range On 0 05-0 4, but for
high temperature the i xperimental valuta, are /uglier than the law Indicates
(32) Andrade, E N da C.
Concept of Creep
Paper from Creep and Recovery, Am. Soc Metal., Cleveland, Ohio (1957)
176-198
Physical distinction between primary, beta, and secondary, k, flow,
behavior of cubic and hexagonal metals, useful characteristics of shear
method; behavior of surface Realms; significance of temperature relative to
melting point
(33) Andrade, E N. da C , and Chow, Y S
Glide Elements of Body-Centered Cubic Crystals, With Special Reference
In Effect of Temperature
Proc Roy Soc. (London) 175 (1940) 290-315
The glide elements of single crystals of bre Na have been determined
at various temperatures Experiments with Na. Fe, and Mn allow that the
sparing of the glide planes increases markedly as the temperature in
raised. It has been (mind that, for equal strain, the crystallite rotations,
an evidenced by the asteriama, are much greater at low temperatures
where the hardening is greater, which supports the view that hardening is
Intimately connected with the rotation of crystal fragments The breaking
up of the asterisms Mtn diecrete smote% which has been found with Na and K
Is shown to be due to recrystallization The general implications of the
results are discisseeti
(34) Andrade, E. N. tla C., and Jolliffe, K. II
The Flow of Polycrystalline Metals Under Simple Shear
Prot Roy, Sot (London) 113A (1952) 3-26
Creep of three polycrystalline metals, commercial Pb, pure Ph, and
Cd, were investigated under conditions of simple shear by a method in
which a constant couple is applied to an asn,ui,is cut in a thicker disk of the
metal For comparison, normal tensile tents were carried mit on wires of
the *ante metals Cause of the elimination of permanent creep under the
conditions of simple shear, and a variety of experimental results explained
in terms of intracrystalline and boundary effects
(35) Andrade, E N da C., and Kennedy, A J
A Surface Effect in the Creep Behavior of Polycryntalline Lead
Prot Phy. Soc (London) 6413 (1951) 363-366
Results of some experiments with pure Pb containing 0.057e Te
This alloy undergoes transient creep only, I e the creep equation is
= fo (I I 00/3). Plots of /3 ogainst stress were prepared for wires of
fonr different diameters, grain size being very nearly the same in earls
cane The plots did not coincide, but could be made to do no if corrected
on the assumption that a thin surface layer was offering effectively no re-
sistance to creep. This surface effect is interpreted as brills disc to a
lesser constraint on the surface region as regards slip. X-ray studies
revealed a change in type of atructure 0 1 mm belnw the aisrface
(36) Antlrade, E. N da C,, and Roscoe, R
Glide in Metal Single Crystals
Proc. Phys. Soc (London) 49 (1937) 152-176
Experiments carried not on hardening and recovery of Cd crystals,
and in Pb crystals on the sparing of glide planes which lass been ahown to
be independent of a range of factors, and so to have a physical signifi-
cance. The results relating in hardening can be explained by a hardening
on individual glide lamellae which is proportional to the glide, anti it is
suggested that the mechanism of the permanent hardening Is in only a ro-
tation of cryatallities in the lamellae
A 4
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(37) Andres J It , and Lee. II
The Work Hardening and Aging id Steel
J. Iron Sleet lent 145 (1942) IS IP-211I,
Experimental work was iintlertaken with a view of proving or diaprov-
mg the hypothesis that the colt) working if iron and steel gives rise to the
formatinn of the alpha phase It is allow,, neat tarbon is dissolved ;luring the
work-hardening ploress, and that tinder certain conditions both angtenite and
martensite may form A new theory nf quench aging is suggested, and the
siiturit id strain aging is also dealt with Ti,,' close correspondence between
the changes produced in steel by cold work anti by quench hardening In strong
evidence in favor of the suggestion. made. A new theory of the yield in
developed. while the changes undergone daring the extension of the tensile
test piece are shown 10 be dependent upon the total elongation and the condi-
tion of the carbide in the steel,
(38) Andrew., R P
A Correlation Between the Tensile Test and the Creep Tent
Abs Dinaert? linty Cambridge, No. 1953 (1950-1951) 73 pp
Results of tensile testis on high-pority Al were prediited from restilts
of creep tests tinder various. loads Thr very close agreement between the
predicted and actual experimental results aupporta the hypotheni? of a unique
relationship between sirens, strain, and rate of strain within the ranges
studied, in the rase of high-purity Al The maximum load in the tensile
tents mere:teed linearly with the log rate of extension, and the relation be-
tween the applied load and log time to fracture, in the creep tests, was also
linear, the two gradients were equal in magnittide but opposite in sign Thin
relationship between the gradients agreed with the result that the creep
curves for various loads coincide when plotted with the nonslimenitional
absciaaae t/TF, where I is the time and TF the 11111,- 10 frartnre
(39) Andrews, K
Principles Involved in the Formation of iron Alloys
Metal Treatment and Drop Forging 19 (1952) 425-489
Empirical relationships developed from a study of experimental facts:
(I) principles of classification, (2) semiquantitative study of phase diagrams
from a thermodynamic viewpoint, (31 explanation of formation of alloys and
their constitution and properties In terms of electronic theory of atomic
structure, anti (4) application of thermodynamic methods to the transforma-
tion in binary and ternary alloys
(40) Arantes, A A
Limitations in the Use of Hardness Teats an a Method of Investigation of the
Mechanical Properties of Tempered and Annealed Steels (In Portngeme)
Bol da :senor brash! Metals 12 No. 4 3 (1956) I 35-14 3
Analysis of the relationships between hardneas, tensile strength,
impact valise, and wear resistance of various steels showing the inadequacy
of hardness tests as a method of investigating mechanical properties, even
when the chemical composition of the steel and its /moony curve are known
(41) Arbtin, E., Jr , and Murphy, C
Correlation of Vt1 kers Ilardnes. Number, Misdoing of Elasticity anti the
Yield Strength for Ductile Metals
USAEC Pub! ISC-356 (1n53) 28 pp
Indentation, scratch, and dynamic methods of hardnenn tenting are
reviewed The Inveatigation is honed on a hypothenin of Carrion. [Meek
Eng 46(1924) 638], that a relationship of the form It CE ml " exists be
tv.een the hardness comber If, 1.11C M011011111 of elantitity L, and the c oiti-
pre/onion elastic limit L. The materials studied included Al beast., Co,
Mg, steel, Ta, Sn, and Zn, and covered a wide range of moduli of elasticity
DPN were determined, lifting three different loads anti three loading times
for each material the mechanical propertien measured were the modnion
of elasticity, the yield point, and the idtimate tensile strength Strength-
hardness relationships are pretiented for three range* of modulus of
elaatitity No discernible relationship exinted between the yield point and
the DPN The foregoing relationehipn appear to he in general agreement
with similar resulta presented elsewhere the prearnt eye:Mons were
limited by the small number of points lined, and by the accuracy of the
ha rdness values
(42) Ardley, 0 W,
On the Effect of Ordering Upon the Strength of Cii3Aii
Acta Met. 3(t955) 525-532
Single crystals of Cu3Au were obtained with different domain
-4 size and degree of long-range order The critical resolved shear strength
and electrical resistivity were measured at room temperature The re-
sistivity measurements enabled the domain sloe to be deduced, and, as this
increased, the strength first increased and then decreased to a value lower
than the initial (disordered) strength The strength decreased as the long-
range order increased The change in strength of crystals equilibrated at
375 C and then quenched to lower temperature was Alen obtained as a fun,
lion of time at the lower temperature. The relaxation times for attaining
the equilibrium long-range order at different temperatures were in agree-
ment with previous e?timates. The variation of strength with domain size
was found to fit a theoretical relation given by Cottrell Some results were
also obtained on the change of strength with tempeatures as a crystal
annealed to equilibrium at 300 C was heated to 680 C The strength in-
creased with temperature up to the critical temperature, dropped sharply
during disordering, and then increased with temperature again up to a
maximum before finally decreasing The Maximum strength in the dis-
ordered region was etrain-rate dependent, and jerky flow was observed
below this temperature It is believed that this is a strain-aging effect.
(43) Ardley, G. W , anti Cottrell, A II
Yield Pointe in Brass Crystals
Proc. Roy Soc. (London) 2I9A (1953) 328-341
It is suggested that the yield point in these crystals is caused by N
In the material The conditions for producing yield point in fcc metals
are discussed in relation to the "atmosphere" theory of the yield point,
which supposes that dislocations in crystals become anchored by solute
atoms, which segregate to firm, and that the material gives way sud-
denly and softens when the e dislocations are pulled away from their
atmospheres at the beginning of plastic deformation
i
(44) Arkharov, V , Borisov, 13 S., and Mardeehev, S.
Diffuelon-Caused-Hardening as a Factor of High Temperature Deformation
(In Russian)
Research on Heat Resistant Alloys 2 Moscow (1957) 120-124
Pointe out that diffusion interchange does not necessarily relieve
etraln. Only the movement of an atom from an Interstitial site to a
vacant lattice site actually causes relief of local microdietortion Reasons
that at high temperatures diffusion-caused-hardening can become an impor-
tant factor In high-temperature strength
(45) Arkharov, V I , Koleenikov, G. N , and Orion', A. N
Possible Development of the Dislocation Theory (In Russian)
Doklady Akad. Nauk S.S.S R 92 no. 4 (October 1, 1953) 751-754
(NSF tr-212)
A theory of edge dislocations is developed. designed to eliminate two
fundamental difficulties of the current theory, viz , the very high energy
required to produce a dislocation, and the very large number of disloca-
tions which must be present in a crystal before the application of forces
The theory is bared upon conelderation of a system of dislocations comprising
two pairs of edge dislocations (4- and -) and eight pairs of screw disloca-
tions (left- and right-handed), arranged so that the dislocations in different
slip planes will, pairwiee, have common frontal planes The activation for
the generation of a dislocation is broken up into elementary parts, and equals
the self-diffusion activation energy, and in the crystal, before the applica-
tion of deforming forces, there is a sufficiently large concentration of self-
-
diffusion-type disturbances in the regularities of the lattice The ideas
developed are somewhat more general than those due to Seitz (Phys Rev
79 (ii) (1950) 8901.
(46) Aroesta, It
Theory of Melting and Yield Strength.
Phy. Rev (1955) 1723-1724
FUrth'e theory of rupture strength (Phil Mag 40 (Ser 7) (1949)
1227), based upon fusion, has been criticized (see Frenkel, "Kinetic Theory
of Liquids", London: (1946) 101) as yielding fortuitous results largely be-
cause rupture strengths are associated with surface phenomena The theory
is now related to yield strength, which Is less surface-dependent. Fiirth's
theory assumes that melting is due to the breakup of a block structure, it is
here assumed that the mechanism is dependent not only on block size but
also on the width between blocks where the atoms are misfits, and may be
expected to enter the mechanism first
05
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(47) Auld. 3 Ii . Coyle, P.. A , Marshall. A hi ? and McKinnon, 11. A
X-Ray Microscopy of As-Grown and Deformed Single Crystals of Aluminum
Trani,. AIME 20.9 (1957) 360-361
The microfocus X-ray method of Schulz (Ibid. 200 (1954) 10821 WAS
used to study the slip deformation and strain hardening of single crystals
of pure Al broth in the state as grown by the strain-anneal method, and
after they had been subjected to increasing amounts of plastic strain Al-
though many a?-grown crystals showed no sign of any macroscopic defect.,
some showed defects similar in structure to those reported by Kelly and
Wei obid 201 (1955) 10411 to occur in crystals grown from the melt,
while others showed a more regular appearant
(48) Averbach, B L.
The Structure of Solid Solutions
Mass Inst. Technol , Dept Met (USAEC) NY0-7054 (Nov 1955) 36 PP
Sizes of atoms in solution, strain energy, strength, local atomic
arrangements, changer in vibrational spectrom in alloy formation
(49) Averbarh, B L , Bever, M B., Comerford, M R.. , and Leach, 3. S.
LI.
X-Ray and Calorimetric Investigations of Cold Working and Annealing of
a Gold-Silver Alloy
Acta Met 4 (1956) 477-484
The energy stored in filings of a 75Au-25Ag alloy ia released during
the recovery and recrystallisation processes Most of the stored energy
tn associated with the presence of low energy boundaries introduced dur-
ing deformation; average specific interfacial energy for these boundaries
wan calculated
(50) Averbach, It. L..)1L-ohen, M , Allen, S , Comerford, M F., and
liousisa, C
Fundamentals of Cold Worktng and Recryetallization
Mass lost Technol (USAEC) NY0-7075 (June 1955) PP 3
An X ray study of recovery and recryetallisation in an alloy contain-
ing 755"'s Au, 25% Ag was combined with calorimetric data to obtain perti-
nent Information on mechanism by which energy in stored in deformed
metals
(51) Bacon, R , and Smith, C S.
Single Crystal Elastic Constants of Silver and Silver Alloys
Acta Met. 4(1956) 337-341
The elastic constants of single crystals of Ag and of dilute alloys
of Mg, Zn, Pd, Cd, In, and Sn in Ag have been measured by the ultra-
sonic pulse-echo method. For all aolntes except Pd, there is a large
fractional decrease, upon alloying, in the shear constant (C11 - C121/2,
and a relatively small decrease in the short-range crystal forces because
of a local weakening of nearest-neighbor repulsive bonds. and an increase
in the long-range electrontattc forces because of the Increase in the
average ion-core charge The two shear constants increase upon alloy-
ing with Pd, and a nimilar interpretation is not inconsintent with this
rental
(52) Bailey, It W
Note on the Softening of Strain:Hardened Metaln and Its Relation to Creep
I Inst Metals 35)1926) 27-43
Postulates strain hartientngn and nimiellancous partial softening by
annealing as a basin for explaining the mechanisms of creep
(53) Bailey, R W
Utilization of Creep Test Data in Engineering Design (Creep of Lead
Tubes Under Compound Stress)
Proc Inst. Mech. Engrs (London) (1735) 131 131-26')
A rational treatment of the subject of creep is attempted, particu-
larly directed to the requirements of design General expree?lons for
reep under any system of stress are given, which include the case of
tension, and permit the reaults of tensile creep teats to be utilized in
the design of parts under complex stress General relationships are
suggested for creep in the threction of three principal stresses, X, Y,
and Z (conaidered positive when tensile) representing any stress system
The expressions are similar in form, e.g , the tensile creep rate Cx in
the direction of X in represented by
St
s k IX - VIZ (7. - X.1;1 1;.: (V 41] "' [(2: V/".`"' it
where A, ro, and n are toistants. Ft, ?ire pie tennfon 11' s 0, 7. 0)
Cx AX. Valises of it. and n are der, :col. distriblition Id sirens
hopes:lard engineering parts undo r nttfilptrjiiii sires, in determineil 1 he
paper .41so in, hides resells for the axial, r rcineferential (1%1,0 11111
tharrietral creep if pipes littler ondrincel I nt,rnjl pressore anti IIXI tl
stress, and internal pet-trent, an,1 axial tor
(Si) Bailey. II, W
A Critical F.sanonatron of Prot-4.414re, 41,41 III Britain and the United
States In Determine Creep Stres?en for ilie 13,?sign of PlIVer Plant
for laing I if,. at Digit Temperateres
J Appl Mechanics 21 ( Ins II jot.12.4
l I,',,,, fit,,!., of extrapolating I reel, Slat Advocate, his
method lit maintaining , reel, stress ron?lant anti y.lrying temperature
Gives his expression for the rreep iirve fuhich int hides a (rctor
for ''thermal .0 tern") dcfilt A,,", 'I', uhere , II strain. 4, is stress t
i.
time and A, is, anti p are material constants, ill, values depen,fing on
the particular ntage of creep. I he oeffir lent A ontains the thermal-
at tom t
(55) Bainbridge.It , 1.1, C ? and I awards, F. It
Itect.nt Observations on the &lotion if Small Angle Dislotation Brurntiarren
Ai ta Met Z (1954) 122-333
The motion of small-angle disloration boundaries in sinc crystals
was investigated in the temperate,: range -196 C to 400 C Boundaries
were rondo to new,. by the applicafion of a shear stress at ting in the di-
rection Of the Burgers vector of the edge dialocations i ornprising the
bertindary The character of this motion YAS friend tn nary markedly with
temperature Motion proceeded at a I onstant rate under a constant stress
at elevated temperatirren At 320 C movement was ilincontint ..... rt. the
boundary advant ing through an appreciable distance during each pimp.
At -196 C, rnotinn appeared to be steady, however, at both 20 and -196 ( a
4.4111ilnunsonly int reasing stress was retpfired for continued moverrient The
bnimelary angle remained ronntant during motion at high temperature At
lower temperatnres, the magnitude of the angle decreased, anti the
boontlary ansumed a more complex shape thrnirgh interaction with stele -
lural defects Small-angle boundaries of like sign in proximity to one
another cnalesred rinring short-time annealing treatments at 400 C to
form a single boundary Closely spaced binintlaries of like sign rofild be
made to imite at any temperature by the application of a stress thus
establishing a mechanism fnr the formation of a substrie lure in the
absence tit appreciable thermal energy Boundareen of !milky ntgii were
iinited through their strena-induced motion and the angle of the resultant
boundary equalled the algebraic sum of the angles of the constituent
boundaries. The results of boundary motion expertments were t ompa red
with the reaultn of simple shear test? nit single crystals of zinc 1 he
niniliaelti es noggented that motion of dirilorationa through the structiiral
barriers of the crystal rather than generation of new sIllf aliens may be
the factor which determines the yield strength
(56) Bakarian,P W and Mathewnon, C 11
Slip and Twinning in Magnesium Single Crystals at Elevated Temperatnres
rrann Alls1E IV (1941) 226-254
The mechanism of plastic deformation in bigh-purtly magnehnorn op
to Ino C wan investigated in .I nf rompression tents Inc-holing rep
resentative orlenta(ionn of the f ryntal edit 05 ill the sterns asia Tht,
renults indicate that slip on the banal plane in the original material
twinning on the (102) plane, slip on the basal plane if the twinned regions,
and fr.tiore (or ? leavage) along the (101) plane are the only mechanisms
operating tiering compression stressing. prior to the entrance of genre
it
crystal breakup. The dominance of these several met lianistes has been
classified into three fields of orientation
(57) Pinkish, It , and Robertson. W
Strut iiire-Depencient Chemical Rear t inn and Noirleation of Fracture in
CuyAli Single ('ryntals
Acta Met ,1 (1956) 142-351
Cu is selectively removed front Cu3An by FeCI3 from imp, rfertions
originating thirmg crystal grouth Selective removal of Cli also net urs
from local areas in slip clustern ajsparently unrelated to growth imperfections tinder applied stress, both sites of rent tone become net lei of
(racier,' I rat ka which propagate only in the presence of stress and rea
gent. the path la normal to the stress axes and is independent of crystallog-
antis. This 111,11,1i/, that the deformation at the leading edge of the
crack in IllgIlly complex It in proposed that fracture occurs as a reset(
of selective removal if Cu from the tone if complex deformation at the
leading edge of the crark
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(Ih) 11,11414 At
A Study Ili %ark 11 traveling and Reannealing ni Iron
J Iron Steel inst 151 (1915) 1#11 7411'
A rnite-tron wires sere gfien 14 different tlegree? nf work harden-
ing and annealed In v.1,1111111 I he tollouing propertie? were determined:
Ft irolie?ss. I 1,1111i al resistivit), elaslir 1 front the bend test. plastic.
,ilteliate stress oniferre elongation. springiness. thermoelectric
properties and ItIll rest,. Inre I he shiners 505(50 ell by reannealing con-
firm the preseme of three tdienoinctia strain aging, re, ryetalliration,
and ? rystal griiwth 1111. main Singe that of rerrystallination uf a work-
rilened wire ix shoun by all the tors, s utu ilf rterrIng in the same range
of tereperatoren ;uteri, ontlary to Ile findings of ranunann), this
suggests that all the thanges .1r, i resell of the sante fundamental cause
The re, ryslalliration r nigi is lowers 1 alien the degree of work hardening
is an, . 41 From the , hang,. in, tie' Lour.. of 110. iircen uhich charar-
terire unrk-liardeneil irlin three regions of plash, sty sere disfinguished
Ball, C
Di?tribillion? of 1/1sloi MO% Ill MI I.11.1 II
PIIII Mag 2 11n571 n77-98;
Tlie analysis described in I [Mrs, h and Sall dud ? 46 (1955)
134 31 is extended to biltinda ries r nioasning dislocations of three systems,
and it is shown that die retain-in axis if ?iit lu a boundary I an lie anywhere
In the I/11r fare of a I one A vinglr bo ..... tars, plane rerresponds to eat h
direction of the rotation axis 1 he theory cannot explain the experinien
tal on tosserientations in /al uniesn slip occurs in nonbasal
iii
rerfions or ori planes not hitherto reported. at high temperatures The
experimental evidence for is i metals is tripoli. iently prer1.e to tent the
theory
160IJ mar c
Thr Flow Stress of Pnlyr rystalloce Aluminium
Mag 2 see 8 (1957) 1011
An X-ray n ii rnhearn Ito !minor wan tined to meager, mulrgrain sire
and range a 1111.14,10 Wail Ill, to olio ,,rigoia/ grain in a serien of speci-
mens, tenni!, tests twilit; 5 arri.41 out in meaner,. flou nte, ss her, in
a strong correlation hem's "ems flew stress. anti soligrant .1
11I411 n sn I-112 giving a giralgItt line The sir, owl, el 110 boundaries
does not appear to depend on botindary angle fho variation ID( o with
temperature was also measured and, from nos, it is soggented that
strength i ontrolling hit site is elastic interat tioe
(611 hill C .1
Nature and Effect of &dint rot tore in Poly, rystalline Alnruirenti
Paper from I3alp5_atinm iLtili1101.5basiitial Pestyeeiteni,f Cryittala
John Wiley and Sons Int NI?Y/ Yerk (1'1571 II, t5i1
Sob 41 I ?
gia 4r creases with
a limiting cal.,. whit!, increases linearly
tion The types of stresn free 10% angle
withont climb with 111.4101 armlet of 455e, tw
alrillated If clinkn, miens of tinly one or
possihle rotation axes are 100), 1 1111. ?
using di formation and approat he
uithIi,. temper:11,1re of deform,
lienintlaries that an his fornied
o Inc three fly., nut hat, been
two systo net are present. the
snot in the pi . (110)
(62) Ball. C and 113rarh. P II
Siirfa, e Distribution of Dinh. atinnn 3n Mo tals
Phil Mag 46 set. 7 (1955) 11,11-1352
It is shown that a linundary rontaining /I tilt's,, nyntervis
ran. in general feirm on two, and only two, planes whit I, have simple
indices and that the rotatinn axes are simple crystallograplii, dire,
terns. Most of the experimental evidence on nusorientationn in deformed
ryslais ,.Ift snhal.sinn,i in terms of .111111 armpit, liniendarien
(611 W Itosi, F D and Seigle. I I
Self -Diffosion of Metals ind Assn, cited Pheresinena Progress Report
IISAF:C Pub! SIP. Ill (1954) 4
rise stress /strain lharacteristics of mingle m ryntalm of dilute alloy.
nf Cii Al, Co-Cle, and Cu Si were determined an a ben lion of ctincentra-
tion The critical shear stress increaned connelerably with alloying con-
tent, Bet yield point wax hem. r defined at higher alloy taintents and with the
highest rontents a double yield wont was (mind Strain aging neggented
that this yield phenomenon 'sin St result of the Cottrell met lianistn ,if
anchoring diking at ions by solnte atoms All "tiny crystals displayed a
tun-stage hardening process. t ii tracterired by a low linear hardening rate
follnued by a high parabollt rate. for similarly orientated I ryntaln the
extent of the first stage increased IT1.1rkedly with alloy content whereas the
shear hardening of both stages decreased Frit. similarly ore ntated
rystals of identical solute t.01111.ntranon the two-stage hardening curves
shciued that in the order Cii-Ge, C,1-51, Cu Al (I) the critical shear
sir. ss de. reased, (2) the extent of the first hardening stage 'Inc reaned,
end (31 the degrer nf strain hardsning in hoth stages int red flea The most
signal, ant differen, e? or , orred ill going from Si to Al It appearn that
cariations tn the d ffusmn rates of the solutes may be a significant factor
III xplaIning tio so, iii If, rein es
(64) Balluffi, R W., Rost, F D , and Seigle, I.. I.
Self-Diffusion of Metals and Ae?ociated Phenomena Progren. Report
USAEC Publ. SEP-151 (1954) 5
A study has been made to entablieh a /InrCIIIII4/11 between the
?tre??-strain characteristic s and the nature oil slip-hand formattion
in metal single crystals, the effect of carrables influt nt ing plash.
flaw, 1141.11 all oriental ill,,. romprisition, 41111 temperature have also
been inveattgated A study if the effect of r rysial nrieniation IIn the
plastic properties of Cu and Ag allOWI'll Ill stress-strain there, ?
terietic? are strongly dependent MI rientatioa, and at f ert.ion orienta-
lions two stagen cif slwat hardening or cur; 12) low, linear hardening
rates occerred when Wan on a melr system. a high. paratioliC
rate Wall always alll.01 with Shp III several nysiteme. the greater
the number of internecting slip planes, the greater the degree of
shear hardening and (I) the first Plage of low hardening rate is
terminated by the occorrent e of secondary slip. and the extent of this
stage der rea.? as the initial orientation of the crystal approaches a
stage for multiple glide It appeared that the onset of stage two was
stsanciated with a renolveil shear stress which was independent of
crystal orientation
(65) Balluffi, 16. W . and Seigle. I. I.
Growth of Voids in Metals During Diffusion and Creep
Acta Met 5 (1957) 449-454
A study was made of the thermodynarmc conditions under which
void? may grow from suitable nuclei in a metal by a mechanism of
vacancy aggregation under the influence of stress and vacancy super-
saturation The results are applied to the intergranular void forma-
tion observed during creep experiments and also void formation during
Kirkendall diffusion In the cane of creep it is postulated that nuclei
may grow into voids by receiving etreria-motivated vacancy currents
from nearby grain-boundary source. In the case of Kirkendall diffu-
sion it Is shown that the development of stress in the diffusion zone
depends directly upon the existence of a vacancy supersaturation
(66) Barnes, R. S.
The Climb of Edge Dislocation. in Face-Centered Cubic Crystals
Acta Met 2 (1954) 380-385
An edge dialocation of type a/2 may be made in a fcc
crystal by inserting two (114 planes perpendicular to the Burgers
vector. The atomic movements involved in the climbing of such a
dislocation from one {114 plane to another produce extension or
shrinking of the extra (t10). planes It is concluded that climbing is
still possible, though more difficult, when the dislocation is dissoci-
ated into Shockley partials A straight line will be unable to climb
unless the environment contains a large excess of point defects, but
this restriction is removed if the dislocation pos.e.ses jogs Surface
markings may be produced by dislocations climbing out of the crystal,
and it is suggested that this may be the explanation of some anomalous
...lip'i steps observed by Forty and Frank [Proc Roy Soc (London)
2I7A (1953) 2621 The possible importance of climb in various
physical phenomena Is stressed.
(67) Baron, II. G
Stress/Strain Curves of Some Metals and Alloys at Low Temperatures
and High Rates of Strain
J. Iron Steel Inst. 182 (1956) 354-366
Tensile tents were carried out on some metals and alloys at
temperatures of 20, -78, and -196 C, using strain rates of 10'3 and
1042 sec The strain rate and temperature have a marked effect on
F'-
and the softer ferritic steels In dynamic tests below a critical tempera-
ture, all the ferritic materials exhibit a type of premature failuro
which the strain is entirely localized to a short neck. It In shown that
this is a consequence of the adiabatic nature of this test The variation
of upper yield stress with temperature and strain rate appears to obey
an activation energy relationship over part of the range of stress Within
this range the activation energy for yielding varies with the stress in the
manner predicted by Cottrell and Pithy The divergence from theory at
low stresses may be due to simultaneous strain aging, while at high
stresses, brittle fracture of twinning eventually intervenes before the
upper yield point is reached
168) Baron, V. V ,and Savitsky, E M
The Influence of Temperature Upon the Strength of Brittle Metallic
Materials (In Russian)
Doklady Akad Naiik S S S.R 94(21(1954) 269-272
The ultimate tensile strength and compreastve strength of Si and
Ge and of brittle metallic compounds of the NiSi, NizSi, Cup CoSi,
and CoSii types were determined at various temperatures up to their
melting point In contrast to the ductile metals, the strength of which
decreased with temperature according to an exponential law, the brittle
Declassified in Part - Sanitized Copy Approved for Release ? 50-Yr 2014/01/27: CIA-RDP81-01043R003200240005-1
A-7
material.* show.